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Algebra for Primary Students
Developing Relational Thinking in the Primary Grades
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Relational Thinking
Students who can express a number in terms of other
numbers and operations on those numbers hold a
relational understanding of the number.
Understanding numbers relationally helps students use
mathematical relationships to solve problems.
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Algebra for Primary Students: Relational Thinking
With a partner…
• Describe how students might use relational thinking to
respond to these number sentences (note not all sentences
are true).
–
–
–
–
–
–
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A.
B.
C.
D.
E.
F.
G.
H.
I.
37 + 56 = 39 + 54
33 – 27 = 34 – 26
471 – 382 = 474 – 385
674 – 389 = 664 – 379
583 – 529 = 83 – 29
37 x 54 = 38 x 53
60 x 48 = 6 x 480
5 x 84 = 10 x 42
64 ÷ 14 = 32 ÷ 28
Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in elementary school.
Algebra for Primary Students: Relational Thinking
3 2003.
Rank the following from easiest to most difficult:
• 73 + 56 = 71 + d
• 92 – 57 = g – 56
• 68 + b = 57 + 69
• 73 + 56 = 71 + 59 – d
• 92 – 57 = 94 – 56 + g
• 68 + 58 = 57 + 69 - b
Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in elementary school.
Algebra for Primary Students: Relational Thinking
4 2003.
Why Relational Thinking?
• Facilitates children’s learning of arithmetic, making it richer
and easier.
• Algebraic reasoning integrally bound with learning arithmetic,
not a separate topic.
• Smoothes the transition to formal algebra
Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in elementary school.
Algebra for Primary Students: Relational Thinking
5 2003.
Composing, Decomposing and Recomposing
Unit Lengths
Naming the Units:
• How might you describe their lengths individually?
• How might you describe their lengths comparatively?
Using the Units:
• Use the lengths to compose different lengths.
• Use the lengths to compose equivalent lengths.
• Decompose a length into 2 lengths, 3 lengths, etc.
• Use the lengths to find similar differences.
Discuss with a partner some of the Big Ideas that students can
experience using these materials.
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Algebra for Primary Students: Relational Thinking
Sample Title
Probably the major conceptual achievement of
early school years is the interpretation of numbers
in terms of part and whole relationships. With
the application of a Part-Whole schema to
quantity, it becomes possible for children to think
about numbers as compositions of other
numbers. This enrichment of number
understanding permits forms of mathematical
problem solving and interpretation that are not
available to younger children.
Resnick,L.B. A developmental theory of number understanding: In H:RGinsburg(Ed.), The development of mathematical thinking.
New York: Academic Press,1983., p. 114
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Algebra for Primary Students: Relational Thinking
Sample Title
“Composing and decomposing numbers is
another approach to addition and subtraction,
one that is often used alongside with counting
strategies.”
Learning & Teaching Early Math: The Learning Trajectories Approach, Clements and Sarama, 2009,p. 81
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Algebra for Primary Students: Relational Thinking
#1 - How many?
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Algebra for Primary Students: Relational Thinking
#2 - How many?
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Algebra for Primary Students: Relational Thinking
#3 - How many?
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Algebra for Primary Students: Relational Thinking
#4 - How many?
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Algebra for Primary Students: Relational Thinking
What did you just do?
You quickly determined the quantity of a small
number of objects without counting.
"Subitizing is a fundamental skill in the development
of students' understanding of numr(Baroo,
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Algebra for Primary Students: Relational Thinking
Round Two….Ready?
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#5 - How many?
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Algebra for Primary Students: Relational Thinking
#6 - How many?
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Algebra for Primary Students: Relational Thinking
#7 - How many?
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Algebra for Primary Students: Relational Thinking
#8 - How many?
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Algebra for Primary Students: Relational Thinking
#9 - How many?
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Algebra for Primary Students: Relational Thinking
What did you do?
Count groups with an understanding that “1” can
be a number of objects.
3 x 4 = 12
3 groups x 4 circles in each group
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Algebra for Primary Students: Relational Thinking
“Unitizing underlies the understanding of place
value…” It is a huge shift in thinking for children,
and in fact, was a huge shift in mathematics,
taking centuries to develop.”
(Fosnot & Dolk 2001, p11)
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Algebra for Primary Students: Relational Thinking
More on Unitizing
• It is vital that children are able to see a group(s) of objects
or an abstraction like ‘tens’ as a unit(s) that can be counted
(Clements & Steffe).
• Whatever can be counted can be added, and from there
knowledge and expertise in whole number arithmetic can
be applied to newly unitized objects.
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Algebra for Primary Students: Relational Thinking
Unitizing and Fractions
Can you see fourths? One-fourth? Two-fourths, Three-fourths, Fourfourths?
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Algebra for Primary Students: Relational Thinking
Implications for Teaching and Learning
• Use relational thinking tasks to build
understanding of big, overarching
mathematical ideas (e.g. equality, place value,
fundamental mathematical properties,
arithmetic)
• Encourage relational thinking as students
generate conjectures, share ideas and critique
others’ ideas.
• Let students manipulate materials and create
models to support their understanding.
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Algebra for Primary Students: Relational Thinking
Implications for Teaching and Learning
“The tasks that students engage in significantly
influence what they learn, but equally
important are the interactions that they have
about those tasks.”
• Use questions that focus on the important
mathematical ideas…
• “Is that true for all numbers?”
• “How do you know that is always true?”
Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in
elementary school. 2003.
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Algebra for Primary Students: Relational Thinking
Mathematical Thinking for All Students
A defining feature of learning with understanding:
Knowledge is connected and integrated.
Students make sense of new things they learn by connecting
them to what they already know.
Developing relational thinking with arithmetic serves as a basis for
learning algebra
Primary students CAN engage in algebraic reasoning. Relational
thinking activities opens up opportunities for ALL Students to
interact with the important ideas that transcend mathematics.
Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in
elementary school. 2003.
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Algebra for Primary Students: Relational Thinking
THANK YOU!!
Judi Laird
Vermont Mathematics Initiative
University of Vermont (UVM)
[email protected]
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Algebra for Primary Students: Relational Thinking