teresa maguire, alex neill, jonathan fisher february 2007 supporting professional development in...
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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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
P.D. on the ARB websiteP.D. on the ARB website
Research
Concept maps
Assessment strategies
… and the Teacher information pages
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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 …
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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 **
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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.
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Assessment strategiesAssessment strategies(under ARBs & Assessment)(under ARBs & Assessment)
Assessment strategies
Concept cartoons
Drawing
Multiple choice questions
Multiple choice items for group discussions
Predict, Observe, Explain
Matching
Mathematical Classroom Discourse ***
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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.
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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.
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Equality problemEquality problem
7 + = 10 + 2
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Solve this equationSolve this equation
2x – 6 = x + 4
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Additive IdentityAdditive Identity
What happens when I’m added
to a number?
Who am I?
What am I?
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Wagons and hand-holdingWagons and hand-holding
7 + 28 + 13 + 12 = 7 + 13 + 28 + 12
= (7 + 13) + (28 + 12)
= 20 + 40
= 60
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Commutativity and AssociativityCommutativity and Associativity
a + b = b + a (a + b) + c = a + (b + c)
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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.
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Fractional Thinking is …Fractional Thinking is …
… Not just about Pizza
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Fractional Thinking Concept MapFractional Thinking Concept Map
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Partitioning: examplesPartitioning: examples
Share this shape evenly amongst 4 people
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Partitioning: examplesPartitioning: examples
Share this shape evenly amongst 3 people
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Partitioning: examplesPartitioning: examples
Share this shape evenly amongst 3 people
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Partitioning: examplesPartitioning: examples
Share this shape evenly amongst 7 people
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Partitioning: examplesPartitioning: examples
Example: Share 2 cakes equally amongst 3 people
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
What is partitioning?What is partitioning?
Partitioning involves the ability to divide an object or objects into a given number of non-overlapping parts.
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Part-whole fractions: examplePart-whole fractions: example
What fraction is shaded?
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Part-whole fractions: examplePart-whole fractions: example
What fraction is shaded?
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Part-whole fractions: examplePart-whole fractions: example
How much of the square is shaded?
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Part-whole fractions: examplePart-whole fractions: example
How much of the shape is shaded?
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
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)
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Animations – fractions of a setAnimations – fractions of a set
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Animations – immediate resultsAnimations – immediate results
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Exploring Fractional ThinkingExploring Fractional Thinking
Teresa Maguire, Alex Neill, Jonathan FisherFebruary 2007
Check it out on the websiteCheck it out on the website
Research
Concept maps
Assessment strategies
… and the Teacher information pages