Making Math a Delight with Denise Gaskins

Pam Barnhill [00:00:01]:
Are you ready for homeschooling to feel joyful again? Do you long for support as you learn alongside your kids? Welcome to Homeschool Better Together, a podcast about building a homeschool experience that works for your family. I’m Pam Barnhill, and it’s time to step out of the overwhelm and into the wonder. Let’s do this. Denise Gaskins is a veteran homeschool mom of five who has a passion to help parents and children see the beauty and wonder of math. She is the author of a dozen playful math books like Let’s Play Math, How Families Can Learn Together and Enjoy It. And her math you can play series covers various topics across the curriculum. Denise’s projects like tabletop math games and math journaling books make creative math easy. She creates open and go activities for students from preschool to high school, and you can find her online at denisegaskins.com.

Pam Barnhill [00:01:05]:
Hi, Denise, and welcome back to the podcast. It’s good to have you back again.

Denise Gaskins [00:01:10]:
Hi, Pam. I’m thrilled to be here.

Pam Barnhill [00:01:13]:
Well, I love to talk to you about math. And the reason I love to talk to you about math is because I always learn something new, something I never really had thought much about before or never really considered before. And so you you keep me on my toes. And I think you’re gonna do that today as well because we’re talking about traditional math instruction and what might be wrong with it. So go ahead and and tell me. Explain that a little bit. What’s wrong with traditional math?

Denise Gaskins [00:01:44]:
Well, you know, our own school experience led most of us to think that what math is all about is memorizing and following specific procedures so that we can get the right answers. But, really, that kind of math is obsolete in our modern world. We have too many tools that can get right answers. The math that matters today is our ability to reason about numbers, shapes, and patterns, and then our ability to use the things we know to figure out something we don’t know. And we adults, you know, we have good intentions. We we want to prepare our children for anything they might face in the future. So we think that we can teach them these rules, these algorithms, these step by step procedures that will let them solve any math problem. And then they just have to listen to us and do what we say, and they’ll be good.

Denise Gaskins [00:02:43]:
The trouble is there’s just too many rules. We pile do this on top of do that until the whole thing is ready to crumble like a tower of Jenga blocks. And the thing is that kids really don’t need to know in advance how to solve every problem they’re gonna face in life. They don’t have to know in advance how to solve 3,762 divided by 58 without a calculator. What they do need is they need to be able to make sense of 36 divided by six and then be able to use that knowledge to reason about related problems like 36 divided by 12 or 3,600 divided by 60, which is a good estimate for that first problem that we said we would use a calculator on. Real mathematical thinking says, I know this thing is true, and I know that thing is true. And when I put them together, I can figure out how to approach this harder problem. For instance, I know there are six sixes in 36, and 12 is twice as big as six, so there must only be three twelves in 36.

Denise Gaskins [00:04:02]:
And that that reasoning, that making sense and figuring things out, that’s the kind of math our kids need to learn.

Pam Barnhill [00:04:10]:
Okay. So instead of spending our time teaching these kind of step by step processes and these step by step methods, we really should be spending our time thinking about reasoning. That’s the stuff that the kids need.

Denise Gaskins [00:04:26]:
The key in math lessons is always to have our kids thinking, and we’re interested in in what they’re thinking about, not in necessarily them following specific procedures. And they don’t have to do what the book is expecting. They don’t have to solve a problem the way we would solve it, but they need to solve a problem in a way that’s true, that makes sense, that they can justify, that they could say, I know this because what matters in math is the journey. It’s how we get to the answer. It’s not the answer itself because the calculator can give us an answer. It’s how we reason to that answer.

Pam Barnhill [00:05:07]:
Okay. So let me let me just ask you a question. If I have a kid who well, and I had a kid like this, I was all ready to teach him long division. Right? I was all ready to teach him the steps of long division because that was the only thing I knew how to do, was to teach him the steps. He kept solving the problem so quickly in his head that I could never show him the steps. Right?

Denise Gaskins [00:05:31]:
Uh-huh.

Pam Barnhill [00:05:32]:
Okay. So, like, I was like, I don’t know how to teach this kid long division because he keeps solving the problem. Now, in my defense, I knew that we were eventually gonna reach a problem that he couldn’t do in his head. Right? Mhmm. I mean, I didn’t think he was a savant or anything like that. Eventually, he was gonna need in my mind, he was gonna need that algorithm and I needed to be able to show him. But it was okay for him to be doing that. But should he have been able to tell me how he was doing that? Is that something that you would expect? Because he was, you know, sixth grade at that point.

