Teresa Maguire, Alex Neill, Jonathan Fisher February 2007 Supporting professional development in...

$: Teresa Maguire, Alex Neill, Jonathan Fisher February 2007 Supporting professional development in algebraic and fractional thinking Workshop presented at.$
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007

Supporting professional Supporting professional development in development in

algebraic and fractional algebraic and fractional thinkingthinking

Workshop presented at

National Numeracy Facilitators Conference

February 2007

Teresa Maguire and Alex Neill and Jonathan Fisher


P.D. on the ARB websiteP.D. on the ARB website

Research

Concept maps

Assessment strategies

… and the Teacher information pages


Research The ARCT (Assessment Research for Classroom Teachers) project promotes classroom-based research as an integral part of developing ARB resources. ARB resources are based on our own, national, and international research. We present research information in formats relevant to a variety of audiences.

We provide evidence-based information to support teachers' assessment practices. Research by curriculum bank

Science Maths English

•Inter-relationships (ecology) common alternative ideas thinking about systems •Vocabulary Scientific vocabulary Prefixes and suffixes Language barriers •Investigations Science investigations Why "dissolving" is a difficult idea

•Maths journalling *•Vocabulary and comprehension Language barriers Prefixes and suffixes

ResearchResearch


Journalling in Mathematics *

What is journalling?The benefits of journallingTeaching tips for journallingWriting prompts for journallingReferences What is Journalling?Journalling involves students writing about their learning in mathematics. What they write …


ResearchResearch

Self-regulated learningThe focus of the ARCT project's research for 2004 was self-regulated learning. This was used to inform the

development of new resources for the Assessment Resource Banks. Support material for teachers is published as it becomes available.

Research conducted into self regulated learning in 2004 follows: Reflecting on reflective journalling (maths)

Self-regulated learning in the mathematics class **


Researchers: Charles Darr and Jonathan Fisher **Context: proportional reasoning (Year 7)

Using thinking models to represent proportional relationshipsThinking models help students to form a representation of a problem situation. They can involve concrete objects, or be more abstract. Thinking models used included double number lines, geometrical shapes, cuisenaire rods, and decimal pipes.

The double number lineOne of the most successful models was the double number line. The double number line allows the elements in a proportional relationship to be modelled on a two-sided scale.

To read more, see this PDF document.


Assessment strategiesAssessment strategies(under ARBs & Assessment)(under ARBs & Assessment)


Concept cartoons

Drawing

Multiple choice questions

Multiple choice items for group discussions

Predict, Observe, Explain

Matching

Mathematical Classroom Discourse ***


Mathematical Classroom Discourse ***

When to useDiscourse can be used ………

The theoryWhile classroom discussions are nothing new, the theory behind classroom discourse stems from constructivist …

How the strategy worksWell-designed distractors provide alternatives that identify particular misconceptions. Providing ……

What to doIn order for discussion to take place, classroom (sociomathematical) norms need to be …..

NM1199, NM1201, and NM1221 ask students to identify which cartoon characters are estimating and …

Examples of ARB resources

Selected referencesBurns, M. (2005). Looking at How Students Reason. Educational Leadership, 63 (3), pp. 26-31.


Teacher information pagesTeacher information pages

Task administration

Answers

Calibration easy (60-79.9%)

Diagnostic information (common wrong answers and misconceptions)

Diagnostic and formative information

Next steps

Links to other resources/information and to concept maps

To do


Concept mapsConcept maps

This is a series of framework statements that have been developed in significant areas of mathematics.

They:

provide information about the key mathematical ideas involved

link to relevant ARB resources, and

suggest some ideas on the teaching and assessing of that area of mathematics.

Computational estimation**** Fractional thinking


Introduction to computational estimation

Background

Research Expanding students' repertiore of

estimation strategies

Computational estimation ****

Directory of ARB resourcesIntroducing students to estimation

Estimation assessment

Language of estimation

Types of estimation

Concept Map - exampleConcept Map - example


Algebra is…Algebra is…

“The language of arithmetic is focussed on answers.

The language of algebra is focussed on relationships.”

MacGregor, M & Stacey, K. (1999) “A flying start to algebra. Teaching Children Mathematics, 6/2, 78-86. Retrieved 17 May 2005 from

http://staff.edfac.unimelf.edu.au/~Kayecs/publications/1999/MacGregorStacey-AFlying.pdf

Love is….2 + 2 = 3 + 1.


