1 Algebra for Primary Students Developing Relational Thinking in the Primary Grades

1

Algebra for Primary Students

Developing Relational Thinking in the Primary Grades

2 Algebra for Primary Students: Relational Thinking

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.


With a partner…• Describe how students might use relational thinking to

respond to these number sentences (note not all sentences are true).– A. 37 + 56 = 39 + 54– B. 33 – 27 = 34 – 26– C. 471 – 382 = 474 – 385– D. 674 – 389 = 664 – 379– E. 583 – 529 = 83 – 29– F. 37 x 54 = 38 x 53– G. 60 x 48 = 6 x 480– H. 5 x 84 = 10 x 42– I. 64 ÷ 14 = 32 ÷ 28

Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in elementary school. 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



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



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.


Sample TitleProbably 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


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


#1 - How many?


#2 - How many?


#3 - How many?


#4 - How many?


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,

14

Round Two….Ready?


#5 - How many?


#6 - How many?


#7 - How many?

18Algebra for Primary Students: Relational Thinking

#8 - How many?


#9 - How many?


What did you do?Count groups with an understanding that “1” can be a number of objects.

3 x 4 = 123 groups x 4 circles in each group


“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)


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.


Unitizing and Fractions

Can you see fourths? One-fourth? Two-fourths, Three-fourths, Four-fourths?


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.


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?”



Mathematical Thinking for All StudentsA 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.



THANK YOU!!

Judi LairdVermont Mathematics Initiative

University of Vermont (UVM)

[email protected]

mailto:[email protected]

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1 Algebra for Primary Students Developing Relational Thinking in the Primary Grades