Episode 54: Teaching Math
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Welcome back to A Delectable Education, the podcast that spreads the feast of the Charlotte Mason Method. I'm your host, Emily Kaiser, and I'm joined, as always, by Nicole Williams of SabbathMoodHomeschool.com and Liz Kutryl of LivingBooksLibrary.com.

Today is the beginning of our series on math, a subject that a lot of us struggle to convey that awe and wonder that we want and desire to be present in every subject that we teach in a Charlotte Mason education. So let's first talk about why math seems so intimidating to homeschool moms. I've been giving this a lot of thought just because for me personally, it's very intimidating to

Even though I did very good in math in school, I got to higher levels than the average high schooler at my age. And then I went on to college and did really well in math. And so I don't know why it is such a hang up for me. It's a different thing when we teach it to someone else. It is. But there's also the thing that I have realized that

My math education did not include an understanding of math. I can read any directions. I can put anything together. I'm good at following directions. And that's what my math education

education was. And there is definitely a part in there for that, but that is not how Charlotte Mason wanted them to start. And so I think that that gives me a sense of insecurity. I think there's also so many programs out there that all say we're the one it's our way or no way. And, and,

And then there's the realization that maybe not any one program is perfect and has everything. And so we get this sense that we don't really know where to put our trust, which program to get started in and stick with, or can we just stick with one? And so I think there's a, it's a complicated issue for a lot of people. Some people did very poorly in math.

And they think that because of that, they won't be able to teach their children well. But to them, I would say, you've probably come a long way. So maybe give yourself a chance. And you're still probably ahead of your children. Right.

Well, you know, this is a subject that Mason doesn't have a lot to say on because she didn't think she had really that many new ideas. She just said, you know, we just do math. Like, you do math, you know. But she does mention a few specifics. It was interesting as I was rereading through the portions where she does talk about math. She seemed to be facing the similar idea in her day. Math was being upheld.

upheld as this preeminent subject and it was kind of forcing out other areas of the curriculum. And I think we see that today too, right? I mean, and that's probably why a lot of us do feel insecure about teaching math is because we hear about its importance and, you know, we need to have these children who are very great mathematicians. And so that kind of doesn't give us any grace, I guess, to get it wrong or to struggle ourselves. Yeah.

And Charlotte Mason did think it was a worthy subject. Before we start diving into the wonders about math, she did give us this warning. She said, arithmetic mathematics are exceedingly easy to examine upon. And so long as education is regulated by examination, so long shall we have teaching directed not to awaken a sense of awe in

in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems. So we see...

That might also be a tendency that we have is we know we can evaluate our children's progress very precisely in math, but that derails our main intent of inspiring their awe and wonder at this natural science. Charlotte Mason said that the chief value of math was that it teaches children to

that there is absolute truth in the world. She said, the use of the study in practical life is the least of its uses. The chief value of arithmetic like that of higher mathematics lies in the training it affords to the reasoning powers and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders.

I think that's really interesting and probably something that most Christian parents haven't contemplated about why math is important. That because there are specific answers, right? There is a right and a wrong about math that.

It teaches our children that there is a right and a wrong about absolute truth is right there. Right. She also says that it changed trains their reasoning ability. And she goes on to say, never are the operations of reason more delightful and more perfect than in mathematics. By degrees, absolute truth unfolds itself. We are so made that truth, absolute in certain truth is a perfect joy to us.

And that is the joy that a mathematics affords. So it's joyful. It brings joy to the student when they achieve a right answer that they see that it's right, but more than just their own ability or achievement in that problem, but that it's teaching them a deeper truth that there is truth. And it's beautiful when something all fits together and works out, you have that pleasure in something coming together so perfectly. And to know that two parallel lines will never meet and that,

All those other laws of mathematics. It is a subject that addresses knowledge of the universe. And just this morning, my son was a little bit awed himself to think that all these huge planets spinning in this endless space out there, that God must be a mathematician because it's all in order and everything.

Thanks, Nicole, for writing that program. It makes it so clear when you put it that way, why it is important, her principle, that they don't go back and correct a math problem. Because this idea of right is right and wrong is wrong, if you're looking at math from this perspective of...

Right.

ties together because I've always thought, why wouldn't they go back? Of course they need to go back and fix it. But when you think about it from that perspective, it makes perfect sense. And leave it to Charlotte Mason to not

give us the pragmatic reason for doing a subject, but to go back to the philosophical principle, what is, what is the true? And this is, she really says, this is why I think it's important to keep math on the program, right? I think she was contemplating ditching it, but said, no, this is the only lesson that gives us this idea. And so it is its place in the curriculum.

So it teaches us truth. It helps a child develop reason, logic through working out math problems, but also it helps him in general develop his intellectual powers. She said, let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be excellent.

