Montessori math

“Children’s early understanding of patterns, like visual repeating patterns, is predictive of their later math learning of numeracy.”

Dr. Elida Laski,
Boston College’s Thinking and Learning Lab

Math is all around us. When it comes to teaching it, the challenge lies not in creating the opportunities, but in recognizing them. Counting grapes while they disappear is a fun one! Around the age of 4, a child’s mathematical knowledge takes a big leap forward. They go from counting and recognizing numbers, to understanding 1:1 correspondence and being able to complete simple addition.

As parents, we can support them in this leap by providing things to count, line up, and compare. Lovevery’s Montessori Math Bars & Number Tiles for months 43-45 are a great place to start. On this episode, My New Life Host Jessica Rolph is joined by the principal investigator for Boston College’s Thinking and Learning Lab, which studies cognitive development with a primary focus on mathematical knowledge: Dr. Elida Laski. Dr. Laski is also on the editorial board of the Journal of Montessori Research.

Transcript:

Importance of math skills

Jessica: So on a previous episode, a researcher at I-LABS at the University of Washington shared that preschoolers math skills predict both third grade reading and third grade math scores. Why are math skills so foundational when it comes to learning? 

Elida: There is an increasing amount of research that shows that the processes that are related to both later literacy and math are the same.

So I did a project with a student, Melissa Collins, and we show that there’s these two key cognitive processes mapping symbols to their reference. So knowing that a symbol the numeral for three refers to three objects is the same thing you have to do in early reading when you have to look at this arbitrary squiggle line that represents an A and know that represents the sound, a, right. And so they share the same processes, but in some ways math being more concrete, more hands-on, it’s kinda easier to harness those and start working on developing that cognitive process. Another one is the idea of part whole relations, which doesn’t come necessarily easily for children. And yet it’s a part of both early literacy and early math. You know, to know that five can be broken up into different parts. It could be a set of three and a set of two, it could be a set of four and a set of one really builds this understanding of part whole relations. And when we are beginning to sound out words and to synthesize them, the individual phonemes are parts of a larger word. And so you see that the same process is involved in each. And so by supporting early math, we’re actually activating the same kinds of processes that will support literacy as well.

How to expose our toddlers to math

Jessica: This is fascinating to know. I did not know the nuance between how, why literacy was connected to math learning. So I love knowing this and connected to music too. I will say music learning is also associated. How can we think about exposing our toddlers to math? We wrote a book called Baby Math, and we included patterns and we included placement like you’re before me, you’re behind me. We included empty and full. So containment concepts, even some geometry for babies help give parents the understanding of how math is encountered in their everyday life in the real world.

Elida: Yeah. And we did some studies looking at opportunities for informal math done in the home. And that’s one of the things that just like your intro said, there are more opportunities than sometimes parents are aware of. And that’s one of the biggest differences is in how much math parents engage in the home is just sort of being more open and looking for those opportunities. There is usually when we’ve done surveys with parents and we ask them what they think of as math, they tend to really focus on the counting and the numeracy aspects. And yet there’s increasing research that these other aspects of math are important. So there’s a researcher named Bethany Rittle Johnson, who has shown quite convincingly that children’s early understanding of patterns, like visual repeating patterns, is predictive of their later math learning of numeracy.

If you think through that intuitively, there’s a lot of patterns in early numbers. So counting by fives is a kind of pattern, place value and the fact that you’re always increasing by tens is another kind of pattern. And so by exposing children to early visual repeating patterns like red, blue, red, blue, it also begins to activate the kinds of processes that will support them in later math. So there’s a lot of sort of more math opportunities beyond just counting that matter for building that early math foundation.

Jessica: Yeah, and I was thinking too, noticing patterns like stripes or a pattern even on a carpet. Being able to also understand where you are in space. So for toddlers really understanding, this is behind me, this is in front of me, above, below. Does that all count as math learning?

Elida: Math vocabulary, spatial reasoning. They all seem to really contribute to the kinds of foundational math, the foundation that’s important for later math.

How to help your child learn math

Jessica: Yeah. And then this role of manipulatives getting our whole kind of body involved with math learning can be so powerful. Talk to us about how math learning can become tangible for young children.

Elida: So there’s two parts to your question there. So one is getting the whole body involved and we know from cognitive science that there are a lot of connections to this embodiment idea. And that using your fingers to do addition actually activates neurons in the mind that then help you sort of map the representation of different numbers, that hopping along a number line can be more effective than just playing a board game, sitting still and just moving your hand. So there is something about engaging the whole body that just provides an extra cue for encoding and representing the number as a concept. But there’s also this idea of having an external representation or concrete way of seeing a concept come alive, if you will. And that’s where the use of physical materials to model numbers, to model math operations really can be important for helping to sort of help children visualize and create a concept of a math idea.

