Problems in postmodern education

MEDIUM-LENGTH: What happens when teachers abdicate the position of intellectual authority in mathematics?

Take that look of worry
I'm an ordinary man
They don't tell me nothing
So I find out what I can

- “Take Me Home”, Phil Collins

There have been many changes in education since I was a child, and this is especially true for K-12 education. Now more than ever before, the focus seems to be on helping children make meaning of things for themselves, as opposed to teaching them a curriculum which they are supposed to simply accept and absorb.

Although this seems like (and is) an effective method in the social sciences and humanities, and also a boon for critical thinking skills development, I find myself skeptical of a new approach to teaching mathematics in my home province of Ontario: “discovery math".

So-called discovery mathematics, generally speaking, seems to be founded on the insight that with any math problem - take 44 + 67, for example - different people will compute the answer in different ways:

  • Some people might start with the “ones” column, add the 4 and the 7 together, making 11, then carry the 1, add it to 4 + 6, and get 111. This is what I was taught in school.

  • Some people might start with the “tens” column, adding 40+60 to get 100. Then I add 7 + 4 to get 111.

Although this seems like a trivial detail, in an educational context this becomes a crucial issue. If different people use different mental processes to solve the same problem, which process should we be teaching? And, most importantly, if we are teaching one or some processes to the exclusion of others, are we structurally disadvantaging the students who are predisposed to think differently?

The answer has been “discovery mathematics”, which is a style of teaching mathematics that lets students come up with the answer in their own way:

“Students could have 2x5 memorized; they might know that 2x2 is 4, so doing that twice would be 8 and add another 2; they might do 5+5, or draw five pictures of two objects, or two pictures of five objects, and so on and on. The main point is students are still expressing that they understand how to multiply two numbers, but they're doing it in a way that makes sense to them.”

- From “ELI5: WTF is ‘Discovery Math’?” (Reddit)

In theory, this is a wonderful practice. No student is being disadvantaged, and different perspectives and styles are being taken into account. However, in practice, there seems to be considerable debate over whether or not this approach is actually helpful.

Comparing postmodern vs. pragmatic approaches

Going back to my example of 44 + 67, let’s compare the two approaches again. First, here’s the first approach broken down in terms of “mental operations”, or computational steps, required to solve the problem with each approach. The key thing to remember is that each step uses mental resources and takes time.

  • Adding ones column: one mental operation.

  • Carrying the 1: one mental operation (two total)

  • Adding the tens column: one mental operation (three total)

  • Putting the number together: one mental operation (four total)

For purposes of this discussion, we will assign a value of “4” in terms of mental resources that this method requires. Now, on to the second:

  • Add tens column: one mental operation

  • Add ones column: one mental operation (two total)

  • Putting the number together: one mental operation (three total)

It would seem like the second approach is more efficient. But what about 253 + 463? The first approach requires 4 steps. The second runs into an issue, where students would have to backtrack, carry the 1 in the tens column, re-calculate the hundreds column, and then go on to solve the rest of the problem. For even larger numbers, like 1124 + 992, the second approach continues to run into difficulties.

Taking this to an extreme, if we were to solve 1124 + 992 by adding them up in a “1+1+1+…” fashion, that would require 2,116 mental operations.

Some ways of thinking are “better” than others

It would seem, at least to me, like the issue with discovery mathematics is that it trades efficacy for inclusivity. The fact is that mathematics exists in its current form, and has been taught a certain way for millennia, because tens of thousands of people have laboured their whole lives to perfect the discipline and its teaching. Discovery mathematics seems to be a new (postmodern) invention that is not holding up in early testing, despite great promise.

Another problem case for education: wisdom

While researching other matters, I came across a humorous video featuring Sadhguru, whom I had never heard of before but is a highly-respected spiritual leader in India. The topic? Recognizing when a teacher has genuine wisdom, which is a problem for followers in every major religion.

Here’s the relevant insight:

“You go and sit in front of your kindergarten teacher. Do you know how knowledgable he or she is? No, but he or she knows ABC. You do not know, so you sit.

So, right now we are talking XYZ. You do not know, so you sit. So what does it matter if I got acknowledgement from somewhere else or not? You want to learn XYZ, you sit here. You don’t want, you go.”

(audience laughs)

This remark actually got me thinking about the concept of this article, as enlightenment is a special boundary case that tests the current educational model’s postmodern assumptions.

On one hand, teachers will generally be quick to praise indigenous belief/knowledge systems, Buddhist teachings, world religions, and may even grudgingly acknowledge that Judaism and Christianity have some value to offer. But on the other hand, the way that knowledge and wisdom are imparted in these traditions is very different from how education is done in schools:

  • Rote memorization of religious texts, catechisms, rules, and precepts

  • Assumption that the teacher knows best, limited ability to challenge & question

  • Taking certain principles as axiomatic (that is, “on faith”)

  • Acceptable and unacceptable methods of inquiry (reason vs. faith)

This is a far cry from postmodern education, which encourages students to be skeptical of what they are taught, to use their own methods of inquiry, and to challenge ideas they find problematic.

In many ways, teachers are erasing themselves from the learning context in Ontario, and sometimes intentionally so. However, what they fail to recognize is that the curriculum (or at least the old one) had genuine value and that the memorization of multiplication tables and endless practice problems and tears are worth it, as they result in a refined, efficient, mathematical mind… which is the point of mathematics education in the first place.

Who is teaching?

I suspect that the heart of Ontario’s mathematics education issues rest upon who the teachers of mathematics are.

It seems to me that a lot of teachers, at least in my area, did an undergraduate degree in the humanities or social sciences before entering teacher’s college. This means that they have limited (if any) experience with advanced mathematical concepts like analysis of variance, special numbers (Pi, e, i), differential equations, or the applications of those concepts.

When asked “when will we use this in real life?” by students, they will not have a compelling answer because they do not know for themselves. When a student comes up with a different way to “do math”, the teacher will not have the expertise in the subject to understand the difficulties that different approaches will encounter later on, and therefore will defer to the student’s best judgement - thus depriving them of a learning opportunity.

In mathematics, as in enlightenment, I suspect that rigorous application of oneself to existing methods will prove to yield better fruit than potentially specious self-discovery. However, time will tell.