Recap
In the previous post, from two weeks ago, I had addressed a development in Maharashtra, according to which they have lowered the pass mark in mathematics and science from 35 to 20. I had concluded that post with three questions. First, what is the relationship between a pass mark in an exam related to the purpose of primary and secondary education? Second, how does the mathematics curriculum reflect or fail to reflect this purpose? Third, how are teachers being equipped to promote student learning? I will deal with the first two questions in this post. The third I am shelving for now. So let us proceed to address the first two questions.
The Dictates of Prior Knowledge
Let us be very honest. Apart from probably giving students some choice of a second and/or third language, most primary school education is quite rigid. I do not mean that primary school teachers are not able to be quite creative in their classes. Of course, they can. And especially in the last school I worked at, I saw this on a regular basis. However, there are certain curriculum requirements that teachers are expected to fulfill. For example, in the context of mathematics, the students are expected to learn addition by a certain grade. Similar standards are set for the other operations and concepts. Indeed, while many schools may move aspects of the curriculum around to suit their specific contexts, every syllabus document released by boards for subjects in grades 11 and 12 has a section called ‘Prior Knowledge’ or something similar. In other words, every board expects every student to have acquired some body of common knowledge before continuing with mathematics in Grades 11 and 12.
This is not true for most other subjects. In English, there is some sort of unwritten rule that students should have studied English language till Grade 10, but even here there is no clear indication of the body of knowledge that a student is expected to have before embarking on a study of English in Grades 11 and 12. Of course, conscientious teachers know that, if a student is expected to be able to comprehend unseen passages of a complex nature, then the student should have been introduced to many grammatical and rhetorical devices prior to that. In other words, even though the Grade 11 and 12 syllabus may not be explicit about prior knowledge, the syllabus itself would be impossible to complete without quite a bit of prior knowledge. Of course, in the case of mathematics the situation is quite different, where most boards list a set of concepts that students are expected to know before commencing studies in mathematics in Grades 11 and 12. What we see, then, is that, in the case of the first language and mathematics, there is an understanding that students should have some knowledge before continuing studies in Grades 11 and 12.
What this means is that there is some sort of consensus, even if unwritten, that students of English and mathematics ought to possess some body of knowledge by Grade 10. This is, in a very real sense, non-negotiable or the idea of requiring prior knowledge would be pointless.
The Purpose of Primary and Secondary Education
Granted, then, that students are expected to have some body of knowledge by Grade 10, what can we ascertain concerning the purpose of primary and secondary education?
Presumably, education serves the role of helping people navigate through life. I mean, if education mostly produced social misfits we would question its role even more than some of us do. Hence, in some way we are declaring that knowledge of the first language and mathematics is essential for a person to navigate life.
Most students see through this ridiculous assumption. When they ask me why they are studying geometry or algebra and where they would use it, they reveal the lie we tacitly tell them, namely that what we are teaching them in primary and secondary school is essential for navigating life. You see, knowledge of how to prove two triangles are congruent or how to solve a quadratic equation is something very few people would ever need in life. I am a mathematics teacher and I can assure you that, outside my classes, I have never needed to solve a single quadratic equation. Neither have I had to demonstrate the congruence of any pair of triangles. In other words, as a mathematics teacher, I can assure you that the content we teach the students in primary and secondary school is mostly useless.
Mind you, I said that the content is useless. However, mathematics is much more than the content we teach students. Rather, just like in a language, mathematics is primarily concerned with the development of skills. Just as it is pointless for someone to learn the meanings of thousands of words, as required by many ridiculous standardized tests like the SAT and GRE, without becoming adept at using those words appropriately and regularly in relevant contexts, so also it is quite pointless for someone to learn hundreds of mathematical formulas and algorithms without knowing when to use which ones and how.
So what are the skills that we hope to develop in students by insisting that they study mathematics in primary and secondary school? Unfortunately, here we encounter a difficulty. High school mathematics curriculums are guided by what universities say they require students to have completed before beginning their programs. In some cases, this concerns the content of mathematical learning in high school. So, some universities may require the student to have studied some calculus by the end of Grade 12. Others may insist that some algebra or geometry is covered. In other cases, the universities specify the number of class hours (or credits) the student should have devoted to the study of mathematics in high school. So, some universities may require the student to have had 8 semesters of mathematics with at least 3 hours of classes per week or something of the sort.
