spring 2020 volume 61 issue 1 · teacher's near miss by susan robinson 27 the montessori...

SPRING 2020

Volume 61

Issue 1

Where children are offered freedom, ample opportunity for mastery, and are treated with dignity and respect, we might expect the need for traditional discipline to almost fall away.

Rebekaah Stenner

Contents

IN EVERY ISSUE

07 President’s Message

41 Cartoon

48 Problem Sets

51 Math Links

52 BCAMT

Spring 2020 | Volume 61 | Issue 1

31 42

38

09 Touch, Tap, Grasp and Zap: New Ways to Learn Multiplicatoin

By Sandy Bakos and Canan Güneş

15 Slow Reveal Graphs

By Chris Hunter

19 Maybe It's Actually a Good Thing if I Don't Understand My Kids' Homework

By Louise Struthers

23 A Curious Student and a Teacher's Near Miss

By Susan Robinson

27 The Montessori Method and BC's Redesigned Curriculum

By Rebekaah Stenner

33 MathematicalVisualizationin the Classroom: A Brief History

By Foster Matheson and Egan J Chernoff

37 Polishing: A Fresh Take on the Review Process

By Marc Husband

42 How Many? A Review from the Kitchen Counter

By Nat Banting and Henry Banting

45 How Many? A Professional Learning Series for K-2 Teachers

By Janice Novakowski

Errata Fall 2019In the issue of Vector, the article “Using Self-Determination Theory to Engage Learners” was written by Jeff Irvine, not Jeff Irving. We apologize for the misspelling of Mr Irvine’s last name

The views expressed in each Vector article are those of its author(s), and not necessarily those of the editors or of the British Columbia Association of Mathematics Teachers.

Articles appearing in Vector may not be reprinted without the explicit written permission of the editors. Once written permission is obtained, credit must be given to the author(s) and to Vector, citing the year, volume number, issue number and page numbers.

NoticetoContributorsWe invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable materials written by BC authors on BC curriculum items. In some instances, we may publish articles written by persons outside the province if the materials are of particular interest in BC.

Submit articles by email to the editors. Authors must also include a short biographical statement of 40 words or less.

Articles must be in Microsoft Word (Mac or Windows). All diagrams must be in TIFF, GIF, JPEG, BMP, or PICT formats. Photographs must be high print quality (min. 300 dpi).

The editors reserve the right to edit for clarity, brevity and grammar.

NoticetoAdvertisersVector is published two times a year: spring and fall. Circulation is approximately 1400 members in BC, across Canada and in other countries around the world.

Advertising printed in Vector may be of various sizes, and all materials must be camera ready.

Usable page size is 6.75 x 10 inches.

Deadline for SubmissionsSpring: FEB 1 (for peer review, December 1) Fall: SEP 1 (for peer review, July 1)

AdvertisingRatesPerIssue$300 Full Page $160 Half Page $90 Quarter Page

Membership EnquiriesIf you have questions regarding membership status or have a change of address, please contact Michael Pruner, Membership Chair: [email protected]

2019/20 Membership Rates $40 + GST (BCTF Member) $20 + GST Student (full time university only) $65.52 + GST Subscription (non-BCTF)

Cover Art: The cover image was created by Clayton Heights Secondary School student Jamie Kim, and was inspired by the work of Vincent Van Gogh. Jamie completed an AP Calculus class last year in order to learn more about graphing, the result of which is her image “Starry Night.” You can watch the creation of this image on YouTube (https://youtu.be/Fltmc98B1gQ).

Photos taken by Sean Chorney and Susan Robinson

VECTOR EDITORSSean Chorney Simon Fraser [email protected]

Susan Robinson Gulf Islands School District [email protected]

BCAMT EXECUTIVEDeanna Brajcich, PresidentSooke School [email protected]

Susan Robinson, Vice PresidentGulf Islands School [email protected]

Colin McLellan, Secretary and Listserve ManagerRichmond School [email protected]

Jen Carter, TreasurerVernon School [email protected]

Michael Pruner, Membership ChairNorth Vancouver School [email protected]

Minnie Liu, Vancouver School [email protected]

Amanda Russett, Kamploos School [email protected]

ELEMENTARY REPRESENTATIVESAdam Fox North Vancouver School [email protected]

Debbie Nelson Comox Valley School [email protected]

Alex SabellSurrey School [email protected]

INDEPENDENT SCHOOLS REPRESENTATIVESDarian Allan Collingwood School West [email protected]

Richard de Merchant St. Michaels University School [email protected]

SECONDARY REPRESENTATIVESRon Coleborn Surrey School District [email protected]

Josh Giesbrecht Abbotsford School District [email protected]

Chris Hunter Surrey School District [email protected]

POST-SECONDARY REPRESENTATIVEPeter Liljedahl Simon Fraser [email protected]

NCTM AND NCSM REPRESENTATIVEMarc Garneau Surrey School District [email protected]

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Contributors

Ray Appel Ray has taught grades 2-8, been a mathematics/science helping teacher, spoken across Canada and the US, and has authored mathematics curriculum. He is currently retired, delving deep into reading, painting, drawing, walking, website design, a bit of mathematics consulting, home building and more! His website, zapple.ca is used by thousands of educators worldwide.

Sandy Bakos Sandy Bakos is currently a TTOC for Coquitlam School District and a PhD candidate in mathematics education at SFU. Prior to that, she taught kindergarten–grade 6 in northwestern Alberta for 16 years, and participated in a teaching exchange in Australia for a year. She is a strong supporter of elementary teachers and the diverse skillset they bring to the classroom and is interested in ways that digital technologies can be used for teaching/learning mathematics in elementary schools.

Nat Banting Nat is a mathematics teacher and PhD student currently on faculty in the Department of Curriculum Studies at the University of Saskatchewan. He shares his teaching practice across the country through various writing projects, speaking opportunities, and social media platforms—

blogging at natbanting.com/blog and tweeting as @NatBanting.

Henry BantingHenry is a kindergarten student at Sylvia Fedoruk School in Saskatoon, Saskatchewan. He likes to play with math tiles and hockey cards, and played on the Wild Stallions hockey team this year.

Egan Chernoff Egan is an adjunct professor in the Faculty of Education at Simon Fraser University.

Richard DeMerchant Richard teaches middle school at St Michaels University School in Victoria. He enjoys a good problem and shares them with his students both in class and as a Math Challengers Coach. Richard currently serves as the Independent School Representative for the BCAMT.

CananGüneşCanan Güneş is a research assistant and PhD candidate in mathematics education at SFU. Prior to that she completed her Master degree in Primary education at Bosphorus University and taught mathematics both at elementary and middle school levels in Istanbul, Turkey. Enriched from both her own teaching and learning experience, her research interest constitutes teacher learning, online

learning, activity-based learning and technology integration in mathematics education.

Chris Hunter As a numeracy helping teacher in Surrey, Chris Hunter collaboratively works with–and learns from–teachers of mathematics from Kindergarten to calculus. He tweets at @ChrisHunter36 and blogs at reflectionsinthewhy.wordpress.com.

Marc Husband Marc is the Program Coordinator for Mathematics, Science/STEM & Robotics for the Toronto District School Board. His classroom-based research investigates using student ideas as a resource for learning mathematics in schools, teacher education and professional learning. Marc earned his PhD in Mathematics Education from York University in 2019.

Foster Matheson Foster has a Bachelor of Arts in English from the University of Saskatchewan. He continues to focus on mathematical visualization in the field of Architecture, where he now pursues his education as an Architectural Technologist.

Debbie NelsonDebbie is an educator living and working on Vancouver Island. In a position as Curriculum

Support Teacher in her district, she works with dynamic educators to explore thinking and reasoning routines in math class. Her most rewarding work in mathematics education has been the professional development of K-7 teachers and the networking and sharing of resources to support student learning.

Janice Novakowski Janice is currently a district teacher consultant working with K-12 educators and students in the areas of mathematics and numeracy in the Richmond School District. She is also a member of the BC Numeracy Network and facilitates the BC Reggio-Inspired Mathematics Project. Janice is interested in curricular structures and pedagogy that are inclusive and responsive to all students.

Susan RobinsonSusan teaches high school mathematics on Salt Spring Island. She is interested in expanding her repertoire of assessment strategies and finds assessment to be like swimming in the ocean–you just have to dive right in and keep your eyes wide open, as every time you enter the sea you are going to see something new.

Vector • spring 20206

Rebekaah Stenner Rebekaah is the mathematics coordinator for the Mission School District, and a former high school mathematics teacher. She is also a PhD student in mathematics education at Simon Fraser University. Rebekaah's passion is empowering students through giving them opportunities for success in mathematics.

Louise StruthersLouise teaches at Byrne Creek Community school in Burnaby. She recently completed her Masters in Teaching Secondary Mathematics at Simon Fraser University. Louise appreciated the opportunities the master’s program provided to think about teaching mathematics and issues in mathematics education.

David WeesDavid works as a Senior Curriculum Designer for DreamBox Learning, working remotely from his island paradise. He has worked as a teacher in NYC, London, Bangkok, and Vancouver, and has six years of experience as a mathematics coach and eight years of experience with the International Baccalaureate Middle Years and Diploma Years Programs. David has his Bachelor of Science (Mathematics), Bachelor of Education (Secondary Mathematics), and Master of Educational Technology, all from the University of British Columbia.

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President’s Message

During the last few weeks, educators have experienced the power of

reaching out to one another more than ever before. Change is a constant in education, especially in this technologically-rich

world, but we never foresaw the need to implement new

platforms of education as quickly and as critically as we have

experienced.

As challenging as this has been for teachers, students and parents, we have all ventured into new arenas of connectivity and learning opportunities. Through all this, we have expanded, and of course are still expanding, our repertoire and filled up a new toolbox of mathematics pedagogy. Moreover, we have had the chance to reflect on what we each believe is essential learning and to reaffirm our caring role in society.

Personally, I have found a new passion for how I teach mathematics by experimenting with new ways to motivate my students from a distance. Teachers rely on classroom energy, peer inspiration and the genuine collaboration of student experiences. Transforming the learning routines I use every day in the classroom to the medium of distance learning has confirmed how powerful these routines are for learning the content and strengthening the competencies of our curriculum.

Within the mathematics education world there is a focus on the equity of mathematics learning. Offering learning opportunities without face-to-face instruction has put equity issues in the forefront. It has become abundantly clear that students across BC have significant differences in their lives which make online instruction challenging and inequitable. The following position statement from the National Council of Teachers of Mathematics provides a plethora of ideas on which to reflect:

Practices that support access and equity require comprehensive understanding. These practices include, but are not limited

to, holding high expectations, ensuring access to high-quality mathematics curriculum and instruction, allowing adequate time for students to learn, placing appropriate emphasis on differentiated processes that broaden students’ productive engagement with mathematics, and making strategic use of human and material resources. When access and equity have been successfully addressed, student outcomes—including achievement on a range of mathematics assessments, disposition toward mathematics, and persistence in the mathematics pipeline—transcend, and cannot be predicted by students’ racial, ethnic, linguistic, gender, and socioeconomic backgrounds (https://www.nctm.org/Standards-and-Positions/Position-Statements/Access-and-Equity-in-Mathematics-Education/).

Speaking of the many transformations in mathematics education during the Covid-19 pandemic, BCAMT is proud to present our

2020 Fall Conference, Transformations, on October 2020 at Guildford Park Secondary School.

Given the changes in our curriculum, we have chosen the theme “transformations” for this year’s conference. Our goal for this conference is to support teachers with this ongoing journey by continuing the conversations about teaching, learning and assessing.

BCAMT’s past president Michael Pruner will be the keynote speaker. Michael is looking forward to sharing some of his insights and personal stories

around transforming the mathematics classroom through active participation and collaboration.

Have you considered being a speaker? We are calling for speakers to introduce ideas, strategies, and approaches that will support transformations—both big and small—in our students and in our teachers. Every change starts with a single step. We are looking for you to share your ideas that will:

• transform the mathematics education experience for teachers and students;

• explore ways of teaching, learning and assessing in the classroom;

• change teacher and student mindsets about learning mathematics;


• reflect on teachers’ own identities and beliefs about mathematics instruction;

• expose student mathematical understandings. Although the status of all PSA conferences in October is unknown, we continue to plan and organize another successful experience for you. I hope to see you all there, in person, eager to continue the transformations we are making while being at home during this challenging time.

BCAMT Connections Pinterest PageThe BCAMT PSA has developed a Pinterest page to house many of the electronic resources available for teachers. Inspired by the current educational needs, the page provides quick direction to many of the websites we, as executive members, use on a regular basis. You will find grade-sorted resources, plus number routines and other organizations’ websites. I hope you will find it useful; it will continue to expand as we add more useful resources.

Introducing the new BCAMT PresidentI’d like to formally congratulate Susan Robinson on being elected BCAMT President. Susan is a mathematics educator in the Gulf Islands School District and has been involved in educational leadership through Island Numeracy Network on Vancouver Island and the BC Numeracy Network. She has proven to be a dedicated and innovative executive member, and we look forward to her leadership.

Educational Leaders Opportunity Are you a leader of mathematics education in your school or district? Reaching out to those teachers who are leaders in mathematics in our province is an important goal of our PSA and we would love to add new leaders to our growing group. Our 2020 Summit will occur the day before the BCAMT Fall Conference. Emails will be sent to your district and information will be available on our website for registration.

Awards and Grant NewsEvery year the BCAMT Executive awards teachers for the inspiring work they do with their students. We continue to ask our members to honour a colleague by nominating them for one of our prestigious awards: the Ivan L. Johnson Memorial Award, Outstanding Elementary and Outstanding Secondary and Outstanding New Teacher awards. Please visit the BCAMT website for more information and to complete the Google Form.

One of the most exciting ways the BCAMT PSA encourages professional growth and financially supports new initiatives is through our awarding up to $8 000 in grants each year. This year we considered almost 30 applications and we were able to jumpstart the following initiatives:

• Christine Hulme (West Vancouver), Building Thinking Classrooms

• Sarah Baxter (Surrey), Shaking It Up in Kindergarten

• Amanda Culver (Victoria), Reggio-Inspired Mathematics

• Heather Mulholland (Surrey), Messy Mathematics

• Tisha Witt (Smithers), Designing Mathematics for All Learners

• Denise Underwood (Kamloops), Collaborative Number Sense Building Parties

• Roberta Toth (Vanderhoof), Integrating Regio-Inspired Mathematics

• Kristine Chambers (North Vancouver), School-Wide Mathematics Inquiry: Number Talks

Moving Forward I have thoroughly enjoyed leading this association these past three years as president and look forward to many more years on the executive. The BCAMT Executive is a collection of educators who are passionate about mathematics education, and who have been supportive, generous and forgiving as I learned the ropes. I am grateful for the professional and personal growth that comes with leading a group of committed educators. Most of all, it has been wonderful to meet so many mathematics educators across the province, Canada and the United States.

In this time of uncertainty, we have seen how generous and sympathetic educators can be to colleagues, parents and students. We have found a plethora of ways to reach out to help calm worries and provide learning opportunities in new ways. For the time being, please enjoy your summer, recuperate from this year of unknowns, and perhaps in August take some time to read a book or listen to a podcast that will incite your passion and innovation for the new school year.

