2020 | issue 1 teacher of mathematicswismath.org/resources/documents/wtm...

WISCONSINMATHEMATICS COUNCIL, INC.2020 | Issue 1WisconsinWisconsin

TEACHERTEACHERof MATHEMATICSof MATHEMATICS

Wisconsin Teacher of MathematicsThe Wisconsin Teacher of Mathematics (WTM) is the official journal of the Wisconsin Mathematics Council

(WMC). Annual WMC membership includes a 1-year subscription to the journal. WTM is a forum for the exchange of ideas. The opinions expressed in this journal are those of the authors and may not necessarily reflect those of the Council or editorial staff.

Journal Governance

WTM is overseen by an Editorial Board, which consists of the Editorial Team and the Editorial Panel. The Editorial Team, led by the WTM Editor, leads the review, decision, and publication process for manuscripts. The Editorial Panel reviews manuscripts as requested by the editor, assists in setting policy for the journal, and gathers feedback from readers. Serving on the Editorial Panel is a 2-year commitment.

The Editorial Board is chaired by the WTM Editor and has two regular meetings during the academic year: a virtual meeting in Fall/Winter and an in-person meeting at the WMC Annual Conference in May.

Editorial Team

• Joshua Hertel, Editor, University of Wisconsin–La Crosse

• Matthew Chedister, Associate Editor, University of Wisconsin–La Crosse

Manuscript Submission GuidelinesThe WTM Editorial Board encourages articles from a broad range of topics related to the teaching and learning of mathematics including submissions that focus on:• Engaging tasks that can be implemented in the preK–12 classroom,• Connecting research and theory to classroom practice,• Showcasing innovative uses of technology in the classroom,• Work with preservice teachers in the field, and• Current issues or trends in mathematics education.

Other submissions not focusing on these strands are also welcome. Articles should be 2000–4000 words and follow APA 6th Edition guidelines. Figures, images, and tables should be embedded. Submit articles in .doc/.docx format to [email protected].

mailto:[email protected]

Wisconsin TEACHER of MATHEMATICSWisconsin TEACHER of MATHEMATICS

Table of Contents

2020 | Issue 1

Editorial 1

President's Message 2

Learning Trajectories: A Road Map for PreK-3 Mathematics Instruction DeAnn Huinker and Melissa Hedges

3

Algebraic and Geometric Misconceptions and Misinformation in ClassroomsCatheryne Draper and Johnny W. Lott

11

The Mathematics of the AvengersDavid Ebert

17

Exploring Cross-Sections in a Computer Environment – Using 3D BuilderSengfeng Liang, Brandon Freeman, and Laura Leahy

19

Examples of how to Incorporate the GAISE Report Recommendations into TeachingAngela L.E. Walmsley

23

Addressing Early Algebra Misconceptions and Misinformation in ClassroomsCatheryne Draper and Johnny W. Lott

32

1 Wisconsin Teacher of Mathematics | 2020, Issue 1

Greetings!

Welcome to another issue of the Wisconsin Teacher of Mathematics! 2020 has certainly presented us with a number of challenges as we strive to provide quality mathematics instruction in amidst ever changing situations. During this time, we invite our readers to enjoy articles including:

• DeAnn Huinker and Melissa Hodges discussing the importance of learning

trajectories in the early elementary years.

• Catheryne Draper and Johnny Lott identifying common misconceptions in algebra

around absolute value and parallel lines.

• David Ebert connecting the plot of the popular movie Avengers Endgame to a

discussion of probability.

• Senfeng Liang, Brandon Freeman, and Laura Leahy examining how 3-D Builder can

be used to explore cross sections.

• Angela Walmsley discussing how to incorporate ideas from the GAISE Report into

teaching.

From

the E

dito

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om th

e Edi

tors

Joshua Hertel Editor

Matthew Chedister Associate Editor

Jenni McCool Associate Editor

2Wisconsin Teacher of Mathematics | 2020, Issue 1

Colleagues,

This edition of the WMC journal is being published in unprecedented times for teachers. We are working at home due to the COVID-19 pandemic. We are connecting virtually with colleagues and PLC teams to brainstorm new ways to approach teaching, assessing, and connecting with our students. I thank you all for your continued commitment to supporting ALL students to learn important mathematics, mathematics that can certainly change and impact lives and maybe, just maybe, mathematics that can also change the world.

This edition’s articles address learning for students from 4K to high school. They address different domains of mathematics: geometry, algebra, statistics, basic number and operation sense. You may choose to read only the ones that appear from the title to apply directly to the grade levels and topics you teach. I’m going to encourage you, however, to at least skim the others. I found that by reading them all I was reminded and affirmed that my goals to know my students and their mathematical understandings combine to help me provide access points and engagement for all learners. Once they’re engaged, I can move students forward. I can help them make connections between what they already understand and new learning. It’s my professional obligation to do everything I can to make this work for every one of my students.

I started my career in education over 30 years ago. For the first couple of years I focused on teaching and not as much on learning as I could have. Then I was blessed to work under the supervision of a principal that helped me focus on the evidence of student learning. I twisted an old saying to describe my goal to reach every learner, “You can lead a horse to water, but you can’t make him drink, but you CAN make the water so sweet and enticing that he can’t resist it.” The articles in this journal can provide ideas to make the mathematics sweeter for your students. Enjoy!

Working together to help ALL students learn important mathematics,

LoriLori Williams

WMC President, 2018-2020

President's MessagePresident's Message


Learning Trajectories: A Road Map for PreK-3 Mathematics Instruction

DeAnn Huinker & Melissa Hedges, University of Wisconsin-Milwaukee

“The learning trajectories, first and foremost, have been invaluable in my teaching. For the first time in 15 years I have a clear idea of how, when, and in what order students learn different mathematical ideas and skills.” –Special Education Teacher

“Understanding the individual levels and what is expected at each level has been eye opening for me. Being able to see where my students are strong and where to move those students that are struggling has made my math teaching more effective for each student.” –Kindergarten Teacher

Children’s early mathematics experiences form the foundation for their future as mathematics learners. In fact, early mathematics is a significant predictor of later academic success in elementary school, and even into middle and high school (Duncan et al., 2007; Ritchie & Bates, 2013; Watts, Duncan, Siegler, & Davis-Kean, 2014). Surprisingly, early mathematics not only predicts later success in mathematics, but also predicts later reading achievement even better than early reading skills (Duncan et al., 2007). This evidence highlights the importance of mathematics learning in preschool and the primary grades.

Given the critical role of good beginnings in mathematics, the National Council of Teachers of Mathematics (NCTM) and National Association for the Education of Young Children (NAEYC) took the position that early childhood programs should “provide for children’s deep and sustained interaction with key mathematical ideas” (NAEYC/NCTM, 2010, p. 6). Furthermore, they recommended that educators use learning trajectories to inform and guide their teaching of mathematics:

Articulating goals and standards for young children as a developmental or learning continuum is a particularly useful strategy in ensuring engagement with and mastery of important mathematical ideas. In the key area of mathematics, researchers have at least begun to map out trajectories or paths of learning—that is, the sequence in which young children develop mathematical understanding and skills. (p. 6)

Through a three-year professional development project, a group of PreK-3 teachers in the Milwaukee area had the opportunity to study research-based

learning trajectories and begin using them as a road map to guide their mathematics instruction. As noted in the opening quotes from participants in the project, the teachers found the trajectories invaluable for informing their teaching. In this article, we discuss learning trajectories in mathematics, provide an example, and describe resources for getting started.

What are Learning Trajectories?

Mathematics learning trajectories are research-based frameworks that detail the likely progressions, over many years, of children’s development of big ideas in mathematics (Clements & Sarama, 2014). Children’s advancement along these learning pathways does not just occur with maturation, but is the result of appropriate learning experiences. At the early childhood level, the prominent researchers in the field are Doug Clements and Julie Sarama. They developed learning trajectories in specific conceptual areas for number and operations, geometry, and measurement (see Figure 1).

A complete learning trajectory is comprised of three components—(1) an overarching mathematical goal, (2) a developmental progression of children’s reasoning, understanding, and abilities, and (3) aligned learning activities. The trajectories answer the four key questions listed in Figure 2.

The first component of a learning trajectory is the mathematical goal or big idea that anchors the learning. A big idea is defined as “a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole” (Charles, 2005, p. 10). At the early childhood level, big ideas need to be consistent with children’s thinking and generative of future learning (National Research Council, 2009).


