AQ Primary-Junior Mathematics, Part 1 Winter 2012SESSION 7 – March 20, 2012

- Exploring the Big Ideas and Representations of Addition (Whole Numbers) - Concepts, Algorithms, and Mental Math Strategies- Exploring Algebraic Reasoning through Generalized Arithmetic -Using principles of structured problem solving and learning theories to design three-part problem solving mathematics lessons, using bansho (board-writing) - lesson plan template and assessment for learning seating plan tool

Design and Implementing a Problem Solving Lesson

For Tuesday, March 20, 2012TREATS: Molly and JoeDUE: Green is upREAD AND RECORD: Check wikispace. Read 4 articles a) Describe 2 characteristics of each

learning theorya) Explain how the learning theory can

be used to analyse the components of teaching and learning through problem solving.

MATH TO DO – Butterfly Problem

Upcoming AssignmentsMarch 27 – Math Task 2April 3 - Annotated BibliographyApril 10 – Lesson plan analysis, learning theories paper

Readings (PDF copies on wikispace)

March 20Complexity Theory (first 2 articles only)

Post your name under which co-teaching name on wikispace

Technology Webquest (May 1, 2012)FN, PRT SC to copyscreen

Right ClickMouse to …Paste

Double Click Left Mouseto … Crop

Applet – interactive Mini softwareProgram on theinternet

National library Of manips

NCTM Illuminations

Nelson Math

Independent Math Study Topics• Lesson Completion - April 10 – collaborative lesson plan analysis• Molly and Joe, Jennifer V – gr3 and gr 4 - division of whole

numbers (number facts, 2 digit by 1 digit)• Margaret, Emilia, Melissa, Angela – gr 5 gr 6 (comparing fractions,

equivalent fractions, benchmark fractions) IPAD APP – Math Motion

• Jennifer D and Diana – Gr1 - addition of 2 digit numbers; Christina F – Gr2 - counting money

• Antonia and Christine – Gr2 – subtracting with regrouping• Start looking for journal articles or books NOW – focus on the

mathematics• Course outline (NCTM, Gazette/Abacus – KKZ can get them) …

others – use the York On-Line Library access

Independent Study• 3 lesson plans (3 part lesson plan, bansho plan, assessment for

learning tool) – 3 different types of plans for 3 consecutive lessons• Justify the design of your lessons in two ways:- math content literature review (annotated bibliography becomes your

literature review) – analysis of details of lesson math content- Learning theories (learning theories paper (1st draft) – look at

behaviourism, constructivism, social constructivism, communities of practice and complexity – OVERLAP – analyze the details of your lesson pedagogy

- Work samples from classroom and from colleagues- Independent Study – literature review (content articles for planning

lesson); learning theories analysis (pedagogical justification in the design of lessons; 3 lessons; work sample analysis – mathematics analyzed in the solutions

Annotated Bibliography (4 articles)due April 3, 2012 (Literature Review)

• Reference of the article: Ma, L. (1999). Knowing and doing elementary mathematics. Mahwah, NJ: Lawrence Erlbaum Associates Inc.

• Description of the mathematics and mathematics pedagogy in the article (6 to 8 sentences – focus and specific detail (no generalities)

• Description of how the mathematical details are informing the mathematics content and design of your lessons (6 to 8 sentences)

Problem Solving –Community of Practice• What community practices support the use of teaching and

learning through problem solving in mathematics?• “Wenger (2007, p. 3) – “It needs to develop various resources, such as tools,

documents, routines, vocabulary, and symbols that some way carry the accumulated knowledge of the community.”

• Work collaboratively (individually) in solving a lesson problem (one lesson problem per lesson) – there is whole discussion and analysis of solutions (consolidation)

• setting classroom routines for success in learning through problem solving – active listening, respectful talk (“I agree, I disgree because … please clarify … ask question … ); collaborative structures – eye to eye, knee to knee; think pair share; one stay the other stray

• problem criteria (details of the problem WE are going to use) – During part • Identifying success criteria in relation to the lesson learning goal (at the end –

Highlights/Summary)

Complexity Theory - Characteristics• Complex learning system• Internal diversity – pool of possibilities; members

contribute in different ways, get collective knowledge (robust –diversity, rather than the sameness; source of the intelligence); creativity needed – invent ideas from our different experiences, perspectives, backgrounds, interests, schema;