Denise Gaskins [00:06:07]:
Yeah. He should be able to tell you how he’s thinking. And to put our thoughts in words is hard. That’s something that takes practice that kids need to work at learning on how how to explain their thinking. And sometimes their explanation is to just sort of wave their hands a little bit and expect you to read their minds. But learning to put our thoughts in words, it helps us to think more clearly. We’re forced to confront, what is it that I really mean here? So, yes, that’s a that’s a valuable skill for him to practice, not only for math, but for all things. Right.

Denise Gaskins [00:06:43]:
But but, yes, in math, he needs to be able to justify his answer. He needs to be able to say, this is my answer because whatever.

Pam Barnhill [00:06:52]:
Okay. So the way I kind of picture this kind of going in my head, if I had another little one here at home, is we would throw a bunch of blocks out on a table, and we would start talking about things like we’ve got 36 cookies, and we want, you know, our aunt Tiffany to have some cookies and uncle Mike to have some cookies and mom to have some cookies and dad to have some cookies, you know, and so on. How are we gonna make sure that we divide these cookies fairly? And then we would start playing with that.

Denise Gaskins [00:07:27]:
That’s a great way to start with younger kids. With with older kids, you know, we don’t really have to just completely change our math curriculum. That’s not what I’m saying. What we do have to consider is how we can use whatever curriculum we’ve decided we like to help our children learn to reason. So the trouble is most of our textbooks work by giving the students a method, a rule, some procedure to follow, and then giving them homework to practice on. But whenever we give the kids a rule to follow, we’re actually giving them permission not to think. All they have to do is follow instructions. But human minds only remember what we think about.

Denise Gaskins [00:08:10]:
So if you imagine all the times you’ve said, but we just did that last week, why can’t you remember? It’s because he wasn’t thinking deeply enough to get it stuck in his mind. The deeper we think, the stronger we learn. So one possible way to use our math curriculum to get to that deeper thinking is to skip the lesson part. Don’t give them the rule in advance. Let’s go to the problems and pull them out and say, you’re a creative kid. You’ve got a mind. What will you do with this problem? Listen to their ideas, see how they can apply what they already know, talk, you know, carry on a discussion, See what they can figure out based on the concepts they’ve learned before, and that will help build true fluency once we get them reasoning about the subject and not just following our instructions.

Pam Barnhill [00:09:04]:
Okay. So start with the problems as opposed to starting with the instruction and the examples. Let’s go back to little kids right now. And I know that there’s, this idea out there that you have abstract and you have concrete and little kids don’t necessarily do the abstract very well. So you’re gonna start with the concrete with manipulatives with the little kids with something like an addition problem as opposed to just showing them, like, the eight and the plus sign and the seven on the page. You’re gonna start with some manipulatives. Right? Not just throw that problem on the whiteboard.

Denise Gaskins [00:09:41]:
Well, you’re talking about different things here when you talk about concrete, abstract, and symbols. Right. And what what the little kids don’t understand as well are those symbols. K. They can do a lot of abstraction if if I mean, they have never seen a dinosaur, but they can tell you a lot about dinosaurs Or whatever they’re interested in, they can abstract. What they don’t really know is the language of the symbols, and so they forget what that plus sign means or what that minus sign means. So, yes, concrete things are useful, especially for very young children. But a lot of times, what we what helps is to let them think of a story in their head that like the one you made about sharing cookies.

Denise Gaskins [00:10:29]:
You don’t. For a very young child, you may need the blocks to count. But for an older child, you know, once they’re in school age, they can imagine that story in their head. A lot of times, they can think about, okay. How am I gonna split up these cookies? Cookies are something I know. They’re they’re abstract in that I don’t have them in my hand right now, but I know cookies. I can think about cookies. So we can replace this symbolic language with a story language, and kids can identify with the stories.

Pam Barnhill [00:11:03]:
Okay. Okay. That’s making sense. That’s making sense. So you’re saying that strong math skills do not come from these algorithms that we memorize, but instead from the reasoning that we practice as we’re just playing with the concepts of things like addition, subtraction, multiplication, division, and then perimeter area. I’m forgetting my math concepts. Estimating.