Number & Operations Geometry Measurement1. Counting

2. Subitizing

3. Comparing Number

4. Composing Numbers

5. Adding & Subtracting

6. Multiplying & Dividing

7. Patterning

1. Spatial Visualization

2. Spatial Orientation

3. Shapes

4. Composing 2D Shapes

5. Composing 3D Shapes

6. Disembedding Geometric Figures (2D)

1. Length

2. Volume

3. Area

4. Angle & Turn Measurement

Figure 1. Learning Trajectories in Early Mathematics (Clements & Sarama, 2019).

Key Questions Learning Trajectory Component

Where am I trying to go with children’s mathematics learning?

The goal of the trajectory identifies the big mathematical idea children need to develop for a strong foundation in mathematics.

Where are children now in their mathematical thinking?

What is the next important mathematical idea to target?

The developmental progression describes successive levels of children’s reasoning, understanding, and abilities along a continuum from informal to more sophisticated mathematical thinking.

How can I foster children’s mathematics learning along the continuum?

The learning activities, matched to each level, help solidify important mathematical ideas and prompt children toward the next level on the trajectory.

Figure 2. Guiding Questions.

The second component is a developmental progression of children’s thinking. Each trajectory describes qualitatively distinct and successive levels of increasing sophistication in children’s mathematical reasoning, understanding, and abilities. The Clements and Sarama trajectories describe 5 to 24 levels of children’s mathematics learning from birth to age 8. For example, counting has 21 levels, comparing has 24 levels, and shapes has 21 levels. The large number of levels for these trajectories may surprise you, while also giving a sense of the complexity and depth of young children’s mathematics learning. As remarked by a teacher in the project, “The math standards indicate where children are supposed to be by the end of the school year, but the learning trajectories tell us how to get there.”

The third component of a learning trajectory is aligned learning activities, which includes critical conversations and mathematical discourse. The activities suggest pivotal experiences for children at a specific level. The experiences aim to develop and consolidate the mathematical ideas (i.e., concepts, abilities, reasoning) at that level and lay the mathematical foundation for the next step in children’s learning.

When using trajectories, it is important to keep several key points in mind:1. Developmental progressions describe typical

learning sequences or continuums. Each child is a unique learner drawing on one’s own mathematical, cultural, and linguistic strengths, so each child’s path will be unique as to sequence and pace.


2. Movement along the learning continuums hinges on instructional experiences, not maturation. Children only advance through engagement with appropriate and challenging (i.e., thought-provoking and sense-making) learning activities, tasks, and discourse

3. Each progression includes developmental milestones. Certain levels reveal major cognitive shifts in children’s understanding and require more instructional time and engagement.

The Composing Learning Trajectory

The big mathematical idea that anchors the composing trajectory (see Figure 3) is that a quantity (a whole) can be broken into parts (decomposed) and parts can be combined (composed) to form a whole (Clements, 2004). Notions of composing and decomposing are used throughout mathematics in every domain at every level (National Research Council, 2009). Composing abilities contribute to children’s conceptual understanding of the operations and arithmetic properties (e.g., associative), use of flexible strategies for single-digit and multidigit computation, and understanding of place value and our base-ten numeration system.

Levels 1-3 of the composing trajectory indicate how toddlers begin to intuitively explore ideas of parts and wholes, but that the real work starts with Level 4 as pre-kindergarten and kindergarten students learn to name parts of a whole to 4, and then to 5. Next children progress to composing numbers to 7 (Level 5) and then to 10 (Level 6). Anchoring numbers to five and ten across these levels are key milestones for children (Huinker, 2012) and align directly to the Wisconsin State Mathematics Standards K.OA.3 and K.OA.4 to decompose numbers within 10.

The project teachers found “hidden part” activities particularly valuable for their students at Levels 4-6. The main idea is for children to decompose quantities in various ways by visualizing little numbers hiding inside bigger numbers.

My students seem to struggle with seeing equations that look like 8 + [ ] = 10 or 10 = 8 + [ ]. The hidden part activities help them see the connection between the equations and the objects that we used in activities, like bears in the cave or finger flash. –Grade 1 Teacher

In intervention groups, we work with missing part baggies. The students begin with a certain number of counters. One student hides part by putting them in the bag and the other student works to determine the missing part. These missing part activities have made a huge impact on my students. I have noticed that students who have the ability to determine a missing part for smaller numbers can also decompose and compose larger quantities. –Grade 2 Teacher

Levels 6-8 highlight children’s development across ages 6-7. In other words, children are simultaneously developing the ideas across these three trajectory levels. Level 7 involves knowing teen numbers as composed of tens and ones and aligns with Standard 1.NBT.2 on understanding place value. Level 8 focuses on the development of flexible, derived strategies, and aligns with Standards 1.OA.6 and 2.OA.2 on using strategies to add and subtract within 20. At Level 9, children use flexible strategies to solve all types of problem situations and then extend their strategies at Level 10 to solve two-digit addition and subtraction problems.

Most children follow this natural developmental progression in learning to compose and decompose numbers. It is important to keep in mind that the ages listed are only an approximation, because the age of acquisition depends heavily on children’s engagement with appropriate learning experiences. Also note that children are concurrently working across many of the learning trajectories listed in Figure 1. The earlier levels on the composing trajectory emerge from ideas developed in the subitizing trajectory and the later levels are associated with the adding and subtracting trajectory.

Web Resources for Getting Started

We hope we have piqued your interest in learning trajectories and that you are eager to explore resources and tools to use with your students. Three early math websites offer easily accessible games and activities to help teachers target children at different working levels: (1) Learning and Teaching with Learning Trajectories, (2) Zeno Math, and, (3) Young Mathematicians. Designed for busy teachers, each website provides learning-trajectory aligned games and activities and feature links to materials and resources (e.g., lesson plans, game boards, cards, most as PDF files to download). Best of all they are free!


Level Age Description Example1 Actor on Parts 0-1 Infant displays actions that show intuition about

parts and wholes such as gathering objects together.

Infant mimics caregiver sounds and shows interest in watching different arrangements of objects, such as stacking a set of cups or blocks.

2 Parts Combiner

1-2 Toddler realizes that sets can be combined in different orders and nonverbally recognizes parts and wholes.

When given 4 yellow blocks and 3 green blocks, the child intuitively appreciates building with “all the blocks” includes both the yellow and green blocks.

3 Inexact

Part-Whole Recognizer

3-4 Child understands that a whole is bigger than the parts, but does not accurately quantify the whole set. Child may be able to subitize parts up to three.

When given 3 yellow blocks and 2 green blocks and asked “how many blocks altogether,” child might name a “big number” such as 5 or 10.

4 Composer to 4, then 5

4-5 Child can name parts of a whole to 4, then to 5, in different ways. Child can compose the whole given the parts or decompose the whole into parts. Using five as an anchor is a key milestone.

Child is shown 5 blocks, 2 blocks are secretly hidden, and then child is shown the remaining 3 blocks. When asked, “How many are hiding,” child accurately responds that two are hiding.

5 Composer to 7 6 Child knows number combinations to 7. Can name parts given the whole, name a part given the other part, and name the whole given both parts. Child anchors the quantities six and seven to five (e.g., 7 is 5 and 2 more).

Child can make 6 as “bunny ears” by holding hands above her/his head and showing fingers on each hand to total 6 fingers. Child draws on mental images of the quantities.

6 Composer to 10

6-7 Child knows number combinations within 10. Can name parts given the whole, name the other part given one part, and name the whole given both parts. Using ten as an anchor is a key milestone.

Child draws a numeral card, from a set that includes numbers 0-10, and names the number partner that would make ten.

7 Composer with Tens and Ones

6-7 Child understands teen numbers as tens and ones. First child thinks about 15 as composed of ten ones and five extra ones. Later understands 15 as one ten and 5 ones. Child starts to understand two-digit numbers as tens and ones.

Child shown a filled ten frame and 4 extras, readily says it totals fourteen because it is ten and 4 more.

8 Deriver 6-7 Child uses flexible strategies and derived combinations to solve problems within 20, both addition and subtraction. Child can break apart and recompose numbers, keeping track of quantities.

Child shows 8+7 on two ten frames, then moves 2 counters from the 7 to make a ten and says, “I made the 8 into a 10; it’s now 10 and 5, so 15.”

Child solves 15–7 by decomposing 7 and subtracts it in parts, 15–5=10, 10–2=8.

9 Problem Solver

7 Child solves all types of problem situations (i.e., join, separate, combine, and comparison with unknown in all positions) within 20 using flexible strategies and some known combinations. Child begins to solve problem situations with larger quantities.

