Let's Listen to Math

by Anne White

This post combines some of our favourite things: Denise Gaskins' approach to teaching mathematics in a way that reflects Charlotte Mason's principles; the New Mason Jar podcast; and the chance for thoughtful educators to dialogue about ideas.

Your assignment (and I do hope you choose to accept it) is, first, to listen to the podcast episodes (here and here) where Cindy Rollins and Dawn Duran interview Denise. 

“[A child’s] education should also offer him passage into wide-open spaces full of treasures to be encountered, appreciated, and enjoyed no matter how he spends the laboring portion of his days.” (Karen Glass, Much May Be Done With Sparrows)

Then come back here and consider these discussion questions. Maybe you would like to get a couple of friends together and talk about them. Or you can post your answers in the Comments section.  

Introductory Questions

1. What is one way that you used math today? (Outside of teaching it.)

2. You are ten years old and you’ve finished your arithmetic worksheet. The teacher says you can go to her shelf of special activities and choose a game to play.  Would you pick something that looked like a math game? Why or why not?

Discussion Questions

1. Denise Gaskins’ website, and one of her books, are called “Let’s Play Math.” Before listening to these interviews, what might you have assumed about her approach to mathematics? How did that change after hearing the podcasts?

2. What does math class at your house look like? Are you interested in exploring more of the "play math" suggestions (games, journalling, reading library books) with your own students?

3. Which of the following quotes make the most sense to you? Are there any that you disagree with?

"Math should be like a nature walk."

"Thinking hard can be fun."

"The 3 R's of math are to Recognize and Reason about Relationships.”

"What children need most are a few basic principles and the ability to reason, to draw their own conclusions about how numbers, shapes, and patterns work.”   

4. The podcast presenters suggested that when Charlotte Mason said that, in her time, the standard approach to math teaching did not need to be “fixed” because it wasn’t “broken.” Do you think our view of mathematics (and math teaching) has become more broken since then?

5. You have a friend who sees herself as more of a math person than someone who enjoys literature and art. Is there a way you could give her an understanding of C. M. principles through the lens of mathematics?

Synthetic Thinking and Math

by Karen Glass

Education is the science of relations. That’s the principle that underlies what I call synthetic thinking (explained more fully in Consider This). The principle applies even to arithmetic, which is, although we don’t usually fathom the reason, one of the liberal arts. An art is not made up merely of information to master--it is meant to used. Arts are “practiced.” We want to foster a relationship between students and the world of mathematics, and that is most readily accomplished when we treat mathematics as an art to be practiced.

There is going to come a time, in math, when a child is going to have to sit down and work through some complicated equations. That is either going to be a challenge met with confidence and a lift of the chin-- “I can do this!”--or with boredom and despair. We often speak of wanting children to love reading, love literature, love books, maybe even love history or science. We rarely speak of wanting to them to love numbers, and this is probably a reflection of the reality that few of us formed that relationship in our early years of education. Whether or not that relationship is formed will determine the response a child--and even an adult--brings to those complicated problems.

The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders. There is no one subject in which good teaching effects more, as there is none in which slovenly teaching has more mischievous results. 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. (Home Education, p. 254)

We often speak of a "Charlotte Mason education" being a paradigm shift, and nowhere is that shift greater than in the area of math. Charlotte Mason knew that math was about much more than getting the “right answer.” There is a relationship between math and the natural life of men, and it was this relationship that she wanted to foster first.

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. (Home Education, p. 254)

In practice, this means that children should begin with what we call “word problems,” and those problems should be based upon real-life experiences that the child might expect to occur. For the smallest children, these math problems occur easily in course of living.

Home life is full of easy little arithmetic problems that bring the importance of numbers, as well as concepts such as quantity, equality, and one-to-one correspondence within the grasp of even quite young children. A family of four is joining us for dinner. How many chairs to do we need to add to the table? I can only find three clean spoons--how many will we need to wash so that we have enough? There are six cookies left in the box. How many can each child have?

Older children can figure how much five cans of corn will cost, or whether there is enough money for everyone to get double-scoop ice-cream cones, or if singles will have to do this time.

Older children may be given more complex, multi-step problems, such as “Joe gathered 87 walnuts and Tim gathered 28. They plan to share the nuts with three friends. How many will each of the five boys receive?” Charlotte Mason says that a child will perceive exactly what must be done in order to solve the problem, although “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.” (Home Education, p. 255)

The more occasions a child has to use math in real life--and that might include playing games in which counting, adding (or subtracting) points, or other arithmetic plays a part--the more likely he is to develop an interest in and a relationship with math.

Math is needed for cooking, for science, for travel, for planning and purchasing, and the more integrated a child’s exposure to math is, the greater will be his appreciation for it. Once that appreciation is established, the extra effort needed to memorize math facts or unravel complex equations will be entered into more willingly. The child has no need to whine, “why do I have to learn this?” If he has developed a synthetic understanding of math, he already knows the answer to that question, and will likely also work out the answer to the arithmetic problem at hand.

How to be a better teacher than Miss Perkins (it's not hard)

by Anne White

I discovered a short story, "Against the Odds" by Martin Gardner, in his book Are Universes Thicker Than Blackberries?  It was first published in the College Mathematics Journal in 2001, but (according to this summary and the Amazon link) Blackberries is the only other place you're going to find it.  I found my copy at the thrift store, but you might check the library.

The plot is almost too predictable, too simple.  Luther Washington is a young African-American boy (so he is described), around 1960, who has a gift for abstract mathematics.  He runs up against a female (white) teacher who doesn't know much more math than her students do and thinks he's just showing off.  Luckily, he eventually gets (more or less) sent to the principal, who does know something about math, and who puts the boy on the road to a college scholarship.  Ten years later, after earning his Ph.D., Luther wins a major mathematics prize.  The teacher, now retired and married to the basketball coach, doesn't recognize the newspaper photo of her former student but mumbles something about "affirmative action."

