More than Just Numbers: Montessori Pre-Math

The most recent post in my early1 math and language series was all about math skills. In this second part of that post, I’d like to look specifically at number sense.

Rote Counting

A simple dice game in our house

Those of us who value concrete, hands on learning don’t typically like the idea of learning anything by rote, but in this case it’s important. Before children can gain a meaningful understanding of quantities, numbers, and the relationships between them, they need to learn basic counting. Children tend to pick this up in their own time, but we can certainly offer some support along the way.

Counting can be a natural part of baking together, grocery shopping, and many other everyday activities. My Montessori teacher trainer emphasized the importance of counting with children and doing so in a pleasant tone, as though we’re enjoying ourselves – and hopefully we are! This suggestion stuck with me and I have always made a point of putting it into practice. It sounds silly, but I think it was actually the beginning for me of learning to love math again after years of dreading it in school. Like many things, an enjoyment of numbers can be contagious.

Coming up with ways to practice counting is pretty easy, and since we’re talking about rote counting here, it doesn’t require any dedicated materials, but here are a couple of basic ideas. I’m certain that you have your own, and I’d love to hear a few in the comments.

Dice Games

This is one of the many simple, yet clever ideas I learned from my former mentors. Not only does it make for great rote counting practice, but it’s an excellent introduction to the idea of turn taking in the context of a simple, non-competitive game. When children first get to the age where they’re interested in joining the games of older siblings and parents, the concept of each person having their turn can be tough! All you need is a basket, a collection of small objects, a die, and one dish or cup for each person playing.

In the photo above, I’ve used small craft store acorns and two apple shaped dishes – items I had collected for fall activities, but you can use anything you have around the house from buttons to bottle caps to pieces of pasta. Our current game set-up uses a numbered die with the numerals 1-6 because Annabelle recently demonstrated that she could recognize these numerals. For a child who doesn’t know them yet, however, a standard die with dots is perfect. On their turn, each person rolls, counts the dots on the die, and then counts the corresponding number of objects into their dish. Young children will make mistakes and there’s really no need to correct them. This is all for fun and practice. When all of the objects are gone from the basket, the players can count together as they place them all back in. There’s no need to declare a winner, since the goal is just to count and have fun.

Count through your day

Counting can be a natural part of most any everyday activity. You can run the comb through your child’s hair (or better yet, count with them as they do it), put on two socks and two shoes, walk down all (insert number) steps, put ten raisins into the oatmeal, and the list goes on. Any time you’re dealing with multiple objects and doing so wouldn’t be a distraction, try counting!

When my daughter was an infant, there were few things she liked less than car rides and I was constantly looking for safe ways to soothe her while on the road. She must have been six or seven months old when, one night, we were driving home and singing to her just was not working. For some reason or another, I tried counting aloud and she relaxed almost instantly. I got to 100 and she was asleep. After that, counting became my default soothing tool and it continued to work for several weeks. Now that she’s older, we still make a fun game of counting together in the car sometimes. Somehow I doubt this is a universal tool for soothing fussy babies, but counting games in the car are great for toddlers and preschoolers.

Use Songs and Rhymes

Songs like “Five Little Monkeys”, or at this time of year, “Five Little Pumpkins” can be so much fun for toddlers and preschoolers, and as a bonus they help familiarize children with counting. I find that singing in the car, (quietly) while waiting in lines or waiting rooms, and during other activities that are otherwise difficult and not-so-stimulating for young children is a great tool for keeping things from deteriorating. There are some wonderful counting books out there, too. The most recent one to join our collection was 1, 2, 3 Dinosaurs Bitewhich I love. There is no sense in rushing or drilling, but I believe there’s much value in sharing a love of numbers with young children through song, literature, and other everyday experiences.

One to One Correspondence

Any parent of a toddler knows that when children first begin to count, they’re not fully aware of the idea that each number corresponds to a discrete unit of something. They love to point while counting, and may reach 20 while, apparently haphazardly, pointing at the same three objects over and over again. This is because they have not yet developed an understanding of one to one correspondence. This typically happens on its own, depending on the individual, sometime around age three or four, and there are plenty of activities we can offer as tools when we see an interest.