Child solves the word problem, “How many pieces of fruit are in the bowl that has 9 apples and 7 pears?” Child explains, “9 and 1 is ten and 6 more is 16.”

10 Multidigit Computation

7-8 Child solves two-digit addition and subtraction problems by incrementing tens, combining tens and ones, or using flexible strategies as appropriate for the numbers in the problem.

Child solves 38+24 by reasoning, “30+20=50, and 8+4=12, so 70 and 12 is 82.”

To solve 65–27, child might reason, “65 minus 20 is 45, minus 5 is 40, and minus 2 more is 38.”

Figure 3. Composing Learning Trajectory (Adapted from Clements & Sarama, 2019).


Learning & Teaching with Learning Trajectories Website: www.learningtrajectories.org

Developed by Doug Clements and Julie Sarama, started in 2017, this site is a web-based tool for educators to learn about the development of children’s mathematical thinking and how to teach mathematics utilizing learning trajectories. Teachers are provided access to the 17 learning trajectories listed in Figure 1. Each level of each trajectory is accompanied by examples of children’s thinking, videos of children working at that level, and applicable instructional tasks.

The site offers two ways to view the trajectories, by age/grade or by content domain. For example, by clicking the button for four-year olds, each of the 17 trajectories highlight the levels for typically functioning four-year olds (see Figure 4). To sort by a content domain, one simply selects a domain and each applicable trajectory will come to the top of the page for easy exploration. Teachers can also create a class roster of their students, assign computer-based tasks, and monitor student progress on a selected trajectory.

Instructional suggestions include small group and whole group activities, some accompanied by a video of children engaged in the activity. For example, the Composer to 7 level suggests the activity, Break and

Make. The teacher selects a target number, say six, and builds a train with connecting cubes. She places the six train behind her back and breaks it into two parts. She shows the children one part, keeping the other behind her back, and asks them to figure out how many cubes are hiding. A brief lesson plan is included, along with a video showing a teacher facilitating this activity with a group of children.

We highly recommend this website as the source for digging deeper into the learning trajectories. Be sure to watch the videos on the homepage. Doug Clements and Julie Sarama offer valuable guidance on how to register for an account, create a class roster, and utilize the website. The “resources” link at the bottom of the web page takes you to suggestions for working with a variety of learners, professional development modules, and links to other web resources. Zeno Math Website: https://zenomath.org

Founded in 2003 and originally created by Norman Alston and a group of parents at Wilder Elementary School in Woodinville, Washington, Zeno’s mission centers on buildings children’s excitement and confidence in mathematics across PreK-Grade 5. The activities and resources are located under broad categories—enriching early learning, engaging

Figure 4. Excerpt from the Leaning and Teaching with Learning Trajectories Website.

http://www.learningtrajectories.org https://zenomath.org


families, and math resources. We suggest you start with the Toolbox to find the activity bank, board and games, math book suggestions, and more. A nice feature of the activity bank is to sort according to Common Core State Standard, grade level, math concept, or type of activity.

Zeno Math features thirteen early learning games specifically designed for children ages 3-5 and appropriate for classroom or home use. The games are accessible, easy to implement, and require few materials. The instructor card for each game shows alignment to learning trajectories (Clements & Sarama, 2019) and Teaching Standards GOLD curriculum objectives. It also gives suggestions for math talk and differentiation based on the age or ability of the children, organized under Seed, Sprout, and Bloom categories. To ensure the materials are accessible for a diverse community, instructions are provided in numerous languages (e.g., Amharic, Arabic, Chinese (Traditional), English, Russian, Somali, Spanish and Vietnamese).

One of our favorite activities is Peek-A-Dino (https://zenomath.org/activities/peek-a-dino/). It engages children in recognizing small quantities quickly without counting, referred to as subitizing. A collection of baby dinosaur counters is placed under a cloth and after a quick peek, children are asked to name how many they saw, make a matching set using their own counters, and describe how they saw the collection of dinosaurs. In the Seed version, children subitize quantities of one and two, in the Sprout version they subitize quantities up to four, and in the Bloom version they subitize quantities up to five. Expectations increase for math talk and part-whole reasoning increase as children move from Seed to Sprout to Bloom.

Young Mathematicians Website: http://youngmathematicians.edc.org

Developed under the guidance of Kristen Reed and Jessica Mercer Young at the Educational Development Center (EDC), this website offers a variety of math games and activities for teachers and parents of children ages 3 to 6. Some are quick games that use everyday materials; others use a game board and require more extended play. The games engage children in problem solving, puzzling, and discussing strategies as they play, and they collectively focus on counting, operations, algebraic thinking, and geometry. The games place an explicit focus on developing children’s persistence,

executive functions, and social-emotional skills through teacher questioning and modifications for a wide range of learners. Extensive directions, tips for classroom or home use, and variations for play accompany each game. Learning progression tables delineate how skills specific to each game are likely to grow and progress. Though not an exact match, the progressions closely align with the trajectories developed by Clements and Sarama (2019).

One unique aspect of this site includes 13 bilingual (English-Spanish) printable mini-math books (http://youngmathematicians.edc.org/math-books/mini-books/). The books are meant to give families a fun way to help their young learners deepen their mathematics by reading the books together and practicing the math at home. Each book includes a short activity for parents and children to do together, and a note for parents that summarizes the key math concepts featured in the book and an explanation of why they are important for young children’s mathematics learning. Best of all the books are easy to make. Simply photocopy the book template on a double-sided sheet of paper, fold, cut, and staple. In addition, the site includes two printable mindset mini-books, Keep Trying and Grow Your Brain (http://youngmathematicians.edc.org/mindset/minibooks/).

The game Jumping on the Lily Pads is a current favorite of ours and a group of early childhood teachers (http://youngmathematicians.edc.org/jumping-on-the-lily-pads/). Children take turns rolling a dot cube and moving a frog along a game board. Children have a chance to practice early number concepts (e.g., cardinality, one-to-one correspondence, reading numerals, subitizing) as their frogs jump on numbered lily pads to the pond and then home again. Three versions of the game board are included: a five path, a ten path, and an 11-20 path. Suggestions are given for teacher questioning strategies as children play the game. A video shows a preschool teacher working with two of her students. The teacher in the video models important questions that push the children’s thinking and shows a playful approach for engaging her young learners.

Summary

The project teachers began to use learning trajectories to inform many aspects of their daily work with children and families. First and foremost, teachers

https://zenomath.org/activities/peek-a-dino/http://youngmathematicians.edc.orghttp://youngmathematicians.edc.org/math-books/mini-books/http://youngmathematicians.edc.org/math-books/mini-books/http://youngmathematicians.edc.org/math-books/mini-books/http://youngmathematicians.edc.org/mindset/minibooks/http://youngmathematicians.edc.org/mindset/minibooks/http://youngmathematicians.edc.org/jumping-on-the-lily-pads/http://youngmathematicians.edc.org/jumping-on-the-lily-pads/


used the trajectories to place students at working levels on various trajectories and then plan appropriate whole group and small group instructional experiences for children at those levels. Second, teachers used the trajectories to monitor student learning and growth and report on progress in conferences with parents and guardians. Third, teachers of students with special education needs used the trajectories to write specific and stepwise mathematics goals for Individual Educational Plans (IEP). Finally, teachers used the trajectories as diagnostic tools to identify specific gaps in mathematical understanding and plan intervention supports. We leave you with these final comments from the project teachers on how the learning trajectories provided them with a clear roadmap to guide student learning in mathematics.

My work with the math learning trajectories, specifically the Subitizing and Composing trajectories, helped me to see student growth over time. I used the trajectories to determine the level that the students were at and then showed how they grew mathematically over time. In my smart goal, I stated that I wanted 80% of my 12 targeted students to develop the ability to subitize to 5. While all of my students are not there yet (as of March), my students are showing growth in their ability to see groups and in their ability to decompose numbers to five. –Kindergarten Teacher

When I first began as a math interventionist I felt very overwhelmed in how exactly to tackle the instructional gaps of students entering my room. Now I consistently use the trajectories to assess students and determine where they are and plan my instruction from there. –Math Interventionist

My instruction has become more intentional with the direction and guidance of the learning trajectories. Through my introduction to the counting and subitizing trajectories I created and implemented an assessment tool... This school year, I was able to assess my students in the Fall, giving each student a working level, which I then used to inform my instructional planning. –Kindergarten Teacher

Being a part of the Strong Start Math Project has been a wonderful experience. I have grown as a teacher. I am more aware of where my students are coming from and what my next steps are once I figure out what mathematical abilities each child brings into my class. . . . Learning about the math trajectories was very helpful to be able to see what a child is capable of and what I, as their teacher, needed to do to get them to move to the next level in that specific trajectory. –Grade 1 Teacher

What I enjoy most about the learning trajectories is that they are a positive tool that labels a child’s development on a progression and does not discount the wealth of knowledge they bring with them to kindergarten. –Kindergarten Teacher

Note: The “Starting Students Strong in Mathematics: Strengthening Teacher Mathematical Knowledge and Instruction in Grades K-3” project was funded by a U.S. Department of Education Mathematics and Science Partnership (Title II, Part B, Project No. WI151205); administered by Wisconsin Department of Public Instruction; and awarded to University of Wisconsin-Milwaukee under direction of Dr. DeAnn Huinker and Dr. Gabriella Pinter.

