Teaching Mathematics part III - good teaching continued

Last time I discussed two steps that math teachers should take in helping students pursue math as a delightful subject to study.  Let's talk about two more important things teachers should do.

3.  Demonstrate the hows and whys.

This is very much related to what I mentioned in my previous post about introducing the concrete before the abstract.  In doing so, these concrete ideas must be demonstrated to the children.

“The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved."  (Charlotte Mason, Home Education, p.255-256) - emphasis mine

I remember as a teacher, teaching that "mile wide and inch deep" curriculum in public schools, really only having enough time to just tell my students what to do in order to solve a problem because, like I said in my previous post, we had to cover a certain amount of material before the end of the year (really, before the big test).  But this method only allows for a "monkey see, monkey do" kind of learning, with the students just memorizing steps, not gaining deep understanding.  And this results in children quickly forgetting the material.

I needed time to give my students the opportunity to go behind the scenes, so to say, and discover the how and why of the material.  And sometimes - very rarely - we would have a little time to do that.  But, honestly, it would not go over well.  My students (remember they were 8th and 9th graders) either 1) did not care about the how and why and just preferred me to tell them what to do, or 2) got lost - the demonstrations and proofs of the mathematical processes would completely go over their heads.

So that told me that they were not used to this - to getting underneath the surface and grasping the rationale of the material.  And this not only results in a lack of understanding the material, but a lack of understanding math for what it is.

For example, if children could see how multiplication works it will provide for a much greater understanding than just memorizing the multiplication table will ever do.  And a firm understanding of the early concepts and how they work will have a domino effect - it will make it easier for the child to comprehend more and more mathematical processes because he is being trained to think, not memorize.

"In introducing these new processes of multiplication and division, the principle of developing ideas before words should control, and the knowledge already gained should be used as a stepping-stone to the acquisition of the new ideas.  For example, the child already knows that 2+2+2=6; also, that 3+3=6.  With this knowledge as a basis, it is very easy to show him that three times 2 are 6, and that two times 3 are 6.  Thus, the child's knowledge of addition is used in teaching him multiplication.  This seems much more rational that at once plunging into the multiplication table, and, by dint of incessant repetition, memorizing the combinations of a host of factors and products.  By this first method, the child learns how the product is formed, and why 3x2 or 2x3=6.  By the second method, he simply remembers - if he can remember - the formula."  (Eclectic Manual of Methods, pg.124) - emphasis mine

Even though these two quotes are about multiplication, the idea still applies to any mathematical topic.  Taking the time to demonstrate and allow the students to discover the hows and whys of these mathematical processes is so important in deepening their understanding and developing an appreciation for the logical nature of mathematics.

4.  Focus on real, contextual problems.

Mathematical drill is important.  Students need to practice being accurate and quick in their calculations.  However, too much focus on drill will not give us the results we want to see in our students.  Instead of encouraging deep understanding and a sharpening of a child's reasoning abilities, it only encourages rote memorization.

"Multiplication does not produce the 'right answer,' so the boy tries division; that again fails, but subtraction may get him out of the bog.  There is no must be to him; he does not see that one process, and one process only, can give the required result.  Now, a child who does not know what rule to apply to a simple problem within his grasp has been ill taught from the first, although he may produce slatefuls of quite right sums in multiplication or long division."  (Charlotte Mason, Home Education, p.254)

So how do we really train a child's ability to reason and problem solve, and encourage deeper understanding?

“How is this insight, this exercise of the reasoning powers, to be secured? Engage the child upon little problems within his comprehension from the first, rather than upon set sums. The young governess delights to set a noble 'long division sum,'––, 953,783,465/873––which shall fill the child's slate, and keep him occupied for a good half-hour; and when it is finished, and the child is finished too, done up with the unprofitable labour, the sum is not right after all: the last two figures in the quotient are wrong, and the remainder is false. But he cannot do it again––he must not be discouraged by being told it is wrong; so, 'nearly right' is the verdict, a judgment inadmissible in arithmetic. Instead of this laborious task, which gives no scope for mental effort, and in which he goes to sea at last from sheer want of attention, say to him–– 

'Mr. Jones sent six hundred and seven, and Mr. Stevens eight hundred and nineteen, apples to be divided amongst the twenty-seven boys at school on Monday. How many apples apiece did they get?'

Here he must ask himself certain questions. 'How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy's share.' That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly; the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.” (Charlotte Mason, Home Education, p.254-255) - emphases mine

In addition to being able to add and subtract, for example, children need to be able to apply these processes in order to solve problems.  This not only strengthens a child's reasoning skills and builds up his ability to do mental math (which is so important in training the child to think), but also keeps him interested in the work.

“That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education.” (Charlotte Mason, Home Education, p.254)

As usual, let me reminisce!  When I would assign problems to my students, which ones do you think 90% of them would skip?  That's right...the real world, contextual, word problems - the problems in which they had to think and figure out how to apply what they knew.  It was disheartening.  And it wasn't just my students - this was the norm, at every school and for every math teacher I ever worked with.  The kids claimed they had no idea how to go about solving problems like these.  They wouldn't try.  Many of them wouldn't even read the problems (I'm sure there was some laziness at play here as well).  It was apparent that this was not something they were used to.  They were used to being drilled, not to think.

"Consider how mathematics was introduced to children a few years ago.  To pupils of Junior School age only arithmetical ideas were presented.  Algebraic and geometrical ideas were thought to be beyond the understanding of young children.  Great emphases was laid on mechanical work and learning by rote." *

Let me stop here and say that this is what I'm talking about.  Our children are not given the time to think and discover the ideas of mathematics, and so we're left with teaching rote memorization.

Continuing on...

"...What was the result of this preoccupation with mechanical 'sums'?  It was true that a few pupils attained an almost machine-like accuracy and came to enjoy the subject in which they could achieve such perfection.  However, the majority managed only partially to understand the elementary mathematical concepts.  There were some, indeed, who gained no real knowledge of the subject; who remained completely mystified by the language of number and size.  The unfortunate pupils in this latter category usually developed an early antipathy toward mathematics.  Everyone who has been involved in the teaching of 'Secondary Stage' maths will have encountered the child, otherwise able, who has no understanding of Number." *

Wow.  This whole paragraph - all of it - describes my experience teaching math in the public school system to the T.  The kids had no knowledge or respect for mathematics.  They saw math as confusing.  They disliked the subject.  They didn't know how to think.

"...The basic ideas of mathematics must be understood if we are to be able to follow modern science, but undue emphasis on 'mechanical drill' will not achieve this end." * - emphasis mine

There must be a balance of drill and solving real, contextual problems - aka word problems; problems in which the child must apply what he has learned.

(*Quotes taken from the PNEU article Knowledge of the Universe by G.L. Davies)

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So to summarize, the successful teaching of mathematics requires, above all else, a teacher who understands how to teach mathematics.  And this good teacher will:

• Give the individual student plenty of time to develop the ideas of mathematics.
• Focus on the concrete before the abstract.
• Get under the surface and allow the student to understand the hows and whys of mathematical processes.
• Include plenty of word problems for the students to solve to encourage deeper thinking.  

These are the things to not only think about and remember as we teach our children math, but also as we search for an appropriate math curriculum.  

And hopefully, as a result, the child will be trained to think, will gain a solid understanding of mathematics, and will develop an appreciation for it as a delightful subject worthy to be studied.

Next time, I'll outline the first year of mathematics instruction.  And you may be surprised to see how much Ray's Arithmetic mirrors CM's thoughts on math.

Other posts in this series:
Part I - why study math?
Part II - good teaching
Part III - good teaching continued <---- you are here
Part IV - laying the foundation