## Contexts

This discussion centers around the term *context*. In the educational literature a context is usually any place or situation outside the classroom. The debate focuses on whether learning in the classroom can be successfully applied to some real-life context.

## Jean Lave

The key researcher in this debate is Jean Lave who looked closely at mathematical skills learned in different contexts. Her research, notably with Liberian tailors, young Brazilian merchants, dieters following recipes, and shoppers trying to determine the best value, revealed that each of these groups found ways of solving mathematical problems without using the general math techniques taught in school. Lave published her findings in her 1988 book *Cognition in Practice: Mind, Mathematics and Culture in Everyday Life.*

## Learning Transfer Theory

It is an important assumption in our education system that general skills taught in class can be applied to solve problems in other contexts of real life. This model is referred to as* learning transfer theory* and is based on three occurrences:

- the requirements of the task are presented in previous learning
- the learner can retrieve the information from the previous learning
- the information can be translated to fit the current situation

Lave’s research calls Learning Transfer Theory into question.

## Contextually Constructed Mathematics

She gives this example of a *contextually constructed mathematics*:

*The subject is a twelve-year old boy manning a stall selling coconuts.*

*Customer: How much is one coconut?*

*Boy: 35.*

*Customer: I’d like ten. How much is that?*

*Boy: (Pause) Three will be 105; with three more, that will be 210. (Pause) I need four more. That is . . . (pause) 315 . . . I think it is 350.*

The quickest way to arrive at the answer is to use the rule that to multiply by 10 you simply add a zero—so 35 becomes 350! However, the boy probably often finds himself selling coconuts in groups of two or three. So he needs to be able to compute the cost of two or three coconuts. He needs to know that 2 x 35 = 70 and 3 x 35 = 105. Confronted with the unusual request for ten coconuts (made by a researcher posing as a shopper) the young boy resorts to his own familiar form of calculation. First, he splits the 10 into groups he can handle, namely 3 + 3 + 3 + 1 and then has to calculate 105 + 105 + 105 + 35. He does this in stages. He first calculates 105 + 105 = 210, then performs 210 + 105 = 315. Finally, he works out 315 + 35 = 350. This is clearly not the way most of us have learned to solve this kind of problem. The boy has developed a mathematics that suits the context in which he uses it.

Taylor [1989]^{1} compared students’ responses to two questions on fractions. The first question asked the fraction of a cake that each child would get if it were divided equally among six, the other asked the fraction of a loaf if shared among five. One of the students in the study responded with a different choice of methods in response to the two words “cake” and “loaf’. The cake was regarded by the student as a single entity which could be divided into sixths, whereas the loaf of bread was thought of as something that would always be divided into a number of slices. The student therefore imagined the bread cut into a minimum of ten slices with each person getting two-tenths of the loaf. This example illustrates the use of context-based math and at higher skill levels than in the abstract mathematical procedure, which gives the answer one-fifth.

Lave also investigated how value shoppers in a supermarket arrived at the best choice. After establishing the techniques these shoppers used she gave them a more conventional type of math test but involving the same calculations used in their shopping. For example, subjects in the supermarket were asked to compare an item priced at $4 for a 3 ounce pack versus an item priced at $7 for a 6 ounce pack. Some of them compared the ratio 4/3 against 7/6 and determined that the former is larger and hence the poorer value. On the “math test” the subjects were asked to circle the larger of 4/3 and 7/6. The test results differed markedly: a score of 93% in the shopping context and 59% on the “math test”. In both cases the mathematical concepts being tested were the same. Interestingly, the subjects did not make much use of the unit price printed on the label, nor did they calculate it. These results suggest that subjects, while shopping, were not using the same procedures learned in school.

Subsequently, the same testing was conducted in the subjects’ homes. Sample products in bottles, cans, and packages were shown. The scores went back up to 93%.

## Contextualizing

By the 1970’s some employers were expressing dissatisfaction with school graduates’ inability to transfer mathematical skills learned in school to the workplace. Some education systems responded by trying to bridge the gap between school mathematics and real-world mathematics. They built lessons around activities like budgeting, bill-paying, banking, salaries, income tax, and reading electricity meters. There are some suggestions that this approach engages and motivates students. Nonetheless, there is still considerable evidence that students perform differently when confronted with abstract versus contextualized calculations, even when the same underlying mathematical concept or procedure is involved. In fact it is likely that the above tasks will seem more real to the adults that design them than to students. They may be perceived as simply another mathematical exercise and even be barriers to understanding.

The problem with this kind of contextualizing lies in the determination of what exactly is *real*. First of all, if the context presented with the task does not conform to the learner’s reality it will be of little value in connecting school math to real-world math. Furthermore, learners will just see such problems as school problems in disguise. In one example cited by Boaler [1993]^{2} students were asked to determine the number of buses needed to carry 1128 soldiers, where each bus can hold 36. The most frequent response was “31 remainder 12”.

Constructivist theory suggests that each learner constructs her own knowledge. It is therefore unlikely that one task context could be sufficiently meaningful to all students. William [1988]^{3} suggests that “open beginningness” is the answer: activities start with a context but are sufficiently open that students can follow their own directions, into their own contexts and choices of procedures. In this way learners can derive personal meaning from the process of developing their own context and then using their own methods to apply the underlying mathematical principles. Boaler sees contexts as useful motivators but which can only affect transfer favorably if they make mathematics more meaningful to the individual.

Boaler proposes that students should be encouraged to bring their real-world mathematics to the mathematics classroom. This is the beginning point from which can come, through discussion and analysis of methods generated by students, an understanding of mathematics in both specific and general situations.

- Taylor, N. (1989) “Let them eat cake”: Desire, Cognition and Culture in Mathematics Learning, in Keitel, C., Damerow, P., Bishop, A., Gerdes, P.

(Eds) Mathematics, education and society. United Nations Educational Scientific: Paris [↩] - Boaler, J.
*For the Learning of Mathematics*13,2 (June, 1993) FLM Publishing Association,Vancouver, British Columbia, Canada [↩] - Wiliam, D. (1988) Open Ends and Open Beginnings. Unpublished paper [↩]

I agree of course with what is said in this article. I teach Communications and as part of the course, we write 3 letters, a Thank-you, a Negative and a Positive letter. These are real-world letters, the kind of letters that students themselves may want to write one day.

But since the article is based on mathematics, let me say a couple of things about mathematics in the classroom.

Firstly I believe that the times-table should be learned by everybody. By rote. When a students are asked what is 7×9 then 63 should pop into their head. Without this automatic response I believe people will have a disadvantage all through their lives.

Secondly I believe that students need to have a cake right in the classroom to learn fractions and get to eat a slice as reward for dividing it correctly. A loaf of bread should also be brought in. Students should be given exercises to mentally imagine numbers … taking 20 from 100 … they should have a mental image, just as if we say to them … now think of a tree. Exercises that give them mental images of the numbers that are being manipulated. The whole class should visit a supermarket and buy items that need to be mentally summed before reaching the checkout … give or take a few dollars but roughly how much have we spent? Which is the better deal, 450 ml item at this price or 900 ml item at that price? And so on.

I once sat in a Montessori class in Auckland and when it came time to learn about planetary motion, every student was given a hammer and nails and string. They hammered two nails in their desktop … imagine the din and into their DESKTOPS and illustrated how an elliptical orbit actually looks. I bet they really learned some things that day (besides how to hit nails on the head 🙂

So yes, active learning should be active.