References

Charles, R. I. (2005). Big ideas and understandings as the foundation for elementary and middle school mathematics. Journal of Mathematics Education Leadership, 7(3), 9-24.

Clements, D. H. (2004). Major themes and recommendations. In D. H. Clements, J. Sarama, and A. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education, (pp. 7-72). Mahwah, NJ: Lawrence Erlbaum.

Clements, D. H. & Sarama, J. (2014). Learning and teaching early math: The learning trajectory approach (2nd ed.) New York: Routledge.

Clements, D. H., & Sarama, J. (2019). Learning and teaching with learning trajectories [LT]2. Marsico Institute, Morgridge College of Education, University of Denver. Retrieved from: http://learningtrajectories.org/

http://learningtrajectories.org/ http://learningtrajectories.org/


Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C., Klebanov, P. Pangani, L.S., Engel, M., Brooks-Gunn, J., Sexton, H., Duckworth, K., & Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428-1446.

Huinker, D. (2012). Structuring number knowledge with anchors to five and ten. Wisconsin Teacher of Mathematics, 64(1), 4-8.

National Association for the Education of Young Children (NAEYC) and National Council of Teachers of Mathematics (NCTM). (2010). Early childhood mathematics: Promoting good beginnings. [Updated joint position statement]. Washington, DC: NAEYC. Available from https://www.naeyc.org/resources/position-statements/

National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. Washington, DC: National Academies Press.

Ritchie, S. J., & Bates, T. C. (2013). Enduring links from childhood mathematics and read achievement to adult socioeconomic status. Psychological Science, 24(7), 1301-1308.

Watts, T. W., Duncan, G. J., Siegler, R. S., & Davis-Kean, P. E. (2014). What’s past is prologue: Relations between early mathematics knowledge and high school achievement. Educational Researcher, 43(7), 352-360.

https://www.naeyc.org/resources/position-statements/ https://www.naeyc.org/resources/position-statements/ https://www.nap.edu/download/12519/


Students can develop misconceptions in mathematics caused by definitions in instructional sources, texts, or even possibly teacher errors. Misconceptions can also be generated when students attempt to fill in their understanding gaps. In this article, we consider two topics, absolute value and parallel lines, where instructional sources may cause issues. We offer teaching options that we have used successfully in our classroom to avoid or correct student misconceptions.

The suggested instructional remedies cited in this article include examples of both verbal and visual instructional tools and methods. Verbal strategies to address misconceptions involve attending to the actual words and the accuracy of definitions or theorems involved. Visual corrections include teaching methods that allow student observations of concept relationships and tactile hands-on experiences to illustrate those concepts. Strategies used here were chosen based on the importance of Silverman’s (2000) visual learning characteristics and Sword’s (2000) description of students who prefer visual patterns of geometry and moving relationships of physics to understand concepts. Examples are seen in student instruction on absolute value and parallel lines that follow.

Absolute Value

In our classrooms, we have identified student misconceptions about absolute value that we believe have arisen because of textbook definitions being misinterpreted, misleading, or misunderstood. Consider

a correct absolute value definition in Figure 1 found in many middle school and high school textbooks:

Figure 1. Absolute value definition.

This definition may be the first two-part definition students have seen and that alone can be confusing. Some authors of texts have tried to help students understand absolute value by associating it with a

number line as seen in the definition in Figure 2 (Berry, Milou, Schielack, Wray, Charles & Fennell, 2017).

The first sentence in the definition in Figure 2 is mathematically correct and is equivalent to the definition given in Figure 1. However, the second sentence in the definition of Figure 2 is not correct. Distance is not always a positive number because the distance from 0 to 0 is 0, a non-positive number. This is noted in the sentence above the number line of Figure 2. In this text, students see three pieces of information regarding absolute value: two that are true and one that is not. This instructional resource, with its conflicting information, has misinformation that will lead to student misunderstanding causing misconceptions that may cause problems in mathematics in the future.

The issue of texts and definition of absolute value is not a new problem in mathematics in the 20th and

Algebraic and Geometric Misconceptions and Misinformation in Classrooms

Catheryne Draper and Johnny W. Lott

Figure 2. Absolute value definition from a Grade 6 text (Pearson, 2017, p. 80).


21th centuries. Figure 2 shows an example from 2017 but the absolute value of a number as presented by Rosenbach, Whitman, Kirchner, and Sloan (1939) in College Algebra as a number without its sign also had issues. Using the Rosenbach, et al., definition, if we considered only the absolute value of a numerical value, then the directions that the removal of the positive or negative numeral sign and the absolute value symbol “| |” is a shortcut that is true. This same result is shown in the definition of absolute value seen in Wikipedia in 2019 as follows: In mathematics, the absolute value or modulus of a real number is the number without the sign.(https://simple.wikipedia.org/wiki/Absolute_value) (It should be noted that later in the Wikipedia version, the definition from Figure 1 is shown).

These mechanistic definitions provide little or no understanding of the meaning of absolute value. For example, if there is a variable inside the absolute value symbol there can be issues. For example, consider |x – 4|. If students remove the absolute value symbolism and write the result as x – 4, then the result when x = -4 would imply that | –4 – 4| would be –4 – 4, or –8. Clearly that is not true because using the numbers alone and doing the computation inside the absolute value symbols, we have |–8| = 8. As students see, here is a definition that does not work in all cases and as a result, is a faulty definition. Looking back, students do not know whether or not the value of x – 4 is positive, negative, or 0 because they do not know the value of x. Simply removing the absolute value symbol and using x – 4 without a sign is problematic. Not knowing the value of x – 4 forces us to consider the following (which is in effect using the correct definition of Figure 1):

|x – 4| = x – 4 if x ≥ 4

(in this case, x – 4 ≥ 0, or x ≥ 4)

|x – 4| = –(x – 4) if x < 4

(in this case, x – 4 < 0 so that – (x – 4) > 0, or x < 4)

Absolute value may be further examined by finding solutions to the equation y = |x – 4|. In Figure 3, there are two parts of the equation to consider:

y = |x – 4| = + (x – 4) when x – 4 ≥ 0

y = |x – 4| = – (x – 4) when x – 4 < 0

Figure 3. Example solution strategy for an absolute value problem.

Unfortunately, many students do not realize that there are two cases when the value of x – 4 is not known. If students remember only that absolute value is always positive, exactly as the sixth grade text definition, the 1939 text, and the 2019 Wikipedia definition seem to show, they might reasonably deduce that the positive “+” symbol represents the same meaning as the absolute value “| |” symbolism. This has been observed in the authors’ classes at all levels, including the collegiate level. In these cases, students may see and read mathematics, but the mathematics is inaccurate or incomplete. To avoid student misconceptions, teachers have a duty to correct what students see and read.

A related issue is that, as shown above, text examples and teacher examples of absolute value often begin with numerical examples. Like most mathematics, new material is frequently based on student understanding. With these numeric examples, there is likely very little, if any, way for a teacher to diagnose confusion or misconceptions since variables and absolute value graphs appear later. However, even hearing that absolute value makes a number positive can later lead students to believe that |x| behaves the same as +x during early stages of learning about functions. This misconception about symbolism and meaning can easily lead to long-term confusion, especially when students begin to graph functions such as the two graphs for f(x) = x and g(x) = |x| shown in Figure 4. Misconceptions about symbolism might lead a student to think that the two functions act the same, but they clearly do not.

Furthermore, when a variable is involved in the absolute value expression, issues arise about the domain of the variable and whether the variable represents a positive or a negative value. For example when y = –x, many students interpret this function as having a domain of negative numbers (in other words “x will always be a negative number”) thus inviting more misconceptions of misunderstood symbolism. This misinterpretation of symbolism compounds (and confounds) the absolute value meaning as well as other symbolism when a negative sign is involved.