As obvious as the sprouting of seedlings in the spring. So just this little daily, there are not long lessons. I mean, even in the upper years, math is kept to a 30-minute lesson maximum, right? Even in high school. These small lessons, but every day that we make constant progress because we're building on our ability, continuing to go on from there.

You know, it just dawned on me that when people ask what book or what level of math my children are at, I should just say they are making excellent progress in reasoning, insight, readiness, accuracy, and intellectual truth. See what they say? Oh, jeez.

Math should be our favorite subject. I think these are all values and virtues that we do want our children to have. I mean, parents are always asking me about character training. Well, here you go. So, okay. So, Nicole, you just alluded to some very specific things about Mason's ideas about math. Let's talk about those. Like the tips that she gave us as teachers or the criteria, rather, that she had for

So, Mom, do you want to tell us a little more about what she meant about wrong is wrong and not correcting? Because I think that is, like Nicole said, this shocking idea to most of us. Yeah. Yeah. Well, right. Wrong can never be made right, she said. Some of the things she says about teachers, we've talked a lot about the role of a teacher throughout the podcast. So when we get to the subject of math,

We have to do a little bit of a double take. She says, there's no one subject in which a good teacher affects more as there is none in which slovenly teaching has more mischievous results. And you kind of cringe and think, yikes. That means I have to be good at math to teach it. But I honestly know many mothers who are excellent in math and even math.

do it for, use math as the main thing in their career, who thought that would be their easy subject to teach. And they soon found out that they were not good math teachers. So I just want to encourage you, if you're not strong in math, you can be a good math teacher. I can honestly tell you that my last three children,

have said that math is their favorite subject. If they're asked by someone, what is their favorite? And that is just hilarious to me because I grew up hating it and I did well in it, but I never felt confident in math. But I think, mom, you years ago, like maybe five or six now, took Mason's

I did. Admonition here to teach her to heart and just said, you know what? I am going to become a good math teacher and start educating myself. And I think that made a huge difference. Well, it was the beauty thing. She talked about how math was beautiful. And I thought, I have never thought of math as beautiful. I did hear a lecture once and I thought the person was nuts because they were just enraptured about math. And I just couldn't get there with them. But yeah.

I do think this is, she said, math is a subject that depends on the teacher rather than on the textbook. And we do know that Mason used living books for most of her curriculum. There is no living book for math because math is,

is not a language that we speak or in words. It is a symbolic language. It is its own language, certainly, but not in the sense that we think of living books. And so she said it was dependent on the teacher, not the book. And I think that we're all, as Nicole pointed out, looking for the right book to substitute for us books.

And so I have come to the conclusion that this is one subject that you just may need to study at least enough to keep ahead of your children. And moms always talk about pre-reading history and geography books and things like that, which we're not talking about that today. But I am saying that I think in math, you definitely have to be a little bit on your toes and seeing what's coming up next.

And we have to put our own fears and insecurities aside and just get brave about approaching it if that is intimidating subject for us. So the teacher, she said, was responsible for giving the child the inspiring idea of

And I honestly don't think that growing up I ever thought about the ideas of math. But she said there is a captain idea in every math function and problem. And that quickens a child's imagination that a wise teacher will see that a child gets it right.

I'm still trying to figure that out. I don't have any quick solutions for any of you out there, so don't write me and say, well, how do you do that? I have had a few insights, but I do know that she didn't believe in the output showing us what our child could do. The first thing we do have to know as teachers is that our child...

This is one subject where maximum attention and concentration is absolutely necessary. So this is, like Emily said, why the lessons have to be short. And I think that if you're using a textbook, you have to realize that you may not be able to finish that lesson, that we need to use the time, not the pages or the number of problems.

It is an effort-filled subject for most children, and so they need a break. So don't continue on and on with a math lesson because they have only done three problems. If they've done them well and they've gotten them right and they're still smiling, then

I would say that's a successful math lesson. Would you, Nicole? Absolutely. She said it shouldn't be taught at the expense of other subjects. And our culture likes it, as Emily said, because it can be tested and we can get the right answers. But I think we can err in two directions. We can find ourselves as with our children doing math that we're driving and pushing and we're moving them too fast. Right.

because maybe it's halfway through the year and we're only on lesson four. That's a little bit extreme perhaps, but or we see that other children are skipping along next door in that same book and they're almost done or something like that. And we start to push our little children because we feel like they're not moving fast enough. Math is definitely a subject where we have to let children move at their pace because there's absolutely no good thing that happens when we

Move a child along who hasn't grasped the truth of the previous thing because math is an orderly subject. And so it needs to be built on incrementally. You can't go on to long division if you still don't understand how to borrow numbers when you're subtracting. I know. I just we hear about, well, my kid takes like an hour to complete their math lesson. It's like, well, this is exactly why Charlotte Mason said we need it.

hone the habit of attention and use short lessons because they're not get, they're not understanding their math. Nothing productive. It's not. Yeah. So if they don't understand a concept, you know, spending 50 more minutes. So this is where you as a teacher can help your child get it right, because it is your job to help them understand the concept. And she did believe, you know, their reason wants to know why before it works hard on solving something. Yeah.