However, in what way does this equip a student who plans to study history or art? I mean, if there is some link between the study of mathematics and the study of history or art, what is it? Will understanding how to differentiate a function and hence interpret the mathematical idea of ‘change’ help the student improve his/her understanding of ‘change’ in history? Will a focus on geometry allow the student to better appreciate and produce art? If it quite obvious that the mathematical content will not help a student except in mathematical contexts. Yet, many universities require some high school mathematics even for programs in fields where mathematics is not required. Indeed, some exam boards require the study of some mathematics in order to award the student with the highest possible diploma or certificate after Grade 12. Why is this the case?
Major Skills Developed Through Mathematics
Unless we are able to determine some skills that are developed while studying mathematics, which are also useful in other fields, making the study of mathematics mandatory in high school will be an arbitrary decision that serves no better purpose than to fill the students’ time with irrelevant study. I do think there are many critical skills that are developed while studying mathematics and that these skills can be transferred to other fields. Here, let me briefly describe three.
Mathematics develops the skill of problem solving. Please note that the skill of problem solving is markedly different from the task of solving problems. Mathematics makes us ask, “What have I been given?” and “What have I to find?” and “What are the concepts or ideas or formulas that link what I have been given to what I have to find?” and “What is the best route to get from what I have been given to what I have to find?” and “Is the answer I have obtained reasonable?” All of these questions help in the analytical process required for problem solving. It enables the student to break down a massive problem into bite size pieces.
The second skill that mathematics develops is logical thinking. Quite often, when I read or watch some piece of argumentation, I come across statements that have the form, “It is known that, if proposition A is true, then proposition B must be true. Since A is false, therefore, we can conclude that B is false.” In case you do not find a problem with this line of reasoning, I invite you to read it again and think of different propositions (A and B) where there is some sort of causal relation between the two. In the mathematical context, we can think of A being the proposition, “This quadrilateral is a square” and B being the proposition, “All the interior angles of this quadrilateral are right angles”, we can see that proposition A implies proposition B. That is, if a quadrilateral is a square, all its interior angles will be right angles. However, a student of mathematics knows that if a quadrilateral is not a square, that is, if A is false, then it does not follow that its interior angles are not right angles, that is, that B is false. After all, the quadrilateral could very well be a rectangle!
The third skill that mathematics develops is a sense of justice. Yes, you read that right. A student of mathematics knows that, given two positive quantities, x and y, such that x is less than y, doubling them or tripling them only expands the gaps between the resultant numbers. In other words, 2x and 2y are further apart than x and y. And 4x and 4y are even further apart. As an example, suppose x = $50,000 and y = $1,000,000. The difference between x and y is $950,000. But if we doubled this the difference becomes $1,900,000. In other words, just giving everyone double their income actually hurts those who are poor because the rich are now able to have a much, much higher standard of living, which makes it all the more difficult for the poor to make ends meet.
Appraisal of Curriculums
Unfortunately, mathematics teaching is so heavily focused on the procedural aspects of mathematics, which are necessary, mind you, that there is no room to step back and appreciate what one has learned. Our curriculums are so bloated with non-essentials that teachers in high school are almost always scrambling to complete the syllabus, leaving little or no time to actually hone the skills that mathematics helps students develop. With regard to the first skill, we spend so much time giving students more and more problems to solve that we do not leave any room for allowing them to ask the questions I highlighted above. Where the second skill is concerned, the questions framed in textbooks and exams are so ridiculous that they further the misconception that mathematics has no relevance on its own in the real world. Please note the italicized words. Due to its power, mathematics has found uses in almost every walk of life. Hence, its usefulness for its own sake has somehow taken the backseat. Most damning, however, is my conviction that few, if any, teachers are even aware of the third skill that I have highlighted here. That mathematics has something to say about truth and justice is something that probably has not even entered the minds of most teachers, let along curriculum designers.
As mentioned earlier, very few of the mathematical concepts are relevant to students except in the mathematics class. On account of this, what mathematics education in primary and secondary school must aim to deliver is not familiarity with these concepts but the development of skills that transcend the realm of mathematics. That must be the purpose of mathematics education in primary and secondary school. Therefore, given the fact that our primary and secondary school mathematics curriculums are obscenely bloated with non-essentials, I can confidently state our curriculums do not reflect the purpose of mathematics education.
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