Thinking Aloud

Rohit Dhankar

[This is an old article slightly edited in response to a friends question, which also prompted me to put this on the blog.]

Many of us have heard people suggesting that mathematics should be taught through ‘experiments’ in a mathematics laboratory. These people probably believe that the fear of mathematics can be overcome by this attempt. I would argue in this short article that a good mathematics teacher should take the idea of mathematics laboratory with more than just a pinch of salt. That is, one should not, perhaps, discard the idea summarily; it may turn out to be useful in some ways and some cases. But a serious examination of: one, for what; two, how much; and three, in what manner a math-lab can be useful is called for. In this article I will try to raise a few questions that may contribute towards that end.

First, let us examine the uses of experiments we make. Here I am not talking of ‘experience’, what is being discussed at present is ‘experiments’. We shall come to experience and its role in mathematics teaching towards the end of this article, presently we shall focus only on experiments. I think all the uses of experiment can be grouped as follows:

1. Illustration/demonstration of an idea, of a fact, or a claim.
2. Verification/proof of a principle, rule, claim.
3. Investigation into a phenomenon that may lead to formulation of new knowledge.

Let’s try to understand all this through some examples. Let us take our first example from science, suppose we want to study pendulums. A teacher may be interested in demonstrating to her students what is meant by ‘oscillation’ and instead of giving a verbal description first she illustrates an oscillation by actually swinging a real pendulum. Here the students are actually forming a new concept, they see a phenomenon mark it mentally, with beginning and end and give it a name. It is true that showing a swinging pendulum in this manner does not constitute a proper experiment. But a laboratory may be used in this manner, to illustrate, that’s why this example is taken here. The teacher may want to demonstrate the power of a particular magnet and may demonstrate it by lifting a weight heavy enough that her students might have found difficult to believe earlier. Here it is a simple demonstration of strength of that particular magnet. That is what I meant by illustration and demonstration. A laboratory is often used profitably in this manner.

The second group I have called verification/proof. Suppose our teacher now wants to show her children that the period of a pendulum is independent of the bob weight. (Period is time taken in one oscillation, and bob is the body hanging at the end if the pendulum string.) Now she will have to conduct a proper experiment. She will have to keep all other variables (length of the string, calmness of the air in the room…) constant and change only the weight of the bob. And each time she swings her pendulum the time period should remain the same. This would prove the time period to be independent of the bob weight. Here, no new concepts are developed. A relationship between already known concepts of bob weight, time period, length, oscillation, etc. is established and proven to be true. Knowledge is basically relationship between concepts; this it proves to students veracity of a new, perhaps for them, knowledge claim with acceptable evidence in scineces.

The third group mentioned above is investigation. Suppose some student or the teacher herself or any one, gets interested in knowing how the pendulums behave. This person may start observing pendulums, conducting experiments with them and noting down the observations she makes. It is certainly possible that this scientist of ours starts with very few and simple notions concerning pendulums; may be just pendulum, swing, string and bob. While observing the behaviour of pendulums she may feel a need to describe, analyse and explain that behaviour. This will necessitate development of concepts like an oscillation, time period, amplitude and so on. Secondly, it would need formulation of relationships between these concepts, generalisations and testing of the generalised rules/principles. Here the experiments are not done to verify something already known, rather they are designed either to observe something new or to test a hypothesis which is not yet accepted to be true.

I would say that almost all experiments and laboratory uses are covered under this somewhat rough classification. Now let us see how experiments may help in mathematics teaching. To understand it properly we will take two examples, first, study of triangles and second, learning numbers and four fundamental operations.

Let us begin with triangles. Can we illustrate any mathematical concept – as we did with oscillation? Perhaps yes, we can illustrate a side of a triangle, an angle and the triangle itself. But is there a significant difference between the ‘oscillation illustrated’ and the ‘triangle illustrated’? Or is there no difference and they are exactly the same? I think there is a difference in “illustrating” a pendulum oscillation in a laboratory and “illustrating” a line or a triangle.

To begin with oscillation is a physical phenomenon. In a laboratory we deliberately create a situation in which that phenomenon (oscillation) can be closely observed, and point out to the observer features of it we want to emphasise. This helps the learner in acquiring a concept of that phenomena and associating a name with it. The concept illustrated here demarcates a part of observable behaviour of a physical object, and that behaviour is pointed to the learner directly, without use of any symbols, signs or representations.