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Touch, Tap, Grasp and Zap: New Ways to Learn Multiplication

by Sandy Bakos and Canan Güneş

Finding effective ways to help elementary school students access mathematical ideas and to firmly grasp and hang onto them is imperative for building a solid foundation of number and number operations. Integrating manipulatives into mathematics lessons is one way that many elementary teachers do this. In this case, tactile manipulative experiences may also be accompanied by mathematical expressions written on paper or a whiteboard. During this type of learning experience, the mathematics may temporarily fade into the background while students engage with the manipulatives and only later be associated with the manipulative activity through purposeful teaching. The increasing availability of touchscreen technologies are providing alternative ways for younger students to interact with digital objects and their symbolic representations. In addition to allowing children to create mathematical objects directly on a screen, some of these digital technologies also enable tactile experiences that are very different from those provided by physical manipulatives (Bakos & Pimm, 2020). A new and free multi-touch application called TouchTimes (Jackiw & Sinclair, 2019; hereafter, called TT) is one such digital technology that is providing teachers with another set of tools to help build a more solid foundation for students when learning multiplication.

Building Multiplicative ThinkingAlthough a variety of approaches and strategies can be used by elementary teachers when introducing children to multiplication, repeated addition remains the most commonly used method (see Figure 1). The use of repeated addition is encouraged by the British Columbia mathematics curriculum and is included in most textbooks; however, this way of thinking about multiplication becomes problematic when students encounter higher level mathematics that requires a direct capacity to think multiplicatively (e.g. Siemon et al., 2005). Therefore, multiplication appears to be a critical point in student learning, possibly even a gateway for future mathematical competence. As students’ ability to make sense of number and quantity progresses to larger numbers, fractions,

ratios, percentages, decimals and proportions, repeated addition is no longer sufficient for navigating these increasingly complex mathematical ideas (Brown, Küchemann & Hodgen, 2010). As mathematician Keith Devlin (2008) argues, “[m]ultiplication is simply not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not” (para.3). He explains that although repeated addition gives the same results as multiplying natural numbers, it does not make it the same. He illustrates this point by pointing out that riding a bike or driving a car will both get you to work but are very different processes. Rather than rely so heavily on repeated addition, it is essential that young learners acquire and develop multiplicative reasoning, which enables them to work flexibly and efficiently with “the concepts, strategies and representations of multiplication (and division) as they occur in a wide range of contexts” (Siemon, Breed & Virgona, 2005, p. 2), both in school and throughout our daily lives.

Figure 1: Multiplication based on repeated addition

This article has been peer reviewed.


Figure 2 compares additive thinking, which involves repeatedly adding groups of the same size, and multiplicative thinking, which begins with individual dots unitised into groups of three (the first many-to-one unitisation), followed by the groups of three combined in a second many-to-one unitisation in which four units of three become one product. Referring to these unitisation actions, Davydov (1992) described multiplication as a transfer of unit counts in measurement. The unitisation establishes a many-to-one correspondence between the units and it constitutes a functional relationship between the variables of multiplication (Vergnaud, 1983). The meaning that children as young as eight-years-old give to the composite units (a unit made up of multiple copies of a smaller unit) they construct when making sense of 4 × 3 is essential to their developing concept of multiplication (Steffe, 1994). Therefore, learners are thinking multiplicatively if they can simultaneously think about units of one (the three is becoming one unit) and units of more than one (the four groups of three) (Downton & Sullivan, 2017).

Figure 2: Additive and Multiplicative Thinking

In order to build the foundation for multiplicative reasoning that is so critical to future mathematical understanding, alternative approaches need to be developed and implemented with children in the early grades. Based on the transfer of unit counts articulated by Davydov (1992) and the functional relationship identified by Vergnaud (1983), TouchTimes is one such digital tool designed to help young learners develop multiplicative thinking in age appropriate and mathematically sound ways.

What is TouchTimes?Perhaps the best way to get a sense of what TouchTimes is, would be to download the app and try it out. Much of the ensuing description will be easier to follow and make more sense after experiencing the app for yourself. TT is a gesture-based, multi-touch iPad application designed to provide young children with easily accessible, multiplicative (rather than additive) ways of experiencing multiplication. By creating and transforming objects directly on an iPad screen, children are able to engage with multiplication through a combination of gestural and visual interactions while receiving direct feedback in the form of pictorial

representations of the objects created on the screen and their corresponding numerical expressions. In addition to its visual and symbolic elements, learners’ active bodily involvement in the creation of multiplicative relationships allows for both static and dynamic experiences of multiplication.

TT currently has two complementary worlds, named Grasplify and Zaplify, each of which provides a different context for learners to experience multiplicative thinking (Figure 3). Both worlds embody the coordination of units in visually unique ways, while highlighting different properties of multiplication. The multiplication in Grasplify involves a distinct role for each hand, which are specific to the function of multiplicand and multiplier, whereas Zaplify emphasizes the commutativity property of multiplication and the symmetry of multiplicative factors.

Figure 3: TouchTimes

The Grasplify WorldIn order to visualise the Grasplify world, it may be helpful to watch this short video (m.youtube.com/watch?v=L3BRXZfBbZo) prior to reading the description that follows. When first opened, Grasplify displays a blank screen divided in half by a vertical line (Figure 4a). Coloured dots (termed ‘pips’) will appear on whichever side of the screen is touched first (the left side in Figure 4b). When users tap their finger(s) opposite the centre line, enclosed bundles of pips (termed ‘pods’) appear (Figure 4c) that reflect duplicates of the original pip configuration (both in colour and shape). Continuous screen contact must be maintained for the pips, though only a finger tap is needed to create pods, which remain even after pod-fingers are removed. However, the content of all pods will adjust to any changes in the pip-creating fingers (for example, if a finger is added or removed), and if all fingers are removed from the screen, both pips and pods disappear, effectively multiplying by zero. TT displays and automatically adjusts a number sentence that corresponds with the pips and pods on the screen. Designed to be

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symmetric, if pips are created on the right-hand side of the screen (rather than the left), the numerical expression will also appear starting from the right and ending with the product at the left (e.g. ‘12 = 3 × 4’), which also creates opportunities to discuss equality with students.

Figure 4: (a) Initial screen of Grasplify; (b) Creating pips; (c) Creating pods; (d) Two composite units

The Grasplify world embodies the coordination of Davydov’s (1992) double many-to-one abstraction as students must first create a unit (the multiplicand) with the fingers of one hand before they are able to create a unit of units (the multiplier) with their other hand. For example, in 3 × 4 (see Figure 4d), the distribution of the three pips across every pod represents the first many-to-one unitisation, and the four pods which have been encircled into a larger group of 12 (the product) represent the second many-to-one unitisation. This ordering of the multiplicand × the multiplier is the opposite of what we usually see in textbooks, and for those of us used to explaining 3 x 4 as ‘three groups of four’, this ordering may seem ‘wrong’ at first. However, this order in TT is intentional and grounded in approaches to early mathematics based on measurement and ratio, where the unit quantity is identified before the number of units.

The Zaplify World When users first enter the Zaplify world, seven fingerprints appear on the screen. Three of which are aligned vertically on the left edge of the screen and the remaining four are aligned horizontally on the bottom edge (Figure 5a). After a few seconds, the fingerprints fade away and a diagonal line appears between the lower left and upper right corners of the screen (Figure 5b). Both the fingerprints and the diagonal line are designed to demonstrate where users should

place their fingertips on the screen.

Figure 5: Initial screen of Zaplify (a) Fingerprints; (b) The diagonal line

In the Zaplify world, users encounter two different types of objects: lightnings rods and orange sparks. Lightning rods are created in either a horizontal or vertical direction depending on where the user touches the screen. These seem to emerge directly from the tips of a user’s fingers (Figure 6a and 6b), whereas the orange dots are created indirectly, appearing wherever two perpendicular lightning rods intersect and create a sizzle (Figure 6c).

Zaplify highlights the binary nature of multiplication in multiple ways. The first relates to the orientation of the fingerprints and the diagonal line that separates the screen into two areas. The second involves the difference between the orientation of the rods, where the horizontal rods represent one factor and the vertical rods represent the other factor as two distinct entities. And finally, the emergence of the sparks is based on the joint actions of two perpendicular lightning rods.

Figure 6: (a) Horizontal lightning rods; (b) Vertical lightning rods; (c) Orange sparks


The screen works in two modes: locked and unlocked, though it remains in the unlocked mode unless the padlock icon found in the bottom right hand corner of the screen is touched. When locked mode is engaged, the lightning bolts form a grid and orange dots replace the sparks at each point of intersection. Once created, locked mode objects will remain on the screen, even after users lift their fingers, enabling larger factors to be more easily created. Conversely, when in unlocked mode, the objects of Zaplify appear and remain visible only while users’ fingers maintain screen contact. If a finger(s) is removed, the associated object(s) disappear. Factors can be increased or decreased without resetting the screen, which makes it easier to follow consecutive changes in the multiplication while building knowledge of the multiplicative relationship between the factors and the product. Unlike the asymmetric nature of Grasplify, however, the distribution of one factor over the other factor is symmetrical in Zaplify. In other words, both factors are distributed simultaneously. As seen in Figure 7a, four vertical lines are distributed over three horizontal lines, and three horizontal lines are distributed over four vertical lines. Any change to one factor influences the product with respect to the other factor. For example, when the product 12 is made with four vertical and three horizontal lines (Figure 7a), creating one more vertical lightning rod does not increase the product by one, rather it increases by three, which corresponds to the number of horizontal lightning rods (Figure 7b).

Figure 7: (a) Distribution on both factors; (b) The result of increasing the factor (4) by one

In addition to multiplicative relationships, Zaplify embodies the properties of multiplication. The orientation of the lightning rods and the dots constitutes an array model. This grid-like representation, together with the symmetrical relationships between the factors in Zaplify, make it easier to introduce not only the commutative property (3 × 4 = 4 × 3) in Figure 7a but also the associative property ([3 × 4] + [3 × 1] = 3 × [4 + 1]) of multiplication in Figure 7b.

Ideas for Using TouchTimesTT provides young learners with opportunities to visualise, identify, physically experiment with and communicate about multiplicative relationships. Rather than channelling learning towards predetermined outcomes, TT encourages learners to make choices while engaging in exploration in an environment structured by mathematical rules. Encouraging open exploration is one of the best ways to support learners when first introducing the app. This is an ideal opportunity for students to make predictions, record questions (Figure 8) and explain what happened in TouchTimes during different situations.

Figure 8: Student questions and noticings

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Although open-ended exploration with TT can engage children in thinking about multiplication, using the app in conjunction with tasks is necessary to encourage the development of multiplicative thinking. In order to assist teachers with ideas for using TouchTimes, a variety of tasks are being developed which can be found in the Grasplify and Zaplify sections of the TouchCounts website at http://www.touchcounts.ca/touchtimes/. These tasks prompt children to use their fingers purposefully while thinking about what they are creating and seeing on the screen. For example, asking children to double (or later halve) a number in either Zaplify or Grasplify is one such task, and is a nice place to start. The concept of doubling is a familiar notion for many children, and this type of task provides a nice bridge from addition to multiplication. Learners can experience doubling as more than repeated addition when they see and experience the functional relationship between the multiplication factors in TT. Moreover, it allows for multiple entry points by allowing learners to explore different ways of achieving the task by doubling either factor. Each method of doubling creates an opportunity to discuss with students the relationships between the objects of TT, the numerals that correspond with these objects, and the method used to double the number (Figure 9). Going back and forth between Zaplify and Grasplify also provides opportunities for meaning making across models, while encouraging students to make connections between different meanings of multiplication.

Figure 9: Discussion on doubling

Another task for use in the Grasplify world is to ask pairs of students to figure out how to make a single pod of five with one finger. This focuses children’s attention on changing the number of pips, which emphasises multiplication as a change of units, which is very different than repeated addition. The task also enables students to develop fluency in performing the unitising action with different numbers using one 1-pod finger. We have found this task to be surprisingly difficult for children to do and to understand initially. In their first attempt, many children will place a single pip-finger down and then tap five times sequentially with a pod-finger, in a one-to-many action that makes five 1-pods (Figure 10 a) rather than the one 5-pod the task asks for. Instead, students must first generate five pips in order to create a single pod of five

with one touch (Figures 10b and 10c), thus creating a many-to-one relationship.

Figure 10: (a) Five 1-pod finger touches; (b) One 5-pod finger touch; (c) Another one 5-pod finger touch

In the Zaplify world, asking students to make only one dot is another interesting task. Children generally respond to this task by initially placing only one finger on the screen (Figures 11a and 11b), which suggests more of an additive approach. In order to successfully achieve this task, students must place a finger in each of the two separate areas of the screen (identified by the finger prints and diagonal line) in order to create two perpendicular lightning rods. As a result of this two-finger placement, a single target


object (orange dot) will be created (Figure 11c). This experience is contradictory to their previous experiences where two fingers are generally associated with two target objects. Therefore, this task may challenge children’s additive approach to quantity, while building a basis to think multiplicatively.

Figure 11: (a) Pressing only one finger; (b) Pressing only one finger; (c) Making one dot with two fingers

These types of tasks are consistent with the BC mathematics curriculum, and are designed to promote mathematical reasoning, flexible thinking and personal strategies as children visualise to explore mathematical concepts while building their understanding of mathematical processes. TouchTimes is an effective tool that can be used to help children develop multiplicative thinking in a way that quite literally allows children to touch, tap, grasp and zap multiplication with their fingertips.

ReferencesBakos, S. & Pimm, D. (2020). Beginning to multiply (with) dynamic digits: Fingers as physical-digital hybrids. DEME, 6(2), page numbers to be determined.

Boulet, G. (1998). On the essence of multiplication. For the Learning of Mathematics, 18(3), 12–19.

Brown, M., Küchemann, D. & Hodgen, J. (2010). The struggle to achieve multiplicative reasoning 11–14. In M. Joubert & P. Andrews (Eds), Proceedings of the Seventh British Congress for Mathematics Education (BCME7), (30)1, pp. 49–56. University of Manchester, UK: BSRLM.

Davydov, V.V. (1992). The psychological analysis of multiplication procedures. Focus on learning problems in mathematics, 14(1), 3-67.

Devlin, K. (2008). It ain’t no repeated addition. Retrieved from: https://www.maa.org/external_archive/devlin/devlin_06_08.html

Downton, A. & Sullivan, P. (2017). Posing complex problems requiring multiplicative thinking prompts students to use sophisticated strategies and build mathematical connections. Educational Studies in Mathematics, 95(3), 303–328.

Jackiw, N. & Sinclair, N. (2019). TouchTimes [iPad application software]. Burnaby, BC: Tangible Mathematics Group. (https://apps.apple.com/ca/app/touchtimes/id1469862750)

Siemon, D., Breed, M. & Virgona, J. (2005). From additive to multiplicative thinking. In J. Mousley, L. Bragg & C. Campbell (Eds), Proceedings of the 42nd Conference of the Mathematical Association of Victoria (pp. 278–286). Bundoora, Australia: MAV.

