## Saturday, January 17, 2009

### M4A1bc & M5A1a - symbols such as □ and Δ

M4A1. Students will represent and interpret mathematical relationships in quantitative expressions.
b. Represent unknowns using symbols, such as □ and Δ.
c. Write and evaluate mathematical expressions using symbols and different
values.
M5A1. Students will represent and interpret the relationships between quantities algebraically.
a. Use variables, such as n or x, for unknown quantities in algebraic
expressions.

In the last entry, I discussed mathematical expressions as the language of mathematics and how important it is for students to learn to write and read mathematical expressions, starting when students start studying addition formally in Grade 1. I also discussed it may be possible for students to represent situations involving missing addend situations using mathematical expressions using a box, like 5+[ ]=8 (or 5+?=8, 5+__=8, etc.). In grades 1-3, those symbols are used as place holders for particular values. However, in Grade 4, students begin the next phase of using symbols to represent numbers and quantities, that is, the concept of variables.

As we consider teaching of this complex idea (variables), it may be worth noting a progression across grades:
Grades 1 - 3: writing math sentences with numbers, and occasionally with place holders
Grade 4: writing math sentences with symbols like □ and Δ.
Grade 5: writing math sentences using letters as symbols
Some people may wonder what's the point of using symbols like □ and Δ in Grade 4. Why not just use letters since that's what is typically done in higher math? Although there are probably many reasons for using symbols like □ and Δ, one possible reason is the principle I have observed in many Japanese curriculum materials: do not introduce a new representation and a new concept simultaneously. Although the idea of using letters to stand for numbers may be straightforward to those of us who already learned the concept, I'm also sure that you have heard people say how they were confused by the idea of using letters in math sentences. This suggests that use of letters to represent numbers and quantities isn't that simple. So, it may not be a good idea to introduce both letters as representations and the concept of variables at the same time. However, since we do need symbols to talk about variables, the natural choice seems to be to use something familiar, symbols like □ and Δ.