Denise Gaskins [00:11:34]:
Yeah. Yeah. The more that we use it, the more that we think about it, the more that we make sense of it, the more that our children are making sense and figuring things out, the more deeply they will learn. Just following the steps of an algorithm doesn’t really communicate the meaning. And there is a lot of meaning in those algorithms. Those algorithms were wonderful inventions when they when they were invented. They were the things that helped the Bob Cratchits of the world get through their days work at the county house. They they were the foundation of, you know, the economic system and the banking system and industry, and they helped early engineers figure things out.

Denise Gaskins [00:12:15]:
But the way they help is by making you not have to think. When you are working through an algorithm, you’re not really thinking about what those numbers mean. You’re just following steps. You’re you’re doing this rote procedure. You can get through a lot of calculations very quickly that way. In the days before calculators, that was wonderful. But it’s not what we need for education. For education, we need to get the kids thinking about what those numbers mean, how they relate to each other, how they interact, what happens when you put an eight and a seven together, whether it’s eight plus seven, eighteen plus seven, twenty seven plus eight.

Denise Gaskins [00:13:00]:
The same sort of thing happens each time. It’s like, oh, you get up to the next ten and five more. You know? Right. They need to build those relationships, make friends of the numbers in a sense, get to know them.

Pam Barnhill [00:13:12]:
So you’re not saying that we don’t need the algorithms anymore simply because we have a calculator. All throughout history, people learning math didn’t need the algorithms because they needed to learn to reason. The people who needed the algorithms were the people who needed to get to those answers quickly. And now we don’t need the algorithms because we have other ways to get to the answers quickly, but we still need to learn that mathematical reasoning that everybody’s needed to learn throughout history.

Denise Gaskins [00:13:42]:
That’s exactly right. The reasoning has always been the important thing. When we relied on the algorithms, we we didn’t have calculators. The algorithms took up so much time that we could kind of we almost forgot about what was the important thing. Now we have freedom. It’s sort of like now we’re not doing our laundry by hand. We’ve got a machine that does it. You know? Now we’re not doing these calculations by hand.

Denise Gaskins [00:14:09]:
We have machines that do it that frees us up to think about the reasoning, which is has always been the important part.

Pam Barnhill [00:14:17]:
Oh, interesting. Okay. Alright. So homeschool moms, you know, they worry just a little bit. Mhmm. I could just I and you’ve already addressed the number one question they’re gonna have is, oh my goodness. Do I have to completely throw out my curriculum? But they’re also gonna worry that their kids are going to miss something by doing things this way or that, like, someday the world’s gonna explode and there’s no longer gonna be a calculator. Or they’re gonna have to walk into some tests somewhere.

Pam Barnhill [00:14:49]:
And homeschool moms are gonna be judged because their kids don’t know these algorithms. So what do you what do you say to that mom?

Denise Gaskins [00:14:57]:
Well, let’s talk about what we could do when our kid gets stuck on something.

Pam Barnhill [00:15:01]:
Okay.

Denise Gaskins [00:15:02]:
Because what our kids really need to learn is how to handle it when you get stuck, when you run into something you don’t know, when you find a gap, or maybe it was even something that you did learn last year, but you’ve forgotten it. What can you do when you’re stuck? And as parents, the first thing that we can do when our kids get stuck on something is to celebrate because now we’ve got a learning opportunity. If our kids never get stuck in math, if they’re only doing math they know how to do, they really aren’t learning. You could give a sixth grader a first grade textbook, and he’d never get stuck. He might get bored to tears, but he wouldn’t get stuck. But in a sixth grade book, he may get stuck a lot, and that’s good. But the thing we have to do when our children get stuck is remember that the wrong thing is telling them how to solve the problem. That’s sort of like saying do it my way.

Denise Gaskins [00:16:00]:
We don’t want them doing it my way. We want them to think and reason and do it the way that makes sense to them. So we don’t wanna just give them a rule or procedure to follow. What we need is to teach them how to handle being stuck. And, you know, that’s a life skill that goes a lot farther than just math. What do you do when you don’t know what to do? And there’s a natural cycle of learning that I like to teach the kids that that will serve them all of their lives. I call this cycle notice, wonder, create. And that basically means notice, pay attention, examine the problem, look at what’s in front of you.

Denise Gaskins [00:16:44]:
You open your eyes and see what you have. So to notice, you might take turns with your child pointing out details about the number, the shape, the pattern, whatever the problem is that you’re stuck on. And it’s a good idea to write down this noticing because you don’t know in advance which detail is gonna lead to a new insight. And when we write down what our children say, that’s another way of validating that we think their thoughts are worth listening to. And then wonder, ask your children to think about the possibilities. How is this problem connected to other things that they’ve learned? What questions can they ask about the things they’ve noticed? And write these down. And then finally and it it’s not a three step process that goes back and forth. The noticing will make you wonder.

