The Beauty of Math

When we think about math, we often think about how we use it. And when our kids say, “What about math? Why am I learning math?” we point to the reason behind it and the logic behind it and, as Charlotte called it, the truth behind mathematics. Let me share with you a little of what she said in Ourselves, Book I. She says, “Never are the operations of reason more delightful and more perfect than in mathematics.”

It’s easy for us to focus on that. Yes, it’s absolute truth. It’s going to stay that way. But if you look in Volume 6, A Philosophy of Education, she also talks about the beauty of mathematics. I don’t usually point out things around me as, “That’s the beauty of mathematics.” In fact, I’m not sure I know how to do that. So let’s talk to a couple of experts on that today. Joining me today are my friends and the authors of the Practical Geometry series. We have Tabitha Wirges and Dr. Julie Ryle with us.

Sonya: I’m so glad you are joining us.

Tabitha: Yes, thanks for having us.

Sonya: We love the Practical Geometry series. It is helpful for so many people. What we want to talk about today is math around us. We often think of it in terms of the practicality of it, the truth of it: that it will never change. When I’m baking, I can count on the fact that a half cup of flour is the half cup of flour or whatever it is. You can tell I don’t bake a lot. What uses only a half cup of flour? But you get the idea.

Tabitha: Or adding on your fingers. There are five here. There are five; there are always five.

Sonya: There you go. Making sure I have them all. So, we often think of it just in those terms. But Charlotte also talked about the beauty of mathematics. Whenever I think of that, I think, well, you have to have a mind like you, the PhD in math, in order to think this is a beautiful thing. But you were telling me earlier that there’s beauty that any of us can recognize in math around us. So let’s talk about that a little bit. All right?

Tabitha: Well, I want to start off by saying the beauty in math around us is not just recognizing it, but knowing that it exists around us. We like to put it in our math lessons and just leave it there. We’ve got to add our numbers up over here, or learn our multiplication tables, or learn how to construct a triangle. We have our little place for math in our days—only during our school days.

Sonya: Or only when I see numerals.

Tabitha: Or only when I see numerals, exactly, yes. In the case of baking, you’re like, “Oh, right, it’s here,” but you leave it there and don’t do much more with it. But math exists all around us. What we study in our books during our math lessons is something that is always happening in the world around us. It’s just a part of the world that God created. That’s the beauty and the absolute truth that Charlotte’s talking about when she says that it’s beautiful. The absolute truth of it is beautiful, but also that it’s just here, it’s everywhere around us.

Sonya: So we really just need to have eyes to see it, kind of like nature study.

Julie: Exactly, and that’s a great place to even begin to see math—in nature. Our Creator has woven this into all of His creation around us, and it’s so cool to be able to see it. That’s one thing that we love, and that we’ve started doing ourselves. We are starting to put numbers in our nature journals. We look for measurements; we look for patterns; we look for ways to include numbers and encourage our students to do that.

Tabitha: There are so many ways that you can note it in your nature journal. You can count the number of times that you heard a bird calling when you were listening on your back porch. You could see how much rainfall you got when you measured the amount of rainfall that just happened, and record that in your nature journal. That was math when you were measuring there. If you wanted, you could take a count of how a plant was growing from your garden.

Sonya: Or how many more weeds were in the garden today than yesterday.

Julie: That would be difficult to count. 

Tabitha: That might grow exponentially, rather fast. 

Julie: But it could be keeping a frequency chart, say, of the bird calls you heard. If you know a few bird calls, you can say, “Well, let me write down the birds that I’m hearing right now.”

Sonya: Red-bellied woodpecker. I heard one a lot today.

Julie: Or chickadees. Just use tick-marks to count them, and you’ve created a frequency chart in your nature journal. And one thing this does is that it takes math out of that bubble of only during our 30-minute math lesson time and takes it out into our regular world; this is where it is. And it reduces the fear of math, because a lot of us do have a math phobia. A lot of homeschool moms have told me over the years that they suffer from math phobia.

Sonya: Which is not our own fault.

Julie: Not at all.