The teacher instructional resolution must address the absolute value definition with students’ discussions and interpretations. As any misinterpretations, misunderstandings, and misconceptions arise during discussion, teachers have teachable moment opportunities to redirect thinking. In those teachable moments, teachers might try a visual and physical

https://simple.wikipedia.org/wiki/Absolute_valuehttps://simple.wikipedia.org/wiki/Absolute_value


lesson using a classroom activity described in Wade’s (2012) The Mathematics Teacher article. Wade’s goal was to illustrate how absolute value connected rate, time, and distance by assigning a walking/running task for her students. The students were to collect data about the distance and time from the front and back stairwells to her classroom using a “sprint to class before the bell” (pp. 193-4). The classroom was located 16 ft from the front stairwell and 24 ft from the back stairwell. Students walked (or sprinted) from the front hall past the classroom to the back hall and vice versa. The task was to collect data and use the measured distances and times to derive an equation relationship between distance and time. Several examples emerged from the different data depending on rates. One of the graphs comparing distances and times is shown in Figure 5. The graph in Figure 5 has the classic v-shape of an absolute value that illustrates a student’s walking path versus time.

Avoiding shortcuts intended to simplify definitions and clarifying possibly incorrect definitions from texts are important instructional strategies. This extra

emphasis on avoiding misinformation, misconception, or misunderstandings may sound like much-ado about nothing but can be very valuable to a confused student.

Parallel Lines

We identified another example of definitions and theorems from an instructional source that can seed student misconceptions in two different definitions of parallel lines. The first definition comes from Plane Geometry by Hart (1950), and the second example comes from Mathematics for Elementary Teachers with Activities (2018) by Beckmann.

In Plane Geometry (1950), Hart wrote (p. 77):

Intersecting lines are straight lines that meet.

Parallel lines are straight lines in the same plane that do not meet even if extended infinitely far.

Theorem: If two lines are parallel to a third line, they are parallel to each other.

Figure 4. Confusion of f(x) = x and g(x) = |x| as different functions.

Figure 5. Comparison graphs from Wade’s classroom activity, page 194.


In Mathematics for Elementary Teachers with Activities (2018), Beckmann wrote (p. 455):

Two lines in a plane that never intersect (even somewhere far off the page they are drawn on) are called parallel.

When two lines in a plane intersect, they form four angles.

To avoid misunderstanding, misconceptions, and students’ incorrectly formed notions, we consider these definitions carefully and ask following: Do intersecting lines have the same meaning in both definitions? In Hart’s definition, intersecting lines are straight lines that meet. In Beckmann, intersecting lines are lines that form four angles. In both cases the implication is that intersecting lines meet in a single point.

Hart goes farther in his theorem: If two lines are parallel to a third line, they are parallel to each other. There are embedded implications in these definitions. For example, consider the algebraic pair of linear equations: 2x + 3y =14 and 4x + 6y =28. These linear equations represent lines; in fact they represent the same line with slope of –(2/3). So could a line be parallel to itself? With Hart’s definition, the answer is no; a line certainly meets itself, and from a set point of view, it certainly intersects itself. At issue is that nothing in either definition mentions that the lines must be distinct. If the lines are not distinct, then using Hart’s definition, we do not know whether a line could be parallel to itself. Because his definition did not take care of this case, we could amend his definition to state that two lines in a plane are parallel if they do not meet in a single point. This definition allows two lines to be parallel if they have no points in common or if they have all points in common and keeps a part of his definition of not allowing lines to be parallel if they intersect (or meet in a single point.).

Why is the previously discussed distinction important? Consider Hart’s theorem (p. 77) in a slightly different form:

If a is parallel to b and b is parallel to c, then a is parallel to c.

According to de Villiers, Govender, and Patterson (2009) in Defining in Geometry (p. 194), for a definition to be true, it has to be true for all possible replacements for a, b, and c in the set being considered. Does that happen here? Think about the following specific case:

If a is parallel to b and b is parallel to a, then a is parallel to a.

With either of Hart’s or Beckmann’s definition and a true hypothesis, the only way that the theorem can be true is if the conclusion, a is parallel to a must be true. Though the intent in the definitions seems to be that two lines parallel may not have a single point in common, the theorem demands that a line be parallel to itself with all points in common. Beckmann’s addition of “when two lines in a plane intersect, they form four angles,” calls for more discussion because “distinct lines” are certainly implied and straight angles are not considered in this statement. The statement is an add-on to the definition in much the same way that there was an add-on to the definition of absolute value in Figure 2.

A basic misconception about parallelism perpetuated by the texts is Hart’s theorem that cannot be true if the given definitions are true. The theorem is only true if the definitions are re-written. Logically, theorems that have true hypotheses and true conclusions are true. In this case, we certainly can have a true hypothesis with “If a is parallel to b and b is parallel to a.” However, with the definitions given, we cannot have “a is parallel to a.” Here we have a true hypothesis and a false conclusion so that the theorem cannot be true. The logic used in this analysis is something that every mathematics teacher must know and it is important that students understand it.

Several things could be done to help this situation. First, the definition of parallel lines could be changed in both books to allow a line to be parallel to itself, and then there is no issue with the theorem being false. Second, lines a, b, and c could all be distinct, but if that is the case, then that condition must be added to the definitions.

So in this case, a misconception or misinformation is transmitted to the student from the texts and the definitions themselves. Whatever the case, if a teacher is not aware of the issues here, students may have mistakes in the mathematics.

A Classroom Tool for Use with Parallel Lines

For those students who benefit from visual activities of sorting, classifying, and learning information from within the broad category of linear equations, The Algebra Game (Draper, 2016) provides a lesson on parallel lines in the Linear Graphs section of the


program. The lesson activity allows students to first identify, then sort out the particular pattern represented by parallel lines from one deck (or several decks) containing many different equations of lines. A section of the Parallel Line Literacy sheet from the program is shown in Figure 6(b) along with the activity card shown in Figure 6(a)

Students typically sort, classify, write, discuss, and share their observations in groups and then with the entire class. That much-needed student discussion piece included in the program materials provides a valuable opportunity for teachers to listen as students describe their understandings of parallel lines to other class members. Teachers might also suggest a discussion about definitions of parallel lines and how a line can be parallel to itself. This discussion could start in groups but eventually should be with the entire class.

Teachers might also introduce other equations to encourage even more discussion. These equations could be ones that may look different but have equivalent expressions for slopes, such as in the above example pair of equations 2x + 3y =14 and 4x + 6y = 28 , or expressed in slope intercept form as and

. These equations have the same Coordinate Pair cards and are represented by the same graphs, a circumstance that students question, talk about, and eventually resolve with the awareness that a line can be parallel to itself.

Misconceptions, misinformation, misunderstandings, and misinterpretations all too often go undiagnosed until test results are reported. Sometimes, even then, the posttest evaluations may not expose the actual misinterpreted part of a definition or misunderstanding of a concept. As in these topic examples about parallel lines, verbal student conversations, debate, and even argument can expose previously unaddressed issues before a test for further mathematics study.

Summary

The examples of absolute value and parallel lines in this paper show how, where, and when students have either misunderstood instructions, have developed misconceptions in mathematics, or have been taught ambiguous theorems or definitions through texts that led to later difficulties. Avoiding misconceptions and misinformation requires ensuring correct information from either teachers or texts as well as long-term

mathematics connections that do not contradict original information. Using different modes of instruction and encouraging student dialogue can help avert these issues. Additionally, appropriate tools, models, and activities can be carefully chosen to avert misconceptions or misinformation that can lead to student misunderstanding.

As teachers, we must guard against introducing ideas in ways that can leave gaps in understanding as well as errors in textbooks or other tools that lead to misconceptions. Only by carefully examining students’ work and listening to their explanations of understandings can we know what they are really thinking. Given that misconceptions and misconstrued student ideas can lead to major mathematics learning difficulties, they must be addressed when they arise.

ReferencesBeckmann, S. (2018). Mathematics for elementary

teachers with activities. Boston: Pearson Education, Inc.

Berry, R. Q., III, Champagne, Z., Milou, E., Schielack, J., Wray, J. A., Charles, R., I. & Fennell, F. (2017). enVisionMATH 2.0 (Grade 6). Boston: Pearson Education, Inc.

de Villiers, M., Govender, R., & Patterson, N. (2009). Defining in geometry. In T. V. Craine & R. Rubenstein (Eds.), Understanding geometry for a changing world (pp. 189-203). Reston, VA: NCTM.

Draper, C. (2016). The algebra game: Linear graphs, quadratic equations, conic sections, and trig functions. Rowley, MA: Didax Education Inc.