Steffe, L. (1994). Children’s multiplying schemes. In G. Harel & J. Confrey (Eds), Development of Multiplicative Reasoning in the Learning of Mathematics (pp. 3-39). Albany, NY: State University of New York Press.

Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds), Acquisitions of mathematics concepts and processes (pp. 127-174). New York, NY: Academic Press.

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Graphs convey information at a glance. They tell stories to the people who see them. Reading graphs means so much more than capturing single data points Individual pieces of information alone don’t tell a compelling story. Like reading text, reading graphs means analyzing and interpreting, inferring and predicting. Graphs ask us to examine the messages that they convey to us as readers and the messages that were intended by those who made them (i.e., to look for misleading graphs and to think about bias). Slow Reveal Graphs, the critical thinking routine described below, can help students become more media literate.

In this article, I’ll invite you to take on two different roles. First, like the students in your classroom, I’ll encourage you to look closely at a graph that I’ll share. Second, I’ll ask you to think about the activity from the perspective of a teacher.

Look at the graph below (Figure 1). What do you notice?

Figure 1

You may have noticed that this graph does not have labels, axes, or scales. And yet, despite the complete lack of context, there are several mathematically interesting features of this graph. Take a moment to think about some of the things you find mathematically interesting before you read my list below:

• there are six line graphs• the six line graphs are made up of two sets of three• both sets of three are made up of two solid lines and one

dotted line• each dotted line is between two solid lines• there’s an overall upward trend• sometimes one line is decreasing while the others are

increasing• there are valleys—especially one dramatic drop• two of the lines intersect at some point• the data is volatile to the left, steady to the right• there are vertical grey bands with varying widths

All of these observations—and maybe more—without knowing what the graph is about! By leaving out pertinent information, I’ve potentially created some curiosity. For example, you might be wondering:

• What do the different lines represent?• What does the vertical axis represent?• Is a comparison being made over time?• What’s causing the overall upward trend?• What’s causing the peaks and valleys?

You might be feeling a need to have your questions answered. Now that you’re hooked, I can gradually provide you with more information (Figure 2).

Figure 2

Slow Reveal Graphsby Chris Hunter


How does this new information change your thinking? The dotted lines can now be explained as black and white populations, graphical representations of the average of women and men. But what is being measured across these groups? Again, I can slowly reveal more information:

Figure 3

Did you anticipate that the topic is life expectancy? Did you consider—and then reject through reasoning—alternative topics?

You might suspect that life expectancy is being graphed over time and have some hunches about the grey bands as important moments in history. I can reveal the dates and events, in turn. Are your hunches confirmed in Figure 4 below?

Figure 4

How does this information change your thinking? What do your initial mathematical noticings mean in context? For example, the “one dramatic drop” in life expectancy was a result of the 1918 influenza pandemic; the intersection of two lines as the life expectancy for black women surpassing that for white men in 1965. Finally, I can share the complete graph, as it appeared in The New York Times (Figure 5).

Figure 5

The text in The Times graph draws your attention to 1982 until the mid-90s, at which time the life expectancy for black Americans decreased due to the crack epidemic. During the Vietnam War, life expectancy decreased for black men whereas it increased for white men. This raises questions about racism and discrimination within the American military: Were African-Americans disproportionately drafted? Were black soldiers more likely to be assigned to combat units? How were rising racial tensions back home reflected in the ranks and on the battlefield in Vietnam? Mathematics can shed light upon these questions. Despite making up about 11% of the civilian population, African-Americans accounted for nearly 25% of all combat deaths in 1965; represented 16.3% of draftees and 23% of combat troops in 1967; made up 2% of the officer corps; and received 34.3% of courts-martial (https://www.nytimes.com/2017/07/18/opinion/racism-vietnam-war.htm).

What story do you think the graph in Figure 5 is telling? Do you agree with The Times’ statement that “[t]wo decades of steady improvements in the health of black Americans have narrowed the gap between black and white life spans”? If not, can you write a different subhead that captures the graph’s main idea?

What’s the story since 2014? How did the opioid crisis affect life expectancies for blacks and whites? How about for women and men? What will this life expectancy graph look like in the future?

What new observations and questions do you have?

Above, I attempted to engage you in a learning experience similar to that of the mathematics classroom. With respect to content learning standards, I suggest this graph best fits functions and

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relations in Foundations of Mathematics and Pre-calculus 10 or graphs in society in Workplace Mathematics 11; slope and rate of change also come into play. Just as importantly, you were engaged in curricular competencies—the doing of mathematics. You noticed and wondered; you analyzed and interpreted; you reasoned to draw conclusions and make predictions.

The activity above mimics an instructional routine that helps students across all grades make sense of data—Slow Reveal Graphs. In this routine, the teacher gradually reveals more and more of a graph, each time asking students to discuss how this new information changes their thinking. It compliments other instructional routines that may already be part of your classroom—Quick Images, Number Talks, Clothesline Math, Which One Doesn’t Belong?, etc. Like these, Slow Reveal Graphs support students and teachers; for both, predictability means paying less attention to lesson logistics and more attention to student thinking.

Jenna Laib, a math coach in Massachusetts, created a website (slowrevealgraphs.com) to help teachers effectively implement this routine. Jenna shares several sequence Slow Reveal slide decks on her site. I will share with you one that she created: a circle graph from Animals by the Numbers: A Book of Infographics by Steve Jenkins.

First, display the “stripped” graph (Figure 6).

Figure 6

Ask your students, “What do you notice?” and record their observations. I prefer to project on the whiteboard so that I can mark up the graph as necessary. Possible observations:

• there are eight circle graphs• each circle graph is made up two colours, yellow and

purple• some graphs are more yellow than purple, others more

purple than yellow

• from top to bottom and left to right the amount of purple is increasing

• the graphs in the top row are less than one-half purple Ask “What do you wonder?” and record students’ questions. Possible questions:

• Why eight graphs?• What do the colours represent?• What is the topic of this set of graphs?• Is something hidden behind the black circles?

Display a bit more of the graph (Figure 7) and ask “What new information do we have? How does this change our thinking?”

Figure 7

Given the two quantities—hours awake and hours asleep—a safe assumption is that a whole circle represents twenty-four hours. Display the hours asleep—8 hours—for the third circle in the top row (Figure 8). This single data point confirms this assumption; students see 8 hours as equivalent to the picture of one-third. This sets up a lovely estimation task: estimate the hours asleep for the remaining seven circles.

Figure 8


Of particular interest are the third and fourth circles in the bottom row. In the third circle, some students might determine the product of three-quarters and twenty-four. Others might add one-half and one-quarter of twenty-four. In the fourth circle, some students might see yellow as one-sixth and subtract one-sixth of twenty-four from the whole; others might see purple as “a bit more” than three-quarters. Reveal the values for each of the eight circles one at a time (Figure 9).

Figure 9

Again, ask “How does this information change our thinking?” Sometimes, I ask “What does this mean for what we’ve noticed and wondered?” You’ll find a question that works for you. Whatever your phrasing, it’s important to orient students to revising their thinking. Students will be guessing what the eight circles represent (e.g., “Different ages? Different days?”). Slowly reveal the different animals (Figure 10). Start with human then dog or cat—students will make comparisons to the people and pets in their lives.

Figure 10

Finally, display and discuss the complete infographic.

This routine playfully compels students to “slow their rolls” and see what they might have overlooked at a glance. Further, through being placed in the position of a reader with a desire for missing labels and scales, students begin to appreciate the reason for attending to these elements when constructing their own graphs to tell their own stories.

As a teacher, this routine empowers me to build upon students’ ideas in ways that typical textbook exercises like “What was the life expectancy for black men in 1970?” or “Which animal sleeps for 2½ hours a day?” do not. Each time that I facilitate Slow Reveal Graphs, students surprise me with observations or questions that I did not anticipate. Their engagement affirms my belief that our students are curious, creative, and capable mathematicians.

ResourcesRelated blog post and complete slide decks for the graphs above: https://reflectionsinthewhy.wordpress.com/2017/09/27/the-new-york-times-whats-going-on-in-this-graph/

The New York Times’ What’s Going On In This Graph? series: https://www.nytimes.com/column/whats-going-on-in-this-graph

Jenna Laib’s Slow Reveal Graphs: https://slowrevealgraphs.com/

Kelly Turner’s Graph of the Week: https://www.turnersgraphoftheweek.com/

ReferencesBlack Americans See Gains in Life Expectancy, https://www.nytimes.com/2016/05/09/health/blacks-see-gains-in-life-expectancy.html

19

Maybe It’s Actually a Good Thing if I Don’t Understand My Kids’ Homework

by Louise Struthers

1See references for a link to the article.

Think about the last time you read about mathematics education in the news, I bet the article wasn’t praising teachers for the exemplary job that they were doing in developing numerate citizens. On the contrary, I suspect it was a “doom and gloom” piece bemoaning the fact that “Johnny can’t add,” that our students do not know their math facts, or worrying aloud about a decline in standardized test scores. If we turned our attention to the author, or primary voice in the article, it was most likely a politician or business leader. Given the overwhelming amount of bad press that mathematics education receives, a newspaper article written by a parent, from a parent’s point of view, in support of a different way of teaching math, caught my attention. I invite you to take a deeper look at the discourse around Canadian mathematics education by examining this article using positioning theory (and no, there is no cartesian plane in sight!).

The article: “Maybe it’s actually a good thing if I don’t understand my kids’ homework1” was written by Edward Keenan, a Toronto Star columnist. In this article Keenan responds to the March 15, 2019 announcement made by Lisa Thompson, Ontario’s Minister of Education, of the new program “Education that Works for You.” This announcement includes a promise to “Undertake curriculum reform that will include a new math curriculum that will focus on math fundamentals for all grades.”

Positioning theory will help us explore the meanings in this article. Positioning theory emerged in the 1980’s in the area of gender studies and has since become an important tool for analysing social discourse in many disciplines. Positioning theory helps us to see how communication shapes identity. In this case we are seeking to understand how the communication in the popular press shapes the public perception of mathematics and mathematics education in our schools.

Positioning theory recognizes that discourse is dynamic and has three parts: positions, storyline and speech/communication acts,

which work together to mutually define each other. Consider overhearing two children interacting on the playground, one child admonishes the other, saying “leave me alone.” If the child has positioned themselves as helpless or as a victim, these words constitute a plea. If the child had positioned themselves as a leader, these same words would be viewed as a demand. The interpretation of this interaction is informed by the storyline: is the helpless child a victim of playground bullying? Is the leader a confident, dominant child who is asserting their right to choose who they associate with and how they play? Positioning is inextricably linked to storylines. When one positions oneself, one is also positioning the other people in the conversation. Through interaction new positions can be negotiated. Positioning is intentional and occurs along a storyline; not all parties involved in the discourse will have equal power in the positioning process.

In his article Keenan writes in the first person, giving him full control over how he positions himself in addressing the reader. This is an example of deliberate self-positioning which occurs in conversations where one wants to express their personal identity.

Keenan positions himself as a knowledgeable and reasonable commentator on the reform of Ontario’s mathematics curriculum. In the article Keenan shares that his children are enrolled in a French immersion program in Toronto, and that he selected this program for his children because he wanted them to “have opportunities and [the] expanded perspective that a second language offers.” With this disclosure Keenan has established his agency; he had (and made) choices for his children about their program of education. Furthermore, he establishes himself as a parent, hence he has a vested interest in the conversation (about the mathematics curriculum.)

In order to position himself as a typical parent with authority to comment on the curriculum announcement, Keenan offers some bibliographic information: his children attend a publicly funded school in Toronto, and he describes his motivation for electing


to have his children in a French Immersion program. Keenan states that he chose French Immersion for his children because of his regret that he lacks facility with French, one of Canada’s two official languages. He offers his inability to speak French by way of explaining his difficulty in helping his children with homework, not just their mathematics homework. Keenan’s thesis is that new ideas are often met with skepticism and resistance, and he provides a personal anecdote to support this, sharing that “when I was growing up when my aunts and uncles complained that the systems of long division I was learning was ridiculous.” This personal story supports Keenan’s argument about new ideas being met with skepticism and indirectly adds to his credibility as a commentator. He learned a basic mathematical skill (long division) under a new (to his parents) system of learning and it hasn’t hampered his success as a functioning adult in today’s world. The reader infers that his is a successful adult given that they are reading his thoughts in a metropolitan newspaper and that he has children who are succeeding in school. The success of his children is another inference; he speaks matter of factly about their education and does not find fault with the education that they are receiving. The biographical information that Keenan provides adds another layer to his credibility as a commentator on the mathematics curriculum.

Having established himself as an individual with agency over his children’s education and having offered the reader insight into his own educational background, Keenan can then present his unique point of view, that he is not intimidated by not understanding his children’s homework. He shares that he often “gets to feel stupid when I try to help them [his children] with their homework, because they need to translate it for me before I can even tell what’s going on. But that’s a good thing.” (Emphasis is mine.) This willingness to be challenged by his children’s work and that “some of the math concepts that seem silly or strange or convoluted to me… will eventually make them better at math than I am” further position him as a reasonable commentator who can offer another perspective to the discussion. It also positions him as a supporter of the current Ontario mathematics curriculum. It is interesting to me that the does not disclose why he hopes his children are better at mathematics than he is.

In the article Keenan refers to himself in the first person singular, through use the pronoun “I.” This establishes himself as a unique person. As a unique person he is able to offer his personal experiences and explain his personal behaviour.

Keenan wrote in response to the March 15, 2019 announcement of the changes that the Ontario Ministry of Education was making to the mathematics curriculum. He is telling the story and as such, he positions the other voices in the discussion. The author positions the critics of the current mathematics curriculum (the one that Ontario is revising) as being uniformed and not credible in that he does not provide expert voices to support their point of view. His comment “everyone is convinced the “discovery math” their kid is learning is convoluted bafflegab producing a generation of innumerate fools” is a general quote with no attribution, containing colloquial, non-academic language, the language of the uneducated and ill-informed. This contrasts with his introduction to the article, where the author positions himself as an informed, educated commentator with anecdotes about the ancient Greeks and their reluctance to accept zero, the introduction of Arabic numerals, and the use of Roman numerals in a comment about the invention of the printing press. His point in these three little stories is that mathematics education has always been in flux and change is not always embraced. This learned history of the mathematics, even written colloquially, further positions the author at a distance from the faceless critic who lacks credibility.

Positioning theory recognizes the three-part structure of discourse: positioning, storyline and communication acts. We just looked at how the author has positioned himself as a reasonable and authoritative commentator while the “other” view he is opposing is faceless, nameless and poorly educated.

The positioning is also informed by the storyline. In the article “A Tale of Two Metaphors: Storylines About Mathematics Education in Canadian National Media” researchers at Simon Fraser University reviewed 71 online national newspaper articles (published between 2013 and 2015, a period that included the release of the 2012 PISA results) and identified four main storylines that are consistent in newspaper articles about Canadian mathematics education:

• there are only two ways of teaching mathematics, discovery and rote learning,

• there is a math war in Canada, between proponents of discovery and rote learning,

• Canadian students are getting worse at mathematics, as measured by student performance on international tests, and

• student success in mathematics is linked to economic growth in Canada.

