Abstraction in Mathematics: Taught or Acquired

Mathematics is considered to be a subject that is considered abstract. But what do we mean by this? 

 Let us take one example. Consider how a child makes the concept of a dog in his or her mind. Initially, the child sees the dog in the real form and notices that the dog has a tail, can bark, walks on four legs and so on. The child will over a while see different types of dogs and then can also relate the pictures of the dog with the real dog. This is one step ahead towards abstraction where the child can connect the picture with the dog. These pictures can get more abstract from very detailed ones initially to more suggestive ones later. 

 Finally, the child can just write the letters DOG to suggest that we show a dog or mean a dog. The letters which have no direct and perceptual connection with the actual dog can also evoke the mental picture of the dog in the mind of the child. This picture is related to all the experiences but is also an abstract form which is just a mental idea. 

Similarly, instruction for children and their understanding of mathematical concepts started from more realistic concrete experiences and move to more abstract forms. This process, however, is not linear. It is rather like a cyclic movement which is upward moving over a while. The movement progresses from more concrete or visual forms to more abstract forms. 

The progress of the concept from concrete to abstract is suggested to be a consequence of many factors. While some assign it to the developmental stage of the child which is determined by age to some believing more explicit instruction and cultural factors for it.  Whatever it is, it remains to be seen in different contexts and situations and research have evidence to suggest the influence of both.  

The instance below is from a remedial class or an extra class with some children who were facing some challenges in their mathematics. This instance shows how fast these children formed a mental abstraction of the concept in just one lesson of about 40 minutes.  

 This is during the lesson where we were exploring the relations between different units like tens, ones, hundreds and so on. 

We gave the children problems in the form of written sentences, like how may 10s will make a 100 or how many ones will make a 100. 

 Some of the children guessed the first one right saying that ten tens make a hundred. However, we weren't sure if they(the children) have actually got the idea or have just heard this somewhere before and said it. 

 We asked the children to tell how they have got this but they weren't very comfortable and confident to give the reasons. One of them said that he just guessed. 

 We planned to help the children by insisting them to make pictures for the above mathematical statement. 

In the beginning, we made a box with ten dots and asked children to count the dots. We then asked them to consider the box as a ten as it has ten dots. So the figures looked like this. 

We used the inductive approach and asked them to show 20 using these boxes. They drew 2 boxes to represent 20, 3 for a thirty and so on for higher numbers till we reached 100. When they reached the number hundred, they could see and tell that ten tens make a hundred. This was followed by giving them more numbers which were a little larger like 200, 370 400 and so on. 

We realized that the children were reluctant and not happy to make the representation of the numbers as it was too time-consuming according to them but we still insisted.  There was also a question following that where the children had to answer a written statement problem after drawing the pictures. For example, they were to tell how many 10s make a 160 or how many tens make a 200 and so on.  

After children had spent some time with drawing pictures and answering the statement problems we saw something interesting. There was a boy who had written the statement first but was drawing the pictures later. For example, he wrote that 17 tens make a 170 and was drawing the picture for that later. I got curious as this was a similar problem which they couldn't answer earlier without looking and counting the pictures. So I asked him why he didn't draw the picture in the beginning and also how he could answer the statements even without drawing pictures. He replied that he could calculate that in his mind and didn't require any pictures now. We realized that he moved to abstraction on his own and could now do the problems mentally.  

We got a little curious to see how others were doing the same problems. It was joyful to see that about 4 children from a group of 8 were not relying on the pictures now. 

This instance made us wonder whether abstraction to things is a necessity for humans to simplify things and we are innately geared for this or is it only through instruction that it can be achieved or is it a mix of both.