Sonya: It probably has a lot to do with the way we were taught—not using Charlotte’s methods.

Tabitha: It could be even how a teacher came at it, even though she, yes, she wasn’t using Charlotte’s methods, so you wouldn’t have seen as much of the beauty as you explored it. You might have had a teacher who said things like “This can be hard…” or might have even said, “Let’s do our math lesson…” and wasn’t really excited about it either.

Sonya: You picked up the atmosphere that that teacher created. That’s a good point.

Julie: Have you ever said, “Oh, I’m just not a math person”?

Sonya: Oh, yes. A lot of people have said that.

Julie: We don’t require ourselves, or our students, to be an artist before we let them practice art, appreciate looking at art, appreciate doing art, playing with those things. It’s the same thing. We don’t have to consider ourselves mathematicians to be able to enjoy math.

Sonya: That is so true. Charlotte even wrote about that. She said something about how you don’t have to be a musician to be able to appreciate and listen to the composers, just like you don’t have to be an actor to appreciate Shakespeare.

Julie: Absolutely. That’s exactly it.

Tabitha: So we’re noticing it, and we like to add in some specifics when it comes to numbers to our math journals. So we make frequency charts, and we graph what’s happening, and we write down the numbers. But in your nature journal, you might just put down that you heard the chickadee, but while you were sitting there listening, you might have noticed that you heard it so many times, or you heard it a lot this morning and not so much yesterday morning. Realizing the difference of a lot and not so much was you picking up on the math that was happening around you. There were numbers that were happening there.

Sonya: So even with this idea of adding it to the nature notebook, we still let each person choose his or her own expression.

Julie: Absolutely. It’s going to be how it speaks to you. I’ll take it even a step further. You were noticing differences in the number of birds you were hearing, but even just categorizing the bird calls that you’re hearing is a math concept.  Sorting and categorizing, that happens all around us every day. It’s taking a moment to realize, “Oh, that’s math. It’s not scary.” This is something that our math lesson is actually going to help us with, to be able to do that in life.

Tabitha: When you want to organize your pantry because it’s just gone to complete awry.

Sonya: Not that we speak from experience, but yeah, if that might happen to someone.

Tabitha: Anytime you’re taking something that’s chaotic and you’re bringing order to it, that’s math. It doesn’t always have to have a numerical value to it. The patterns that you might recognize when you are doing sewing techniques or quilting and admiring the quilting, that’s math right there, noticing those patterns. There’s so much to math that’s patterns and you get to pick up more on the patterns in your lesson, like Julie said, but you can see the beauty of it everywhere else in handicrafts and in your science journals.

Sonya: What are some other places that we can see math around us?

Julie: One is our history. We have a lot of history charts in a Charlotte Mason education. So we have a century chart, and then we might have a wall timeline, and then we have our Book of Centuries. We have all these different ways that we are often using our base-10 number system to organize our history. And so we have this 10 by 10 grid, and we can see what things are happening, and then we expand it out to centuries and into millennia, and we can see all these things happening, and that’s just a great way for us to think about the base-10 number system and how you can look at a decade and a century and a millennium. It’s just really cool.

Tabitha: Putting it in order like that. Even just looking at time; when you talk about history, you’re looking at what the Babylonians did with the number 60; they had a base-60 number system and we still use that today. And so you get to see that when you’re looking at time and scheduling your days; you can recognize the math that’s happening there.

Sonya: So do you point this out to your child and say, “Look, here’s the beauty of math?”

Tabitha: No, I wouldn’t. That takes away from the beauty of math. If you’re sitting there and you notice this in your history chart, and you’re doing your book of centuries and you’re like, “Wow, this happened at the same time that Columbus was exploring America. That’s amazing.” And then if I were to sit there and go, “Oh, yeah, and now we’re going to count that and see exactly how many it is,” and “Do you know that’s just like the math that we were doing in our lesson 30 minutes ago?” 

Sonya: Oh, that just sucked all of the joy of discovery out.