Besides pushing a child too fast, though, the other extreme we can go to is being a crutch for our child. It is difficult and they do sometimes have tears or struggle. And so we as moms also have a tendency to get frustrated because we know that all they have to do is subtract here. Why do they keep adding these numbers or whatever? And so we can tend to get in and say, here, let me just show you how to do this.

And I just went to a conference where I think it was Carrie Forney had us all just in stitches because she kept saying about the need to let them work through a difficult math problem and how she has to hold herself back from saying, okay, here, why don't you just

Never mind. Never mind. You did it. Okay. Never mind. So we do have to use a little bit of self-control and allow our child the time and space they need to work through a problem and figure it out. But they do need to understand that, yes, it is work. And I know one time I put it in terms for my boys that this was moving their mental muscles and, you know,

You know, they know all about bodybuilding and exercise and stuff. So when they realized that they had to make their mind work harder in certain things to make progress, that actually helped them a little bit. She said if a child does not know what rule to apply to a problem within his grasp, he has been ill-taught from the first, although he may produce slatefuls of right sums in multiplication or long division.

So it wasn't the quantity of work that's being done. I think a lot of people like certain math workbooks because their kids fill up pages of material and they have something to show for it. Or those disposable books, you know, we've ripped most of the pages out. So we must be understanding our math. But that has nothing to do with understanding math. I, you know, having taught six children math, have seen some of my children who were quicker at math and faster.

move to high levels of math

and up into college that had the least understanding of what they were actually doing. And sometimes my slower progressing students... Like me? Yeah. Sometimes my slower progressing students have actually solved math problems for someone who's sitting at the dining room table working on probability and statistics by making the simplest observations. I like when Liz caught me counting on my fingers the other day. What are you doing? Yeah.

I said, you don't really do your math that way. So it's not the amount of work that they're producing, but the understanding. And she said we have to secure them in exercising their reasoning powers and engage the child upon all problems within his comprehension from the first rather than upon a set number of sums. Anyway, back to the wrong is wrong and right is right. I think that as a teacher, we...

can't let our child think that this is mostly right either. It is an absolute thing. Mason said nearly right is a judgment inadmissible in arithmetic. And because, you know, it's going to gray up their black and white ideas of absolute truth, right? But what about, you know, so we mark out a problem, we say it's wrong and they aren't allowed to fix it. Well, how do we secure this understanding that you're talking about, mom?

Well, I do think that this process of narrating in math helps because sometimes we can have a child tell us what they do understand about the problem. And maybe they, I mean, a lot of times when something's marked out, my child will say right away, oh my goodness, I forgot about those tens over there or whatever. And they know right away what they did wrong. They don't need to rework that problem. We just give them other problems within math.

that same scope to ensure that they really do have understanding. But when they narrate to us and they tell us what they've learned or show us why 27 years

in three parts equals nine equal piles or whatever, they can show us what their manipulatives and we do see that they're understanding and we can give them further problems that are even a little maybe more complex or... But we can also, as moms, look at the problem that they're getting wrong and if we notice there's something particular happening there, we can give them a more simplified problem that works in that area or, you know, try... Instead of them going back and...

It's sort of messy for them to go back and try to identify their own error. Right. And we're not really asking them to do that. We just want them to learn the principle we're teaching them. And so if we need to go back and hone in on that one area, then we can give them a new problem and let them be successful at it and get it right.

You know, and I think she does have some principles. And we know Mason did use a math curriculum. It wasn't like all of her teachers were, you know, PhDs in mathematics and just devised their own. Whipping up lessons out of the air. Yes, you know, they did follow a math. But she did give us some principles that we follow or should keep in mind as we choose our curriculum. So what are those principles we should look for and keep in mind when we are going about the task of choosing curriculum?