Denise Gaskins [00:17:39]:
The wondering will make you notice. As you work through these things, you begin to create something new based on the ideas you’ve noticed and wondered about. You may create a solution to the problem. You might be able to create an explanation of how to solve it or maybe a new problem, an idea of your own that gets kick started by what you were working on. Maybe just create a journal entry about the things you’ve wondered. As you talk about math with your kids this way, this it’s it’s sort of a natural way that our minds learn anything. You can apply this to literature. You can apply this to history.

Denise Gaskins [00:18:20]:
You can apply it to anything you’re studying, but paying attention, the wondering, and then from that, shaping a new understanding.

Pam Barnhill [00:18:30]:
Okay. Can you give me an example of what this would look like with maybe an a topic that we might teach our kids with math?

Denise Gaskins [00:18:39]:
Okay. I’ve actually got two here. Maybe we want older kids or younger kids first?

Pam Barnhill [00:18:43]:
Younger kids first, and then we’re gonna do older kids. So never fear. They’re both coming.

Denise Gaskins [00:18:48]:
Okay. So let’s think about younger kids. For many children, subtraction with borrowing or renaming. Oh, yeah. That’s like the gateway to thinking about math as nonsense, where the only thing that matters is just memorizing and following the teacher’s rules.

Pam Barnhill [00:19:04]:
Yes.

Denise Gaskins [00:19:05]:
So what if your kids get stuck on a multidigit calculation, let’s say, 431 minus 86, and you don’t wanna just teach them the procedure? How can you help them make sense of this multidigit problem? So we take turns noticing and wondering and writing things down. K. We begin with the problem 431 minus 86. And we might say, it often helps for the adult to start the noticing, give the children an idea. So I noticed the first number is 431. What do you notice?

Pam Barnhill [00:19:46]:
I noticed the other number is 86.

Denise Gaskins [00:19:49]:
Okay. I noticed subtracting 86 is gonna be hard. If we were only subtracting 31, that would be easier. That would get us down to 400. Subtracting 86 is a lot more.

Pam Barnhill [00:20:03]:
Yes. It is. Oh, I have to know this someday.

Denise Gaskins [00:20:06]:
Yeah. You’re done. Now it’s your turn. I mean, you don’t have to strictly take turns with the kids. But since since you’re wanting the kids to think, you’re wanting to draw out how they understand it, you don’t wanna do too much of the noticing yourself. You want to coax it out of them. So

Pam Barnhill [00:20:23]:
Well, it it would also be easier if we were only subtracting 50.

Denise Gaskins [00:20:30]:
Yeah. 50 is a nice friendly number. That’s a nice one to subtract. Or a hundred. A hundred is a nice number to subtract.

Pam Barnhill [00:20:36]:
Oh, a hundred is even easier.

Denise Gaskins [00:20:38]:
Yeah. In fact, I noticed that if we subtracted 31, that would get us down to 400, but that’s too little. We need to subtract more than that. If we subtracted a hundred, that would get us down to 331, but we took too much. So I’m gonna predict the answer is in between those. More than three thirty one, but less than 400.

Pam Barnhill [00:21:01]:
Okay.

Denise Gaskins [00:21:02]:
Now wonder how much less. And you could continue with this. You go on noticing things, wondering things, speculating, creating the ideas until something that your child sees makes enough sense to them that they can jump to the answer. Usually, it will it takes a little bit longer than using the algorithm because you’re not just teaching them to follow your instructions. You’re teaching them the process of how do I think when I don’t really know what to do? How do I come up with something that is new to me? And that’s a bigger idea than just following instructions.

Pam Barnhill [00:21:45]:
So this would never be because my brain over here, I wrote the problem down and I’m looking at it. And my brain over here is going things like, you know, take from one and add to another or make 10 or all of these strategies that I never remember learning in school, but I do remember teaching my kids. But that’s not even where we’re going with this, is it?

Denise Gaskins [00:22:06]:
Well, I mean, it could eventually go to that if that’s what the child likes to do. Some kids really have sort of an accounting brain, and they like those algorithms. But we wanna make sure that there’s understanding behind. I mean, the algorithms are based on how place value numbers work. The fact that that three means a different thing when it’s in the tens place as it would if it’s in the hundreds place or the ones place. And it has there are things to do with how numbers relate. For instance, one way a child might start this problem, but this is what my son would do. Mhmm.

