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Today we want to answer some questions that we have been given about multiplication tables; learning them in a Charlotte Mason way, when to move on and when to stay until the child has learned them completely; all of those wonderful questions that so many of us ask as we work with our child day to day in that mountainous land of mathematics. As we talk about these questions we have with learning the tables and when to move on, we’ve called in the expert, Richele Baburina.
Sonya: Thanks so much for joining us, Richele.
Richele: Thanks for having me. I’m one of the few people, maybe, so excited to talk about multiplication tables.
Sonya: Yeah, that’s right. So talk about them. We need to know how our children should go about learning them in a Charlotte Mason way. And then the details about “When do you move on?” and “When do you stay on the same table?” Those are the big questions we’re wondering about. Where do you want to start?
Richele: All right. I’m going to start with a quote from Irene Stephens, who was the math and science lecturer at Charlotte Mason’s teaching college, which meant she basically was teaching the teachers how to teach math. And so I’m going to start with a quote by her, because I think that that kind of lays out the direction that we’re going to go in today. She says,
There is no royal road to the multiplication table; It must be learned by heart. This is a fact which faces every teacher of elementary arithmetic, and which each must prepare for in the best way possible.
This lays out a few things: One, “There is no royal road,” which means it’s not something simple and easy for every student. “It must be learned by heart.” This answers the question, did Charlotte Mason want her students learning the multiplication tables by heart? The answer is yes. And I would say that that is because it is such a great tool in each child’s tool kit that can be carried throughout all the higher maths as well. “This is a fact which faces every teacher of elementary arithmetic.” So hey, we’re all in this together. “Which each must prepare for in the best way possible.” The beauty of Charlotte Mason and Irene Stephens is that they laid out a very clear road map of the best way possible to do this. And we have put our lessons together in the Charlotte Mason Elementary Arithmetic series so that you can follow that road map with your student.
Sonya: Will you walk us through it—just a shortcut—today? What are the steps involved? A Charlotte Mason way of learning multiplication tables, what are the steps?
Richele: Okay. And for those interested, we have done a post on learning the multiplication tables, and we go into a lot of detail there. So the first thing is the way that we introduce the ideas of multiplication. And we start by introducing the idea of multiplication as quick addition. So we’re giving our students interesting problems for which they already have the tool of addition to solve. But as they are adding more and more times, say the number 4, we have given the number 4, five times. And then we get to show them that there’s actually a shorter method to do this. And we’ll introduce the symbol for times at that point. So even why we say times is going to be evident to the student. And it’s going to help that idea embed in them. So that’s our basic introduction. Once we do that, we’re going to move to the tables. Also, it is important that students don’t memorize tables or math facts without ever having proven them. So this introduction is going to give them a chance to prove each of those facts.
It is important that students don’t memorize tables or math facts without ever having proven them.
Sonya: With manipulatives, with everyday objects, so they can see the “why” behind the formula, if you will, of what they’re memorizing.
Richele: Right, and that really helps give them that hook to “Oh, this is why I’m memorizing it.” So there is not a “Why do I have to memorize these?” So then it gets kind of fun and exciting, because it’s almost like doing picture study. But rather than a great work of art, we’re going to use numbers for what our student is going to be visualizing. So we start out; each multiplication table is taken individually. So we start out with the 2s table. We might write one number 2, and then ask “How many times do you have 2?” “One time,” and then we put a little 1 above it. And 2 taken 1 time is 2, and then the answer goes down below. So we’re basically setting up our little column multiplication problem right there. Then we’ll have another 2 in the center row. Now how many times do you have 2?
Sonya: Two times.
Richele: Two times, so we put a little 2 above that. And what is 2 taken 2 times?
Sonya: 4.
