USING NUMBER SENTENCES TO INTRODUCE THE IDEA OF VARIABLE Max Stephens The University of Melbourne NNFC Auckland Feb. 2007

Opening interview – Year 3 Peter’s Method Peter is subtracting 5 from some numbers. Peter says that these are quite easy to do. Do you agree? 37 – 5 = – 5 = – 5 = 81 But Peter says some others are not so easy, like: 32 – 5 53 – 5 84 – 5

Peter’s Method – Year 3 Peter says, “I do these by first adding 5 and then subtracting 10, like 32 – 5 = – 10. Working it out this way is easier.” Does Peter’s Method give the right answer? Let’s look at the other two questions (53 – 5, and 84 – 5). Can you use Peter’s Method on each of these? Rewrite each question first using Peter’s Method, and then work out the answer.

Peter’s Method – Year 3 Can you write three other questions using Peter’s method? (Ask the student to write three new questions using Peter’s method.) The teacher said to Peter: “Are you saying that your method always gives the right answer?” Can you explain why Peter’s method always works?

Peter’s Method Does Peter’s method work if you are subtracting 6? How do you think Peter would re-write this question? 73 – 6 Can you rewrite the sum first as you think Peter would, and then work out the answer?

Peter’s Method What number would Peter put in the box to give a correct answer? 73 – 6 = 73 +  - 10 Can you write two other examples like this. Rewrite each question first using Peter’s method, and then work out the answer. Do you get a correct answer?

Peter’s Method Peter says that his method works for subtracting 7, and 8 and 9. Can you show how Peter’s method works for these three questions: Rewrite each question first using Peter’s method, and then work out the answer. 83 – – – 9 Can you explain how this method always works?

Peter’s Method – children’s responses Alan, end of Year 2, answered the last question: “For any number you take away, you have to add the other number, which is between 1 and 10, that equals 10; like 7 and 3, or 4 and 6. You take away 10 and that gives you the answer.”

Peter’s Method – children’s responses Alan understands the allowable range of movement of the number being taken away, and imposes a boundary condition, namely that the number being subtracted has to be less than 10.

Peter’s Method – children’s responses Alan also identifies the connection between the number being subtracted and the number which Peter then adds on. He also sees that the initial numbers, 83, 123, or 235, can be left out of an explanation of why Peter’s method always works.

Peter’s Method – children’s responses Zoe (8 years and 4 months and at the start of Grade 3), gave an explanation which focuses on the equivalence between subtracting a number (less than 10) and Peter’s Method: “Whatever the number is you are taking away, it needs to have another number to make 10. You add the number to make 10, and then take away 10. Say, if you had 22 – 9, you know = 10, so you add the 1 to 22 and then take away 10.”

Peter’s Method – children’s responses Tim, (9 years, 1 month at the start of Grade 3) said: “Here is an explanation for all numbers. Whatever number he (Peter) is taking away (assumed to be less than 10), you plus the number that would make a ten, and you take away ten. The bigger the number you are subtracting, the smaller the number you are plussing. They all make a ten together.”

Peter’s Method – children’s responses Kou, (9 years, 6 months at the start of Year 3) explained: “It does not matter what number is being taken away, when the adding number makes ten, the answer is always the same, whether the subtracting number is increasing or decreasing.”

Peter’s Method – children’s responses Tim and Kou discern the pattern of variation between the “adding number” and the “subtracting number” – as one becomes bigger the other must become smaller.

Peter’s Method – children’s responses These four students are also able to ‘ignore’ for the purposes of their explanation the value of the ‘starting number’. They are also comfortable with “a lack of closure” (see Collis, 1975), that is, leaving the expression in uncalculated form.

Peter’s Method – children’s responses Their explanations capture in their own language the equivalence between expressions experts would represent as a – b and a + (10 – b) – 10 Their explanations do not fully capture the range of variation implicit in this algebraic sentence, hence our use of the term quasi-variable.

Keiko’s responses 83 – 7 = – 14 = 76 version 1 83 – 7 = – 10 = 76 version – 8 = –16 = 115 version – 8 = – 10 = 115 version – 9 = – 9 = 226 version – 9 = − 10 = 226 version 2

Susan’s Method – Years 5 & 6 Susan said it this way: “Instead of writing 32 − 5, 32 − 6, 32 − 7, 32 − 8 and so on, I decided to write the symbol ▼ to stand for the numbers 5, 6, 7, 8, and so on. So, I wrote 32 − ▼ (read as: “32 minus some number”) to represent all of these.” Susan then says: “So instead of 32 − ▼ (“32 minus some number”) Peter says 32 +  − 10” (read as: “32 plus some other number minus 10”).

Susan’s Method – Years 5 & 6 Susan then says: How does Peter find the value of the second number  ? What do these two numbers add up to? What can you say about ▼ +  = (Pause for answer to this question. Then ask: ) Could ▼ (“the first number”) stand for a fraction like 7½ or a decimal fraction like 5.2 ?

Susan’s Method Can we look at how Peter’s Method could be used for subtracting numbers like 95, 96, and 97? Suppose Peter had 251 − 95, what do you think he might do to make it easier? What would he do if he had 251 − 96.5? What do you think he would do if he had 251 − 93⅓ ?

Susan’s Method Remember what Susan did before. Now, instead of writing different sentences like 251 − 95, 251 − 96, 251 − 97, 251 − 98, Susan again uses the symbol ▼ to represent all of these. What do you think she would write? (Pause for students to write 251 − ▼.)