Denise Gaskins [00:22:44]:
He would start by saying, oh, 86 minus 31. That’s 55. And you go, what? No. No. We’re taking away the 86. What is this? Well, what he’s doing is he’s saying, oh, if I take away that 31, I’m gonna have 55 left I still need to remove. So he’s thinking in a way that for my brain trained on the algorithm is, like, totally backwards. Like, you can’t do that.

Denise Gaskins [00:23:11]:
That’s not what it is. We’re we’re not subtracting the 31. But he’s saying, oh, no. I know I have to get that whole amount taken away eventually. I’m gonna take off the easy chunk and see how much more I had to go. So that’s the sort of really thinking about what the numbers are actually doing. That’s what we’re after.

Pam Barnhill [00:23:33]:
Okay. Give me the older kid example.

Denise Gaskins [00:23:38]:
Okay. We’ll see how far we can get on the older kid example. It’s a lot harder to do over a podcast because you don’t have the pencil. You don’t have the pencil.

Pam Barnhill [00:23:47]:
It’s okay. You still have been stomped on the younger kid one, so it’s fine.

Denise Gaskins [00:23:51]:
Oh, no. Let’s let’s go for it. Let’s do something really tough here. Let’s divide fractions. Okay. How about if your kid gets stuck on, like, one and a half divided by three eighths?

Pam Barnhill [00:24:09]:
I can’t even see how to work that with the algorithm.

Denise Gaskins [00:24:12]:
Well, you see, that’s a that’s a thing. We’re trying to get it so that they don’t need to remember the algorithm. We need them to be able to think. Because, you know, when you’re taking a test, your mind’s gonna go blank. Sooner or later, you’re gonna be in a test and your mind’s gonna go blank. It’s like, I have no brain here. I don’t know what I’m doing.

Pam Barnhill [00:24:29]:
Right. Right. Right.

Denise Gaskins [00:24:30]:
So we don’t wanna just give them the rule. If we give them the rule, they’ll be able to do their homework. They may be able to remember it to the chapter test. They might even remember it to the state test at the end of the year, but they’re not gonna really understand it. Instead, let’s notice whatever we can about one and a half. Let’s notice what we can think of about three eights. Okay. And let’s let’s ponder a bit what division even means.

Denise Gaskins [00:25:00]:
How can we understand the idea of this amount divided by that amount?

Pam Barnhill [00:25:05]:
Right.

Denise Gaskins [00:25:06]:
So I might start with I noticed that one and a half is more than one, but it’s less than two. Right. I I wonder, will the answer be more than one or less than one? What might you notice?

Pam Barnhill [00:25:23]:
Well, one and a half is three halves.

Denise Gaskins [00:25:26]:
Mhmm. It’s true. One and a half is three halves. And sometimes it’s a lot easier to work with fractions when we get them out of that mixed number and into an improper fraction.

Pam Barnhill [00:25:40]:
I also know that three eighths is something I would never fool with.

Denise Gaskins [00:25:44]:
This is one you’d use the calculator on,

Pam Barnhill [00:25:46]:
Well, it’s just like when you think about it, I mean okay. So 95% of the time I’m dealing with fractions, it’s in a recipe and like you never do three eighths of anything

Denise Gaskins [00:25:58]:
in your recipe.

Pam Barnhill [00:26:00]:
No. But, you know,

Denise Gaskins [00:26:01]:
I noticed that three eighths is almost half and halves are comfortable fractions.

Pam Barnhill [00:26:07]:
Yes. They are.

Denise Gaskins [00:26:08]:
Okay. Three eighths a little bit less than half because four eighths would be a half. Correct. And then I noticed that you said one and a half is three.

Pam Barnhill [00:26:20]:
Three halves. Right.

Denise Gaskins [00:26:22]:
So if three eighths is a little bit less than a half, we’re going to have a bit more than three of them in our one and a half. Yes. So this is thinking about what division means that that division means how many three eighths do we have in this total amount? How many of these chunks do we have in our one and a half? And it’s a funny thing because a lot of kids will think, oh, division, we’re going to have the answer is going to get smaller. But now we just said, oh, no, there’s going to be more than three here. Our answer is actually going to get bigger. So then we might begin to wonder when does division make answers smaller? When does division make answers bigger? There’s a lot of things that we can begin to think about the numbers as we notice. I wonder, could we maybe make up a story problem that would help us to understand what’s going on here? Remember I said earlier that one of the things that really helps us think is to get away from the symbols that our minds don’t naturally think in symbols and bring out stories. Our minds think in stories.

