Education: Are concepts essential? (Warning: long)

I completed my schooling in India (ages ago) before moving abroad for university. In my experience, education in India at that time could have been generalised to ‘drill and practice’ and ‘learn by rote’. There were certain Nationalised exams that had to be passed, competition was immense (considering the population), and so we crammed like hell. Teachers were well aware of this and did their best to steer us towards exam success. 250 algebra problems, all of the same kind, but only with different constants? Pah, child’s play. Just a regular weekend.

And it worked. We passed our exams (mostly).

Throughout my schooling, concepts - as opposed to facts - were given the backseat. Concepts were brought up and explained primarily when bright kids asked questions. Facts, on the other hand, were the mainstay of education. Why is the area of a circle = pi.r^2? Don’t ask! It just is! And we managed.

Those of us who were of higher ability asked around and were told the Whys and Wherefores, or we figured them out ourselves, or we had blinding conceptual flashes of the obvious ten years after by-hearting the facts. The rest just puttered along and went on through life without knowing why teacher insisted A = pi.r^2.

Now, I am a teacher myself. I teach mainly low-ability kids and kids with developmental disorders, but in a normal school. Thing is, I live and teach in an age and in a system where concept is king. You have explain the whys before you get to the hows.

And frankly, I’m a little…underwhelmed.

Yes, concepts are good. But does everyone really need to be burdened with them? I have little ones who are absolutely lost when I tell them that See, when you multiply something by 32, you need to put a ones place zero in the second line because you’re actualy multiplying by 30, not by 3. No amount of concrete materials or diagrams seem to help. Explaining why you’re carrying over a 1 when you add 34 and 9 seems pointless. And God help me if I try to bring concrete materials into the borrowing while subtracting scenario. It doesn’t help with these babies. In fact, it seems to make life more difficult for a lot of them. So, I attempt to give them the concepts, but I also give them shortcuts I learned from my teachers, shortcuts I figured out for myself…just to make life in the real world easier for them.

Sometimes I truly feel that if you empower children with the facts, the ones who are going to do anything long-term with them will figure out the whys anyway. And the rest, who’re going to become functioning members of society that don’t deal with areas of circles on a regular basis, hey, atleast they’ll have the pi.r^2 thing buried in some corner of their mind if ever the need should arise.

See, education resources are limited. The time a child is schooled for is limited. Holding a kid back a grade on account of lack of sufficient knowledge and understanding is becoming increasingly rare. (Heck, I can’t remember the last time I heard of a child being kept back in the system I’m in. Not USA.)

Well, why not maximise the time and resources we have and push facts, not concepts? At least for mid- to low-achievers?

I work in IT and having dealt with people in the ME whose education consisted of learning just the facts, I can say it makes them quite capable of trouble-shooting and fixing items they have come across before, but they are deficient when they have to solve issues that they have not previously experienced. Spending time questioning things from an early age helps considerably once you are older and not just in the workplace.

I focus intently on concepts, as a math tutor.

The reason we teach Algebra in our schools is not because the vast majority of children will have a need to factor trinomials or solve variables. Very few of the students I teach will use Algebra beyond their college courses. The reason that I don’t feel that teaching Algebra is thus a waste, then, is because I use it as a vehicle to teach problem-solving strategies. I force my students to look at a problem in multiple ways before I show them the easy, step-by-step way.

Another reason that I teach conceptually first is that otherwise math becomes a pointless, boring game where the numbers have no real relationship to each other. I frequently take discussions into the history of certain concepts, and sometimes even into the personal lives of some of our greatest discoverers.

Most important to me, though, is that by teaching conceptual framework I can often share the joy of discovery with my students. I give them the opportunity to figure it out for themselves. This is important for two reasons: students are more likely to remember things that they figured out on their own, and they get excited about what mysteries they will uncover next. Students that are excited by or interested in the material learn far better than students that are bored or frustrated. This means that I rarely work with students more than a semester: by the time that I’ve gotten through a class with a studdent, he or she is ready (and, in some cases, looking forward to) for the next batch of discoveries.

Humanist’s response is absolutely correct. Rote memorization of a fact or process is only useful to a student if they can see some purpose for retaining the knowledge. That requires locking it into concepts.

Take the arithmetic process of long division; there is no reason in this day and age why this skill should be memorized by the general population. Even in the rare instance where a calculator is not available and the job must be done, often times an estimated answer will do. Sure students could be forced to learn it, but (1) it will be of little to no value to them, and (2) since it has such little value in the real world the knowledge of the process is inert and inflexible (i.e. they can use it only to answer specifically-phrased division questions on a school test).

Yes, students should know their multiplication tables, and it may be easiest to drill them on it from an early age, but unless that knowledge is applied in some way (i.e. through the exploration and application of mathematical concepts), it is a pointless waste of grey matter. Of course, if the point of education is to just pass some test, it works wonderfully; I guess I prefer that those who have been educated are, indeed, educated:)

When I was at school, we learnt our 11 and 12 times tables by heart. Because at that time, there were 12 pence in a shilling.
We were regularly tested and told how important this was.
Later the UK decimalised the currency.

