Improving Mathematics Teaching with the TI-92

Learning to Solve Linear Equations with the Help of Technology - The Scaffolding Method

by Bernhard Kutzler and Michael Meagher

Tools such as the TI-92 look certain to change the teaching of mathematics fundamentally. But before new curricula are implemented we must continue to teach mathematics according to existing curricula. In this article we show how to include the use of TI-92’s in traditional mathematics lessons - for the benefit of both teachers and students.

We can compare teaching and learning mathematics with building a house: And so on. We call the result of this process the ”House of Mathematics”. Clearly this is a simplified picture. A fairer analogy would be that of mathematics as a set of buildings, perhaps even a village or town. However the concept of a single building will serve as a good picture for understanding the concepts that follow.

Here is an example of how we currently go about teaching and learning.

In order to solve the equation a student has to transform the equation into the form ”x = ...”. This is achieved through the choice and application of an appropriate sequence of equivalence transformations. Typically, the student is advised to ”bring terms with x to one side of the equation” and to ”bring all other terms to the other side”. Therefore we start by subtracting 2x. After choosing this equivalence transformation, we apply it to both sides of the equation i.e. we have to simplify: Now we have to choose another equivalence transformation namely +6 And we apply it to the equation: An analysis of this procedure reveals two alternating tasks: Here, the choice of an equivalence transformation is a higher-level task insofar as it is the essence of the strategy for finding the solution of an equation. It is the new skill which the student has to learn when learning to solve equations. The simplification of expressions is a lower-level task, for which the teacher has to assume that the student is sufficiently well trained.

Therefore, the skill of solving equations, in a simplified picture, can be represented as:

While solving an equation the higher-level task of choosing an equivalence transformation is repeatedly interrupted by the lower-level task of simplifying expressions. The concentration that we wish the learner to have on the higher order task is therefore continually broken. This repeated interruption and change of level is the reason for the mistakes of students who have not developed the lower level skill to a sufficiently high level yet. We demonstrate this with our example: the original equation was transformed into and one has to choose another equivalence transformation. What happens in the brains of so many beginners? Anyone who has taught linear equations will recognise the following argument: ”There is a 3 in front of the variable x. To get rid of the 3 I need to subtract 3”. And, again, the student has to change levels in order to apply this equivalence transformation and inevitably gets the solution ”x = 18”. The reason so many students make this mistake is lack of care in expression manipulation. They have chosen the transformation -3 in order to produce x on the left hand side and therefore do not take sufficient care in the manipulation of 3x - 3 and make the left hand side now x with no further thought. This example is typical of how we teach mathematics. In a first step we teach a task A. In a second step we teach a new task B in order to develop a higher-level skill such as solving equations. But most of the time, training task B requires the students to use (and train) task A at the same time. This method is useless for all those students who have not perfected task A by the time the teacher starts teaching task B - which often is the majority of students. For these students the task A storey remains incomplete. The gaps, or holes, in each student’s storey are different so at this stage, ideally, we would provide individual training for each student. Since this is not possible, for a variety of reasons, how can a teacher proceed with teaching further topics and still meet the demands of each individual student? When building a real house one uses a scaffolding to facilitate the building of a new storey on top of an existing (but still ”wet”) storey. We apply the same idea to the teaching of mathematics. We start by teaching task A in the traditional way. As soon as we start teaching task B, we let the computer solve all those sub-problems, which require task A. Thus the computer serves as a scaffold between the storeys A and B. The idea was first mentioned by Bruno Buchberger. He called it the White-Box/Black-Box Principle. In the following we demonstrate this idea using our example.

We start with the equation

5 x - 6 = 2 x + 15 [ENTER]

A good first step would be to subtract 2x from both sides. On the TI-92 we simply type

-2x

Having started with the binary operator - the TI-92 automatically inserts the previous answer (ans(1)) so

[ENTER]

results in the equivalence transformation -2x being applied to both sides of the equation.

It is clear that the equation has now become simpler and therefore that the chosen transformation was a good one.

The next step is to add 6

+ 6 [ENTER]

Again the resulting equation is simpler.

Let us look at what happens to the student who decides to subtract 3:

- 3 [ENTER]

The student immediately notices that the equivalence transformation -3 did not lead to a simpler equation. On the contrary, the equation is now more complicated. The TI-92 is thus able to give feedback on the quality of a student’s choice of equivalence transformation. Students must now try a different equivalence transformation and will learn by trial and error. Highlight the last answer, then cancel the history pair with CLEAR .

s

[CLEAR]

A good transformation at this point would be to divide by 3.

t... to move the highlighting back to the entry line.

/ 3 [ENTER]

Students can now check their answers using the "with" (|) operator on the original equation:

5 x - 6 = 2 x + 15 | x = 7 ENTER

If the topic being learned is fundamental, for example if it is to be directly tested in an examination in which no technology is allowed, a consolidation step after the aforementioned steps 1 and 2 is recommended (and probably inevitable). In this step the student must practise solving problems without any assistance from the calculator. This approach to learning is known as the scaffolding method: If we wish students to learn a topic which can be divided into lower order and higher order tasks, we can use the technology to build a scaffold over the, perhaps imperfectly learned, lower order tasks, so the students can be supported by the scaffold to concentrate on the higher order tasks. After learning the two tasks separately the students then learn to do them together.

The method of delegating the solution of certain sub-problems to a computer or calculator is established already: since the Seventies the scientific calculator has been a scaffold for the arithmetic storey. On the one hand it saves time and allows more exciting examples. On the other hand it enables students, who are weak in performing hand calculations, to proceed with more advanced topics without necessarily having this as handicap.

This article is based on the booklets ”Solving Linear Equations with the TI-92” and ”Solving Systems of Linear Equations with the TI-92” by Bernhard Kutzler published by bk teachware (e-mail: info@bk-teachware.com) which are available from various distributors. (See the homepage of the bk teachware Series for your nearest dealer.) Thescaffolding method is described in detail in ”Improving Mathematics Teaching with DERIVE” by Bernhard Kutzler published by Chartwell-Bratt which is available in Europe by Chartwell-Yorke (e-mail: info@ChartwellYorke.com) and in the US from WEST (e-mail: west@trib.com).