Denise Gaskins [00:27:29]:
We’re story processing computers or something. I don’t know. Story is our natural like, our brain’s natural language. So perhaps we could think, oh, baby, we have one and a half cups of raisins, and we would have split them into snack bags. And each snack bag holds three eighths cup. So how many bags would be could we get? And there’s a story that our children can begin to to use to figure out the problem. And you go back and forth like this. You keep talking about the problem until something sparks an insight that helps your children figure out what you’re doing.

Pam Barnhill [00:28:06]:
So do you ever come to the exact answer?

Denise Gaskins [00:28:10]:
Oh, yeah. You always you always wanna get an answer. I mean Okay. These these school math problems have answers. You know, we’re not dealing with the kind of the Kolak conjecture or some other math problem, you know, that doesn’t have an answer. We’re dealing with school math. It has an answer. Okay.

Denise Gaskins [00:28:28]:
It’s just that the answer is not really the important part. We wanna get to that answer, but it’s the path we take that matters.

Pam Barnhill [00:28:38]:
So what if I feel totally inadequate to help my child get to the answer without using the algorithm?

Denise Gaskins [00:28:48]:
Well, you know, the wonderful thing about using this notice wonder create cycle of learning is it works even when we adults don’t know what we’re doing. As we discuss the problem with our kids, we begin to build up our own understanding too. With this approach, you don’t get through as many problems in a single lesson. But by examining each problem in-depth and making sense of how it’s working, how the numbers and the shapes or the algebra patterns or whatever you’re studying relate to each other. You and your students are both learning more than you’d ever get from following the memorized procedure. And sometimes you may not be able to figure out your problem without resorting to the textbook rule, and that’s when you go back to the beginning of the lesson, the part that we skipped before and said, okay. What is the rule here? How do they tell us to solve it? And now the thing that you’ve created is a new question. Why does this rule work? Why does this rule do what we couldn’t figure out how to do? So the process of noticing and wondering helps you clarify exactly what you understand and what you don’t understand.

Denise Gaskins [00:30:10]:
And, usually, since math builds on itself, if you’ve learned this stuff up till now, you’ll be able to take that next step. You’ll be able to think, okay. I know how to take, you know, a box of three pounds of raisins and divide them into half pound bags. So I can figure out how to take one and a half cups of raisins and divide them into three eight cups bags. You know? I can use what I know to figure things out. But if we also clarify what we don’t know, and then we have a question, a specific question that we can take to the Internet, or we can ask a friend who knows math, or we could look it up on Wolfram Alpha. Or if we get really desperate, we can go to DeniseCaskins.com, find a contact form, and write a message saying, hey. What’s happening here? I’m stuck.

Denise Gaskins [00:31:02]:
And questions are great. When we find a question, we wanna celebrate that even when the question is just something like, why does this rule work? Because mathematicians know, when we find a new question to ask, that’s how our knowledge grows.

Pam Barnhill [00:31:17]:
So you said we didn’t have to throw out our curriculum, but it sounds like so that it the curriculum would still, for all intents and purposes, be our scope and sequence, but we would be doing far fewer problems by following this approach.

Denise Gaskins [00:31:33]:
Usually, yeah. When I’m doing this with my daughter well, she’s in college now, so I’m not doing it with her anymore. But when we were, we would sit down with the book, and we’d set a timer and say, okay. We’re gonna work a half an hour here, and we’ll see how far we get. And some days we’d get through a whole page of problems. And some days we’d only get through one or two, or we’d get through one and we’d get stuck. And we’d get to the point where, okay, this is what we know. This we don’t know.

Denise Gaskins [00:31:58]:
We’ll look at it again tomorrow when our brains are fresh because we’re starting to feel really tired and frazzled here. But usually, I found that doing this, I could tell what exactly how much she understood. I could tell when we could skip a little bit because she had it, when we needed to dig deeper because she didn’t have it. You sort of pace yourself through the curriculum according to your child’s needs. And usually, we got through about the same amount of material.