“A child’s mind is not a vessel to be filled, but a fire to be set alight.”

I can’t remember who said that, but I know how to look it up. I’ve been educated!

That’s funny, because I learned my 11 and 12 times tables by rote, in the U.S., in elementary school, in 1983, and I’m pretty sure we’ve never had 12 pence in a shilling.

That’s the problem with teaching by rote. Whatever you are teaching is guaranteed to be hopelessly out-of-date. All you are really teaching is how to learn other material by rote.

I don’t see how high school math can be taught by rote memorization. Synthetic geometry requires proofs. That’s the whole point of studying geometry in the first place. Even HS algebra and trig require some conceptual thinking; when students factor trinomials or prove trigonometric identities.

I agree that teaching concepts is important, for the reasons already mentioned. On the other hand, it’s certainly useful to know your multiplication tables by heart: even though I’ve got the concept of multiplication down pagt, it would be irritating if I had to derive the answer to 12x8 every time I needed it (e.g., if I’m at the gym trying to figure out if I’m hitting my target 12 calories/minute burn on the elliptical machine).

It’s not just the smartest kids that need conceptual background. The kids who find math more difficult will appreciate the “scaffolding” provided by the conceptual learning, if it’s done well. I’ve worked with kids who had trouble understanding math, and I absolutley found that working with them on the underlying concept meant that, although they’d never be able to come up with the right answer as quickly as the math-talented kids, at least they’d be able to come up with it eventually. If it’s a memorization thing, they’ll never come up with the right answer.

Daniel

12 x 8
= (10+2) x (10-2)
= (10x10)-(2x2)
= 100 - 4
= 96

I don’t mind knowing the times tables – I know a fair number of individual bits of them past the 12 times table – but remembering them is helped if you know a few rules and relationships, including (x+y)(x-y) = x^2 - y^2.

So, I agree that students who have just learnmed a lot by rote have really learned nothing at all: you need to know how to apply old things in new situations if they are going to be of any use.

True, true! Me, I memorized it, so my thoughts look like:

12 x 8
=96

It’s vital that students be able to figure out a problem in multiple ways (as the OP suggested, to know 8 ways to solve 2+2), but for mutliplication facts and addition facts, I think it’s helpful to have a good chunk of them memorized. Saves time and intellectual resources.

Daniel

Agreed, but sometimes I wish they’d spent the months we spent doing multiplication and division drills on walking around in the park, and instead taught us:

** [1] [2] [8] [=] **

I agree that a base set of memorized facts are good to know, to speed things up. There is another reason to learn concepts though - for checking answers.

In the days of slide rules, you needed to know more or less the magnitude of your answer, because the slide rule will never tell you. In the days of calculators, the answer you get will be precise and incorrect if you somehow keyed in an extra zero, or dropped a digit. When I add a bunch of numbers, I always estimate what the answer should look like before I do it, and check.

I’m happy to say that when my daughter was in hs algebra a few years ago they did teach estimation.

I’ve always found I can do problems a lot faster when I know the concepts - and this extends well beyond math. When I taught data structures, I worked for the entire term drilling in the difference between a variable and a type. It took 3 test questions on three different test for most of the class to get it. But they did.

Anothe rimportant point here is that everybody learns differently. I like to consider myself fairly intelligent, yet as a child I struggled with rote learning. Not becuase of any memory deficit but simply because it bored me and I couldn’t be bothered. If a concept was taught I could usually puzzle my way through the correct answer in almost any subject despite failing tomemorise lists of facts.

There isn’t any single correct way of educating children. The one-size-fits-none approach that we’ve adopted for the last 200 years always assumes that there is simply one way that will ahcive the best results, and as a result the method of teaching changes regularly with no real imorvement in results.

Some students will learn best by rote leanring of fact and mechanical substitution. many will not learn at all. That’s not because they are les intelligent but simply because they are different.

And that’s the crux of the argument right there; some children learn well by rote–that is, through exposure they gain an appreciation for the concept–while others become distracted or confused by going through the motions without understanding the process.

The biggest problem I have with “teaching by rote” is in math and the sciences. Learning to do the problem without understanding it–what we call “plug and chug”–offers zero insight and doesn’t let the student apply the same knowledge to a new problem. Basic science texts are the absolute worst; not only are they full of conceptually vague statements (see Richard Feynman’s reaction to California state-selected science texts) but the information is often just plain wrong, or stated in a way that is prone to conceptual misunderstanding. This leads to mistakes in areas where intuition doesn’t serve as a guide to comprehension.

Teaching “by rote” in the arts and letters isn’t much better–you should learn to interpret what the writer is saying rather than parrot back what you read in Cliff’s Notes–but there is merit in memorization and recitation.

In the end, concepts are essential to a good understanding and a base from which to assimilate new knowledge independently. However, with many applied skills, practice and memorization is required in order to be able to perform the skill without having to refer to another source or derive the methodology every time. But memorization doesn’t equal understanding.

Stranger