Pearson Education. (2017). enVision MATH 2.0 (Grade 6). Pearson Education, Inc., or its affiliates.

Hart, W. W. (1950). Plane geometry. Boston: D. C. Heath and Company,

Rosenbach, J. B., Whitman, E. A., Kirchner, P. & Sloan, S. (1950). College algebra. Boston, MA: Ginn and Company.

Silverman, L. K. (2000). Identifying visual-spatial and auditory-sequential learners: A validation study. In N. Colangelo & S. G. Assouline (Eds.), Talent development V: Proceedings from the 2000 Henry B. and Jocelyn Wallace National Research Symposium on Talent Development. Scottsdale, AZ: Gifted Psychology Press.


Wikipedia. Accessed 2.6.2109. https://simple.wikipedia.org/wiki/Absolute_value

Sword, L. K. (2000). I think in pictures, you teach in words: The gifted visual spatial learner. Victoria, Australia: Gifted and Creative Services.

Wade, A. (2012). Teaching absolute value meaningfully. The Mathematics Teacher 106(3). pp. 192-198.

(a). Activity card from Linear Graphs Teacher Manual, p. 30.

(b). Parallel Literacy sheet from Linear Graphs Teacher Manual, p. 31.

Figure 6. Examples from Draper (2016).

https://simple.wikipedia.org/wiki/Absolute_value https://simple.wikipedia.org/wiki/Absolute_value


The Mathematics of The Avengers

David Ebert, Oregon High School, Oregon, Wisconsin

Spoiler alert! Please see the Avengers movies before reading further!

Avengers: Endgame, released in April 2019, immediately shattered many box office records, and grossed over 1.2 billion dollars in its opening weekend. Prior to seeing the movie, my kids and I watched Avengers: Infinity War, which was released in 2018 and is the number five highest grossing movie of all time. At the conclusion of the movie, my daughter said, “if half of all people on Earth died, it seems as if more of the main characters died than you would expect.” This offered me an outstanding opportunity to share some probability theory with her.

In Avengers: Infinity War, Thanos constructs a plot to kill half of all life in the universe. His motivation is to save life by destroying it; he believes that the overpopulation of the universe is causing its demise, and by destroying half of all life he is making the universe a better place for those who remain. This is actually not a unique theory. In 1798, Thomas Malthus published An Essay on the Principle of Population, which warned of the dangers of a population that grows geometrically while food production could only grow arithmetically. In 1992, the Union of Concerned Scientists offered a warning that we all must stabilize our population, while also managing our resources more effectively. This theme has appeared in many additional works of fiction prior to the Avengers, including in Dan Brown’s best selling book Inferno.

This raises an important point. Would killing half the population really save the Earth, or simply prolong our impending doom? According to Wikipedia, the population of the Earth increases by 1.1% each year. There are currently about 7.3 billion people on our planet. An exponential growth model, y = aebt , can be used to calculate the time it would take to double the Earth’s population, as shown below.

14.6 = 7.3e0.011t2 = e0.011tln 2 = 0.011t63 years ≈ t

According to this model, the Earth’s population doubles every 63 years. If Thanos was successful in his plot to kill half of all humans on Earth, he wouldn’t necessarily save the Earth; he would merely extend life for another 63 years. Perhaps if one of the Avengers was more proficient in advanced algebra and/or mathematical modeling, she could have simply explained this to Thanos and avoided a lot of trouble.

Regardless, Thanos successfully unleashed his plot and killed half of all living beings. This killing is completely indiscriminate, meaning that the probability of any person surviving is completely independent and is equal to ½. The probability of any pair of people surviving is (½)2 = ¼. The probability of any group of three people surviving is (½)3 = ⅛ . Assuming that there are a dozen Avengers superheroes, the probability that they all survive is (½)12 = 1/4096 ≈ 0.0002.

While the probability that all of the superheroes survive is very slim, it is more interesting and realistic to determine the probability of some of the Avengers surviving. To do this, we need to apply the Binomial Theorem. Although the Binomial Theorem appears to be very complex, it can be explained to any high school student using Pascal’s Triangle. Pascal’s Triangle is shown in Figure 1 below.

Figure 1. Pascal's Triangle.


Pascal’s Triangle is generated by writing a 1 at the start and end of every row. Every other number in each row is formed by adding the two numbers immediately above the number. Think of the “1” at the very top as the 0th row, meaning the row with the terms 1 5 10 10 5 1 is the 5th row. Also, think of the “1” at the start of every row as the 0th term. Each of these values in Pascal’s Triangle is also called a combination, and is noted nCr. The Binomial Theorem states that to expand (x + y)n, the powers of x of every term will decrease, the powers of y of every term will increase, and the coefficients will be the values from Pascal’s Triangle. For example, (x + y)5 = x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5.

Let’s apply this result to a probability exercise. If there are five Avengers, what is the probability that any number of them survive? Adding up the numbers in any row of Pascal’s triangle gives the total number of possibilities. Therefore, in the 5th row, there are 32 different possibilities (notice this is equal to 25). The probability of 0 survivors is the 0th term divided by 32. Here are the probabilities of any number of survivors:

0 out of 5 survive = 5C0/32 = 1/321 out of 5 survive = 5C1/32 = 5/322 out of 5 survive = 5C2/32 = 10/323 out of 5 survive = 5C3/32 = 10/324 out of 5 survive = 5C4/32 = 5/325 out of 5 survive = 5C5/32 = 1/32

These probabilities can be added as well, meaning that the probability that at least two out of five people survive is 26/32 = 0.8125.

I used the Avengers: Infinity War poster in my son’s bedroom to determine that there were 28 total Avengers heroes, and 13 of them survived. The probability of this happening is 28C13/2

28 ≈ .139. The probability that at least 13 of the 28 avengers survived is

(28C13+28C14+28C15+28C16+...+28C28)/228 ≈ .714.

To answer my daughter’s original question, the probability that 13 or more of the original Avengers survived is about 71%.

There are some other interesting mathematics that comes about from Avengers: Infinity War. Dr. Strange looked into the future and witnessed 14,000,605 possible scenarios for how the future would unfold, and in only one of these would the Avengers be able to defeat Thanos. How small is this probability? This probability is greater than the probability of picking a perfect NCAA tournament bracket (1 in 9.2 quintillion) or of winning the Powerball lottery jackpot (1 in 292 million), but exactly how small is one out of 14,000,605?

Let’s relate this probability to a stack of pennies. Each penny is 1.52 mm thick. A stack of 14,000,605 pennies would be 21280919.6 mm tall. We can use dimensional analysis to convert this into miles:

So the probability of the Avengers defeating Thanos is approximately the same as the probability of selecting one lucky penny from a stack of pennies over 13 miles high.

Similar analysis can be taken to find the height of a stack of pennies related to the probability of making a perfect NCAA tournament bracket (8.7 trillion miles) or winning the Powerball lottery (over 275 miles).

The next time you are teaching probability, I encourage you to engage your students by incorporating examples from The Avengers, a movie series many of them have seen and/or can relate to. Using pop culture examples and references is an outstanding way to engage our students in mathematics.

Historical note: French mathematician Blaise Pascal wrote a book about the triangle that bears his name in the year 1665. This triangle was known around the year 1000 in Persia, and is called the Khayyam triangle in Iran. It was also known around the year 1050 in China, and is called the Yang Hui triangle there. It was known in the year 1566 in Italy, and is called Tartaglia’s triangle there.


Exploring Cross-Sections in a Computer Environment – Using 3D Builder

Senfeng Liang and Brandon Freeman, University of Wisconsin-Stevens Point Laura Leahy, P.J. Jacobs Junior High School

Geometry learning has an important role in the Common State Standards for Mathematics (CCSSM). Students are expected to have knowledge of 3D shapes as early as kindergarten. Kindergartners should be able to “identify, name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres” (Common Core State Standards Initiative, 2010, p. 9).

The learning of 3D objects is one of the main learning goals in geometry after grade 6. For example, students are asked to find the surface area of prisms and pyramids by decomposing them (Common Core State Standards Initiative, 2010, p. 40). Unlike two-dimensional objects, such as triangles, rectangles, and circles, in which students can examine their properties by drawing and observing shapes on paper, the learning of three-dimensional objects can be hard for students. According to Mithalal (2009), in space geometry, “using drawing is generally too difficult for the pupils” and “They have to break down the drawing into various components, so that they can imagine the shape of the object. In fact, they would have to carry out dimensional decomposition before any visual exploration” (pp. 798). Mithalal (2009) suggested that using 3D geometry computer environments may balance these difficulties.