Richele: Four; and we continue on to either 10 or 12. It’s up to you, what you want to do with your student. Once we’ve done this, the student is going to take time to visualize it. And this is where the picture-study part comes in. He has a written table to visualize, to look at, until he can see that in his mind’s eye. Now the difference is we’re not going to turn it over quite yet. He will have the table to continue to use until he doesn’t need it anymore. We’ll erase a few numbers out of there for him to fill in. And then from there he will say it through again. And then we’re going to give him interesting little problems so he will get to use the multiplication table if he wants to. I might have this out of order. We’ve got it all in order in the books. But then he is going to write the same table out in his math notebook. And the child will work with that until he has achieved a comfort level with it. And you’ll see your students start to not look at the table; or some children, their personalities, they don’t want to look at the table. And so we might give them a little extra time to answer those problems. But they have that table whenever they need to refer to it. So then from there, we have some interesting experimental work in longer multiplication. So that student might be multiplying 21 by 2. And so this adds a little more interest to it. Now we want to keep our questions lively and interesting. So, since each child works at this individually, some children learn their multiplication tables quickly, some take a little bit longer, and some actually take quite a long time. When we progress to the next table or the next concept is going to be individual for each child.
Sonya: I love how learning the tables involves a lot of senses. It is a multi-sensory experience, if you will. You’ve got the kinesthetics, because they can work with items to set it up and remind themselves, “Well, if I do have seven 2s, how much is it?” And they can make dots, or they can use objects. And you’re discussing it with them. They are saying it through as they go. They’re also having the visual picture-study effect, where you’re taking some out, putting some in. So they have the hearing, and the speaking, and the hands-on, and the visual. It’s all intertwined there. So you can customize it. You’re giving your student a chance to use all of his senses, but you could allow him to use the strongest one more.
Richele: So since our children might be more kinesthetic learners, they might want to build each individual table using manipulatives or our everyday objects. Pennies are very good for this. A student who maybe just doesn’t like working with objects might be using just hash-marks in his math notebook, just so that he can see kind of the sense there. But the time where they get to visualize and study that math table, they are looking and discovering, on their own, these amazing patterns, which is going to help them. I don’t want to give away every pattern, but if you allow your student time to look for those patterns, it’s amazing what they’re going to find.
Sonya: So it’s not just, “Look at this and get a static mental snapshot of it and then see if you can recite it.” You want their minds engaged, looking at it like a puzzle almost. “What can I find here? What can I discover for myself?”
Richele: Yes, and this might take place, those discoveries, as they’re writing the tables out. They might see those patterns on their own.
Sonya: All right, so as you were saying, some students will pick this up very quickly, some will not take it as quickly, and some will take quite a while to actually get all of those math facts cemented in their minds. So the question is, does my child have to have memorized the 2s table in complete total before we start going to the 3s table? How long do we wait? When do we know when to move?
Richele: Sure. So maybe for the traditional student, you see he is working comfortably. He no longer needs to use that table. He is ready to move on, and you can see that quite obviously. Second, remember, we are still working with the discipline of good habits. So when we said that there is no royal road, we know that it can be difficult to learn. It is a challenge to learn these tables, especially maybe the 7s and the 8s tables. So we also want to make sure that it isn’t that our student just doesn’t want to put in the hard work. We need to be there watching for that. And then, third, if you feel confident that you have done all of these steps, and it’s growing so wearisome for him, you don’t want to erode his confidence. And so Charlotte tells us, then we can move on to a different concept or a different table. Some children who have a hard time grasping, or just getting into their minds’ eyes that multiplication table, sometimes they might grasp other concepts much more quickly. So we have time to review those tables and to continue to work with those math facts. If we look at our students and say, “It’s time to move on,” and you go to the next concept, we have time of mental math to solidify those facts. We also have a time when we can use our number sentence cards, which is a great resource that Simply Charlotte Mason has. And there are maybe six different problems on each one. I think we have 500 cards for the multiplication facts. And so we can use those as a way to help solidify those facts.
Sometimes children who have a hard time grasping, or just getting into their minds’ eyes the multiplication table, might grasp other concepts much more quickly.
Sonya: Talk about how to use those, but they are not flash cards.
Richele: They’re not flash cards. So each card would have six different facts. So it might be, we’re learning the 2s, it might say 2 times 9, 2 times 7, 2 times 2, 2 times 4, so they’re not in order. So six of those facts are on one card. So we might pull out a couple of cards at the beginning of the lesson to have on hand, say, if Mother gets interrupted, somebody’s at the door, or there’s a diaper to be changed, we can take those cards specific to where our child is, and they can work those problems in their math notebook or the dry erase board, however you wish, as you go to attend to something else. So it doesn’t interrupt the math lesson. Or you can take those and you can use those for your mental math, the oral work, part of your lesson, and you can use those to help you say, “Oh, my child needs help on this math table, or he hasn’t solidified that, but we’ve moved on.” So, take those cards and you can use those to give math questions in oral form during your time of mental math.