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The first three storylines are addressed in “Maybe it’s actually a good thing if I don’t understand my kids’ homework.” Keenan speaks to the storyline, that there are only two ways of teaching mathematics, from a number of different angles. He introduces the change in the curriculum, by quoting Kristen Rushowy, a fellow columnist. He positions her as a neutral party, referring to her simply as a colleague, who reports the Ministry’s announcement of the “back-to-basics” pledge that would emphasise “fundamental concepts and skills” and discourage “discovery math.” Back-to-basics is another way of staying that there is an emphasis on rote learning of algorithms and math facts. In this reporting the “two ways” of teaching mathematics are named but neither is supported or refuted. Keenan explains that discovery math is a term that is not used in our school system and outlines what is encompassed by this approach to mathematics teaching.

As a prelude to exploring the difference between “discovery math” and “back-to-basics”, Keenan references a video which shows a teacher explaining the multiplication of two two-digit numbers while in a parallel frame, another individual uses the traditional algorithm to multiply the two numbers and then has time left over to brew a cup of coffee (https://www.youtube.com/watch?v=PCgo0syhQqU). Keenan describes the difference in the two frames: explaining what the math means versus teaching a student to calculate. I wonder how many of Keenan’s readers sought out the video and viewed it? And if they did, were they able to see beyond the humour and understand his explanation of the two very different mathematical activities that they were viewing?

Keenan concludes his discussion of their being only “two ways to teach mathematics” by commenting that his children have been tasked with memorizing their multiplication tables and that he sometimes finds his children’s textbooks (with their “discovery math”) confusing. By placing these two statements, one after the other, he asks the reader to reconsider the storyline that there are two (mutually exclusive) ways to teach mathematics.

Memorization of tables is often used as code for back-to-basics instruction. Mathematics education researchers analysed reader comments on the same 71 online articles described above, and concluded that “For many readers, learning mathematics in elementary school is synonymous with whole number arithmetic,” and “Readers insisted that computational efficiency is a foundation to other opportunities to learn in mathematics class.”

Computational efficiency being the key to future mathematical success could well be considered an additional storyline in

conversation about Canadian mathematics education. Keenan addresses this storyline in comments about lining up columns when adding decimals, and in noting that his children have been tasked with learning their multiplication tables.

The second storyline in mathematical education is the idea that Canadian educators are engaged in a math war. The “memorization of multiplication tables” is often described as one of the battle flags of this war. The math war storyline is alluded to in his comments that “I have nothing against the ‘basics’ or ‘fundamentals’ that Thompson and her government love so much. I just hope that they get more than that too–the stuff the ‘inquiry-based’ systems and striving for ‘automaticity’ of the new math are supposed to teach.” In this quote he clearly identifies the two sides in the “war” and suggests that there is a middle ground. With this passage Keenan drops his balanced stance and comes across as clearly “new math.” He takes a side in the war. His words are still balanced, stating that he hopes his children get the best of both approaches to mathematics, but in his reference to the Minister of Education, he clearly repositions her as unworthy of respect. This repositioning is done by his referring to her by only her last name, he does not use her first name, title or honorific. Similarly, he does not name the government currently in power. By attaching “her government” to the dismissive reference to the Minister of Education, he shows that he does not find them a worthy contributor to the dialogue about mathematics education.

Keenan acknowledges the storyline “Canadian students are getting worse at mathematics” in explaining the motivation for “Education that Works for You” initiative. As a mathematics educator, I appreciate that while he acknowledged this storyline, he also subverted it with his statement that “our students’ performance in math shows some struggles by international standards.” He did not use the inflammatory language of war or disaster nor does he explore what skills are in decline.

In conclusion, it was refreshing to read a parent’s positive perception of a more inquiry-based method of mathematical instruction. The author positioned themselves as a credible commentator who, while appearing neutral, casts the announcement of education reform (moving the curriculum towards rote learning) in a questionable light. Examining this article systematically, using Positioning Theory, provides a framework for placing the author’s message in the context of larger cultural storylines, and, by focusing on the positioning of different players in the conversation, highlights whose voice is not present. Other authors have noted that much commentary on mathematics curriculum is dominated by


mathematics educators, teachers, and mathematicians. This paper presents the thoughts of a parent (who does not purport to speak for all parents or a collective); despite being a key stakeholder in education, parents are underrepresented in coverage in Canadian national newspapers. Going forward, it would be interesting to seek ways of engaging parents and students who constitute underrepresented voices of significant value in this discourse.

ReferencesChorney, S., Ng, O., Pimm, D., A Tale of Two or More Metaphors: Storylines About Mathematics Education in Canadian National Media, Canadian Journal of Science, Mathematics and Technology Education, Vol. 16, No. 4, 2016.

Harré, R. (2012) Positioning theory: moral dimensions of social-cultural psychology. In J. Valsiner (ed.) The Oxford Handbook of Culture and Psychology. New York: Oxford University, pp. 191–206.

Harré, R., van Langenhnove, L., Varieties of Positioning, Journal for the Theory of Social Behaviour, Vol. 21, No. 4. Pp. 393-407, 2015.

Keenan, E., Maybe it’s actually a good thing if I don’t understand my kids’ homework, The Star, Friday, March 15, 2019. https://www.thestar.com/opinion/star-columnists/2019/03/15/maybe-its-actually-a-good-thing-if-i-dont-understand-my-kids-homework.html, retrieved 20 May 2019.

McGarvey, L., McFeetors, J., Reframing Perceptions of Arithmetic Learning: A Canadian Perspective, ICMI Study 23, 2015. https://

www.researchgate.net/profile/Judy_Sayers/publication/281366454_Foundational_Number_Sense_The_Basis_for_Whole_Number_Arithmetic_Competence/links/55e3f59408aecb1a7cc9dfc0/Foundational-Number-Sense-The-Basis-for-Whole-Number-Arithmetic-Competence.pdf, retrieved 20 May 2019.

Rodney, S., Rouleau, A., Sinclair, N., A Tale of Two Metaphors: Storylines About Mathematics Education in Canadian National Media, Canadian Journal of Science, Mathematics and Technology Education, Vol. 16, No. 4, 2016.

“Back-to-Basics” Math Curriculum, Renewed Focus on Skilled Trades and Cellphone Ban in the Classroom Coming Soon to Ontario, Press Release, Ministry of Education, Province of Ontario, https://news.ontario.ca/edu/en/2019/03/back-to-basics-math-curriculum-renewed-focus-on-skilled-trades-and-cellphone-ban-in-the-classroom-co.html, retrieved on 15 June 2019

Rom Harré Positioning Theory Symposium Bruges 8 July 2015, https://positioningtheory.wordpress.com/what-is-positioning-theory/, retrieved on 20 June 2019.

What is Positioning Theory, https://positioningtheory.wordpress.com/what-is-positioning-theory/, retrieved on 12 June 2019.

Maths vs Making a Coffee, https://www.youtube.com/watch?v=PCgo0syhQqU, retrieved 27 June 2019

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A Curious Student and a Teacher’s Near Missby Susan Robinson

It’s the first day of Precalculus 12 and I have many questions about my students and the mathematical thinking they are bringing with them to class. I set them to work solving a series of equations:

1. 2x+1=02. 4x2-1=03. (2x+1)(x-1)=04. 2x2-3x-2=05. x2=x-56. xy=x

In planning for this lesson, I make a list of what I might see happening as students start working. The linear equation is a warm-up: will they use inverse operations to isolate the variable in a linear equation? will they show the steps they have used or will they just write down a solution? The set of quadratic equations has a progression: will they remember there are two solutions for the second equation, or do they only write down the positive value for x? do they remember the zero product rule? are they able to solve a quadratic equation in factored form by inspection, or do they need to solve the embedded linear equations? are they okay with factoring? will they remember to move terms to one side of the equation? do they remember the quadratic formula?; do they recognize when there is no solution? do these equations seem familiar to the students?

The first five equations in my list are routine equations my students would have had experience with in previous mathematics courses, although it is likely that not all of the students will be equally familiar with the strategies involved in solving these equations. The sixth equation is included because it is non-routine. Students likely have not have seen an equation like this in previous classes, and the equation doesn’t necessarily match equations that are typically solved in a Precalculus 12 class. I’m curious to see what my students will do. Will they divide both sides by x? Will they recognize equation 6 as a second-degree equation and consider bringing both terms to the same side and then factor?

Generally, by the time these students arrive in Precalculus 12, they are competent with solving linear equations. They recognize that they can apply operations to both sides of an equation, as long as they do the same things to both sides, and that this process should eventually isolate the variable. This is a procedure they can rely on, and often use it when they are not sure of what else to do. Sometimes this leads to a solution that looks like this:

(2x+1)(x-1)=0 2x2+x-2x-1=0 2x2-x-1=0 2x2-x=1 x(2x-1)=1

At this point, they have managed to get x by itself on the right-side of the equation, but are uncertain with what to do with the equation due to the presence of the variable in the denominator on the other side of the equation. The procedure falls apart when the equation gets more complicated, and students become unsure of how to move forward when they start going in circles. These students don’t have a deep conceptual understanding of solving quadratic equations—they spend a lot of time trying to work out how to isolate x without seeing that x was already isolated in each of the factors of the given product.

My hope is that students will see connections between the questions—that they will recognize that equation #1 is embedded within equation #3 and that this will trigger a memory that they only need to identify what values of x will give each factor a value of zero. And that finding solutions for the factored form of a quadratic equation will help them to remember that factoring might be useful in the next question, but that factoring won’t always be a useful strategy when they move on to the following question. And maybe, after having solved a few quadratic equations they might even go back to reconsider whether they found both of the solutions to equation #2.


I want to remind them of the connections they might have formed in their previous mathematics classes, and so once they have arrived at the solutions for equation #2:

I ask the question, “Are these the only answers? How do you know?” and hope that someone has something to say about the graph of the corresponding parabola crossing the x-axis at exactly two points.

Equation #6 is a chance for me to see what kind of algebraic flexibility students possess. Or at least, that’s what I was hoping for in my planning. What I hadn’t considered when thinking about this activity was that many students would be able to find a solution to the equation by inspection. Number of students looked at the equation xy =x and saw that if y = 1, then what must follow is x = x, a true statement. They did not see a need to consider possible values of x that might have made the equation true.

I did see several groups work through the solution I was expecting to emerge:

Perhaps this is because of their familiarity with equations such as 2x = 1, where the next step that students consider is dividing both sides by 2. Dividing both sides by x certainly isolates the variable y. But what about x?

Fortunately, at least one group worked through an algebraic approach that involved factoring, and we ended up with this solution on a whiteboard:

xy = x xy - x = 0 x(y-1) = 0 with solutions y = 1 and x = 0

which allowed us to discuss why you might not want to divide by a variable when solving equations.

Gaining insight into my students’ algebraic fluency is one of the goals of this activity. Another goal is to encourage students to use Desmos as a method of checking solutions. This provides

the students with the ability to easily check their work through a visual representation of the equation. One method is to graph each side of the original equation as a system of equations and look for intersection points:

Figure 1

Another method is to type in the equation as given (Figure 2). This representation can lead to some confusion if students don’t make the connection that the vertical lines that appear on the screen are the graphs of the equations of the solutions: and x = 1 . It is useful to compare the two strategies of using Desmos to make sure the students understand the difference between what they are looking at, and the ability to enter the entire equation is the only way to see what is happening with an equation such as xy = x.

Figure 2

The first time I ran through this set of equations with a class, we didn’t spend as much time as I would have liked using Desmos. We ran out of time, and moved on to the next activity before getting to any graphing. But really, I think that I neglected to talk about Desmos because of what I had discovered when I was planning the activity. Okay, I don’t need Desmos to check my solutions for any of these equations. And I knew what I would see if I graphed the first five equations on Desmos. But for equation #6, I was not expecting the graph to look like it did (see Figure 3).

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Figure 3

From the graph, I could see the solution y = 1 , but what about the solution x = 0? Why was I only seeing one solution? And why was Desmos telling me there was a hole at the point (0,1)? Later, when we did look at this graph, some of the students thought that because they had solved the equation by dividing both sides by x, that meant there would be a hole when x = 0. But, this was not a restriction on the original equation, xy = x , so there should not be a hole at this point on the graph.

As I was working on this equation at home, doubt started to seep into my thinking. Was I sure that my two solutions were actually both solutions to the equation? I knew they were okay algebraically, but couldn’t explain the solution x = 0 from a graphical perspective.

I didn’t press my students to follow through and check all of their solutions to the six equations with Desmos. And in doing so, I lost an opportunity to raise a question in class that I didn’t know the answer to, and to give my students a chance to explore and question and try to explain something themselves.

Fortunately, at least one of my students asked the question anyway.

About a month later, I was away from my classes and came back to a note from my replacement. He said I should really check in with Nick about what he was doing with Desmos. It turned out that Nick was equally confused with the graph that appeared in Desmos for the equation xy = x. In his original algebraic work, Nick had divided both sides by x, found the solution y = 1 and thought that x = 0 was a restriction (relating this to a hole, as above). He was confident with this solution, but as he returned to his seat he started looking around the room at other work left on the whiteboards. A solution that caught his attention was one that used subtraction and factoring, and that ended with two solutions: y = 1 and x = 0 . Nick thought it was an interesting approach and wanted to know why x = 0 was a solution instead of a restriction. Nick is fluent with

Desmos, and if he’s solving an equation/factoring/simplifying, he graphs the original in Desmos and whatever he’s working on to see if they match up—if they don’t, he knows he’s done something wrong. He graphed the equation and saw the horizontal line at y = 1, but no vertical line that would represent the solution x = 0. Not satisfied with this graph, he tried something different: What would happen if he graphed yx = x instead of xy = x? And this is what he saw:

Figure 4

There is now a vertical line at x = 0. Here is the solution x = 0, but now without the solution y = 1!

Nick was really confused about this, and so he sent an email to the Desmos support team through the feedback option (available on the online graphing calculator page). The response from the Desmos team had to do with this being a simple implicit equation and so the graph shows the solution that results from the simplest case. In order to see both solutions at the same time, you need a complicated implicit equation:

xy=x+0x3+0y3

Enter this equation into Desmos and we see:

Figure 5

Finally, both solutions are visible, and the Desmos graph matches our understanding of the solutions of the equation. This was


an interesting lesson in how there are humans at work in the background of the Desmos graphing calculator.

My students love to push the graphing capabilities of Desmos. How far can they zoom in on a function before the screen goes black? Can they find an extreme version of an equation that will make Desmos stop working? They are learning to ask questions about what they are seeing in front of them, and to try to find a way to make sense when something catches their interest or just doesn’t feel right.

I have a lot to learn from my students. My original hesitation upon seeing the original graph of the equation xy = x had me questioning what I knew about mathematics instead of pushing me to ask questions about the mathematics I was seeing. Next time, I will be more open to my own uncertainty and allow for possibilities to emerge where students and the teacher can be seen as shapers of mathematics. If I want my students to learn to love feeling uncomfortable while learning mathematics, I too need to spend time with this tension.