• In the problem solving math lesson – differentiated instruction –parallel tasks (lesson problem or is it the practice problem??? supports diversity; promotes diversity – engaging students in their different ways of thinking -> limiting ??? (if numbers beyond the child, so need parallel task); problem that can be solved in different ways - scaffolding when (what is the context for parallel task)

• differentiated responses - everyone solves a problem differently

Complexity Theory – Internal Redundancy• Characteristics – sameness among the agents (students -

ideas and interpretations, manners of representation –notation, models like number line); common ground (like communities of practice – language; teaching routines); compensate for others; important for consistency and robustness - foundational

• Problem solving – three-part lesson (before, during, after (consolidation), after (highlights/summary), after (practice); orient paper landscape; Polya’s problem solving process metacognitively; success criteria (strategies, math examples); use of colour chalk (blue chalk for vocabulary)’ to understand the problem – What information are WE going to use to solve problem)

Complexity Theory – Neighbour Interactions• People hold the ideas interaction of close ideas and

interpretations; juxtaposition of ideas; putting someone in a group doesn’t guarantee anything will happen – not group work; “a sufficient density of diverse thought”; “bump ideas up against one another”

• Problem Solving – selection of the problems on the board – can see a sequence of ideas; putting solutions for consolidation – different solutions – can see differences and relationships (e.g., repeated addition is related to multiplication); read a similar reading and put together a chart of characteristics;

Complexity Theory – Liberating Constraints• Characteristics – relationship between too much and too little;

structure – proscriptive (limits, but not limiting; not prescriptive (I expect and you will do it this way); constraints give guidelines, but not how tos or how expected

• Problem solving – solve in 2 different ways; open ended (different answers, different solution/approach; or open-routed problem (same answer, but different solution/approach); choose how to use it, but have limited choice (conceptual errors, if students choose not useful manipulatives … watching time – to choose carefully); for the practice problem – choose your manipulative; for lesson problem –teacher provides

Complexity – Organized Randomness• Characteristics- Boundaries and rules with flexibility; from

ideas, creativity – you won’t know what is coming; organization –

• Problem solving – what solutions select to put up (and what parts of the solution that we focus on – folding paper); bansho (organization of the randomness); taking advantage of the teachable moments (bumping of ideas – randomness, with organization); board – is the key tool we can use for the organization

• Teacher as consciousness of the collective – responsible for prompting differential attention, selecting among the options for action and interpretation, orients as it contributes to the reworking of the parts that constitute our perceptual whole (lesson learning goal); think in terms of many nested systems (concentric circle); network

Complexity Theory – Decentralized Control• Self-organization; learners organizes itself; no one is in

charge; Stephen Johnson (Emergence – 2003); shared is a synonym for distributed

• Problem solving – students solving the problem (during –before – gives you insight into what might possibly happen – no guarantees; sharing of their ideas (consolidation –liberating constraints – teacher selected who will speak)

Journal Writing- Success Criteria• Topic sentence – directly related

to the purpose and focus of the journal prompt

• Details in your journal are from the journal article, direct quoting, with elaboration and explanation

• Include a mathematics teaching or mathematics learning example, with mathematical details

• Focus on 1 or 2 ideas depth of detail

• Conclusion – that restates the focus of the journal, with 1 or 2 new ideas brought out

a) Describe 2 characteristics of each learning theory

b) Explain how the learning theory can be used to analyse the components of teaching and learning through problem solving.

Davis, B. (2005). EmergentInsights into mathematical intelligence from cognitive science. Delta-K, 42(20, pp. 10-19.

Davis, B. (2005). Teacher as “Consciousness of the collective.’ Complicity: An International Journal for Complexity and Education, 2, pp. 85-88. to math

Kindergarten NSN Expectations• Investigate the idea that

quantity is greater when counting forwards and less when counting backwards

• Investigate some concepts of quantity through identifying and comparing sets with more, fewer, or the same number of objects

• Recognize some quantities without having to count

• Understanding of counting concepts of stable order, one-to-one correspondence

• Estimate number in a small set

• Use, read and represent whole numbers to 10 in a variety of meaningful contexts

• Use ordinal numbers in a variety of everyday contexts

• Understanding of numbe rrelationships for numbers from 0 to 10

Counting Strategies … Counting Principles• Stable Order (always assign numbers in same order)• One-to-One Correspondence (assign one and only one number

word to each object)• Cardinality (last count indicates number of objects counted)• Order Irrelevance (order in which objects counted is irrelevant)• Abstraction (all principles can apply to any collection of objects)