Nicole, how many different curriculums have you examined in the last two years? Well, I've looked at quite a few myself. Well, I think one of the main principles that she has is that they get an understanding of math, that it cannot be like apparently I was taught math where it's just follow the rules and get it right. She wanted them to understand math.

why these two parts made a whole and what was happening when different processes were taking place. Is that what we call concept-based math versus mastery, right? Yeah. And they need both, but does the program that we're looking at allow the child to see, to touch, to understand things in the concrete before they move to the more abstract? Or does it jump straight to

working numbers on a piece of paper. Yeah. She over, I mean, she uses manipulatives and even in upper levels, if it's appropriate, you know, if you're doing a new concept to use something concrete before you move to numbers, right? Right. And is the curriculum you're looking at, does it look like filling out a lot of busy work? Is it just drill? Yes. And production of numbers of problems. Does that seem to be the

the main thing that curriculum has to offer. Or like Nicole said, is it based on understanding some principles? And another thing closely related to that is the way the math lessons are structured in this particular curriculum. Is it going to allow my child to narrate as he goes along? So that is the book going to help him with expressing his judgment and his reason about things?

Does it include story problems or word problems where the child has to work things out in his own mind, even if he takes to paper and pencil to figure things out? Does it allow him to mentally solve math problems? Well, and we've been talking about there being a right and wrong, but in the early years, estimation is really important, which doesn't necessarily have a right and wrong. And so maybe that's why it gets overlooked a lot.

But it is an important concept for the kids to practice. Because they're developing their understanding of their application, their idea of number, right? And is it allowing the child to self-pace? I would say it would be another idea to consider in picking out a program. Yeah, over and over Mason stresses that we need to give children problems which are within their compass, that they are able to maybe

Maybe it's going to stretch them a little bit, but they are, you know, if we explain this as they are understanding, they can work these problems. Right. So is there enough practice so they are secure before moving on or does it very rapidly progress? Or are we trying to put our child in a level that's ahead of where they really are ability wise just because it's their grade?

Right. So the main principles for Mason are, you know, concrete before abstract, moving from the general to the specific, from the simple to the complex. Mm hmm.

With lots of use of manipulatives and concrete objects. And imagery, too, progressing to picturing things in their mind as well. And also mental math, being able to do story problems aloud that they're thinking in their mind without working on paper. So just as some general rules of thumb, how long and frequent lessons were. Math was a daily lesson in all levels of study, starting in Form 1.

First grade, all of form one, first, second and third grade did only 20 minutes of math a day. And they worked on number and sums is what she called math then. So they were they developed their idea of number, like the properties of each number, and then learn to work with that.

more and more in what we would consider arithmetic, basically. So the first graders, she emphasized rapid mental work and also using a lot of manipulatives to understand their number. And then in second and third grade, they built tables like they had multiplication and division tables that they would construct out of objects as they were learning those. And they were actually

were exercised in every single lesson for five minutes on their table. So there was like repetitive drill, I guess, quizzing or what would we call that? Just making sure they keep it. Yeah, like knowing their math facts. They would work on that. And then another important concept in elementary, lower elementary math is working with money.

In Form 2, 4th through 6th grades, they increased their time to 30 minutes and still did daily lessons in arithmetic for all of those three grades. But when they got to upper Form 2 in 5th and 6th grade, they started practical geometry. And that was more hands-on, moving into dealing with some geometric concepts. And we'll talk more about that later in a subsequent podcast.

And also in Form 2, she started assigning a leisure time reading. So we talked about not having living math books as their math lessons, but she did assign them all of her students beginning in Form 2 that they would read number stories of long ago by Smith. And I'll link to that in the show notes as well.

So in Form 3, which is junior high, middle school, 7th and 8th grades, they again just have daily lessons for 30 minutes, and they continue their arithmetic, and they read the number stories from long ago, but they do start geometry and algebra. And Mason thought that those two subjects were sister subjects and should be

taught consecutively with one another that the understanding of algebra helped enforce their understanding of geometry and vice versa. And then for Forms 4 through 6 high school, they still only had 30 minutes daily and continued out

arithmetic, algebra, and geometry. So that's all Mason students did. We, of course, have more higher level math, and you can make a decision about that, but that's what she did in her schools. I've also read where Mason talked about a child connecting with math better by reading about some of the famous mathematicians

through the ages and the way they came to some of their discoveries in the field of math. And I think of how Emily was so profoundly impacted in her 20s by reading Strings, Straight Edge, and Shadow. If I had ever been...

told what a square number really meant and had something to do with an actual square shape, I think I would have gone much further in my understanding of what I was working out. And she did do pre-calculus too. And I know nothing. I don't even know what that subject entails. I don't know how I made it through that class.

So that was just putting in the time and getting the A and not really having a conception of what we're spending our time here doing, which is what we all would like to avoid for our children. Yeah. Well, let's wrap up today. Next week.

We'll be back discussing math for the elementary years. Thanks for joining us today on the podcast. We would love it if you could leave us a review or a rating in iTunes. That helps our podcast out immensely. And we appreciate all of you who have taken the time to go to the trouble of figuring out how to do that. Thank you so much. If you have a question,

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