I have learned there is something to be gained from a slow approach, and there is a lot to be said about students having visual access to one another’s ideas. Having the solutions to these equations on the whiteboards around my classroom left traces of our work to linger along the edges, raising questions even after the lesson was over.

I have also learned that it is not always easy to lead students to be curious about things I find interesting. With my second semester Precalculus 12 class, I was excited to share what we had explored in my previous class and showed them the different versions of the graphs related to the equation xy = x. There was no energy or excitement about the different graphs, but there were a lot of glazed looks around the room. This wasn’t their question, it was mine.

And so, I continue listening for the questions that might be overlooked but that give space for unexpected mathematical thinking to emerge. I want to be a participant along with my students in the mathematical meaning making in my classes, and to provide opportunities for everyone to share in the excitement (Davis & Renert, 2013). In my planning, I make note of what I think might happen, but also leave space to add in notes about what actually took place. I ask questions and share my love of the mathematics with my students but take care to nurture their own curiosities as well. Each day is full of possibility and I trust that my students will take me some where interesting.

ReferencesDavis, B & Renert, M. (2013) The Math Teachers Know: Profound Understanding of Emergent Mathematics. New York, NY: Routledge.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

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The Montessori Method and BC’s Redesigned Curriculum

One of the challenges in mathematics education is that the way mathematics is traditionally taught does not meet the educational or psychological needs of many students. Even with a redesign of BC’s curriculum, mathematics classrooms sometimes can feel as they did before with the teacher as central, tests, textbooks, and students learning mathematics separately from other subjects. Until we change some of these fundamental components, small changes we make in our classrooms may not yield the results we desire. The redesign is a call to action, and in order to follow its recommendations, we need to make some changes in our schools. The Montessori model can inform us about a possible path forward.

Maria Montessori was an Italian physician who, through her observations of “deficient children” in asylums in Rome, came to believe that this “mental deficiency” was a pedagogical problem and not a medical one as was thought at the time (Standing, 1957). This realization altered her life path. “That form of creation which was necessary for these unfortunate beings, so as to enable them to reenter human society, to take their place in the civilized world and render them independent of the help of others—placing human dignity within their grasp—was a work which appealed so strongly to my heart that I remained in it for years” (Montessori in Standing, 1957, p. 29).

In 1899, as a result of Montessori’s work with and advocation for these children, a school was opened for them in Italy, and it was placed under Montessori’s direction for two years. It was this experiment that led Montessori to believe that the methods she used to educate these children could also benefit “normal” children. “I became convinced that similar methods applied to normal children would develop and set free their personality in a marvelous and surprising way” (Montessori in Standing, p. 30).

In this article, I will present some of these methods. Specifically, I will discuss how her methods support our current goals in mathematics education and how they differ from methods more

commonly seen in schools. From what I have read and observed, it seems that when implemented with fidelity, the Montessori method allows for differentiated instruction, collaboration, student choice and autonomy, which all support a deep learning of mathematics. I will also provide a brief overview of some of my own experiences using the Montessori materials.

The PhilosophyMontessori (1964) was deeply disturbed by the education system of her time, as she felt it was inhumane and ineffective:

Today we hold the pupils in school, restricted by those instruments so degrading to body and spirit, the desk—and material prizes and punishments. Our aim in all this is to reduce them to the discipline of immobility and silence,—to lead them,—where? Far too often toward no definite end. (p. 26)

In striving to discover a better way, she knew it would be necessary to begin by learning about children—their needs and natural tendencies—and that the only way she could do this was by observing them in an environment where they were free to make their own decisions:

The fundamental principle of scientific pedagogy must be, indeed, the liberty of the pupil;—such liberty as shall permit a development of individual, spontaneous manifestations of the child’s nature. (p. 28)

The intention from the very beginning was to create an environment that would be suitable for the observation of a child’s spontaneous learning and activity. Montessori had no idea what was to come from this experiment—in the following years she would learn how much the children were capable of and that they would thrive in such an environment.

by Rebekaah Stenner


Student Choice and Autonomy

During Montessori’s two years of experimentation at the first Children’s House, she observed that children were naturally drawn to certain materials in the environment. From this observation, she decided going forward to allow students to select whatever material they wished and work with it for as long as they liked. She moved the materials to low shelves so the children could access them, use them, and return them when they were finished. This freedom of choice became a fundamental component of the Montessori philosophy. Students often chose to work at challenging tasks for extended periods. This has profound implications for us today, as it means that if the environment and expectations are set up appropriately, students can handle freedom. In today’s classrooms, we typically operate as if the opposite were true. From the moment students enter our classroom, we tell them what to do. We direct everything, from where they sit to what they learn to when and how they get assessed. It seems that a natural consequence of this would be students leaving school unprepared to manage their lives, as we have not allowed them to develop autonomy. In a Montessori classroom, students are required to plan their week, set their own goals, direct their activities, manage independently during three-hour work periods, and inform the teacher when they are ready to be assessed on a concept. Lillard (2019) writes:

In Montessori classrooms “children have free choice all day long. Life is based on choice, so they learn to make their own decisions. They must decide and choose for themselves all the time… They cannot learn through obedience to the commands of another.” (Montessori in Lillard, p, 8)

Allowing more autonomy for children had many unexpected positive outcomes, perhaps the most notable being that the students developed an intrinsic motivation which before had been absent. Montessori (1967) described this phenomenon:

This inner liberation is accompanied by a new sense of dignity. From now on a child becomes interested in his own conquests and remains indifferent to the many small external temptations which would formerly have been so irresistible to his lower feelings… I was astonished when I learned that a child who is permitted to educate himself really gives up these lower instincts. I then urged the teachers to cease handing out the ordinary prizes and punishments, which were no longer suited to our children. (p. 59)

What Montessori discovered is consistent with recent research on motivation, which emphasizes the critical role of autonomy, and the lack of effectiveness of external incentives in motivating work on complex tasks (Pink, 2009).

Montessori (1967) also noticed that when given this freedom to spend extended periods of time on tasks, students tended to get incredibly absorbed in their work: “[…H]e becomes so attentive to what he is doing and so immersed in his work that he does not notice what is going on about him but continues to work […]” (p. 96)

Another important benefit of offering extended periods of self-directed work time is that it allows students opportunity for mastery. If students are not given enough time to master new material it can be detrimental to their learning, as mastery is a major factor in student confidence and motivation (Pink, 2009; Ryan and Deci in Lillard, 2019).

Under these circumstances, where children are offered freedom, ample opportunity for mastery, and are treated with dignity and respect, we might expect the need for traditional discipline to almost fall away. Montessori (1967) explains that, “[…] children are very well disciplined indeed when they can all move around in a room in a useful, intelligent, and free fashion without doing anything rude or unmannerly” (p.54).

Differentiated Instruction

In an environment where students are free to move about and choose what to work on, using the traditional approach of daily whole-group instruction no longer makes sense. As an alternative, most lessons are given individually or to small groups, and they are brief, concise, and flexible according to the response of the child.

A concert director must train his musicians one by one if he wishes to obtain from their collective efforts a noble harmony… In the ordinary schools, on the contrary, we place a person in charge who teaches the same monotonous and even discordant melody to instruments and voices of the most diverse character. (Montessori, 1967, p. 113)

This was a major divergence from the accepted pedagogy of the time as traditionally, all students would get the same lesson at the same time, regardless of their interests or ability. Teachers and administrators would demand compliance through systems of

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punishment and reward. The Montessori method was a shift away from this and toward a student-directed learning environment. “Activity has hitherto been the special competence of the teacher, but in our system it is left mainly to the child” (Montessori, 1967, p. 149).

One of the things I struggle with most as a teacher is having to teach content at a certain grade level to a class of students who are all at different levels and often with very different needs. We necessarily leave students either bored, or confused and frustrated. This often leads to students feeling powerless, unheard, and frustrated. School, the way we traditionally see it, does not foster a love of learning. “Most students do not feel capable, confident, and happy in (conventional) schools, resulting in a plethora of “positive psychology” interventions that try to help (Waters in Lillard, 2019, p. 19). On the contrary, Lillard (2019) reports that students in Montessori schools are generally confident in their abilities and happy at school.

The EnvironmentA Montessori classroom looks quite different from a traditional classroom. Some of the most obvious features are flexible seating, order, cleanliness and lack of clutter, and the various materials displayed on low shelves (see Figure 1). Montessori’s intention was to design an environment that would be most suitable for observing the development of a free child.

Our little tables and our various types of chairs are all light and easily transported, and we permit the child to select the position which he finds most comfortable. He can make himself comfortable as well as seat himself in his own place. (Montessori, 1964, p. 84)

Figure 1: A Typical Lower-Elementary Montessori Classroom (https://www.pinterest.ca/pin/574068283751196869/visual-search/)

The MaterialsThe Montessori mathematics materials are very different from manipulatives common in traditional classrooms, primarily because they bring with them a philosophy of implementation and are intended to be used together as a set, in a logical progression from the concrete to the abstract.

In a typical elementary classroom, a variety of mathematics manipulatives can usually be found including base ten blocks, pattern blocks, geoboards, counting bears, snap cubes, linking chains, and geometric solids to name a few. Usually the teacher will introduce a manipulative to help students understand a mathematics concept. For example, when teaching regrouping, teachers often use base-ten blocks. In middle or high school, teachers may use integer chips to help students understand addition and subtraction of integers. Once the teacher introduces the manipulative and models its use, she is then required to distribute it, manage behavior around it, and then make sure it is returned and sorted appropriately. This is a lot of work and can be time-consuming. Moreover, learning the correct use of the manipulative can be difficult for students, and they often struggle to connect the manipulative to the abstract mathematical idea it was meant to represent.

For several reasons, I have come to believe that the use of the Montessori materials as an alternative to traditional mathematics manipulatives may be more practical and pedagogically sound. First, students use the materials over time and for several concepts so that the teaching and learning of the material is worth the time investment. For example, the bead bars are first introduced in pre-school as an introduction to number and counting. Children can then go on to learn composing and decomposing numbers, place value, skip counting, and multiplication with these bars. Second, the materials are designed in such a way as to allow the student to focus on the intended mathematics—there are no extraneous features. Montessori (1967) explains:

How are we to isolate from many qualities one single one so that attention may be focused on it? […T]he objects are identical among themselves with the exception of the variable quality which they possess. (p. 101)

Laski, Jor’dan, Daoust, and Murray (2015) found that research supports this idea: “Recent research, however, suggests that manipulatives that represent real objects may actually impede learning” (p. 4). They go on to hypothesize that it may be the fact that teachers allow students to play with these manipulatives that is


causes them to be less effective. This implies a dual purpose which can be confusing and distracting for students.

Also, there is a progression in the materials from the concrete to the abstract. For example, the progression from the geometrical insets to the shape cards (Montessori, 1967).

Figure 2: Geometric Insets and Shape Cards

(http://www.montessoriequipment.ca/IFIT-Montessori-Metal-Insets-with-2-Stands-p/l.411.1.htm https://www.montessoriequipment.ca/IFIT-Montessori-Cards-For-Geometric-Cabinet-p/s.021.2.htm)

Interestingly, Montessori discovered that it was often helpful for students to use their sense of touch in making this transition to the abstract. She offers an example: “His recognition of these [figures of contrasting shapes] is greatly helped when his visual perceptions are associated with those of touch” (1967, p. 133). This was the inspiration for the sandpaper letters and numbers (Figure 3).

Figure 3: Child working with sandpaper numbers

(https://www.montessorialbum.com/montessori/index.php?title=Sandpaper_Numerals)

According to Laski, Jor’dan, Daoust, and Murray (2015), “Children who attend Montessori programs in early childhood demonstrate high levels of mathematics achievement” (p. 2). They found that current research on manipulatives supports the use of Montessori mathematics materials and, that, “The benefits of the Montessori approach to mathematics learning in early childhood may, at least in part, be due to its effective use of manipulatives” (p. 2).

Personal ExperienceAs part of my research on the Montessori Method, I was able to use some of the materials with students in my school district. Below is an introduction to the materials I used and a brief summary of what happened.

The Bead Bars

The bead bars are primarily used to help students learn multiplication. The bars are coloured beads on wire, representing the numbers 1 to 10. In order to help students differentiate between the bars, each quantity is a different colour; the one-bars are red, the two-bars are green, and so on (see Figure 3).

Figure 4: Montessori Bead Bars (photo taken by author)

I am currently teaching a grade 8 mathematics intervention course and one of my students is only comfortable with mathematics up to about a second-grade level. Before being introduced to the beads, she had no concept of multiplication although she was able to use a multiplication chart to determine products. After a day of familiarizing herself with the beads and the different quantities, I taught her how to use the beads to perform repeated addition. She is now able to use the beads to perform any single-digit multiplication, which means she can now be included in activities the rest of the class is doing.

Although I have not yet had a chance to work with the Montessori checkerboard, it is the next step students would progress to with the bead bars. Students in the 6-9 classroom use this material to learn how to multiply multi-digit numbers. It has also been successfully used to teach struggling seventh grade students multi-digit multiplication in an intervention setting (Donabella and Rule, 2008).

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Figure 5: Student Using Bead Bars to Perform Single-Digit Multiplication (photo taken by author)

The Number Rods

The Montessori number rods are a series of wooden rods that increase in length from one to ten units. Each rod is painted in alternating red and blue stripes to show the different units (Figure 6).

Figure 6: Small Montessori Number Rods (photo taken by author)

These rods are the first material used for counting in the Montessori curriculum. This is because according to Montessori (1969), students tend to have difficulty understanding counting the way we traditionally present it to them – by pointing to various objects in a set saying “one, two, three”, etc. Montessori noted that students would more commonly name each successive object as “one, one, one”. She explains:

The fact that a group is enlarged through the addition of a new unit and that this increasing whole must be considered constitutes the chief obstacle for children of three-and-a-half to four in learning how to count. (p. 263)

I introduced the number rods to small groups of kindergarten students using what is known in Montessori classrooms as a three-period lesson: “This is one, this is two”, etc., followed by “Show me one, show me two”, and finally, “What is this?” (while holding up one of the rods). Once I was confident that students could select or identify the appropriate rod according to its number, I mixed up the rods on a mat and asked them to make a staircase like the one shown in Figure 6.

I had students work with the rods in pairs as there were two sets. Once both students finished making the staircase, I combined the sets and had them try to work together to make two staircases. I was surprised to see that several students counted the sections in order to find out which rod to place next, rather than recognizing that each should simply be one unit longer than the previous one.

The Cylinder Blocks

The cylinder blocks are a set of four wooden blocks containing cylinders of different sizes such that only one attribute varies in each block. For example, one block contains cylinders with different heights but the same diameter. Holding one attribute constant while changing another allows students to focus on the attribute that is changing.

Figure 7: Small Cylinder Blocks (photo taken by author)

I thought a nice collaborative activity might be to mix up the pieces from the four blocks and have students sort them. To my surprise, many of the kindergarten students had difficulty with this exercise.


When they did complete the task, I challenged them to redo it with their eyes closed. No one was able to complete this although a few students attempted it.