The story raised some questions for me, besides obvious ones like "could that ever happen?" Actually, that could be taken either way, as that summary points out: the first part of the story certainly could happen.  It is depressingly realistic about ignorance in teaching that kills the desire to learn.  The second part is, of course, a fantasy, a wonderful Cinderella solution, but one that probably doesn't happen often.  It's nice to see Luther's career success, but it's just a bit of luck, really, that he suddenly gets noticed by the right people, and doesn't have to bury his dreams.

It's with the earlier part of the story that we, as parents and educators, need to concern ourselves.  We can even ignore the question of racial prejudice to some extent, though it is largely what keeps Miss Perkins from seeing Luther's brilliance.  But based on the description of the teacher's limitations (an American-history major who got stuck teaching math), is it likely that she would have been more accommodating of a white student whose talent for math surpassed her own?  Would she have been more accepting of a student who showed great aptitude for American history, or would she have been as narrow-minded about the correct response to questions in history and government?  Is it not also something of a stereotype to assume that an American-history major would not recognize vectors, or that she would be so uninterested in her own teaching subject, however accidentally acquired, that she wouldn't go the library that night, or phone up a colleague, and find out what that confounded boy was talking about?  Or ask him to stay after school and show her how his proof worked?  That's what you'd do, isn't it?

Well, you and I aren't Miss Perkins.  She may be a Dickensian stereotype (especially in the "devout Baptist" part), but she's got enough truth in her to make it worrisome.  Stories pop up in the media about teachers who can't spell, can't punctuate, and yes, can't do math.  More stories come up about families who turn to homeschooling after encountering their own version of Miss Perkins. We know there are problems in the public school system.  We know there are bad teachers.  We know that exceptional learners of all kinds, including gifted students, often get shortchanged by "the system." What does all that have to do with Charlotte Mason?

Just one, maybe two things.  Miss Perkins might seem to be mostly racially motivated, but as teacher-detectives we need to look at what else is going on. Stick with me, class...who can tell what educational principles Miss Perkins violated?  What was her biggest mistake?  Pride?  Sloth?  Misuse of authority?  Not recognizing Luther as a born person?  Not giving him the respect due to his personality?  Messing with Luther's desire for knowledge?  Stomping on his living ideas?  Putting all the stress on the idea of the teacher having to put information into the student's head, instead of recognizing that he could figure things out aside from her?  Thinking that if she didn't know something, there was no way that a teenager (let alone a minority-group teenager) could know it?  (It's always a temptation to think "I can't learn anything from this person, because I'm X and he's Y.")  Ignoring the Gospel command (quoted by Charlotte Mason) to "Take heed that ye offend not--despise not--hinder not--one of these little ones?" All of the above?

I think that list covers most of her educational sins, but there's one other point, and perhaps it is a greater problem for some of us...who are, like Miss Perkins, teaching outside of our own fields, or without any "official" teacher training.  Yes, we have wonderful educational resources to draw on; even "scripted" ones that practically guarantee teaching success without having to have deep knowledge of a subject.

And there's the problem.  We are not teaching machines, any more than our students are learning machines.  What kills learning for the student goes double for us, even if we have such thought-out-in-every-way materials that we can now teach on auto-pilot.  Especially if we have such materials.  Charlotte Mason did not approve of too-elaborate manipulatives and models for students; and, by the same token, she would probably not care for lessons that don't let any unscripted learning sneak in.  Especially lessons that go so far as telling us what we, as well as the students, are to think.

If Miss Perkins seemed determined to shut down Luther's learning, it appears that she had already shut down her own. (Ignorance breeds intolerance?)  Her teaching had been reduced to one-lesson-at-a-time, and please don't ask me any questions that might make me look foolish or take us five minutes over the time limit.  This is what this lesson's about, this is how you do it, and this is the right answer.  Some people say they like math because there's always one right answer...but no, it's not true even in math.

Yes, teaching is "easier" if there are lesson plans, assignments, printable tests with answer keys.  "Open the book and teach" brings high praise from reviewers. Homeschoolers are, proverbially, always looking for "curriculum" that does everything but diaper the baby and cook dinner.

Charlotte Mason would say, run from anything of the sort.

A little help, yes.  As someone who has attempted to "help" by writing several AO study guides, I'm sympathetic to all the reasons of our lack of time, lack of background, having several children to teach, and all the rest of it.  The reason I wrote my first Plutarch study was simply because nobody had written one for me to use.  If there had been a set of notes, I would have used them, but I couldn't find any, so I just kept reading and looking stuff up until it started to make sense.  To keep others from having to reinvent that particular wheel, I put the notes online.  And I've been extremely grateful for other people's work in other areas.  But especially with Plutarch, I can't tell you why he says everything he says, what everything means, or what, exactly, to say next.  Or what not to say.

So if we take Miss Perkins as a cautionary tale, let's be careful about thinking that any written lesson or teacher's manual (short of the Bible!) contains complete and final knowledge of anything; and let's also be open to truth wherever we find it.  Even in a murder mystery that turned out to be so graphic I'd never read it again:

"'My mind would make these magic little leaps. You know what I mean?' I nodded. I knew about minds making magic little leaps."  ~~ Sue Grafton, C is for Corpse

Here's to our students' success.  May we not offend these little ones.  May we not take ourselves and our educational materials so seriously that we close our eyes to curiosity and new ideas.  And may we take our desire to learn...and the humility to learn from each other's magic little leaps...out into the world. Because there are still a lot of Luthers out there.