Cross-sections in geometry

When teachers teach 3D objects, one common approach is to connect 3D objects to 2D objects. There are several ways one can achieve this goal: isometric drawings, nets, and cross-sections (Bassarear, 2012). Students can draw 3D objects by using isometric drawing paper. Students can also connect 3D objects and 2D objects by using nets: they can construct 3D objects (such as tetrahedrons, cubes, octahedrons, and icosahedrons) by taping 2D nets; they can also unfold 3D objects to get 2D nets (such as unfold a cereal box). CCSSM requests that “students work with three-dimensional figures, relating them to two-dimensional

figures by examining cross-sections” (Common Core State Standards Initiative, 2010, pp. 46). Students may have some opportunities to experience cross-sections in everyday life, such as cutting watermelons or cucumbers into smaller pieces. They may get a circle or ellipse (depends on the way they cut). However, students may not have many opportunities to cut other 3D shapes such as cubes, cylinders, pyramids, and cones, and then observe the cross-sections. It may be difficult to find concrete objects that students can actually cut to find their cross-sections. For instance, we asked a group of students if it is possible to get pentagons or hexagons for cross-sections of a cube, and nearly all replied no. But in fact, both pentagons and hexagons are possible cross-sections of a cube. This activity suggests that at least for some students, there is a need to explore more cross-sections of some basic 3D shapes.

The software

As suggested by Mithalal (2009), teachers may use computer environment to teach 3D geometry. The app we recommend using is the “3D Builder.” This free app, developed by Microsoft, allows people to either develop 3D shapes or upload 3D models to the app, modify them, and finally print out the 3D shapes using 3D printers. It works on Windows 10, or Windows 10 Mobile devices. Even though this app is a tool for 3D printing, it also serves as a tool to address the problem of limited opportunities for learning about three-dimensional figures by allowing students to examine cross-sections of 3D shapes. The software is available for download at https://www.microsoft.com/en-us/p/3d-builder/9wzdncrfj3t6.

How to use the software to do cross-sections

By using the 3D builder, students can first insert an object, say, a cube, and then edit the object. One of the edit options is split. This option allow user to split a shape into two disconnected parts and user can choose to keep one of these two disconnected pieces or keep both pieces. Once you click the split button, there will be three circles with arrows that allow students to adjust the position of the cross-section. Then students can split

https://www.microsoft.com/en-us/p/3d-builder/9wzdncrfj3t6https://www.microsoft.com/en-us/p/3d-builder/9wzdncrfj3t6


the object and obtain the cross-section. Students can also choose to keep the top, keep bottom, or keep both. For example, one could find the possible cross-sections of a pyramid.

Step 1: First, click “New Scene,” and you will have a screen look like Figure 1

Figure 1Step 2: Click the “insert” button, and you will see multiple options as shown in Figure 2.

Figure 2Step 3: Choose “Pyramid” and the scene will look like Figure 3.

Figure 3

Step 4: Choose the “Edit” button, and then click “Edit.” The pyramid will become editable and look like that shown in Figure 4. The light blue shape indicates the plane to get the cross-section.

Figure 4Step 5: Students can now adjust the arrows on these three circles to find the cross-section. For example, we manipulated the arrows and found that one possible cross-section is a pentagon as shown in Figure 5.

Figure 5


Step 6: Then we can choose one of the three options in Figure 6: “Keep top,” “Keep bottom,” or “Keep both.” We chose “Keep bottom.” Next, click the “Split” button. Finally, we obtained the new shape with the cross-section shown in Figure 7.

Figure 6

Figure 7

Tasks to explore To help develop students’ learning experience of cross-sections, they are asked to explore the possible cross-sections of a cone, a cube, and a pyramid.

1. What kind of shape will you get if you take the cross-section(s) of a cone?

Students’ exploration of cross-sections of a cone includes circle, ellipse, parabola, and isosceles triangle. For example, one student illustrated how a hyperbola can be found in Figure 8.

Figure 8

2. What kind of shape will you get if you take the cross-section(s) of a pyramid?

Students’ exploration of cross-sections of a pyramid includes square, quadrilateral, isosceles triangle, trapezoid, and pentagon.

3. What kind of shape will you get if you take the cross-section(s) of a cube?

Students’ exploration of cross-sections of a cube is a most interesting activity. Most students have no problems finding squares, non-square rectangles, and triangles. Some students, however, were additionally able to find pentagons and hexagons as shown in Figure 9.

Figure 9


Figure 10

Conclusion Cross-sections allow students to examine the

properties of 3D shapes; however, it may be difficult for students to simply imagine the cross sections of 3D objects rather than visualize them (Cohen & Hegarty, 2007). Without this app, our students usually need to imagine the cross-sections of 3D shapes. They may successfully get the cross-sections of simple situations. For example, students usually can find that cross-sections of a corner of a cube is a triangle. However, when the question becomes more complex their imagination may not give them complete and accurate results (For example, in one class, many students could not imagine the situation when a cross-section of a cube is a pentagon). This app has positively enhanced students’ learning of cross-section in a computer environment. By using the 3D builder app, students can conveniently explore the cross-sections of 3D shapes and try unlimited attempts to find the cross-sections of any given 3D shapes.

References

Bassarear, T. (2012). Mathematics for Elementary School Teachers (Fifth Edition ed.). Belmont, CA: Brooks/Cole.

Cohen, C. A., & Hegarty, M. (2007). Sources of diffi-culty in imagining cross sections of 3D objects. In D. S. McNamara, & J. G. Trafton (Eds.), Proceed-ings of the Twenty-Ninth Annual Conference of the Cognitive Science Society (pp. 179–184). Austin TX: Cognitive Science Society.

Microsoft Corporation. (2017). 3D Builder. Retrieved from https://www.microsoft.com/en-us/p/3d-build-er/9wzdncrfj3t6.

Mithalal, J. (2009). 3D geometry and learning of mathematical reasoning. In V. Durand Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceeding of the 6th Congress of European Research in Math-ematics Education (pp. 796-805), Lyon, France.

National Governors Association Center for Best Prac-tices, & Council of Chief State School Officers. (2010). Common Core State Standards for Mathe-matics (CCSSM). Washington, DC: National Gov-ernors Association Center for Best Practices and the Council of Chief State School Officers. Re-trieved from http://www.corestandards.org/Math/Content/K/introduction

https://www.microsoft.com/en-us/p/3d-builder/9wzdncrfj3t6https://www.microsoft.com/en-us/p/3d-builder/9wzdncrfj3t6http://www.corestandards.org/Math/Content/K/introductionhttp://www.corestandards.org/Math/Content/K/introduction


Today’s students must be able to reason and analyze quantitatively. “…the enormous amount of data available affects the decisions one makes politically, as a consumer, and in one’s career, and in everyday life.”

(Metz, 2010, p.1).

The Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report was first established as a Pre-k – 12 Curriculum Framework in August 2005. An updated version is currently in the works. The same year, a GAISE College Report was also published (and was updated in 2016). The first of its kind, the GAISE documents placed a solid importance of integration of statistics in mathematics classrooms across our country at all grade bands. With the ultimate goal of statistical literacy for all, the GAISE reports aim to arm teachers with necessary knowledge to teach statistics at all levels.

As you probably know, the National Council of Teachers of Mathematics (NCTM) promoted the use of Data Analysis and Probability in its Principles and Standards for School Mathematics (2000). Despite this importance, the American Statistical Association began focusing on guidelines for mathematics teachers with examples to fully incorporate statistics into the curriculum. The GAISE document fully explains the format for problem solving at various educational levels with examples that teachers may use in the classroom.

According to the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report (2016, p.6), the following six key teaching principles are identified as the framework for statistical learning:

1. Teach statistical thinking.

2. Focus on conceptual understanding.

3. Integrate real data with a context and a purpose.

4. Foster active learning.

5. Use technology to explore concepts and analyze data.

6. Use assessments to improve and evaluate student learning.

In addition, the GAISE Report further stipulates some new focuses from their original report in 2005 for our modern society (GAISE, 2016, p.6):

In addition to these six recommendations, which remain central, we suggest two new emphases for the first recommendation (teach statistical thinking) that reflect modern practice and take advantage of widely available technologies:

a. Teach statistics as an investigative process of problem-solving and decision-making. Students should not leave their introductory statistics course with the mistaken impression that statistics consists of an unrelated collection of formulas and methods. Rather, students should understand that statistics is a problem-solving and decision making process that is fundamental to scientific inquiry and essential for making sound decisions.

b. Give students experience with multivariable thinking. We live in a complex world in which the answer to a question often depends on many factors. Students will encounter such situations within their own fields of study and everyday lives. We must prepare our students to answer challenging questions that require them to investigate and explore relationships among many variables. Doing so will help them to appreciate the value of statistical thinking and methods.