Conservation of number – deduce through logical reasoning that the quantity of a collection remains the same when its spatial arrangement and empirical arrangement changed

• Hierarchical inclusion –numbers are nested inside other numbers• Movement is Magnitude – on a number line (0 to 1, 1 to 2, 2 to 3)• Unitizing – counting by a unit of 1, 5s, 10s,

Counting (Learning Trajectory) Clements and Sarama, 2009

• Pre-Counter• Chanter• Reciter• Corresponder• Counter (small numbers to 5)• Counter (to 10)• Producer (small numbers to 5)• Counter and Producer (10+)• Counter Backward from 10• Counter from other than 1• Skip Counter by 10s to 100

• Counter to 100• Counting on Using Patterns• Skip Counter 5s and 2s• Counter of Imagined Items• Counting On – Keeping Track• Counting of Units (Unitizing)• Counting to 200• Number Conserver• Counting Forward and Back

Subitizing and counting are children’s main strategies for quantification

Ontario 3-Part Problem Solving Lesson• 1- Before (Getting Started) - 5 to 10 minutes – revisiting

mathematical ideas and strategies from a previous lesson that relates to the learning goal of the lesson

• 2- During (Working On It) - 15 to 20 minutes - solving the lesson problem in pairs, small groups, or individually

• 3a- After (Consolidation) - 20 to 25 minutes -> coordination of whole class discussion/analysis of student solutions

• 3b- After (Highlights/Summary) - 5 minutes -> recounting key mathematical ideas and strategies related to the learning goal of the lesson

• 3c -After (Practice) - 5 to 10 minutes – solving a problem that is similar to the lesson problem in order to practise applying new ideas and strategies.Pa

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KKZ, 2012 22

Kindergarten Lesson Plans – Analyze and Revise

• Check the curriculum expectations and the types of counting strategies that are provoked

• What landmarks in the learning trajectory for counting (Clements and Sarama) are provoked through the lesson problem?

• What could be changed in the lesson problem to provoke additional landmarks in the counting trajectory?

• What instructional strategies are used to promote a problem solving community of practice? What else could be used?

From Counting to Addition to Same Size Groups

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Lesson

Learning

Goal

Know the curriculum learning goal sequenceto anticipate the range of students’ mathematical thinking

KKZ, 2012 26

BEFORE• 5 to 10 minutes only

• Activating students’ mathematical knowledge and experience that directly relates to the mathematics in the lesson problem

• Includes student responses to a prompt/problem that is similar to previous work to highlight a few key ideas and/or strategies

KKZ, 2012 27

Show 12 in

different ways?

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DURING• 15 to 20 minutes• Understand the problem, Make a Plan, Carry Out the

Plan• Understand the problem - Ask, “What information from

the problem will we use to make a plan to solve it? Explain.”

• Teacher records below the problem the information the students identify in a list.

• Students solve the problem in pairs or in small groups• Teacher circulates to notice the range of mathematics used

by the students in their solutions and to discern the sequence of student sharing of solutions (one solution should relate to and elaborate from the previous solution)

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3 butterflies landed on a bush. 4 more butterflies came. Then 8 more butterflies came.•How many butterflies are on the bush?

•Show 2 different solutions.

DURING (Working on It) Butterfly Problem

KKZ, 2012 31

3 butterflies landed on a bush. 4 more butterflies came. Then 8 more butterflies came.•How many butterflies are on the bush?

•Show 2 different solutions.

Butterfly Problem

KKZ, 2012 32

Which solution should be shared 1st, 2nd, 3rd … ? Sequence shows mathematical elaboration)

1. What mathematics (i.e., concept, algorithm, strategy, model of representation) are the students using in their solution? - How does this mathematics in the solution relate to the

mathematics lesson learning goal?

2. Which solutions are conceptually-based? - Which solutions have an efficient method or algorithm? - Which solutions include or have the potential for a

mathematical generalization?

3. How are the solutions related to each other, mathematically? KKZ, 2012 33

From Emilia’s Class

From Emilia’s Class

From Emilia’s Class

From Emilia’s Class

KKZ, 2012 38

For Wednesday, March 21, 2012TREATS: Molly and Kathy

Upcoming AssignmentsMarch 27 – Math Task 2April 3 - Annotated BibliographyApril 10 – Lesson plan analysis and learning theories paperMay 1 – Technology Webquest and Independent STudy