The House of Mathematics

We can compare teaching and learning mathematics with building a house:

• The teacher starts by teaching arithmetic. This is like building the first storey of a house.
• Then (s)he teaches elementary algebra. We compare this with the second storey: Just as the second storey requires a solid first storey, teaching elementary algebra requires the student to have good skills in arithmetic.
• The next topic, the third storey so to say, could be linear equations. Learning how to solve equations requires a profound knowledge of elementary algebra.

And so on. We call the result of this process the House of Mathematics.

Clearly, the picture of a single building is very much simplified. In fact, mathematics would have to be compared with a whole group of buildings; it is more like a village or a town. But the concept of a single building will, without loss of generality, be very helpful for understanding the following ideas.

Here are two examples of how teaching and learning is achieved today.

Example 1: Solving Equations

For solving the equation

5x-6 = 2x+15

one has to find a sequence of equivalence transformations that transform the equation into the form x = .... 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.

5x-6 = 2x+15 (-2x)

After choosing this equivalence transformation, we apply it to both sides of the equation:

5x-6-2x = 2x+15-2x
3x-6 = 15

Now we have to choose another equivalence transformation, namely +6.

3x-6 = 15 (+6)

And we apply it to the equation:

3x-6+6 = 15+6
3x = 21

An analysis of this procedure reveals two alternating tasks:

• Choosing an equivalence transformation.
• Applying an equivalence transformation.

When looking at the manual skills only, the skill of solving equations - in a much simplified picture - can be represented as:

We purposefully consider manual skills only here. Skills like the conceptual understanding or the recognition of the syntactical structure of an expression are left out for simplicity.

While solving an equation, the higher-level task of choosing an equivalence transformation repeatedly is interrupted by the lower-level task of applying an equivalence transformation:

This repeated change of level is the reason for mistakes for students who have not well developed the lower level skill yet. We demonstrate this with our example: the original equation was transformed into 3x = 21 and one has to choose another equivalence transformation. What happens in the brains of so many beginners? The following argument may sound familiar to you: There is a 3 in front of the variable x. To get rid of the 3 I need to subtract 3.

3x = 21 (-3)

And, again, the student has to change levels in order to apply this equivalence transformation. Why do so many students produce the wrong answer x = 18 then? Quite simply because they do not use enough care in this final expression manipulation. They will argue that they have chosen the transformation -3 so to produce an x = on the left hand.

Example 2: Applied Trigonometry

The following problem is given:

On a hill with a slope of e=30 degrees stands a tree. Below the tree there are two points A and B, A being 15 metres below B. The angle between the line connecting A with the top of the tree and the hill is a=40. The respective angle at point B is b=50. What is the height of the tree?

We assume that the students have learned basic trigonometry, so that they know how to use the theorems of Sine and Cosine.

The strategy for this kind of problem is to find a sequence of triangles, such that in each triangle we know enough parameters in order to compute all other parameters. The parameter which we are looking for (in our example: the height of the tree) must be part of the final triangle. It helps to choose appropriate names for all points and sides.

In this configuration there are three triangles: ABS, AFS and BFS. But in none of these do we know three parameters. In fact, we need to employ geometrical properties like complementary angles or supplementary angles in order to find a suitable first (or next) triangle. And this is exactly what makes choosing the triangles so difficult.

Since angle ABS and angle FBS = b = 50 together make 180, the angle ABS must be 130. We now know one side and two angles in triangle ABS and, therefore, can employ the Sine theorem to compute x and y as

x = 55.525 and y = 66.172.

Since angle BFS = 90+e = 120, we now know one side (namely the previously computed x) and two angles in triangle BFS. As above we can compute t and h:

t = 11.133 and h = 49.114.

The tree is an enormous 49.1 meters high! A retrospective analysis of the above strategy reveals two alternating tasks:

• choosing a triangle and
• computing a triangle's unknown parameters.

Again, the skill of solving applied trigonometry problems - in a much simplified picture - can be represented as:

This example shows even more clearly, how the repeated change of levels causes those with an insufficient knowledge of the lower-level skill to fail: the student chooses a triangle. Now (s)he must descend and apply the very recently learned basic trigonometry skills. This is a difficult task for those who still learn. If the student succeeds, then (s)he must ascend back to the higher-level skill and continue the previously interrupted search for suitable triangles. And many students fail.

How do we teach and learn?