One fact is important for all readers to know and understand: statistics is different than mathematics. In particular, the issue of variability in data is what makes statistics different. This variability leads to approaching and sometimes solving statistical problems differently. Cohen (2015) explains that proficient statistical students should be able to formulate a problem or detect a statistical problem in context. In solving a statistical

Examples of how to Incorporate the GAISE Report Recommendations into Teaching

Angela L.E. Walmsley, Concordia University Wisconsin


problem, students must select the most appropriate statistical model or application. There may not be one “right” answer. All along, a student must consider the variability in the data or the statistical problem in choosing the best way to analyze.

In order to help students facilitate the best approach to working with statistics, the GAISE report provides the following guidelines to assist students in every problem: 1) Formulate Questions; 2) Collect Data; 3) Analyze Data; and 4) Interpret Results. A more comprehensive table of these four steps in the framework can be found in the Appendix.

In addition, students should use this framework as they increase their statistical abilities across three bands labeled A, B and C. The rationale for using bands described in this way is that learning is not necessarily dictated by grade level. A young student could get to Band B while an older student may be starting at band A depending on his or her statistical background and knowledge. The general idea is that students start in band A as a young student (elementary school level) and progress through B (middle school level) and onto C (high school level) by the time of high school graduation; but the authors realized that the ability to do this would differ by school.

The purpose of the remainder of this article is to take an example and discuss it across the three band levels of A, B and C while utilizing the four step framework.

Example: The Ice Cream Problem - Level A

One important aspect of statistical education is for children to recognize the real life, daily importance of understanding data. One way to get small children excited about statistics is for them to see themselves in the data. For example, the teacher of a class of first graders could easily “survey” the class about their favorite type of ice cream. In this example for level A, the question posed is “What is our class’ favorite type of ice cream?” In Level A, students can start to collect data by offering some suggestions: vanilla, chocolate, and strawberry. Some children may ask what happens when their choice is not listed; which can offer another category of “other.” Then children can go to the board and draw an ice cream cone in the category that they choose. This method of data collection becomes a pictograph….a precursor to the bar chart that is popular with categorical data. At Level A, when analyzing the

data, students can answer the question about which is most popular (the mode) and which is least popular. The interpretation of data allows for students to see their own information displayed and understand the class’ choices in some sort of collative manner. Statistical investigations in elementary grades are often framed by the teacher and then answered with data from the classroom. Information stays within the classroom, and the classroom is the population.

Using the four step process of the scenario above, a sample explanation of this type of problem follows:

Step 1: Formulate Questions

A teacher in a 1st grade classroom may wonder about which ice cream flavors are the flavors for the class? After discussion, the question to be investigated could be:

“In our class, what types of ice cream flavors do students like the most?

Step 2: Collect Data

Students then provide some samples of popular ice cream flavors. The class as a group should decide what the options are (including an “other” option since we can’t possibly list all the available flavors of ice cream). The group could decide on the following: Chocolate, Vanilla, Strawberry, and Other. The class could then decide the best way to collect the data. The teacher hands out a piece of paper to each student with an ice cream cone copied on it and asks the students to label their cone and also color it with the corresponding colors to match the cone. They then must come to the board and put their cone in the correct area of flavor choice.

Step 3: Analyze Data

Data on a single categorical variable such as this can be presented nicely in a picture graph as shown in Figure 1. Once all ice cream cones are categorized by flavor, the students can physically move their ice cream cone to the pictograph by placing it above the correct flavor. From this graph, students can have good discussions about which flavor is most popular, least popular, or discuss if there are ties.


Figure 1. Ice Cream Picture Graph.

Step 4: Interpret Results

In this example, students will notice that Vanilla is the most popular choice. Strawberry is the least popular. And chocolate and “other” categories tie. The teacher should highlight that this survey is of our 1st grade classroom. Good questions would be “What would this look like if we surveyed the other 1st grade classroom? What would it look like if we surveyed the whole school? What would it look like if we surveyed only teachers? Students should understand how to essentially tally results, display the data, talk about the data, and then talk about any limitations of the data (the survey results don’t necessarily mean all 1st graders think the same as this classroom).

Students who are competent at Level A are naturally curious about data and how mathematics works in a data driven situation. The traditional mathematics topics of geometry, algebra and number sense can be developed using data. It is important that students are introduced to data early so that they may develop their knowledge of statistics throughout school. As an adult, most people will analyze data more than any other mathematics topic with the exception of the importance of developing problem solving skills – which is also a benefit of statistical investigations.


Example: The Ice Cream Problem - Level B

In Level B, this same data can move from a pictograph to a more formal way to show data. The primary question might still be “What’s our class’ favorite ice cream flavor” but might expand outside the classroom to “What is the entire 6th grade’s favorite ice cream flavor?” In this case, collecting data would require students to create some way to gain the information. Should they survey every single student in the 6th grade and how should they do it? Should they go from homeroom to homeroom? Should they survey only a representative sample? The data can then be displayed in a frequency table where students move from tallying responses to calculating frequencies and then on to calculating percents. Lastly, the data could be displayed in a pie chart or bar chart in order to analyze the data. Students could also color code their ice cream cones by male and female which allows a comparison of groups (how many in each category belong to boys and how many to girls). Furthermore, students might break down the data by class. Let’s say there are four sections of students in the 6th grade. They can show the data overall and then again by section. The importance of relative frequency percentage becomes important here because of the variability of class sizes.

Using the four step process of the scenario above, a sample explanation of this type of problem follows:

Step 1: Formulate Questions

The ice cream question can evolve to middle school where the question might be:

“What’s the 6th grade’s opinions on best flavors of ice cream?

Step 2: Collect Data

In a large school where the entire 6th grade class might be 400 students, the math class assigned to investigate this problem would need to discuss the best way to sample students. Should they try to get data from every single 6th grade student? Do they have the time and resources? Or should they sample a large number of 6th graders to determine? If they sample a large number, a discussion ensues about what the best way of sampling would be. Should they just ask their immediate friends? Or does this create bias?

In discussing the organization of the data collection, the importance of accurate and non-biased data can be analyzed. Students should realize that if they are unable to sample all 400 students (which is quite likely), they need to come up with the best way to sample. They should perhaps use a simple random sample generator and select 100 students to survey. The students could email those 100 students a survey (allowing them the experience of designing an on-line survey). Or, perhaps if there are 16 homeroom sections of 25 students each; they randomly choose 4 of the 16 sections and survey every single person in that classroom (this is called cluster sampling). Discussing the best way to sample leads to rich conversations about representation of the sample to the population of all 6th graders.

Step 3: Analyze Data

In analyzing the data, students first create a frequency distribution from their 100 samples as shown in Figure 2. After, students can create bar charts and pie charts easily using technology (see Figures 3 and 4). The importance of clear graphs should be discussed.

Figure 2. Ice Cream Frequency Table.


Figure 3. Ice Cream Bar Graph.

Figure 4. Ice Cream Pie Chart.

Step 4: Interpret Results

In interpreting of the results, students should discuss how to move from tallying to calculating percentages. The percentages allow for easy discussion about what flavor is most popular and least popular. In addition, students should discuss how graphs are created regarding the importance of looking clearly at labeling and axes designation. Most importantly, the students should start to discuss that these results (if sampled correctly) should be representative of the entire 6th grade population in the school. If they surveyed 100 other students, results should be similar but not necessarily exact. To further the investigation, students could survey 100 other students and compare the findings. Students could also begin to formulate their own questions. Like, “Do the results look different based on gender?” Students can further their investigation by splitting up flavors of ice cream by gender and showing

results in table form (using a contingency table, see Figure 5) and graphically as shown below in Figure 6. This movement towards analyzing and connecting two categorical variables is appropriate for Level B.

Figure 5. Ice Cream Contingency Table.


Figure 6. Ice Cream Side-by-Side Bar Chart.

All of these graphs were created on the artofstat.com website (http://www.artofstat.com) which has free web apps where students can input data and create various graphs as they try to answer questions. In our technology driven world, students should not be confined to hand drawn or calculator resources. Instead, they should utilize easy-access technology which produces clear graphs easily. It allows students the freedom to explore multiple data questions easily.

In middle school, students move away from only teacher directed questions to asking and answering more statistical questions as they collect and interpret data. Most im

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