Equality problemEquality problem

7 + = 10 + 2


EqualityEquality

=


Solve this equationSolve this equation

2x – 6 = x + 4


Additive IdentityAdditive Identity

What happens when I’m added

to a number?

Who am I?

What am I?


Conjectures about zeroConjectures about zero

When you add zero with another number it doesn’t change the number you started with.

a + 0 = a

When you take away zero from a number it doesn’t change the number you started with.

a – 0 = aIf you take away the same number from the one you started with you get zero.

a – a = 0


Wagons and hand-holdingWagons and hand-holding

7 + 28 + 13 + 12 = 7 + 13 + 28 + 12

= (7 + 13) + (28 + 12)

= 20 + 40

= 60


Commutativity and AssociativityCommutativity and Associativity

a + b = b + a (a + b) + c = a + (b + c)


Student’s rulesStudent’s rules

If you add up numbers in a different order you still get the

same answer.

When you add three numbers it doesn’t matter whether you start by

adding the first pair of numbers or the last pair of numbers.1

It doesn’t matter if the numbers are swapped around on each side of the

number sentence. If the numbers are the same, the number sentence will still balance

1 Carpenter, Franke & Levi Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School.


Relational thinkingRelational thinking

For each equation, work out what number goes in the box to make the number sentence true. Be prepared to explain how you worked out the answer.

3 + 16 = + 18

36 + 78 = + 74

57 + = 63 + 51


Introduce pronumeralsIntroduce pronumerals

For each equation, find the number that replaces the letter.

68 + b = 57 + 69

82 – 47 = g – 46

a + 38 = 36 + 59

87 + 45 = 86 + 46 + t

234 + 578 = 232 + 576 + e

92 – 57 = 94 – 56 + h


Repeated pronumeralsRepeated pronumeralsFind the Variables

Find the number that each variable is replacing in the following

number sentences.z + z + z + z = 20

11 = t + t

b = 8 – b

j + j + 3 = 7

d + d – 5 = 13

2 × d – 5 = 13 (5 × b) – 6 = 14


Fractional Thinking is …Fractional Thinking is …

… Not just about Pizza


Fractional Thinking Concept MapFractional Thinking Concept Map


Partitioning: examplesPartitioning: examples

Share this shape evenly amongst 4 people



Example: Share 2 cakes equally amongst 3 people


What is partitioning?What is partitioning?

Partitioning involves the ability to divide an object or objects into a given number of non-overlapping parts.


Partitioning strategiesPartitioning strategies

Simple partitions:

halves, quarters, eighths … (halving)

evenness/equal-sized parts

More complex partitions:

larger number of pieces (e.g., 6ths, 10ths, 12ths …)

odd number of pieces (e.g., partition into 3rds, 5ths …)

Partitioning complex shapes: e.g., two squares


Part-whole fractionsPart-whole fractions


Part-whole fractions: examplePart-whole fractions: example

What fraction is shaded?



How much of the square is shaded?



How much of the shape is shaded?


Part-whole fractions: examplesPart-whole fractions: examplesGiven the part find the whole … or another part

If is 1/4 what fraction is ?

What would a whole look like?

If is 2/3 of all the counters, show how many 1/3 is


Comparing fractionsComparing fractions


Comparing fractions: examplesComparing fractions: examples

Show or explain which fraction is larger.

2/3 or 3/5

1/9 or 1/8

7/6 or 8/9


Strategies for comparing fractionsStrategies for comparing fractions

Attempting to use whole number knowledge

Drawing pictures

Identifying fractions with the same denominator or numerator

Benchmarking fractions to well known fractions

Using equivalent fractions

Less sophisticated

More sophisticated


AnimationsAnimations

Immediate response

Computer medium

High motivation to “do again”

Independent

Formative information for the teacher

Sets of objects: NM0129 (unit fractions) & NM0130 (non unit fractions)NM1231 (unit fractions)NM1232 (non unit fractions)


Animations – fractions of a setAnimations – fractions of a set


Animations – immediate resultsAnimations – immediate results


Exploring Fractional ThinkingExploring Fractional Thinking


Check it out on the websiteCheck it out on the website

Research

Concept maps


… and the Teacher information pages


Assessment Resource BanksAssessment Resource Banks

www.nzcer.org.nz/arb

Username: ARB

Password: guide

Teresa Maguire, Alex Neill, Jonathan Fisher February 2007 Supporting professional development in...

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