That said it can be sort of problematic to stay in that realm for too long [laughter], right? That there you want to also kind of create a bridge between the use of materials and moving carefully to an abstraction. So going from the physical materials to perhaps a pictorial, still symbolic, but representation of those materials that’s now more 2D and can’t be manipulated as easily and that creates a really important bridge to going into just mental math later.

Jessica: Let’s pretend like I have a three-year-old at home. What, what should I do with my three-year-old to bring the best practices around math learning to my child? In addition to getting the Lovevery play kits where we have many different manipulatives and different items that support math learning throughout their learning journey.

Elida: So for three-year-olds, one of the main math concepts, I think three-year-olds are working on mastering [laughter], in terms of their developmental trajectory in math, is counting. Both being able to count concrete objects using one-on-one correspondence. So one word for one object. And then this an idea of cardinality, which is that the last word I say actually represents the number I’ve counted. And it seems probably intuitive, but it is not so intuitive for a four, a three-year-old. So you can ask a three-year-old to count how many. They count, 1, 2, 3, 4, 5, then you ask them, so how many do you have? And often they go back and count 1, 2, 3, 4, 5. So in thinking about those two concepts, one key thing to do is to help them with the count procedure. So using any kinds of objects, helping them find strategies for keeping track what they’ve counted and what they haven’t counted.

And that can be, after you’ve counted, push it aside or let’s take this group of objects and line them up straight in a row. So it’s easier for us to keep track of what we’ve counted. Because those counting errors, like double counting or skipping an object are one of the things most common that we see in three year olds. And then there’s some empirical research that shows that something as simple as labeling the count set after you’ve counted with your child can make a big difference. So if you’re doing a counting book or counting together, don’t just count 1, 2, 3, 4, 5 and assume the child will know that that represents five. But go ahead and just say, so we have five objects and do that extra step of labeling and it helps to build that cardinality concept. So in addition to these little things you would do while you’re counting, of course finding any opportunities to count during snack time, to count when you are lining up the Legos to sort into different sets and to count different sets and parts of sets is helpful as well.

Jessica: I love that. It, it really did inform the design of one of our play kits at the end of two, so we’re getting a little bit a little bit before three, but we’ve got this wooden counting box where you count the pegs while you place them in, and then it says correspondingly five, the number five, and then it shows the spelling of the word five and then the child can crash them out and they all come tumbling out. So they can really get that kind of visceral sense and that physical sense of manipulating one peg going in at a time would then create the set to then crash them out together. So yeah, I would love to hear any other ways that parents can help bring Montessori math at home, like counting parts of the body, whether dressing or, singing songs, or thinking about dates and time. How else can we support our child’s math learning? 

Elida: Besides the counting and knowing the absolute, what we would call absolute magnitude, so what amount does each number represent? 10 is 10 pegs, and you just used an example of how they could visualize that with your counting box. But another thing that we now know is that magnitude comparison, or knowing the relative magnitude of numbers, is really predictive of later math learning. And young children tend to overemphasize the smaller numbers, I think in part because they have a lot more experience with smaller numbers. So the difference between one and three to them seems super salient and obvious. And then nine and 10, even if you just look at them visually, it’s a lot harder to tell the distinction. And they just sort of seem, they’re kind of those big numbers and they group them above. So having more opportunities to compare numbers all along the number of sequence is really important.

So you can do that even when you’re riding an elevator, for example, and you’re going to the fourth floor versus the third floor. And just the sense of we’re going higher and four is higher. Now we have to go all the way to six, how many more floors is that? So this thinking about the relation between different numbers is really important. And in one of your kits, your Montessori math bars that shows the numbers organized as a pyramid, really visually shows that not only do the numbers represent a larger quantity as you move up the count sequence, but it visually shows how much more. That it’s exactly plus one each number you move up in the count sequence. And that’s also really important to start helping children understand it’s not just that later numbers are bigger, but it’s exactly sort of this plus one rule between numbers.

And that’s consistent throughout the whole number string. I will say another thing that Montessori does differently from I think traditional math, and that seems to be quite important, is introducing young children to a far greater range of numbers than we sometimes think of. So there are certain patterns in the number system that are recursive. So once you get past 10, then 11 through 19 is the same units, and then 20 to 21 to 29 is the same. And children can’t really start becoming aware of those if they’re really restricted to small numbers for too long. So not being afraid to count past 10 the first step will then just be rote counting, but that’s the first step for them being able to then anchor that rote counting sequence to some numerical concepts later on.

Subitizing 

Jessica: Yeah, I think that sometimes we forget to expose our children to the concept of a hundred or a thousand or even bigger numbers. So I love knowing that. There’s some research that I’m, I know a little bit about, but maybe you can help us understand around subitizing. So estimating quantities in and a child’s ability to accurately, within within plus or minus a few, understand how much is in, for example, how many black dots are on a piece of paper. I think is the cognitive test that they often give children but that actually also predicts later math success and maybe later school success. Can you talk to this a little bit more?

