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### Multiplication and division inquiry

s designed as a compare and contrast prompt,

Lines of inquiry
1. Extend lists
2. Answer 'why' questions - "If you divide a whole number, how can the answer get bigger?"
3. Start with another number, e.g. whole number or more complex 3.7
When trying to explain why ... "I can almost say it, but I cant say it."

The prompt was inspired by Borasi’s use of two lists that lead, respectively, to 00 = 0 and 00 = 1. Patterns, she asserts, can provide a “valuable heuristic to guide the extension of a known operation to a wider domain” (Borasi, 1992, p. 60).

students would learn to “identify and extend patterns in lists of equations (in order to derive knowledge of operations with negative numbers)”.

The design of prompt 2 encourages students to extend the lists of calculations. This can be done through a process of empirical pattern-spotting in comparing the two lists. For example, Amelia asks if the “one before” the green list would be 4 ÷ 100 = 0.04 “because it’s like the middle numbers sort of flipped round too” (Inquiry 2, lines 91 and 93). She says that the order of the multipliers and divisors are reversed. Adam’s reasoning is becoming transitional because he compares the two equations that form the next line of the lists – that is 4 x 100 = 400 and 4 ÷ 0.01 = 400. He claims that the two are linked:

I’m just saying that if you like look at it, the decimal ones are practically exactly the same as that [blue] one only the other way round. Zero point zero one, if you turn that around you get one hundred, so the answer – four times one hundred. (Inquiry 2, line 61)

Adam’s contribution has the potential of leading to a structural analysis of why multiplying by 100 is equivalent to dividing by 0.01, although we note that he is still a long way from producing a formal deductive argument.

In inquiry 2, he focuses attention on the division side of the prompt by inviting students to compare the two sides:

What I would like you to do is to look at that green [division] one and think about the comparison between the two. Try to explain the green one and look the answers are the same. The answers are the same and the starting points the same. Something’s different, so I’d – you have two minutes to discuss with your neighbour what the pair of you are going to say about the sequence of sums on the right. (Inquiry 2, line 64)

Later, he responds to Sharon by reinforcing the idea of comparing the two sides of the prompt:

Zero point four is linked to forty. The four here is in the tens column and here it’s in the tenths column. So the four’s in a different column. It’s got a different value. So they are linked, which is I think what you [Sharon] were saying. (Inquiry 2, 114)

Throughout the inquiry, students commented on empirical similarities between the two lists without attempting to explain the key concept of division by a number between zero and one. Their note sheets show they pursued a superficial pattern-spotting approach. Indeed, when Zaynab comes to devise an example using the number seven (figure 8.1), her use of the number four in solutions suggests that her thinking involves, at least in part, direct and uncritical transmission from the teacher’s model

Starting with prompt 2, which, as we saw above in section 8.2.1, is designed as a compare and contrast prompt, the teacher seems to give a mixed message. His regulatory statement “Try to explain” the right-hand calculations (Inquiry 2, line 64) is drowned out by the general idea of comparing the two lists. Towards the end of the inquiry, any suggestion of explaining has disappeared as the focus is wholly on repeating the pattern: “I would like you to make up two series like this of your own” (Inquiry 2, line 174)

Inquiries 2 and 6 feature similar prompts. Both take inspiration from Borasi’s exponentiation prompt that uses two lists to arrive at the contradiction that 00 = 0 and 00 = 1 (Borasi, 1992, p. 61):

50 = 1, 40 = 1, 30 = 1, 20 = 1, 10 = 1, hence: 00 = 1;

05 = 0, 04 = 0, 03 = 0, 02 = 0, 01 = 0, hence: 00 = 0.

Justifying the prompt, Borasi writes that “while we may be aware that patterns can occasionally be deceptive, they nevertheless provide another valuable heuristic to guide the extension of a known operation to a wider domain” (Borasi, 1992, p. 60). It was in the hope that students would extend a known operation to a wider domain that the prompts are designed. Prompt 2 aims to make students aware of the consequences of dividing by numbers between zero and one and the link between, for example, multiplying by 10 and dividing by a tenth. The design in ascending and descending powers of 10 also invites students to extend the lists above and below the lines that appear

Resources

A contradiction encountered by extending patterns
Professor Raffaella Borasi, in her book Learning Mathematics Through Inquiry (1992), introduces the students she is working with to the two patterns in the illustration. She writes: "If the heuristic of extending a pattern is applied when we try to evaluate 0
0
, a potential contradiction follows, since two alternative patterns seem both possible and reasonable and yet each suggests a different value for the result." For Borasi, this is a fruitful contradiction because she is running a series of inquiries with two students into what constitutes a mathematical definition.

The patterns can form the basis of an interesting starter discussion with older students in secondary school. However, the discussion can become a bit messy with students taking sides on the basis of what 'feels right' to them. So, typically, they argue 00 must equal zero because 'there's nothing there' or 00 is one because the law of indices would not work if it was anything else. While there might be merit to both arguments, this is an opportunity for the inquiry teacher to remind students of the wider mathematical culture in which 00 is normally defined as one (although I note that Borasi supports the case for it remaining undefined).

One of the most novel responses I have had to these patterns was when a student suggested using a spreadsheet to see what xx equalled as x approached zero. The answer is one, although the spreadsheet, we decided, does not amount to a proof. When students graphed values (to two decimal places) between zero and one, they noticed that values for xx start to rise between 0.36 and 0.37. More broadly, the activity reminds students that they cannot extend patterns unquestioningly.