Pam Barnhill [00:32:28]:
Okay. So you’re gonna yeah. Because normally what we do, what we tell moms who are teaching math, like, the traditional way, traditional being how we’ve done it the past couple hundred years, is that, you know, if your child is able to do it and do it well and seems to understand and do it quickly, you don’t spend a lot of time on that. It’s the same process. It’s you’re just evaluating the reasoning as opposed to being able to follow the steps of the algorithm.

Denise Gaskins [00:32:55]:
Right. What I’m what I’m not doing is having my child go do the homework on their own, bring me back answers to check. Right. We’re talking about them as they do it, and they’re justifying what they’re doing to solve the problem. Or I may be doing a problem. A lot of times we went back and forth. We did what I call buddy math where we would take turns doing problems. So on my turn, I’m justified.

Denise Gaskins [00:33:19]:
This is why I’m doing this step. This is why I’m thinking about it this way. This is what makes me know that my answer’s right. And the child is judging whether that makes sense to them, and then it will be their turn again. We go back and forth. The focus on the thinking on the journey to the answer, not so much on the answer. We assume that if we’ve made this journey and it made sense the whole way, the answer’s gonna be good. So a lot of you know? So what do you tell

Pam Barnhill [00:33:50]:
the mom who has, let’s just say, let’s just say three. But it could be five. It could be six. It could be seven. Kids of varying ages. And she’s also teaching reading and nursing a baby and doing spelling. How does this work for that mom?

Denise Gaskins [00:34:07]:
It gets harder. It does. I had five kids. They were sort of spaced out. Some things we could do together, the history and the read alouds and things we often could do together, or I could split it into, like, older kids are doing this, younger kids are doing that, and not trying to do five different complete curriculums. But in math, everybody is pretty much at their own pace, at their own age, so it’s harder to do that in math. As the children get into high school, they become more independent. They begin to recognize recognize for themselves what they understand and what they don’t understand when they need to ask questions.

Denise Gaskins [00:34:51]:
And once a child can recognize when he needs to ask a question, he can work a lot more independently. The problem is the younger ones, they don’t really know what they understand and what they don’t understand. They have to learn to recognize that feeling of, I’m confused here. I don’t this isn’t making sense. I need to make this make sense. With the younger ones up through even through into middle school as much as possible, we tried to work one on one. Something that helped with the very young ones, it’s easy enough that an older one can do the lesson. So when I was working with one student on algebra, again, my middle school daughter was maybe teaching my son first grade math.

Denise Gaskins [00:35:42]:
You know? Okay. You open up the book and you talk about the problems. She can do that. And that was it. Yeah. Yeah. So it’s basically we’re learning through conversation, which means you have to have some one on one time in order to do this.

Pam Barnhill [00:35:58]:
So when you get to algebra, did it go easier? I know that’s relative, but, like, it it would seem to me that all of that reasoning and talking about numbers through the years would make things like algebra easier.

Denise Gaskins [00:36:12]:
It builds a really solid foundation. If the kids are making sense all through the years, they’re basically learning to do math proofs all along. They’re learning to do algebraic reasoning all along. The the same principles that underlie the younger kid calculations, the arithmetic calculations, are the principles that underlie algebra. Algebra is just a more general view of how do numbers relate, how do they act, how do quantities adjust with each other. So you have built you build a strong foundation of reasoning, and you’re gonna reap the benefits when they get into algebra and beyond. And they reach still often helps. Pardon?

Pam Barnhill [00:36:58]:
Do you reach a point where you you do give them the calculator and let them just use the calculator in order to Like Yep.

Denise Gaskins [00:37:07]:
Like, for instance, you reach that point with high school chemistry. Okay. They’re they’re doing all these calculations of moles and and, you know, parts per however much and grams per centimeter and density and different calculations. At this point, they’re not really learning about what numbers how numbers relate to each other. They’re using the numbers to do something else.

Pam Barnhill [00:37:32]:
Okay.

Denise Gaskins [00:37:32]:
At the point where you’re using the numbers to do something else, the calculator is a tool that helps you keep from making silly mistakes. Okay. So, yeah, the calculator, we don’t wanna use it to replace thinking, but we do want to value it as a tool that helps us do hard problems.

Pam Barnhill [00:37:51]:
That makes perfect sense.

Denise Gaskins [00:37:53]:
Yep. Okay.

Pam Barnhill [00:37:55]:
This is very interesting. Where can moms because I

Denise Gaskins [00:37:58]:
know they’re gonna be I

Pam Barnhill [00:37:59]:
know they’re gonna wanna know. Where can they go and get help with this kind of mathematical teaching?