In illustrating a triangle, we have several choices. We may cut a triangular piece of paper, or tie string around three nails stuck on a wooden board, or place three sticks in a triangular shape, or may draw a triangle on paper with pencil and ruler, and so on. But notice that all these illustrations are ‘make believe’ in a certain sense. They are only representations of a triangle and not the triangle itself. They all are only approximate representations of the concept. A triangle is a closed figure with three straight sides. The sides have only length, unidimentional ‘shapes’, do not have breadth or thickness. They simply are good or bad illustrations of this idea. And are used to ‘abstract out what is common in all of them’, which is just the shape without any material (substance) in it. Oscillation, on the other hand is, it is the behaviour of the pendulum which is being directly observed, that behaviour does not represent something else. In oscillation we have generalisation on a universe of other physical things, and what to bring our approximation of its properties as close to the real phenomenon as possible. In triangle we want to form an abstract idea which has no physical existence, through abstracted from physical things.

Generally, we can say that in science, which is the home of the idea of laboratory, a concept is illustrated directly by the part of natural world it relates to or by an effect of that part of natural world. Oscillation is an example of direct illustration by the ‘part of natural world’ and attraction of a piece of iron towards a magnet is an example of ‘effect’ of the presumed magnetic force. In mathematics illustrations are just representations with the help of visual signs or materials amenable to manipulation, something like a ‘gharonda’ (house of sand village children make after rains) and not the real house.

The second kind of uses we identified for laboratory experiments are verification or proof of a claim. Suppose one wants to prove Galileo’s claim about pendulums that “the period is independent of the amplitude”. Let us take an actual experiment done by a group of people to verify the truth of this claim.

Their results and conclusions follow: “Scholars debate whether he (Galileo) meant that the periods are exactly the same or that they differ very little. As a test of whether they are exactly the same, two pendulums with identical lead bobs were suspended 28.9 cm. They were released at the same time from different angles. One was pulled back about 5 degrees while the other was released from about 45 degrees. The pendulum pulled back five degrees was allowed to travel through thirty cycles, and the numbers of oscillations of the other pendulum during this time were counted. The data is below.

The pendulum that travelled through the larger angle had a longer period. It averaged 29.42 oscillations during 30 swings of the other, and had fewer oscillations in every trial. Clearly, pendulums with different amplitudes do not have the same period. In fact, it appears that pendulums with larger amplitudes have longer periods. The difference is quite small, though. Whether Galileo’s claim is true depends on interpretation of the claim, but the interpretation that identical pendulums of different amplitudes have periods independent of amplitude is false.”

There is a significant difference in the amplitude, the time period is very close, still according to these data it is not the same. The only way to find out the truth of Galileo’s claim is to experiment. No amount of a priory reasoning can prove or disprove it. Also, even after experiment there could be doubts. Here first, the difference is too small to say that time period differs with the amplitude. Second, the design of the experiment itself begs many questions: can one really claim that bob weights and lengths of the pendulums were identical? If one measures them more accurately, they may be found to be different, that may explain the difference in the time periods. If some one else does the same experiment with only one pendulum, pulled back 5 degrees and then 45 degrees the results may be different. But answers even to these questions can be found only through experiments with the real pendulums, not with symbols or signs of pendulums. This is because the relationship between the amplitude and time-period of a pendulum is not contained in the concepts themselves, it is contingent on the properties of the physical world. Logic can not help establish this relationship; observation is a necessity.

Now let’s see how can experiments help in settling the claim that “sum of all the interior angles of a triangle is equal to two right angles”. One can draw lots of triangles and measure their angles against tight angles, one can make triangles of various kinds of material (suitable in varying degree) and again measure or compare angles, and so on. All these ‘experiments’ are bound to give differing results and the differences are going to me more in magnitude than what we have seen in the pendulum experiment above. Shall we come to the conclusion on that basis that sum of angles of a triangle is ‘not exactly equal’ to two right angles? We will not. We will simply ignore the differences and say that the original proposition was correct.

The experiments can not establish a mathematical truth, while they are necessary to establish a scientific truth. But the more important fact is that experiments are not needed to establish mathematical truths, they are through reason alone and investigating relationships between concepts is enough.