Julie: But you’ll find that some of your students might have a natural love for exploring that kind of stuff on their own. So here’s a story. I had my kids out this past year. We had an emergence of two different broods of cicadas. We had the 13-year cicadas and the 17-year cicadas. And you can find articles and read about how those prime numbers—this is just another testament to our Creator—how He created these particular insects to come out this prime number of years so that they don’t end up coming out at the same time very often, so that they’re not interbreeding. Then I started talking to my kids. We’re noticing so many cicadas. 

Tabitha: It’s just the past year when this was happening.

Julie: Yeah, it was this year, we were talking about the dual brood, which is unusual for this year, and how there are 13-year cicadas and 17-year. They just naturally wondered, “I wonder when was the last time both broods emerged?” Then we just started playfully counting back by 13s, trying to find something that would work. We ended up realizing that it was 1803 the last time this happened. That was what we were studying in our history period at the time. And so they were like, “Oh, the last time this happened, so-and-so was alive.” Or, “This was going on.” And I didn’t stop them and say, “You know what? This is our mental math for today. And isn’t it beautiful?” They just had fun playing with it.

Tabitha: There’s also the fact that you’re a math person. That spills over into your home a lot, and so your kids would want to explore that, right? I want to point out, for moms where maybe this is not something that they normally do, but if they start to just recognize when they’re talking to their kids about the 13-year and 17-year cicadas, and they realize that this doesn’t happen very often, just recognizing for themselves that they are finding the appreciation in numbers right then. That is math, and they can just know it for themselves. They don’t have to say, “Well, let’s… ” They don’t have to know that they should count that back, or their kids may not want to count that back and that’s okay. That’s not required to enjoy the beauty of that moment of recognizing the prime numbers that were there, the fact that this doesn’t happen very often, and the underlying beauty of math that was there. As she continues to recognize this over and over throughout her day, and days, and life, and continues on, it’ll start to show and she’ll bubble out from that, and it’ll carry on to the rest of her kids, kind of like mother culture, and how we take in and feed ourselves into all the joys of what we’re trying to teach to our kids, and that comes out so that they enjoy it as well.

Sonya: I love how you are explaining that, because the more that Mom can recognize those things, and that’s just kind of a little gift for herself. It’s not a brick that we’re building towers or anything like that, but it’s one more piece that will help her; it’s one more rung on the ladder to help her climb out of the math-phobia pit.

Tabitha: There we go.

Julie: Yes, I love that.

Sonya: So what are some other ways that you have seen beauty in math around you? 

Tabitha: Picture study. You might not notice math in your picture studies and when you’re exploring art, but there’s so much math when it comes to art. Whenever you’re looking at a picture and you notice the rule of thirds, that is “the golden ratio,” that’s Fibonacci’s sequence, there’s so much math to that. That is all through nature, and you’ll notice the Fibonacci sequence once you dive into that. It’s even recognizing when you see the rule of thirds in your pictures and that they followed the rule of thirds or didn’t follow the rule of thirds; I like this or don’t like this, this is pleasing to my eye because it follows the rule of thirds, you just found math beautiful. Or colors that mix together, you noticed the color scheme and that they go together well, and that’s pleasing to your eye. That color scheme, having that mixed together well, that’s math right there; the way that those interlock when we talk about primary and secondary numbers, that’s math. We could get into the scale of the pictures, that seems a little bit more obvious, right? That’s math. If you’re replicating for your picture study, and you’re wanting to paint it yourself, or try it out, and you’re trying to mix your colors in order to get the right color that you’re seeing in front of you, your color mixing that you’re doing, you’ve got a ratio of your paints. And that’s math. You’re not thinking about the numbers you’re using for it, but you are using numbers, you have numbers that are going on there, there is a ratio happening. Even if you don’t know what the ratio is, it’s there.

Julie: And it could be whatever it is that a mom or a student may already be into. So if it’s music, then you can think about the intervals between the notes, and there’s so much math in music and you’ll have to forgive me, I’m not a musician. I have not had music lessons, but I know that it exists, that it’s there, and you can study that.