Susan’s Method (Having written 251 − ▼, then say:) Susan then re-writes this new sentence 251 − ▼ to show how Peter would subtract numbers like 95, 96, 97,98 and so on. She uses a second symbol  to write  …. (Be careful to read this as: “251 plus some other number.”) Can you complete this sentence?

Susan’s Method How is the value of the second number  connected to the value of the first number▼ ? (Point to but do not verbalize the symbols.) What do these two numbers add up to? What can you say about ▼ +  = Could you use this reasoning to show how Peter would solve 251 − 83?

Susan’s Method – children’s responses Leo (Grade 6) said that Peter’s Method was not confined to subtracting numbers less than 10, or to subtracting numbers less than (and near to) 100, or even to subtracting numbers less than (and near to) Leo said that Peter’s Method and its extensions were instances of a pattern that “always works”.

Susan’s Method – children’s responses When asked how this pattern could be expressed, Leo wrote: –  = + (▲ –  ) – ▲ saying any subtraction can be converted into “an easier subtraction” by choosing a “cleaner” number ▲ (greater than the original number  being taken away), adding the difference (▲ –  ) and then subtracting ▲.

Susan’s Method – children’s responses Satoshi (Year 7) after calculating 83 – 7, 123 – 8 and 235 – 9, wrote a symbolic expression M – n = M + (10 – n) –10 to explain why this method always works. After the second extension, Satoshi wrote: M – n = M + (100 – n) –100.

Susan’s Method – children’s responses Other students in Grade 7 and 8 used x and y to write expressions similar to those used by Satoshi. Some used x and y to show, for example, that 251 – x could be written as: y – 100 where y = 100 – x

Susan’s Method – children’s responses Other students in Years 7 and 8 wrote 251 – x = (100 – x) – 100. It might be thought that these students already have a notion of variable, but we argue that these forms of completely generalized thinking do not emerge out of the blue.

Susan’s Method – children’s responses These symbolic responses from students at the end of primary school and at the start of high school are solidly based on students' grasp of the underlying structure of specific number sentences. Take away the experience of working with specific number sentences, and these elegant generalizations are less likely to emerge

Summary and discussion The number sentences described above can be seen as a kind of “proto-algebra” where number sentences become objects for exploring patterns of variation that can be, but are not necessarily, represented by algebraic expressions. Many number sentences have this potential, but how this potential can be realized depends how students are supported to see these possibilities for variation.

Summary and discussion A first step in looking beyond particular number sentences to seeing generalizable patterns in these sentences is helping students to leave number sentences in uncalculated (unexecuted) form. This may go against the grain for many students who think only computationally.

Developing ideas of a variable A second step is to avoid premature generalization. Even in Peter’s Method, where the aim is to have students see that subtracting 5 is equivalent to adding 5 and subtracting 10, the numbers 5 and 10 are still be seen by many students as particular numbers, and not as a part of a pattern.

Developing ideas of a variable When asked to consider 73 – 6 = 73 +  – 10, some students immediately see the numbers 5 and 5, 6 and 4 as part of a pattern with both numbers adding to 10. Others need further practice applying Peter’s Method to 83 –7, 123 – 8, and 235 – 9 before being able to articulate the relationship between the number being subtracted and the number to be added before subtracting 10

Developing ideas of a variable In extending Peter’s Method, larger numbers are used, as well as a decimals and fractions, in order to illustrate underlying patterns of generalized thinking. Even when students are working with numbers, they still need many carefully selected exemplifications of the underlying general relationship.

Developing ideas of a variable A third step in developing students’ ideas of variable numbers is to acknowledge the importance to students of the boundary conditions implicit in Peter’s Method and its extensions. Many students were able to explore numerically and to articulate important patterns of variation implicit in mathematical relationships, a – b = a + (10 – b) – 10 and a – b = a + (100 – b) – 100, long before they might be expected to know these formal algebraic expressions.

Developing ideas of a variable A fourth step is the use of representative symbolic ‘terms’ as a way of summarizing multiple numerical expressions that students have already met. Working with and on symbolic representations is very powerful in drawing the attention of students to the relationships between numbers and to their boundary values. Even when students were invited to use summarizing symbols, such as  and ▼, these appeared to retain the boundary values of the originating number sentences.

Developing ideas of a variable Simple symbolic expressions, like those using ▼ and , based on number sentences that students have met, enabled children to attend to and explore patterns of variation embedded in Peter’s Method, and to extend its range of application. However, these symbols do not have a meaning on their own, as they will in later algebra.

Developing ideas of a variable Their meaning is anchored in, but not completely restricted to, the specific number sentences from which they are derived and to which students are able to return to confirm the meaning attached to them. Students need a lot of experience before they are able to detach “symbols” from specific number sentences which give them meaning.

Developing ideas of a variable Students need to be able to move forwards and backwards across the bridge connecting number sentences and generalizations that can be derived from them, whether these generalizations are stated verbally or symbolically. That bridge permits “proto-algebra” to take place.

Developing ideas of a variable Some children are never introduced to such a bridge in their school experience of number sentences. Some are able to make a bridge of their own devising. For others, a bridge is made available for a short time, but withdrawn too quickly leaving them to cope with symbols that have lost their connection with number sentences.

Developing ideas of a variable Teachers’ vision has for so long been restricted to thinking of arithmetic primarily as computation. The potentially algebraic nature of number sentences as discussed here can provide a strong bridge to the idea of variable. It can also strengthen children’s understanding of basic arithmetic.

Developing ideas of a variable In the 21st century, the mathematics curriculum in the primary and early high school years must address these two objectives.