This method is useless for all those students, who have not perfected skill A by the time the teacher starts teaching skill B - which often is the majority of students. For these students, learning the next topic is like trying to erect a new storey on top of a fragmentary storey.

Our House of Mathematics as sketched above is the ideal from the teacher's point of view. The student's perspective looks quite different. Over the years, each student builds his or her individual House of Mathematics. Whenever the teacher finishes teaching a certain topic, most students still have a very incomplete individual storey, i.e. their respective storey still has many holes. And, needless to say, each student has his or her own set of holes.

Each student now would need individual training: But there is not enough time for that. How can a teacher proceed with teaching further topics and still meet the demands of each individual student?

Building a house with DERIVE

We apply the same idea to the teaching of mathematics. We start by teaching skill A in the traditional way. As soon as we start teaching skill B, we let the computer solve all those sub-problems, which require skill A. Thus the computer serves as a scaffolding between the storeys A and B.

This idea was first mentioned by Prof. Bruno Buchberger, director of the Research Institute for Symbolic Computation at the University of Linz, Austria. He called it the White-Box/Black-Box Principle. In the following we will demonstrate this idea using our two examples.

We start with the equation 5x-6 = 2x+15. In the following computer dialog the user (i.e. the student) is responsible for choosing an equivalence transformation, whereas the student uses DERIVE for applying an equivalence transformation.

• (A)uthor 5x-6=2x+15(¢)

The (F4) function key copies the highlighted expression into the author line. Choosing the equivalence transformation -2x, therefore, is done as follows:

• (A)uthor (F4)-2x (¢)

This was the choice of the equivalence transformation. Its application to both sides of the equation is done with DERIVE via the ›Simplify‹ command:

• (S)implify (¢)

We choose and apply the next equivalence transformation:

• (A)uthor (F4)+6 (¢)

• (S)implify (¢)

Voilà. A student, who would make the earlier described mistake of trying to subtract 3 would proceed as follows:

• (A)uthor (F4)-3 (¢)
• (S)implify (¢)

The result does not have the expected format x =, so it is obvious to the student that the equivalence transformation -3 was a bad choice and (s)he needs to try something different. This approach has the following advantages:

• The student can fully concentrate on choosing equivalence transformations.
• The student receives immediate feedback on whether or not the choice was a good one, thus (s)he learns by trial and error.

The student chooses triangle ABS and specifies the arguments for the function TRIANGLE as follows:

• (A)uthor TT>triangle(15,x,y,10°,40°,130°)

The ›Simplify‹ command computes x and y.

• (S)implify (¢)

Now the student chooses the triangle BFS.

• (A)uthor triangle(55.5,t,h,120°,10°,50°)
• (S)implify (¢)

Again, the student fully concentrated on finding the triangles.

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 scaffolding 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 having this as a handicap.

DERIVE allows us to apply the scaffolding method to most of the topics treated at school. Investigations in England, Germany and Austria have shown that DERIVE automates up to 80% of what we teach up to A-Level.

Using the computer as a scaffolding, however, is not always sensible. In particular, a storey must have a certain degree of completeness before it makes sense to erect a scaffolding. One such prerequisite may be the conceptual understanding of the respective operation.

The search for conditions under which the use of a scaffolding has a pedagogical justification opens a huge field for future educational research. In fact, this issue shows why using computer algebra systems make teachers even more important - rather than obsolete!

Scaffolding Didactics

In case a skill is crucial (e.g. because it is needed for an external examination), one should add a third consolidation step, in which the student has to solve respective problems without the help of the computer:

In case a skill is utterly important, the teacher could require the student to build the scaffolding themselves. For example, the student could be asked to program the function TRIANGLE.

The opposite case can and will happen as well. Some algebraic manipulation skills may be considered obsolete like, for example, the skill of computing a square root. Hardly anybody teaches the respective algorithm any more. Both teachers and students simply use their calculators.

Altogether we now distinguish five different methods for teaching mathematics. Method 1 is the traditional approach. Methods 2-5 is what we call Scaffolding Didactics. Here is a summary:

The importance of a skill - and, hence, the most efficient teaching method - will depend on the curriculum, on the type of school, and on the teacher. The trigonometry for general triangles, for example, could be taught by Method 3 in one school, by Method 4 in another, and by Method 5 in a third one.

What is so nice about DERIVE is the fact that the teacher can decide how to use it. DERIVE doesn't force us to use a particular method. It simply broadens the range of methods to choose from.