Sonya: So even when you move on to another concept, you’re not abandoning the table. You can still have touches along the way.
Richele: Yes. So we could use those math tables as review. And so you’ve moved on to another concept. Now I know my student has a harder time keeping this table in his mind. So we’re going to use that table work for our time of review. So we have bookmarks in our books that you can use to mark which tables that you want to be sure that you keep reviewing. And so that’s another way. Plus, as you move on in concepts, you’re going to be solidifying those facts. So say you’ve moved to finding the area of a rectangle, your student is going to be multiplying the length of those sides. So we are going to continue to use our math facts.
Sonya: So those are another opportunity to review those as well. I love how this approach, again, respects the child as a person, as an individual. And it holds the child to a high standard. You do need to learn these, that is a given, but it also does not say, “We’re going to drill and kill until you learn it, or else.” It respects the child’s progression, if you will. That’s a wonderful combination.
Richele: It really is. And I have to say, when you said “respect the child,” I just think about my own children. My firstborn had his math facts down very quickly. And my second one was a lot faster at concepts, but he could not retain those facts without that constant 5 minutes of mental math. So we had moved on in concepts, but as we continued with that mental math and time of review, eventually he had them and he did want to learn them because he was so good at concepts that not knowing the math facts, not having it learned by heart, was holding him up from where he really wanted to chew into more difficult ideas.
Sonya: So again, it’s the “guide by the side” as you’re walking with your students through that land. That’s a wonderful way to teach them the math facts. I guess it’s not really even teaching them the math facts, it’s helping them discover those tables for themselves and the relationship between those numbers. And then it’s encouraging them and being faithful to help them solidify those facts in their heads. So important.
Richele: I think one thing I want to add is that if we don’t look at the math facts as just a drudgery, but really a time of exploration and adventure, where they are able to discover some great patterns. And you might see them too, as you start filling out that multiplication table yourself, because it is such a unique way to have every, instead of maybe a hundreds chart, which I don’t eschew at all, but when you see each number with its own multiplication table, and you see these patterns, it makes it quite exciting, because suddenly if you have caught that pattern, you can actually answer maybe your 9 times 14, or 9 times 15, because you understand the pattern. And I’m not saying to give your child just a shortcut. I mean, we have the idea of fast addition; we also have the idea of the way that the child is learning. And when we’re twisting and turning those questions, they’re actually learning their division table at the same time, which makes it kind of fun.
Sonya: Yes. So you could say “9 taken 8 times equals,” and they have to tell you, or on those number sentence cards, I’m thinking it can also say “8 taken how many times equals this number.” And so we’re introducing the division; they’re learning them both at the same time.
Richele: They’re learning division before they’ve ever seen the division symbol. So how many times is 9 in 72? And, really, that just shows that division is a complement of multiplication.
Sonya: Which is another wonderful concept of discovery that they can make on their own. I remember one of my grandchildren was learning the multiplication tables using this approach. And she suddenly looked at her mom and said, but that’s just division. I thought I was going to learn the multiplication and then the division and I’m doing both at the same time.” The light went off; she saw how they were totally inter-related. So that’s what we want for our kids.
Richele: Yes. And they’ll see pretty quickly, I think that 7 times 9 is the same as 9 times 7. So without telling them we have the commutative property happening here; they won’t learn that until much later, but they will already have discovered it themselves. And that’s what Charlotte Mason says makes every child “another Newton.” When they look at a flower, they are making these discoveries that mathematicians made long ago.
Sonya: Yeah, just like in nature study, they discover many things firsthand that they don’t get the name for until later, but they know it, they know it intrinsically.
Richele: So I would encourage the parent to look at this as part of the whole of mathematics; we might just see that 2s table or 4s table, but really it’s part of the wondrous whole. And we want to give our child enough time to investigate and explore the multiplication tables before moving on.
Sonya: Yes, wonderful. And all of these wonderful ideas and helpful tips are in the front of each of the Charlotte Mason Elementary Arithmetic books that you have written. So any of the books that contain the tables will have these reminders in there and the step-by-step is in there as well. So, all that we need is there to help us be a guide to our students. Thank you for writing those, by the way.
Richele: You’re welcome.
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