ConclusionsIf the Montessori model holds such potential, why are we not seeing it adopted more widely? Lillard (2019) suggests that it is because of its “cultural incommensurability.” She agrees with Cossentino that “Montessori requires a paradigm shift in how one thinks about school” (Cossentino in Lillard, p. 17). As we are unlikely to see such a paradigm shift in the near future, it seems more reasonable to think about ways we might incorporate some aspects of the Montessori philosophy into our classrooms. I can envision a school making use of focused manipulatives over a number of years, offering students more opportunity to make their own choices, providing differentiated instruction and assessment to students based on those choices, and creating an environment that is organized, uncluttered, and welcoming to the students.

ReferencesDonabella, M. A. (2008). Four Seventh Grade Students Who Qualify for Academic Intervention Services in Mathematics Learning Multi-Digit Multiplication with the Montessori Checkerboard. Teaching Exceptional Children Plus, 4(3), 1-28.

Laski, E. V., Jor’dan, J. R., Daoust, C., & Murray, A. K. (2015). What Makes Mathematics Manipulatives Effective? Lessons from Cognitive Science and Montessori Education. SAGE Open, 5(2), 1-8.

Lillard, A. S. (2019). Shunned and Admired: Montessori, Self-Determination, and a Case for Radical School Reform. Educational Psychology Review, 1-27. Retrieved from https://doi-org.proxy.lib.sfu.ca/10.1007/s10648-019-09483-3

Montessori, M. (1964). The Montessori Method. New York: Schocken Books Inc.

Montessori, M. (1967). The Discovery of the Child. New York: The Random House Publishing Group.

Pink, D. H. (2009). Drive: The Surprising Truth About What Motivates Us. New York: Riverhead Books.

Standing, E. (1957). Maria Montessori: Her Life and Work. New York: Plume.

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Mathematical Visualization in the Classroom: A Brief History

by Foster Matheson and Egan J Chernoff

IntroductionThe use of spatial visualization strategies in teaching has have been studied in regards to the teaching of mathematics since the late 1970’s (Fennema & Sherman, 1978). Within mathematics, spatial visualization is commonly required and often involves “the ability to mentally manipulate, rotate, twist, or invert a pictorially presented stimulus object” (McGee, 1979, p. 893). In addition to spatial visualization many researchers have supported the claim that other different forms of visualization play a key role within mathematical problem solving, even when such problems are not necessarily geometric or visual in nature (Polya, 1945). Further studies have promoted the idea that the incorporation of visual relationships with the teaching of mathematical concepts is positively related to student achievement in mathematics during elementary and high school years.

As a mathematical visualization tool, diagrams have long been used to assist in the learning of mathematical concepts. Krutetskii’s 1976 research, for example, had an impact in determining that visual-spatial components of learning affected the type of mathematical abilities a student may develop. Some of the basic concepts behind a visual-spatial approach to the learning of mathematics come from the idea that many mathematical concepts are indicative of very real and concrete situations. The addition of 1 plus 1, for example, can be taught through visualization to young children, in which a scenario plays out that involves the combination of two groups of one identical object into one group of two identical objects. Such a scenario can be presented and approached in a multitude of ways and allows students to relate the mathematical concept of 1 plus 1 to relatable, real-world experiences. Visualization in later grades follows the same approach, albeit with more complex mathematical subject matter.

Growth Within the FieldAn important milestone within the study of mathematical visualization in learning came in 1991, when both imagery and visualization were officially recognized as new areas of study at

the Psychology of Mathematics Education Conference in Assisi, Italy. A major study to come from this conference was Dorfler’s research, titled “Meaning: Image Schemata and Protocols,” which posited a theory of image schemata from which all mathematical visualizations occur. Figurative images represent concrete visual perspectives that could be a picture in the mind. Operative images operate within a series of changing events, such as the visualization of a sound wave resonating between two points. Relational images were also fluid in nature, except their fluidity existed spatially, not temporally. In such a case, one could imagine rotating a cube in their mind and being able to spatially visualize it from every possible position. Dorfler’s final image category was that of symbolic images, which exist as representations of something else, such as a mathematical formula. Dorfler’s claim was that each of these four types of image schemata acted as a carrier, or promoter of their corresponding visual representations. Among adolescent subjects, Presmeg’s 1985 research on a highly comparable topic revealed that the figurative, concrete approaches to learner visualization tended to be the most commonly utilized form of mathematical visualization.

Teacher Approaches to Visual MathematicsPresmeg’s 1991 report was titled, “Classroom Aspects Which Influence use of Visual Imagery in High School Mathematics.” Her three-year study was centered around thirteen high school mathematics teachers and their students. The thirteen teachers were categorized into three groupings based on their tendency of incorporating visual aspects within the classroom. These three groups were classified as follows: visual group, non-visual group, and middle group. The teacher participants fell almost evenly into these three categories. Teachers that fell within the visual group tended to make connections regarding mathematical concepts and other areas of thought, presenting such concepts as relatable to a larger whole. The teachers who fell within the middle group tended to focus largely on generalizations. Students in the classes of nonvisual teachers were encouraged to focus largely on pure memorization, resulting in limited mathematical


success. The overall findings of Presmeg’s study was unexpected by the researchers. Based on collected data, it was clear that visual learners mostly benefitted from being taught by a teacher who fell within the middle group. The reasoning behind this conclusion fell towards the idea that many of the students within the visual teacher’s classrooms often struggled with basic generalization skills. Such skills were primarily focused on within the middle group, and as such, tended to promote a strong conceptual base for student’s math knowledge, while also promoting visualization strategies as a secondary focus.

Computer VisualizationsThe 1990’s saw the progression of more accessible and affordable computer technologies. Advancing computer hardware and software made it possible to amplify areas of mathematics in ways never before accessible. In terms of mathematical visualization, the digital rendering of mathematical shapes and images meant that one could not only “externalize” mathematical images of the mind in brand new ways, but also manipulate them through software and share them with others (Palais, 1999). Objects and processes could

both be experienced visually in ways that were clearer and more concrete than ever before, and it was only a matter of time before the incorporation of this technology within a classroom setting would be seriously examined.

By the mid 1990’s studies and journals regarding mathematical visualization in the classroom placed a newfound focus on computer technologies and their potential for advancements in the field of math education. Thomas, Johnson, and Stevenson (1996) examined the use of computers in elementary and high school mathematics education, and concluded that with proper teacher accountability (including student accessibility and cost-effectiveness), such visualization tools could have enormous potential within the classroom. Borba (1995) specifically emphasized computer technologies as a potential approach to visual mathematics learning that does not necessarily rely on student’s prior knowledge of algebraic symbols or equations. Instead, for visual concepts such as graphing, visual interactions can prove cognitively meaningful to students without the need for symbols through the experience of virtual manipulation and

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exploration of mathematical objects. Education software that allowed students to translate, stretch, and reflect graphs, as well as experiment with geometrical objects, soon began to open up new possibilities for both learning and doing visual-based mathematics in the classroom. The effective growth of this technology (and those like it) continues to support elementary and high school mathematics curricula in both visual and non-visual areas of study.

Modern Approaches to Visual Problem SolvingMathematical problem solving is not as visually demanding as tasks in geometry or graphing. Nonetheless, the relationship between mathematical problem solving and visualization is a topic that began to be seriously considered. A 1994 French study by Baldisseri et al. was one of the earliest to examine this relationship within the classroom. Taking place among a group of 47 kindergarten students, Baldisseri’s team investigated visual representations in a scenario where the students were asked to solve the following question: ‘Greta has six balloons. Two of them were deflated. How many balloons does Greta have?’ Of the 47 students, 41 of them produced their answer by drawing only the objects mentioned in the problem (Greta, the balloons, or a combination of the two) without the use of numerical symbols. The remaining six students made use of numerical symbols in their answers: three of these students used a combination of symbols and drawn objects, while the other three students used only the symbols by themselves without an accompanying pictorial representation. The findings of this study emphasized the importance of visual representations in mathematical problem solving for students of this age, and prompted further inquiry.

Such inquiry came with E. Deliyianni et al.’s 2009 study titled “Pupils’ Visual Representations in Standard and Problematic Problem Solving in Mathematics: Their Role in the Breach of the Didactical Contract.” This study aimed to add to Baldisseri et al.’s findings by leading 38 kindergarteners as well as 34 grade one students through a series of questions that expanded upon the problem presented in the 1994 study. The researchers hoped to uncover a discrepancy in the way the two age groups depended on visual representations in order to go about solving a series of word problems. The problems given to the participants were categorized as follows: one standard addition problem (Baldisseri’s Greta problem), one standard subtraction problem, and two “problematic” problems in which there were no clear mathematical solutions. Student’s responses to these problems were analyzed both qualitatively and quantitatively over the course of two weeks, and the results found by the research team supported the age discrepancy that they had expected to find.

The standard addition problem saw 100% of the kindergarten students who answered correctly using some sort of pictorial representation in their answer (with just under half of those answers including numerical symbols), while only approximately 29% of the grade one students included a pictorial representation (with most of these answers being accompanied by a symbolic equation). Both grades’ answers for the standard subtraction problem yielded remarkably similar data. In addition, for both problems, it is worthy to note that the variety of pictures used by the grade one students was far less than the level of variety found among the kindergarteners’ work. The two problematic problems shared very similar results, in that the kindergarten students continued to rely on pictorial representations (both with and without numerical symbols), while the grade one students tended to rely overwhelmingly on the use of symbolic equations (with only about a third of these equations being accompanied by pictorial representations). These results exhibited rather interesting insights in regards to how young students make use of visualization strategies in mathematic problem solving. Further so, the drastic change that the role these strategies take within a single year of educational advancement implies that, as an educator, student age is perhaps one of the most critical factors to consider while taking a visual approach to the teaching of mathematical problem solving. Ultimately, further examination of the role that student age plays in visualization skills associated with mathematical problem solving has the potential to lead to more effective teaching strategies through the advancement of mathematics curricula. As Deliyianni et al.’s study focused solely on students between the ages of 5 and 7, it is clear that there is a place for continued research concerning the relationship between mathematical problem solving and visualization among other age groups as well.

Discussion and SummaryAs can be derived from many of the studies discussed in this paper, there are multiple factors to be aware of when considering a visual approach to the teaching of elementary and high school mathematics. Fennema and Tartre’s study exhibited that students within many elementary-level classes often tend to think either more visually or more verbally when it comes to working mathematically, with age playing determining roles in this relationship. Deliyianni et al.’s research also supports age as a determining factor in how students make use of visual strategies in mathematical problem solving. Such research, along with other studies discussed in this review, exemplify the importance as a teacher of both understanding where one’s student’s fall within this spectrum and being able to instruct in a manner that delivers in accordance to such needs. Consideration age is paramount in


determining how to approach the incorporation of visual learning within the mathematics classroom, yet one must also consider the nature of the content itself that is to be taught as well as the tools that are at one’s disposal (i.e. computer software) in order to teach such content effectively. Many mathematical topics require visualization, such as geometry, while other topics require less, such as many word problems.The final aspect of visualization to be aware of as a teacher is in regard to what types of visual schemata one might be incorporating into the classroom, and how students can benefit from any one type more than another in a given situation. Familiarization with the nature of such aspects of mathematical visualization in the classroom can be beneficial for both teachers and students and are highly worthy of consideration.

Since the late 1970’s the use of spatial visualization strategies has been studied specifically regarding the teaching of mathematics (Fennema & Sherman, 1978). Fennema and Tartre (1985) examined potential discrepancies within visual and verbal mathematic thought processes within elementary-level students, determining that age can play a role in this relationship. The topic of visualization in mathematics was officially recognized as an area of study at Assisi’s 1991 Psychology of Mathematics Education Conference, providing researchers with additional exposure and resources that would prove beneficial for future studies. Among these researchers, Dorfler (1991) solidified a theory of image schemata that would be referenced by future studies. Presmeg (1991) dedicated research towards the topic within the classroom by incorporating variables regarding teaching styles into her report “Classroom Aspects Which Influence use of Visual Imagery in High School Mathematics.” By the mid 1990’s a newfound focus on computer technologies and their potential for advancements in the field of maths education proved influential on studies and journals regarding mathematical visualization in the classroom. Recent studies such as Deliyianni et al.’s “Pupils’ Visual Representations in Standard and Problematic Problem Solving in Mathematics: Their Role in the Breach of the Didactical Contract” (2009) explored a potentially influential relationship between mathematical problem solving and visualization among young elementary-level students. Further exploration regarding influence of mathematical visualization upon student learning remains necessary in order to determine concrete conclusions that might lead to a higher understanding of related pedagogical approaches for mathematics educators.

ConclusionThrough this brief examination of the history of the field of visualization in mathematics, this paper aims to provide exposure to the field in relation to elementary and high school mathematics education. Many findings from research can continue to prove applicable within the mathematics classroom, and with proper consideration, such applications could potentially be used to benefit educators and students worldwide, Findings within the field of visualization in mathematics could continue to prove noteworthy, and potentially pave the way for the advancement of mathematics curricula, both within and outside of The Best Place on Earth, British Columbia.

ReferencesBorba, Marcelo C. (1995). Teaching Mathematics: Computers in the Classroom. In The Clearing House, 68.6 (1995): 333-34.

Deliyianni, E., Monoyiou, A., Elia, I., Georgiou, C., & Zannettou, E. (2009). “Pupils’ Visual Representations in Standard and Problematic Problem Solving in Mathematics: Their Role in the Breach of the Didactical Contract.” In the European Early Childhood Education Research Journal, 17.1 (2009): 95–110.

Fennema, Elizabeth, and Lindsay A. Tartre. (1985). “The Use of Spatial Visualization in Mathematics by Girls and Boys.” In the Journal for Research in Mathematics Education 16.3: 184-208.

Fennema, E., and Sherman, J. (1977). “Sex Related Differences in Mathematics Performance, Spatial Visualisation and Affective Factors.” In the American Educational Research Journal, 14, 51-71.

Mcgee, M. G. (1979). “Human Spatial Abilities: Psychometric Studies and Environmental, Genetic, Hormonal, and Neurological Influences.” In Psychological Bulletin, 86.5 (1979): 889-918.

Palais, R. S. (1999). “The Visualization of Mathematics: Towards a Mathematical Exploratorium.” In Notices of the American Mathematical Society, 46.6 (1999): 647–658.

Presmeg, N. C. “Classroom Aspects Which Influence use of Visual Imagery in High School Mathematics.” In F. Furinghetti (Ed.), Proceedings of the 15th International Conference, 3 (1991): 191-198.

Presmeg, N. C. “Visualisation in High School Mathematics.” In For the Learning of Mathematics, 6.3 (1986): 42-46

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Polishing: A Fresh Take on the Review Processby Marc Husband

Reviewing completed work is an important step in creating anything. The artist stands back from their canvas, the mathematician looks over their proof. But how can teachers make reviews a more significant part of the student’s learning process? After all, the artist often adds a few brush strokes to the painting, and the mathematician may improve their notation. What can we do to support reviewing mathematical solutions beyond simply repeating the solution or lesson contents? This article describes a review process called polishing where students take time to rethink, reproduce and explain their solutions.