Elida: So subitizing is a process where you can sort of fairly accurately count by quickly looking without having to engage the count procedure of one-to-one correspondence. Generally you can only subitize for numbers three and under. It’s something where you can quickly see a set of three and identify it as three. That’s a process that we share with other apes, our ape relatives. It’s a process that we share with other animals, so it seems to just be a biologically evolutionary process. But there’s this other system which we call the approximate number system, whereas we can just look at a set of dots on a screen, and these can be larger numbers, these can be eight, nine going into the teens, and we can have a sense of what that number might be, or at minimum a sense that it’s more or less than another set of dots.

And that approximate number system is also something that we share with apes, which suggests it’s been biologically evolutionary process. And it’s the anchor for which what we call symbolic and exact math is built upon. And so that’s one of the reasons we think it’s so predictive. But just like I was explaining earlier that children are more precise in knowing the difference between smaller numbers, the approximate number system is also more precise for smaller numbers. And it’s also more precise at being able to tell the difference between numbers that are really far apart. So it’s easy for very young children, toddlers, preschoolers, to tell that a set of 15 dots is more than a set of 3 dots. It’s a lot harder for them to tell a set of 9 dots from 12 dots. And if you just think about what 9 dots and 12 dots looks like, that seems pretty intuitive about why that might be. They’re visually less easy to discriminate. And so having lots of opportunities for children not to just compare in smaller numbers, but to refine and to develop some precision of comparing larger numbers and larger sets is really quite critical for later math.

Jessica: This is a really nerdy question. Do those subitizing flashcards work? Where you give your child a flashcard of I don’t know, 9 or 12 dots and see if they can quickly estimate and almost kind of train the brain to be able to estimate quickly. Is that, is that a helpful thing or is that kind of a gimmick? We do not sell these at Lovevery, I will just say.

Elida: No, I would say it’s a bit of a gimmick. So I mean, we already know that an approximate number system exists that does that. Now there is some research that shows that adding some precision to the approximate number system is useful, but not more than actually practicing with symbolic math, with counting numbers and making the connection to numerals and numbers. And so if I was going to invest my time, it would not be in doing those kinds of flashcards, it would be in far more authentic math kinds of activities where we’re counting real things and we’re engaging in comparison of who has more crackers, who has less crackers and organizing things by their order so that you can see their relation to the count sequence. I think in the long run that would be a much more productive use of time.

Tips for parents that have an aversion to math 

Jessica: We have dominoes that you would approve of [laughter] They call them subitizing dominoes, but they’re very concrete and a child can then compare and match quantities when they’re seeing little visual objects in different representations.

What if a parent has an aversion to math and how do they ensure that their child doesn’t pick up on that? 

Elida: Yes, and [laughter], there is a research that’s come out of University of Chicago that shows parents own math efficacy, their own confidence in math can be in some ways implicitly communicated to children. In two ways. One is if I don’t feel comfortable with math, I might be less inclined to introduce math opportunities into the home. But also sometimes we say things, and this would be more true of older children, like it’s okay I wasn’t good at math either, and we would never say that about something like literacy or reading. It’s okay, I wasn’t a good reader. And so it’s changing the, the mindset that that math is something that everyone is capable of barring some severe disorders.

And also just I would tell preschool parents, parents of preschoolers that undoubtedly they can do the kinds of supports that their children need at that point in time and that their fear of math is probably related to concepts that are going to come much later and not to worry about at the time to just show their children and try to engage their children in the love of math early on as much as possible.

Jessica: Oh, this has been so wonderful having you with us today. You are such an important voice for parents, really. Again, really being able to prioritize math as much as we prioritize reading in our homes will make a big difference for our children later on. So thank you for bringing this knowledge to us today.

Elida: Well, thank you for having me.

Takeaways:

1. By supporting early math, we’re activating the same pathways in the brain that will support literacy. For example, knowing that the symbol 3 refers to three objects is the same thing a child has to do in early reading when they look at an arbitrary combination of lines that represent a B and know that it makes the sound buh. 
2. Beyond learning the numbers, exposing children to early visual repeating patterns like red, blue, red, blue, also builds a foundation for later math learning. 
3. There’s research that shows that labeling the count set after you’ve counted with your child can make a big difference. So don’t just count 1, 2, 3, 4, 5 and assume your child knows that represents five. Go the extra step, and say: “So we have five pegs.” This concept is demonstrated in the Lovevery wooden counting box.
4. The Lovevery Montessori math bars not only show that numbers represent a larger quantity as you move up the count sequence, but also visually show how much more. You can reinforce this in the day-to-day, by asking your child: How many more floors do we have to go to get to number 6?
5. Rather than leaning on tools like flashcards, Elida recommends more authentic math activities, where you’re: counting real things; comparing who has more crackers, who has fewer; and placing things in order so that you can see their relation to the count sequence. 

Discover more ways to reinforce early math learning on our Lovevery blog.