Scientists also carry their investigations through laboratory experiments. In science investigation may start with curiosity about something (where do colours come in rainbow?) or to solve some problem (how can rain water be stored for summer months?). In any case the investigation will start with a hypothesis and observation/experimentation will provide relevant data to prove or disprove or to modify the hypothesis in a certain manner. We may want to know how length of the pendulum effects its period. To investigate the matter, we will choose various lengths and measure the time periods for them and will try to see if there is a pattern. Again, the pattern available may not be formulated very neatly as a rule. And may tolerate a certain degree of inaccuracy. Compare it with investigations in mathematics. Suppose that after learning about the sum of angles of a triangle, we want to know the sum of all the angles of a quadrilateral. Do we need to do any experiments here? I do not suppose so. All we need is previously proved results and basic assumptions; and the sum of all the angles of a quadrilateral can be easily deduced from them.

Thus, we can say; if our examples are sufficiently representative, that in science experiments can help in illustrating a concept, demonstrating a fact/rule. Experiments are necessary to establish truth of principles and rules. And are also necessary to carry on investigation for generation of new knowledge. In mathematics experiments are not necessary, nor they provide the final proof of any thing. But that does not mean that the idea of laboratory in unacceptable or totally useless in teaching mathematics? Before we can say something on that issue, we need to consider the role of experience in learning mathematics and science, because laboratory experiments are only a means to have controlled and selective experience.

We know of the physical world through experience only. The sciences develop in order to describe the world, to explain how it functions and then to modify and control parts of that world, as far as they can and want. The concepts of science, therefore, need to represent the world as precisely and as ‘truly’ as they can. If they do not help either to describing the world or to explain it, they are of no value. This holds true for rules and principles of science as well. Therefore, the movement of intellectual activity in science is towards better correspondence with the physical reality. Since the knowledge in science is knowledge of the world, experiments and observations have central place in its methodology as well as in its truth criteria.

Like all forms of human understanding mathematics also begins with experience of the world. The basic concepts of mathematics (number, shapes, and quantity et al) generate out of and are dependent on experience. But once the concepts are formed mathematics has a tendency to create regular, ideal and perfect world out of them. Therefore, may be the idea of a line initially begins with something long and straight, but very soon it becomes only length and its straightness becomes perfect. It matches no reality in the physical universe. And its qualities depend on no reality in the physical world. It becomes abstract. Pure. Until it acquires that typical abstractness it is not really mathematics. Therefore, the intellectual movement in mathematics in the opposite direction: create an idea from experience and then make it perfect in its own ideal world. The mathematical knowledge is knowledge of inter-relationships between these perfected ideas. This knowledge is self-contained in these ideas and logical forms. It does not require physical experimentation; it cannot be gained through physical experimentation.

Now, perhaps, we can say something about the laboratory for mathematics. It may have a limited role. To provide children the initial experiences where they form the intuitive mathematical ideas. But should not be given an impression that any thing in mathematics can be proved through experiments. The ‘experimental verification’ should, at the best, be considered only as a hint that the idea may be worth trying to prove formally, mathematically. If it provides the use of concrete objects only as a first stage in building an understanding of a concept, that may be acceptable. But, to really ‘have’ a mathematical concept the child has to move towards working with the concept in abstract form. To take a simple example: a child who learns to add with the aid of marbles should quickly move to adding on her fingers and from there on she should quickly move to adding in her head without any physical props. If she is not doing that her mathematics learning will remain deficient. The manipulatives (or physical teaching aids) could be a very useful in the hands of a teacher who understands their proper use. But elevating that idea to metaphorical use of ‘mathematics laboratory’ generates a false impression of mathematics, of how it is leant and of how it is created. The best use of the so-called mathematics laboratory in the hands of a good teacher would be to show to the children that it has a limited use in the first steps in learning mathematics and is misleading incumbrance after that.

Bridges and leaps

In mathematics we invent or imagine the fundamental or initial elements of a system: concepts or definitions, axioms/postulates and sometimes rules of inference. After that the system becomes independent of us and has its truths built into it. We, the inventors of it, have to struggle hard to find all the true statements possible in it. And the methods at hand we have are all bound by strict logic. This becomes a self-contained world of ideas.

The laboratory prompts us to make bridges from the reality of our experience to reach this world of ideas and gives us tools of visual models to use there when we reach there. But the nature of this world is such that the bridges always fall short and visual models become limitations on our imagination rather than any help when we actually somehow reach that world. It becomes our limitation, and stops us from hearing the real music of mathematics.

The way out is not making more and more cumbersome bridges; but to learn to take a logical leap into the abstract beauties and exhilarating music of mathematics. The idea of laboratory hampers the preparation for this leap. My personal view is that use the manipulatives wisely where they help; but don’t elevate them to the status of a laboratory, not even metaphorically.

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Originally written in or sometime before 2001.

Slightly edited and last section added on 7^th July 2020