Placing Objects

While on the path to learning that numbers each refer to individual objects or quantities, the numbers themselves tend to be very abstract and therefore not particularly helpful for children as they make sense of the world. Children will first learn to connect objects with other objects. One sock for each foot, one spoon for each bowl, one coat for each hook, and other real world experiences with bring children’s first exposure to the concept of one-to-one correspondence. Activities you can create will simply be those where one set of objects is matched exactly to another set. The activity shown at left is one example, where there are two equal sets – one set of pom poms and one of sections in a colorful ice tray. Exactly one pom pom is transferred into each section. You can do a similar activity, with or without tongs or another tool for transfer, using any container that has defined sections and any objects that fit inside. Placing muffin tin liners in a muffin tin, peat pods into a planting tray, or shoes onto a shoe rock are all great ways of practicing one-to-one correspondence.

Matching Items

The examples above are excellent first experiences since they involve a stationary and clearly defined place to put objects. Before long, this will be easy for the child and it’s then that you can begin to offer two loose sets of corresponding objects for pairing. If your child has a large collection of dolls, you can bring together all of their socks and shoes and invite your child to match one sock to each shoe. The photo at right shows my two and a half year old placing pom poms into disposable muffin liners she took an interest in while we were baking. I gave her exactly one pom pom for each muffin cup so that the sets of objects corresponded exactly. Play food and little plates would be another great thing to transform into an activity for one to one correspondence. Of course there’s no need to step in and constantly steer our children’s play toward a learning activity – especially since play is learning after all. Simply offering equal sets tends to be enough, and the child who is beginning to understand one to one correspondence will likely use them as a tool for pairing on his or her own.

Concrete Experiences with Quantity

Playing with quantity: a leaf stair

As we all know, an ability to count does not indicate a concrete understanding of numbers. Actually recognizing that three is more than two and ten is much more than both two and three, etc, comes after repeated, meaningful experiences with quantity. In the Montessori classroom, we use the Number Rods to fulfill this purpose. This material includes ten wooden rods varying in length from 10cm (shortest rod) to 100cm (longest rod). The first, or one rod is exactly half the size of the second, or two rod and each one is exactly 10cm longer than the one before. Each rod is painted so that it has alternating red and blue sections of exactly 10cm each. We count the sections and use the three period lesson (described here) to connect number to rod, introducing the first by saying “this is one” and the last by saying, “this is ten,” and so on. Not only do the number rods allow the child to see just exactly how much greater ten is than one, and to examine all of the quantities in between, but the child also feels the difference between each rod while carrying them from the shelf to rug and back again, and moving them around on the rug to build them into a stair.

At home, you can certainly create your own materials for this purpose. Deb at Living Montessori now has a roundup post filled with ideas for DIY Number Rods and Alternatives. Counting is a great skill to practice, but developing a concrete sense of quantity is more complex, and anything that can give a visual and tactile sense of different amounts is valuable when children are ready. The number rods are a great starting point because they deal with a “fixed quantity,” so there aren’t loose objects that children have to spread out in a specific way. Less room for error means it’s easy to focus on the quantity alone and less likely that there will be errors and confusion. Later, loose objects are a great, more challenging choice.

Numeral Recognition

Remember how I said at the very beginning of this series that there’s so much more to early math and language skills than learning letters and numbers and that, in fact, there’s not much point in worrying about those things early on because they’re far more complex than we think? This post should give you a sense of what I mean when it comes to numbers. Knowing the numerals is really not all that beneficial until children can count, understand one to one correspondence, and have a concrete sense of quantity.

By the time a concrete sense of quantity has developed, many children will have learned to recognize the numerals already. Others may not have taken an interest, but there are a number of ways to introduce them and create an invitation for practice. In the Montessori classroom, we use the Sandpaper Numerals for this purpose, introducing them in the same way as the Sandpaper Letters, using a Three Period Lesson. For those whose children are doing their early learning at home, Deb at Living Montessori Now has a great post with ideas for DIY and Inexpensive Sandpaper Numbers.