The Well-Accepted Practice of Reviewing Completed WorkResearchers and practitioners agree that reviewing completed work enhances mathematics learning. In Polya’s early work (1973), he saw looking back at mathematics work as “reconsidering and reexamining the result and the path that led to it” (p. 14), and said that “some of the best effects may be lost if the student fails to reexamine and to reconsider the completed solution” (p. 6). Current standards and curricula authorities consider reviewing and analyzing student work to be a key practice for supporting students in developing their mathematical skills and knowledge (British Columbia Ministry of Education, 2016; NCTM, 2000; Ontario Ministry of Education, 2005). Reviews are an opportunity for students to analyze and critique mathematics work—a facet of the core competencies for critical and reflective thinking recommended in the BC curriculum (British Columbia Ministry of Education, 2016). The question is what teaching actions support students in actually rethinking their work.

Teachers may implement reviews in a variety of ways. They may ask students to review or look over their work on math tasks or tests to judge the reasonableness of their solutions, or to check their work for the correct answers. Teachers also may reiterate the correct solution and the main points of the lesson, or lead a class discussion to summarize the lesson. The pitfall is that reviews can become repetitions, rather than opportunities for students

to rethink their solutions to support understanding. My own experience illustrates how teachers can avoid this trap and enrich their practices using polishing activities.

My Experience with PolishingI have used several types of mathematics reviews, both in my early teaching work with elementary students and more recently with teacher candidates and in-service teachers. These approaches have always included going over completed solutions with a goal of consolidating students’ understanding and providing opportunities for checking their work. When reviewing content with my middle-school students, I simply asked my students to look over their work to see if what they had done made sense. Later I encountered polishing when reading an article about how Japanese teachers use a practice called Neriage or “polishing up ideas” in classrooms. According to Takahashi (2008), after students have worked on a task the teacher selects a few solutions to orchestrate a whole-class discussion. In this context, the teacher polishes the work by highlighting the parts of the student work that are closely aligned with the goal of the lesson (Takahashi, 2008). More recently I used a different form of polishing when performing “Math Jams”—an interactive method where students review a peer’s work and revise it to improve the clarity of the peer’s mathematical idea. In this setting, polishing is described as “adding or deleting text to make the thinking more clear and concise” (Norquay & Rapke, 2017, p. 29).

Fundamentally, polishing activities support student learning by revisiting and improving completed student work with the intention to share it with others. The details of these activities may vary in terms of who selects the work to be reviewed, who leads the discussion, and who does the polishing. The teacher and student roles in polishing can be adapted to suit a teacher’s practice. In my version of polishing, students polish their own work consistent with my emphasis on students doing and discussing mathematics in class. In the following sections, I discuss more specifically how students can polish their own work.


How Polishing Differs from Typical ReviewsPolishing is a classroom activity for improving how students show and explain their mathematics work to others that also causes them to think more deeply about their solution strategies. It is rethinking their own work, with the incentive of sharing it with fellow students, that differentiates polishing from teacher-led reviews or lesson summaries where students are listeners, rather than doers. When rethinking and rewriting a solution for others, students are critically analyzing and reproducing their own work. This enables them to clarify, reinforce and possibly correct their own understanding of the mathematics involved. In looking at other students’ work, they may also learn by comparing different solution strategies and alternative ways of presenting the mathematics.

The polished version of a solution is typically more neatly written with a clear sequence within the work, along with additional annotations or illustrations to help the reader. It also includes corrections when the student notices errors in their original solution. Importantly, the polished solution need not be short, elegant or even logical; it is simply the student’s best work on that mathematical task at that time.

In my teacher education classes, teacher candidates erase their original solutions and create new polished versions that they share with their classmates, often gaining new insights during the process. I studied this type of polishing in my PhD research about how new elementary teachers could deepen their mathematics understanding using the recommended strategies for teaching their future students. The example of polishing below draws on the data from the professional development course that was the setting for that research.

An Example of Polishing in a Professional Development CourseThis professional development course was a ten-day elementary mathematics course designed for 15 new graduates of the teacher education program at a large urban university in Toronto. The format and content of the course reflected the active-learning approach that I have used in my university methods courses for teacher candidates and my professional development courses designed for in-service teachers. The participants worked in small teams investigating elementary-level tasks. In addition, they acted as co-teachers, read about related research and looked back at their mathematics work each day.

During the course, the teachers were often asked to polish their work after they had generated multiple solutions for a math task.

As the final step in the lesson, they selected their preferred solution strategy, erased all of their blackboard work, and rewrote the selected solution so other participants could make sense of their ideas. A good example of polishing occurred on course-day six, when participants worked on a task called “ribbons and bows.” This task investigated the inverse relationship between division and multiplication using two units, metres of ribbon and number of bows, stated as follows:

You have six meters of ribbon. Each bow requires 5/6 of a meter. How many bows can you make? How much is left over?

Many of the participants used two methods to solve this task; some began by drawing diagrams, and others by using the division algorithm for fractions. When two different answers emerged—7 1/5 and 7 1/6—they all worked to understand and resolve how the two answers could make sense. Pat and Moira’s work and interactions illustrate how course participants generated multiple solution strategies, selected one to polish, and grew their understanding of mathematics while revising, expanding and explaining their mathematics solution.

Developing several solutions. To start, Pat and Moira worked on the ribbons and bows problem using a number line to represent the six metres of ribbon. They divided the line into one metre units, subdivided the metres into 1/6 pieces and then counted by fives, circling groups of five 1/6s to show how many 5/6 metre bows could be made. After 40 minutes of working on the task, Pat and Moira’s board space was full of different solution strategies for dividing 6 metres of ribbon into 5/6 metre bows and determining the remainder, as shown in Figure 1 below.

Figure 1: Pat and Moira’s Blackboard Solutions

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In addition to the number line, they used dots in an array to count the number of bows, and the algorithm for dividing fractions. On the top left of Figure 1, their number line solution resulted in 7 bows with 1/6 metre of leftover ribbon. Towards the bottom right of Figure 1, their division algorithm yielded 7 1/5 of a bow.

Polishing their preferred solution. After nearly an hour of work, I asked the whole class to determine how much more time they needed before sharing their solutions with colleagues. They laughed when I suggested that someone else might be able to make sense of their messy board work, and asked for more time to clean up their work and make it clearer. Then I asked them to spend about 20 minutes polishing one of their solutions, making it “beautiful” like they might expect from a future student.

Moira and Pat decided to polish their solution that used dots organized into an array. As Pat was redrawing the small circles, she started talking to Moira and recording her work on the board. Her polished diagram included 6 horizontal groupings of 5/6th metre, plus one vertical grouping of 5 leftover 1/6 metre sections, bringing the total number of bows to 7.

At this point, I saw Pat and Moira at the board polishing their work. Pat was saying and recording “7 bows,” when I interjected:

Marc: So that’s one sixth leftover, right? (pointing to the ungrouped circle at the end of row 6) And, if you put that into one bow?

Pat: Oh, 1/5th of a bow.

Moira: Yeah, see we had it! We didn’t know why it worked.

My question provoked Pat to notice that 1/6 of a metre was equal to 1/5th of a bow. Moira recognized that they had found this solution in their previous work without understanding how the 1/5th was connected to the 1/6th. Pat later confirmed in her journal that this question enabled her to make the connection between the two units: “the 1/5 represents the leftover ribbon that makes one bow.”

Pat and Moira completed polishing by adding another vertical grouping of four circles with one missing circle at the bottom. The missing circle represents the one piece of ribbon that would be needed to complete another bow— 1/5th of the bow. Their polished solution, shown below in Figure 2 to the right of their original solution, now included the 1/5 of a bow not present in their original image, in addition to the 1/6 metre of a ribbon included originally. This new version now clearly illustrated how the task could have two answers depending on the units—it was either 7 1/6 metres of ribbon or 7 1/5 bows. The polishing exercise enhanced Pat’s understanding of the connection between the two units, as she said, “I was able to compare both the 1/6 and the 1/5 visually.”

Figure 2: Polished Version

Improving their understanding. The polished result enabled course participants to solve the puzzle of having two different answers, 7 1/5 and 7 1/6. Some gained new insights by having a fresh look at their own solutions, and others found a peer’s solution to be more helpful than their own. For example, when Barbara reviewed Pat and Moira’s polished solution, she said that seeing the 1/5th drawn “separately” was a moment where she made the connection between the two different units, the ribbons and the bows.

This example illustrates how polishing fostered mathematical understanding in a specific course setting: a professional

Figure 2: Original Array Solution


development course for new teachers that was highly interactive and focused on connecting their mathematical ideas. Erasing and rewriting their work near the end of the lesson provided both the time and motivation for the teachers to rethink and present clear solutions that their peers could follow easily. Sometimes participants even colour-coded their notes with diagrams to make it easier for the reader. The participants were fully engaged in doing mathematics. In fact, their interactions and results were similar to those you might expect from school students.

How Polishing can Enrich Mathematics Teaching PracticesUsing polishing ensures that students spend enough time developing and expressing their mathematical ideas. Even after students think they have finished their work, there is often more to be learned. Polishing offers them a way to find errors, correct misunderstandings, and potentially see connections between their solution methods and other students’ methods. When student interaction is maximized, it is also an opportunity for them to rewrite, revise and explain their solution to others—essentially rethink the question and the mathematics they applied in solving it. With polishing, a teacher creates a thinking classroom, where students are “thinking collectively and learning together, and constructing knowledge and understanding through activity and discussion” (Liljedahl, 2016, p. 362).

Incorporating polishing into a teacher’s practice. Polishing reviews could be particularly beneficial for new mathematics teachers or those seeking an alternative to traditional whole-class review discussions. It provides an approach for focusing on student work, whether the teacher leads the process or students take the lead. Using the different forms of polishing could, in fact, provide a step-wise approach to increasing the focus on student work and student engagement in classroom activities. A teacher could start by leading the process using examples of student work, then progress to inviting students to present their own work, and finally asking students to both polish their own work and present it to classmates. This transition to polishing also supports students as they become accustomed to sharing and discussing their solutions in the classroom.

To add variety to their classes, teachers can ask students to polish each other’s work, while being sensitive to students’ readiness to having their solutions revised by their peers. To maximize polishing benefits, however, I recommend that students polish and present their own solutions, as described in my research example. This ensures that the process focuses on the students’ own work,

and challenges them to notice possible improvements and see connections between solution strategies.

As teachers gain experience with polishing, they learn, as I have, that careful task selection can turn reviews into a powerful addition to their teaching practice. In addition to supporting the goal of the lesson, a task ideally draws out potential points of confusion and, as Boaler (2015) suggests, elicits a range of solution strategies from informal to formal to enhance the learning process for everyone. Then polishing student work can reveal gaps in student understanding and help them make mathematical connections. In the research example discussed in this article, the task was not just about finding out how many bows can be made out a length of ribbon—a division of fractions exercise—it investigated the more easily missed point about the relationship between two units, ribbons and bows.

When used routinely in a classroom, polishing can help students develop positive habits and attitudes related to mathematics, like: looking over work to make sure it is understandable, discussing solutions with peers, seeing corrections as part of the process, and seeking clarity in thinking and recording solutions. Arguably, a polishing review process may also diffuse anxiety and build student confidence in their mathematics skills, contributing to their mathematics identity and to their future pursuit of the subject.

Using polishing to rethink and improve mathematics solutions adds specific activities to the well-accepted review practice and fosters engagement in classrooms at all educational levels, regardless of the teacher and student roles being used. With polishing, reviews can become the highlight of a mathematics lesson, rather than just a repetition of a mathematics solution. Taking a fresh look at solutions and improving them is invigorating for students and their teacher because it builds on students’ work and promotes an evolution of their thinking live in the classroom.

ReferencesBoaler, J. (2015). Mathematical mindsets: Unleashing students' potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: Jossey-Bass.

British Columbia Ministry of Education (2016) Critical and Reflective Thinking. Retrieved from https://curriculum.gov.bc.ca/competencies/thinking/critical-and-reflective-thinking

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.),

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Posing and Solving Mathematical Problems: Advances and New Perspectives. (pp. 361-386). New York, NY: Springer.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.

Norquay, N., & Rapke, T. (2018). Math Jams: Students analyzing, comparing, and building on on another’s work. OAME Gazette, 56 (3), 25-30.

Ontario Ministry of Education. (2005). The Ontario curriculum, grades 1–8: Mathematics (revised). Retrieved from http://www.edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf

Polya, G. (1973). How to solve it. Princeton, NJ: Princeton University Press.

Takahashi, A. (Eds.). (2008). Beyond show and tell: Neriage for teaching through problem-solving–Ideas from Japanese problem-solving approaches for teaching mathematics. Paper presented at the 11th International Congress on Mathematical Education, Monterrey, Mexico.


How Many? A Review from The Kitchen Counter

Reviewed by Nat Banting & Henry Banting

Most of the children’s books in our home spend their time on the shelf next to Henry’s bed, with the ones earmarked for tomorrow’s bedtime routine laid out on top. A select, well-loved few are scattered across the floor close to the window, the location where they were last read. In our collection, you will find many counting books, one of which is Christopher Danielson’s How Many? How Many? is a different kind of counting book. While the book is an amazing tool for classrooms and can be purchased with a teacher’s guide, it can also sponsor rich mathematical conversations at home. Fittingly, then, this review was woven together from a conversation between us (a father and son) while reading together at our kitchen counter—a setting that I believe Danielson would approve of. Here, we offer two themes that consistently stand out for us as we read the book together. Henry’s unedited contributions are provided in italics, and Nat’s commentary is interspersed throughout for the sake of clarity1

Less is MoreUnlike other counting books, How Many? doesn’t count for the reader. You don’t need to read the whole book. So, you can just find a page to start counting, even in the middle or with eggs. Well, I can count other stuff if somebody tells me to or, if I want, I can see one handle, four of these lines with the pan, or two bowls. Danielson crafts possibility into each page and harnesses the sensory experience with a simple invitation: how many? With so many possibilities, relationships emerge organically and so eight eggs come, like, in four packs, and when they’re cracked they come into eight. All of them are cracked. The minimalist query of “how many?” invites explosive potential, gives the book inquisitive longevity, and leaves room for discoveries upon re-reading because I can even think there’s five eggs, so these, the middle of the egg yolks is five. As the children play—yes play—with numbers and units, they re-count and justify their new ways of seeing the image on the page. The

1 Clarity, of course, for the adult reader. Henry sees no need for Dad’s commentary.

Two pages from How Many? Images provided courtesy of Stenhouse Publishing.

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structure becomes so familiar that when a new page has grapefruits, I’ll just count them and then I can see how much. So there’s six ones and they’re all cut opened, so that makes 1-2-3-4-5-6-7-8-9-10-11-12 of the grapefruits. Danielson’s unobtrusive stance combines simple invitations with rich opportunity to sponsor multiple counts and re-counts. Each page acts like an entire counting book, where a child could easily count both six and twelve on a single page because, like, all six of them are twelve when they’re all cut up. In this way, less direction results in more opportunity.

