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

2 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. Algebra for Primary Students: Relational Thinking

3 With a partner… Describe how students might use relational thinking to respond to these number sentences (note not all sentences are true). – A = – 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 Algebra for Primary Students: Relational Thinking

4 Rank the following from easiest to most difficult: = 71 + d 92 – 57 = g – b = = – d 92 – 57 = 94 – 56 + g = b Carpenter, Thomas P. Thinking mathematically: integrating arithmetic and algebra in elementary school Algebra for Primary Students: Relational Thinking

5 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

6 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. Algebra for Primary Students: Relational Thinking

7 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 Algebra for Primary Students: Relational Thinking

8 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 Algebra for Primary Students: Relational Thinking

9 #1 - How many? Algebra for Primary Students: Relational Thinking

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13 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, Algebra for Primary Students: Relational Thinking

14 Round Two….Ready?

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20 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 Algebra for Primary Students: Relational Thinking

21 “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) Algebra for Primary Students: Relational Thinking

22 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. Algebra for Primary Students: Relational Thinking

23 Unitizing and Fractions Can you see fourths? One-fourth? Two-fourths, Three-fourths, Four- fourths? Algebra for Primary Students: Relational Thinking

24 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. Algebra for Primary Students: Relational Thinking

25 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 Algebra for Primary Students: Relational Thinking

26 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 Algebra for Primary Students: Relational Thinking

27 THANK YOU!! Judi Laird Vermont Mathematics Initiative University of Vermont (UVM) Algebra for Primary Students: Relational Thinking