Connecting Numeral with Quantity

In the Montessori classroom, once children have developed a concrete understanding of quantity and have also learned to recognize the numbers 1-10, we begin to offer materials that allow children to connect concrete quantity with numerals. We use number cards to label the number rods and introduce the spindle boxes and the cards and counters as well. Again, the number rods and cards are a great first step since the quantity (rods in this case) is fixed and not moving. For those learning at home, a fixed quantity is recommended as well and you can refer to the link for DIY and Inexpensive Number Rods.
The spindle boxes contain labeled sections and loose “spindles,” which these days are simple wooden dowels. Each section is labeled with a numeral and the child counts the corresponding number of spindles into each section. Working with the loose quantity makes these a bit more challenging and also a great next step for the child who has mastered the Number Rods and Cards. Of course Deb has a post with ideas for DIY Spindle Boxes, too.
The Cards and Counters deal with loose number cards and loose counters, so both the quantity and the numeral are “loose,” which makes this a still more challenging activity. It’s also an introduction to odd and even, but that’s a different post altogether. At home, you can create your own number cards using anything from plain old paper to squares of balsa wood, and you can use any small objects you have around the house as counters.
The above only scratch the surface. There are many more Montessori math materials and they’re all amazing, but these should serve the purpose of showing how shallow a foundation in math can be when we focus only on teaching numbers – and how silly it is to waste the valuable time of busy toddlers and preschoolers with such an abstract concept. They get there in time, and there’s much more to preschool math than numbers!

This post is linked up with Montessori Monday at Living Montessori Now.

  1. It occurred to me recently, when I came across posts with reading and math activities for the very young, that some people may get the wrong idea when I talk about “early” math and language. Just to clarify, I’m not talking about teaching math and language “early.” In fact, I sort of advocate the opposite. I mean “early” as in developed before more complex and less concrete math and language skills. If that makes any sense. Hopefully you catch my drift.

14 thoughts on “More than Just Numbers: Montessori Pre-Math

  1. Lisa Nolan

    These are just great! So important to include activities for indirect preparation for math in the later years! They really helped my son with Down syndrome when he was younger! He’s 8 now and loves (simple) math worksheets (he’s now in public school), and he uses the Addition Strip Board to do the homework problems!

    Reply
    1. melissa Post author

      The addition strip board would be such a great tool for homework help at that age. What a great idea, Lisa!

      Reply
  2. Amy @ You Shall Go Out with Joy

    This series has given me so much to think about in terms of how children learn about math and reading. Thanks so much for sharing it!

    I was just thinking the other day about Gus’s understanding of numbers and counting. He’s a great counter (i.e. reciting the numbers in order), but when he is actually counting objects, he says a number for each object, but not in the proper order (1, 3, 6, 5, 10!!!). I thought it was really fascinating to be able to see how the different aspects of counting haven’t really come together yet, especially since I myself am quite good at counting by now, so I have a hard time separating the numbers from the objects, if you know what I mean. Anyway, I think he might be interested in trying out some of these activities.

    Reply
    1. melissa Post author

      That is so interesting, Amy! It sounds like Gus has figured out one-to-one correspondence before rote counting. I love seeing the different ways development around these concepts can progress. I often get caught up in thinking of these things as linear, but they absolutely aren’t!

      Reply
  3. Angela

    These are all wonderful ideas, thank you! It is true that children learn so much simply from being involved. My just turned 2 year old started to count while grocery shopping this week…she made it all the way to 20! It seems that her brothers constant counting to 100 has paid off, I was pleasantly surprised :)

    Reply
  4. Amy W.

    Wow, Melissa, what a rich post! There is SO much information here! I’m going to have to use some of these ideas to prepare that Montessori inspired play environment we were talking about ;)

    Reply
  5. Discovering Montessori

    I loved your language series and I am definitely learning a lot reading your post on maths. Thank so much for helping me to remember to stay with the most concrete materials as possible so that I am able to remain true to the child. Please keep writing these post! I am learning so much.

    Reply
  6. Deb @ Living Montessori Now

    Awesome post, Melissa! It’s wonderful that you’ve found so many ways to naturally include counting activities in your daily life. I love your ideas for car rides … my daughter needed lots of activities for car rides when she was little, too. Thanks so much for linking to my posts and for linking up with Montessori Monday. I featured your post at the Living Montessori Now Facebook page and pinned your pre-math posts to the Montessori-Inspired Activities and Ideas Board at http://pinterest.com/debchitwood/montessori-inspired-activities-and-ideas/

    Reply

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