Monday, June 15, 2015

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)

~ ~ ~ ~ ~ ~

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


  1. Angela,
    After being so eager to read these posts a few weeks ago, I completely forgot to check my sidebar and read them! How did that happen? Well, probably because the one year anniv of my Dad's passing was last Friday, so I have been sooo pre-occupied with thoughts of him and his last days on this earth. Also, we are praying about selling our house and moving, and because of that possibility, we sold all of our dairy goats. So much sadness and change in recent days. :( But I will catch up...eventually.

    1. I understand - it sounds like an overwhelming time. We, too, are trying to sell our house and move. My husband's job is over an hour away, but we haven't had much luck. It's been for sale for almost a year now. It takes a lot of patience and faith to figure out where God is leading you. I haven't been on my blog much, either, what with this whole job/house thing, preparing to begin homeschooling in a couple of weeks, plus a baby on the way in 3 months! Ahhhh! I'm not saying I've been doing the best job of keeping my cool (but I can blame hormones right now, right? ;)), but deep down I have to believe that God knows the plans He has for us. I'll be thinking about you in the midst of this time :)

  2. It's such a joy when children are taught this way from the beginning. There is nothing more delightful than a little one discovering on his own that 2+2=4, just from playing with manipulatives. The excitement! The joy! And that knowledge is his own now.

  3. Another great post!

    Maybe you can help me with my oldest - we have always used manipulatives to demonstrate concepts concretely before getting abstract, but lately he's been telling me that he doesn't like to use them because they make him feel stupid. I'm thinking that what he means is that make it so easy to help see the concept that he feels like he didn't get to do any of the work himself - the c-rods did it for him. But I think that he could use a little help with the concept of long division and how it's related to subtraction.. he just won't have it though. Do you have any suggestions as to how I might be able to work on his conceptual understanding perhaps without manipulatives?

    1. Thank you for the encouragement!

      I'm sure you know this, but remember that teaching concrete first and explaining how mathematical processes work doesn't necessarily mean that we have to use manipulatives to achieve those things. I think in the later years manipulatives can sometimes get in the way, depending on the student. Some kids like them, some don't. Some find them useful, some don't. If they seem to be getting in the way of your son's learning, then I say put them up for awhile! Each child learns differently. I hardly ever used manipulatives when I taught because like I said, I felt they got in the way and it was like they over-complicated the math - even though you'd think it would be the other way around.

      Anyway, how does your son do with simpler division? Not 12/6 or something, but more like 484/4, for example? He should be able to do a problem like that in his head before moving on to problems with a remainder, and then eventually problems requiring long division. If he understands HOW to the simpler problems work then he should be able to move on to more difficult problems, slowly graduating toward long division.

      Have you gone through the short division section of Ray's Practical Arithmetic book, pages 54-57? It does a good job of breaking down how it all works - demonstrating the how and why concretely. That should definitely be mastered before attempting long division, which begins on page 59. And again, the book does a good job of explaining how long division works.

      It really takes practice. Practicing many problems, constantly talking about what's really going on and why we do each step, until the process is mastered and understood, rather than memorized. Goodness, I hope that's helped some! Feel free to email me if you have any other questions (about this or anything else!) - anwilhite at gmail dot com.

      And hoping everything goes smoothly when you finally have your sweet baby! :)

  4. Angela, I recently stumbled upon your blog and am encouraged by your math series. We've been homeschooling eight years and have a homeschool graduate. However, math has always been problematic in our home. I totally agree that the math teacher is super important and I suck as a math teacher, LOL! I don't have any real adversities to math. I didn't struggle with it in school and I don't love it or hate it. I definitely understand the importance of math and I love nature, puzzles, patterns, and symmetry so I'm thinking I see the beauty in math. I just have no idea how to get this across to our children.

    We started homeschoolling with a traditional textbook and when they began to struggle, I would jump ship and go to another textbook. Our current school age kids will be in 5th & 6th grade this fall and I have no idea how to undo their math phobia/hate and how to start over with them and a "living math" approach.

    I'm very eclectic in my approach and love Charlotte Mason's methods. We use this method in pretty much every area of our homeschool life, unfortunately, with the exception of math. Any ideas on how to right my wrongs and start off fresh this fall?


    1. Thank you for visiting!

      Here's what I would do:

      I would first assess where my kids are mathematically - which topics they're confident and strong in and which they struggle in. For the topics in which they're struggling, I would go back and reteach - start over completely with the topic if you have to. Focus on the concrete first, make sure the child understands the why of the process, and include lots of real world problems. There's no point in moving on to more difficult material if previous material is not understood.

      I would also keep the teaching and explanations as simple as possible. There are some curriculums out there that over-complicate things. Math is beautiful enough and difficult enough without all the fluff - manipulatives, projects, etc. That's why I like Ray's Arithmetic. It's simple, straightforward math, focusing on all the things I've written about so far, which gives the child an opportunity to be successful and understand. I think that's where a good attitude about math comes from first - success and understanding.

      I would also incorporate some time for playing math games together, maybe once per week or two - I'm sure you can find lots if you google or search on pinterest. I remember my classes always loved playing games - they were so competitive and really got into it, working as hard as they could, even the kids who struggled or had negative attitudes about math.

      And, I would try to incorporate some living math books, especially biographies of the great discoverers of mathemetics. "How interesting Arithmetic and Geometry might be if we gave a short history of their principal theorems, if the child were meant to be present at the labours of a Pythagoras, a Plato, a Euclid, or in modern times, of a Descartes, a Pascal, or a Leibnitz. Great theories instead of being lifeless and anonymous abstractions would become living human truths each with its own history like a statue by Michael Angelo or like a painting by Raphael." (CM, Towards a Philosophy of Education, p.110-111).

      Here are a couple of links with lots of living math books:

      Lastly, the parent's attitude is also very important - it rubs off on the kids. Not saying your attitude is negative! Just something to keep in mind. I saw lots of inadvertent negativity from parents when I was a teacher and I could tell it made a difference in the attitudes of the kids. Stay positive and encouraging. It'll go a long way.

      And of course, not everyone will develop a love of math. We're all different. What we want more than that is for our kids to be successful and confident. Let them make the discovery of the nature of mathematics. We can nudge them in the right direction, but we can't force them to see or appreciate something. That's for them to decide - which I think is the whole concept of CM's method of presenting children with a feast of ideas and letting THEM find meaning; letting THEM learn to think and ask questions and make discoveries.

      Okay, that's probably enough! I hope I've helped. Feel free to email me at anwilhite at gmail dot come if you have any other questions :)

      (PS - I may include this information in a future blog post!)

  5. Thank you, Angela. That is very helpful! I have a few more questions, so I'll try to email you when I can think straight enough to write them out - the baby has arrived and lack of sleep pushes all coherent thoughts out of my brain :)