Tabitha: That one gets really fun because you can look at the sound graphs for it sometimes, for particular notes. With frequency charts, you can actually see the graphs for that and that’s very mathematical. So if you get a chance to do that, that’s beautiful. But it’s also just listening to the notes; if a frequency is off, it’s not going to sound pleasing to your ear. But when it’s in tune, and it comes in tune with another instrument or another note; or maybe you’re singing and you start harmonizing together, because those frequencies are walking in together, that’s what makes it sound beautiful. And so now you just said that math sounded beautiful.

Sonya: Yeah, and even a circle of fifths that you have in different keys. I never thought of that before, but yeah, that’s math.

Tabitha: Or when you’re reading music, right? So it can be in 4/4, and so you have your quarter notes and your whole notes and your half notes and there are fractions right there.

Julie: And then you can take it to upper level and the distances between the frets on a guitar, or on another string instrument, that is logarithmic. And so we’re getting into that in Algebra II / Trigonometry and we can use that as an example. But it’s something that a musician would have already noticed about his instrument; I have this irregular spacing between my frets, but there is a very mathematical reason for that. There’s a reason that it goes that way.

Sonya: In my grand piano, the wires are different thicknesses, different lengths. Sometimes there’s more than one to a hammer. I never thought of that as being math.

Tabitha: The ratio, correct me if I’m wrong, but I believe the ratios for your piano note keys were actually created by Pythagoras. He’s known for doing that, anyway. I don’t know. He’s known for doing quite a few things. I think it might have been one of his followers.

Julie: Reading about the history of math—that may sound like a very dry subject to read as mother culture—but there are some great books out there that make it very living and very entertaining. So to read some books on the history of math is a great way for a mom to get that mother culture into her and you’ll start noticing more things around you.

Sonya: Okay, you’ll have to give me some of those titles. We’ll share a list of living math history books for everyone.

Julie: It sounds like an oxymoron.

Sonya: No; well you gave me that great book on the history of cancer and I never thought that could be living, but oh, it was so living so now I’m a believer. 

Tabitha: The math ones are really fun because they will have beautiful pictures in there; they’ll have the sunflower in there, and you’ll have the swirl drawn that you’ll naturally see in the sunflower, but they’ll draw it there and then they’ll show you why that’s Fibonacci and the golden ratio right there in a sunflower.

Sonya: So much beauty around us. Any others that you want to give for examples?

Tabitha: I did want to mention, of course science, you can notice the math that’s happening in science, but leading into that, math is the language of the sciences, and when you’re studying science, it’s exploratory, you’re trying it out, you’re seeing what’s happening. I want to give that feeling into the math subject as well, because it’s the language of the sciences. You’re using math to explore your science that you’re studying right then. That’s one way to consider the beauty of math that’s happening. Also, it’s realizing that if math is the language of the sciences, and science is how everything around us works, then there’s an underlying science-works. This is how the plant grows, this is how photosynthesis happens—down to the electrons and the molecules. The underlying base underneath that is your math. You needed the math in order to recognize the science that’s already happening. Math is already happening all around you. It’s part of this world. That’s how it works.

Julie: The last thing we want to do is put more of a burden on mom to be looking for these things and to be searching them out. That’s not what we’re saying at all. We’re just saying to recognize what’s already there. It’s there, all around us, all the time. And if you could open your eyes to that and see it, I do think that it will reduce the fear of math, because you’ll see it’s here, and it’s okay to play with it and just enjoy it.

Sonya: That’s wonderful. So often we think, well, I need to learn this so I can pass it on to my kids, and we feel pressure; instead of just, I want to explore that for myself. I can continue to grow as a person and as a parent as well.

Tabitha: And that’s where it comes. To bring math into your mother culture, it doesn’t have to be that you get better at trig. It can just be that you recognize that the flower petals that you thought were pretty had three petals, and you realized that was math that you just thought was pretty. So it can be that simple. Don’t worry. Math mother culture.

Sonya: Yes, that’s what it is, that’s what it is. Thanks so much for helping us, for opening our eyes, encouraging us to open our eyes, to the beauty of math around us, not just the truth, which is all around us, but the beauty of it as well. We appreciate that.

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