Space for StoryThe images in How Many? encourage kids to look and count them in my brain, then say it out loud if I want; however, the images also build on each other in ways that support a connected narrative. The eggs are covered in pepper and I think a little bit of salt, and it is details like these that draw out curiosity from the child. We can count that there are two spices, but we then encounter a problem, so I can’t count all the salts because there’s too many dots. This leaves us to wonder if they would put salt on those eggs so it’s impossible

Henry and Nat talking mathematics with the help of Nat’s prototype of upscale pattern blocks (http://natbanting.com/upscale-pattern-blocks/).

to count it. As the book unfolds, so does the imaginary narrative between the child and the author. Sequences of pages include similar images, like the three consecutive pages containing eggs followed by the three showing arrangements of grapefruit. The correlation makes it seem like they finished their other pack of eggs, and their other pack just had one. And they got a new pack and they just used four. Luckily, they are left with more eggs to crack, because oh my, they cut up way, way too much grapefruit for juice. Intrigue is added as the counting is enveloped in the developing story, which might include having maybe two people with just four eggs but six grapefruits. I don’t think that will work. The book comes to fruition with a wide-angle image of a kitchen containing all the motifs imaged throughout. This is a beautiful way to end because it unites the emerging narrative and requires the readers to check to see if our countings change. While the front cover advertises a counting book, and it certainly is that, this book is also, undoubtedly, a story book.

Suffice to say that the book receives our highest recommendation. The interaction begins just by counting and looking. Find which


page you like and which things, I think, you actually like to count. Placing the onus of what to count on the child grants parents a glimpse into which distinctions are relevant for the child, and it is through the distinction of different units that this book contains the potential for rich, connected mathematical conversations. It is filled with nuanced brilliance, and I really like it because it’s it so, like, cool and fun. The thinking and engagement sponsored by the

book is evidenced through spontaneous episodes of counting, like the one we had after we drew this picture of the biggest hockey rink with two benches. As you look at the picture, please ask yourself a simple, yet powerful, question: How many? If you need a place to start, perhaps begin with all the guys, so this one’s faceoffing. So 1-2-3-4-5, so there’s five guys each. 1-2-3-4-5.

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How Many? A Professional Learning Series for K-2 Teachers

Reviewed by Janice Novakowski

How Many? A Counting Book Christopher Danielson Stenhouse, 2018 Image source: stenhouse.com

K—2 teachers in the Richmond School District participated in three afterschool professional learning sessions focused on big ideas of mathematics. Each teacher received a copy of Christopher Danielson’s book How Many? A Counting Book, as well a copy of the accompanying teachers’ guide.

Between sessions, teachers were asked to read specific sections of the teachers’ guide as well as use some of the images for unit chats with their classes. A unit chat is a type of number talk that focuses on both “how many” and how many of what–the unit that is being counted. For example, a picture of three dice with five pips showing on each could have a count of three dice or fifteen pips.

How many?

During sessions, the teachers shared their reflections and learning from their readings and experiences. Teachers brought photographs and video clips of their students engaged in unit chats and other counting routines such as counting collections, choral counting and count around the circle. Teachers shared and discussed how their students responded to these routines and what they learned


about their students and their understanding of our number system and counting. These counting routines were a focus of this series and formed an instructional focus to support a district early numeracy assessment.

Throughout the time we were meeting, our district was piloting a new K-2 numeracy assessment tool to support instructional planning, and teachers used the How Many? teachers’ guide to inform their observations and next steps with their students. For example, most of the assessment is based on performance tasks and teachers’ observations and the teachers’ guide provided pedagogical content knowledge for our teachers to assess and respond to where students were in their understanding of early number concepts.

Teachers considering the possibilities that emerge with the question “How many?”

The How Many? teachers’ guide contains a chapter on important number learning concepts such as quantity, number language and numeration, as well as the mathematical understanding involved with subitizing, counting and place value. Another chapter provides many suggestions for how to use How Many?

with children and in the classroom, and includes prompts and questions to nudge student thinking and communication. A final chapter titled, “Children are Brilliant Mathematicians” shares several stories of interactions with children that focus on their mathematical thinking and weaves in some very big mathematical ideas, including a final story that highlights “counting is real mathematics.”

As suggested in Chapter 3 of the teachers’ guide, teachers and their students took their own How Many? photographs and we collated these to create our own digital How Many? book. The collection we have so far can be found here, best viewed via Chrome:

How Many? Book Creator book https://read.bookcreator.com/VAl7lS8GQqQclrmGIqJ8ZnlVyZB3/hEFsi-BJRtSX1Q9zHB4NNQ

These photographs were also used to create posters in both English and French for classrooms and schools to encourage families to talk about mathematics together.

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How Many? is now available in many of our schools’ libraries and as part of our district’s Math Play Space.

A blog post about this professional learning series can be found here: https://blogs.sd38.bc.ca/sd38mathandscience/2018/12/19/big-mathematical-ideas-for-k-2-2018/


Problem SetsContributed by Debbie Nelson and Richard DeMerchant

K–3 Problem SetOpen conversations with students will bring any one of these primary problems to life in the classroom. Use the resources you are familiar with: a number path (K–1), a number line (2–3), counters, Rekenreks, blocks, dry erase boards for drawing pictures, and offer a classroom walk-about to contemplate different possibilities.

Consider posting a visual for the first three problems as a way to provide story context to bring student thinking and community to the forefront. Discussions in mathematics can improve students’ capacity for mathematical thinking and reasoning. After students have worked together to understand and represent their thinking, teachers might ask: Why is the solution important? Can you explain it to others? Is there another way you might solve it? What might you change to solve a similar problem? What tool supported you and your partner’s thinking? How can you show your model?

When facilitating these discussions, invite the children to share the question which they found most interesting or the question which was most challenging.

The Bear and the Eagle A bear and an eagle are having a contest to see who can catch the most fish. The bear catches three salmon and six trout. The eagle catches six salmon.

How many trout would the eagle have to catch to have the same number of fish as the bear?

Source: NCTM Problems of the Week, https://www.nctm.org/pows/

Yummy Gummy BearsThree friends are walking home from school. One of them has 13 gummy bears left in her lunchbox. She wants to share them with her friends. She wants each girl to get the same number of gummy bears.

How many will each child get? Will there be any left over?

What if they have 17 gummy bears? How many will they each get? Will there be any left over?

How many gummy bears would mean there are no gummies left over?

Source: Adapted from NCTM Problems of the Week, https://www.nctm.org/pows/

Some Birds’ Eggs and a Monkey MunchingThree birds lay some eggs. Each bird lays an odd number of eggs. Altogether they lay 19 eggs. How many eggs does each bird lay?

Three monkeys ate a total of 25 nuts. Each of them ate a different odd number of nuts. How many nuts did each of the monkeys eat? What other ways are possible?

Source: BEAM Stem, https://www.stem.org.uk/primary-maths

A Number Path

Use a number path to show how to illustrate 8 + 7. Use a different number path to show how to solve it another way.

Share and show your thinking for 15 – 9.

Describe another number path.

Grade 4–12 Problem SetSuppose you are working on a unit on adding which could include whole numbers, fractions, decimals or polynomials depending on the grade level. You might use a quiz to check the understanding of students. No surprise–the students follow the procedures they recently learned and come up with the solution. However, imagine another scenario where a problem is presented which is out of context. It is not tied to a particular unit of study. Students must dig into their mathematical tool kit and pull out a tool that will work

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Problem 1: Perimeter and AreaCreate two rectangles so that the first has exactly twice the perimeter of the second and the second has exactly twice the area of the first.

Source: Orlin, Ben. (2018) Math with Bad Drawings: Illuminating the Ideas that Shape Our Reality. Black Dog and Leventhal Publishers.

Problem 2: DivisorThis puzzle is a variation of Sudoku. In this variation the numbers 1 to 9 are placed in each row, column and block just as in a standard Sudoku. An additional challenge comes from the consideration of divisors: A< B indicates that A divides B, and A<B indicates that A is less than B.

Source: Source: Su, Francis. (2020). Mathematics for Human Flourishing. Yale University Press

Sudoku variation puzzles can be found in a series of books, Brainfreeze Puzzles, by Philip Riley and Laura Taalman. Visit https://www.gathering4gardner.org/g4g10gift/puzzles/Riley_Philip-Sudoku_Variation_Puzzles.pdf for examples of other Sudoku variations.

Problem 3: Piece of CubeThere are five people wanting to eat a cube piece of cake. The cake is homogenous on the inside and has equal-depth delicious icing on all five of the exposed sides. How can the cake be cut so that everyone has the same volume of cake and the same surface area of icing? Extension: How could the cake be cut for six people? For n people?

Source: Parker, Matt. (2014) Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux.

Problem 4: Area MazesDetermine the value of the length of the side marked by a “?” in the figure below. If your calculation creates a fraction or decimal, STOP and look at it another way. Area mazes can be solved using whole numbers only. However, do not assume that every length or area in the puzzle must be a whole number.

Source: Inaba, Naoki and Murakami, Ryoichi. (2017) The Original Area Mazes. Gakken Publishing.

Problem 5: Chess CoinsWhat is the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most four coins? Note: Only one coin is allowed per square.

for them. One student may decide on one way to solve the problem while another student may choose a completely different approach.

For several months this year, the Middle School at St Michaels University School in Victoria posted problems on a bulletin board for students to solve each day. For those times when school was not in session we posted in a Google Classroom, which has been particularly effective during the pandemic. A leaderboard where students earn points for solving problems was created to acknowledge the hard work of students and give recognition to those who put in the effort. Many students enjoy the competition and there is a sense of accomplishment when they solve a problem they consider “hard”.

Encouraging students to play with mathematics outside of the classroom is as important as the work that is completed in the class. There are many times at break or lunch when I could see students asking each other to work together on a problem or verify their solution. It warms my mathematical heart.


Extension: What if each row, each column, and each long diagonal contains at most five coins?

Source: https://brilliant.org/

Problem 6Using the digits 0 to 9 at most one time each, fill in the boxes to make one function with no real roots, another function with one real root, and the last function with two real roots.

Source: https:www.openmiddle.com

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Math Links

Spring 2020 Mathematics Websites

https://wcln.ca

The Western Canadian Learning Network (WCLN) is a consortium of school districts that work together to produce online learning resources for teachers. Until the end of the COVID-19 crisis, the resources from the WCLN website, which are aligned to courses in the BC Math Curriculum, are available for free for any school in BC.

https://dreambox.com/canadaDreamBox is an online mathematics learning platform for students from kindergarten through Grade 8. Parents who sign up before April 30th can get a free 90-day trial for DreamBox, so this is a good resource to share with parents who may be looking for more independent math activities they can do at home. Disclaimer: I am a curriculum designer for DreamBox.

https://www.illustrativemathematics.org/distance-learning/Illustrative Mathematics, which creates high quality and open-source mathematics resources for teachers has created a page where they pool resources teachers may find helpful, especially those new to online learning. They also describe strategies for using their free curriculum resources with students.

https://twitter.comMany of you likely miss chatting with your colleagues and are looking for support and ideas to help you through this crisis. If you go to Twitter and search for #MTBoS (Math Twitter BlogoSphere), even if you don’t have an account you can see what mathematics teachers around the world are saying. Since nearly all of them have recently moved to online teaching, you can find out what they are learning about this new experience.

https://www.nctm.org/100/Since the National Council of Teachers of Mathematics was forced to cancel their annual conference, they have opted for inviting 100 speakers, on 100 consecutive days, to give their presentations online, for free. Every presentation will be recorded and available up until the October conference in St. Louis.

Links selected and described by David Wees (http://davidwees.com). Previous websites can be viewed here: http://davidwees.com/m/mathwebsites

Vector • Spring 2020 52

The BCAMT sponsors awards in three categories (Outstanding Teacher, Ivan L. Johnson Memorial, and Service) to celebrate outstanding achievements of its members. Winners are honoured at a BCAMT conference and receive a commemorative plaque.

AWARDS & CRITERIA

Outstanding Teacher Awards (Elementary; Secondary; New Teacher with less than five years teaching experience)

• shows evidence of significant positive impacts on students, staff and parents; • has initiated innovative and effective programs in their classroom, school, district, or province (teacher research,

technology, active learning, assessment, etc.);• has and continues to demonstrate excellence in teaching mathematics regularly in British Columbia (teaching

style, knowledge of the curriculum, current curriculum trends, etc.); • has made contributions to mathematics education at the school, district or provincial level. This may include

workshops, seminars, conferences, community projects, curriculum development, publishing etc.);• is not a current member of the BCAMT Executive.

Service Award

• has provided extraordinary service to mathematics education as an active member of the BCAMT for a significant period of time

Ivan L. Johnson Memorial Award

The Ivan L. Johnson Memorial Award is awarded in honour of long-time BCAMT executive member Ivan Johnson. Ivan donated money to the BCAMT for an award in which the recipient will receive significant funding to cover costs of attending the NCTM Annual Conference.

• inspires teachers to try new ideas that improve the quality of mathematics education; • consistently seeks ways to innovate practices in the mathematics classroom; • actively engages in professional dialogue involving mathematics pedagogy; • is not a current member of the BCAMT Executive, but is a member of the BCTF.

Note: Nominees for the BCAMT Outstanding Teacher Awards will automatically be considered for this award. Previous winners of BCAMT Outstanding Teacher Awards may also be nominated. Recipients of this award are expected to contribute an article to Vector.

Teachers Awards Information


SELECTION PROCESS

• All nominations are reviewed by the BCAMT Awards committee who recommend the recipients to the BCAMT Executive for ratification;

• Each nomination is considered for two years, after which time the application can be re-submitted with updated information.

HOW TO NOMINATE

Required documentation:

1. Nominators will fill out a short nomination form at www.bcamt.ca/nominations2. Nominees will be contacted at a later date and asked to provide a CV and 1-2 reference letters from a colleague

or administrator.

Deadline: March 31, 2021


Each year the BCAMT offers grant funds to its membership to be used for furthering mathematics education in BC. These initiatives must meet the Goals and Objectives of the BCAMT (see www.bcamt.ca/goals-and-objectives), and are not meant for individual professional development.

We encourage you to apply for a grant, with a maximum amount of $1,500, using the form on our website: www.bcamt.ca/grants.

All applications must be received no later than November 15, 2020. Applicants will be informed about funding after approval by the BCAMT Executive.

The BCAMT values the sharing of ideas and expects that successful applicants will:• Write a 150-word summary explaining the initiative and results and include photos suitable for publication.

Funding for successful grants will be distributed as follows:• If the initiative requests for $1000 or less, 100% of the funds will be provided upon approval by the BCAMT

executive.• If the initiative requests more than $1000, 75% of the funds will be provided upon approval by the BCAMT

executive, with the remaining funds being released upon completion of the grant requirements (see above).

Application steps:1. Complete the Grant Application Form found at www.bcamt.ca/grants.2. Upload or email a one-page rationale for the funding request that outlines:

a. The goals of your initiative (SMART goals: specific, measurable, attainable, realistic, timely).b. Details of your initiative, including estimated number of teachers and / or students affected (breadth of impact) as well as likelihood of success (depth or duration of impact).

3. Upload or email a detailed budget (expenses, other funding etc.).

We look forward to receiving your application!

BCAMT Grants 2020

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spring 2020 volume 61 issue 1 · teacher's near miss by susan robinson 27 the montessori...

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