Denise Gaskins [00:38:06]:
Well, I mentioned my blog. They can get on my blog and and get my contact form and send a question to me. I have a lot of resources on my blog at DeniseGaskins.com. They can search how to homeschool math. They can search specific topics. This whole notice and wonder and create idea, I’ve just gone through a series of blog posts that talked about a lot of different tough topics in math and how you can reason through them. So there’s a topic on the fraction division where we talk first, we talk about here are some things you might notice and wonder, and then here’s some ways you would reason to the answer. And I think I gave five different ways that a child would be able to think through that division and make sense of it.

Denise Gaskins [00:38:55]:
You know, some by drawing pictures, some by making analogies. Just so we go through, you know, hard subtraction, long division, adding fractions, different tough topics and say, if we’re not using the methods, then what will we do? If we don’t wanna teach the methods, and that’s the name of the series, if not methods, then how can we think? How can we reason through these problems? So I think your listeners might find that useful.

Pam Barnhill [00:39:26]:
Okay. And we will link that blog post, and then you have something else for us as well, don’t you?

Denise Gaskins [00:39:30]:
Well, I also do have a store where I’ve got books. I I’ve got my anchor book is Let’s Play Math. It’s been around. I think we’ve been around ten years going on or maybe more. I don’t remember exactly. That’s at my publisher store, tabletopacademypress.com. But you can find it. You don’t have to remember that.

Denise Gaskins [00:39:47]:
You can find a link on my blog. So I’ve got books. I’ve got books of math games. I’ve got all sorts of enrichment stuff, a lot of stuff there that you can find at the blog.

Pam Barnhill [00:39:58]:
Okay. Tell us about your Kickstarter.

Denise Gaskins [00:40:02]:
Okay. My newest series of math books is called math journaling adventures, and it’s a great supplement that will work alongside any math program. We’re coming to Kickstarter with that next week, but you can use the prelaunch link, which I will supply to you to make sure that Kickstarter notifies you as soon as that launches. The math journaling prompts are a way of drawing out your child’s thinking about math. We use games, art, number patterns, problem solving, writing activities, historical research, all sorts of different explorations to help our kids get deeper into the ideas of math. There are so many different ways to learn how to think about and make sense of numbers, shapes, and patterns. So I hope your your listeners will check out those books.

Pam Barnhill [00:40:52]:
Oh, that sounds awesome. And, yes, we’re gonna put the link for you in the podcast show notes so you can check out that Kickstarter. It sounds like a wonderful way to do math that I never got to do with my kids. I just love that. I love journaling anyway. So math journaling sounds really cool. Well, Denise, it sounds like the thing that I’m really taking away today is to talk math with your kids.

Denise Gaskins [00:41:18]:
Yeah. I’d say that that is the thing. So often our experience of math has been you send the child off to do homework problems on their own. They come back and you tell them what they got wrong. We really want to draw out their thinking more, and that only happens through conversation. So, yes, talk math with your kids. You know, our children won’t build robust thinking skills if we force them to walk with crutches. Every time we say use this method, follow these steps, we’re teaching them to be mathematical cripples.

Denise Gaskins [00:41:54]:
Students that are trained in these procedures, they can pass tests because tests only focus on the answers, and we want to get at the reasoning behind the answers. That’s what mathematics is all about. Let me tell you what master teacher w w Sawyer once said. He said the main purpose is to get students into the habit of thinking and to show them that they can think for themselves, and that’s the kind of math our kids really need to learn.

Pam Barnhill [00:42:22]:
Oh, I love that. Like I said, I always learn so much from you every time. Every time I talk to you, you challenge everything that I know about math, which is not a lot. So it’s pretty easy to do, but you always make me think of, boom, how could it be different? And I really, really do appreciate that and all the resources that you’ve created for moms. So we’re gonna drop the links out for those math games and the math journaling books, the Kickstarter and the publishing website, all of those things so that you can find those. And I know that everybody’s gonna have a lot to think about just like I did. So thanks, Denise.

Denise Gaskins [00:43:01]:
Thank you, Pam. I really enjoy talking to you.

Pam Barnhill [00:43:04]:
That’s our show for today. Be sure to follow, subscribe, and leave a review so you never miss out on the wonder of homeschooling better together. To stay connected and learn even more about the homeschooling better together resources And to join our free community, visit hsbtpodcast.com. Until next week, keep stepping out of the overwhelm and into the wonder.