Can ‘number sense’ be taught?

First graders with poor “number sense” rarely catch up in math skills, concludes a University of Missouri study. But it’s not clear how parents or preschools can teach number sense.

What’s involved? Understanding that numbers represent different quantities — that three dots is the same as the numeral “3″ or the word “three.” Grasping magnitude — that 23 is bigger than 17. Getting the concept that numbers can be broken into parts — that 5 is the same as 2 and 3, or 4 and 1. Showing on a number line that the difference between 10 and 12 is the same as the difference between 20 and 22.

Factors such as IQ and attention span didn’t explain why some first-graders did better than others.

Math learning disabilities often aren’t diagnosed till fifth grade, much too late, says Dr. Kathy Mann Koepke, of NIH’s National Institute of Child Health and Human Development.

David Geary, who conducted the Missouri study, thinks parents can help children develop number sense before they start school.  NIH’s Mann Koepke urges parents to talk to young children about “magnitude, numbers, distance, shapes as soon as they’re born.”

– Don’t teach your toddler to count solely by reciting numbers. Attach numbers to a noun — “Here are five crayons: One crayon, two crayons…” or say “I need to buy two yogurts” as you pick them from the store shelf — so they’ll absorb the quantity concept.

– Talk about distance: How many steps to your ball? The swing is farther away; it takes more steps.

– Describe shapes: The ellipse is round like a circle but flatter.

– As they grow, show children how math is part of daily life, as you make change, or measure ingredients, or decide how soon to leave for a destination 10 miles away,

However, researchers don’t really know why some kids get that 3, three and xxx are the same thing and others don’t. Children with poor phonemic awareness need to work harder to distinguish the sounds in a word. Perhaps some kids need to work harder — or differently — to see mathematical relationships.


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  1. cranberry says:

    “Children with poor phonemic awareness need to work harder to distinguish the sounds in a word. Perhaps some kids need to work harder — or differently — to see mathematical relationships”

    Kids with dyslexia need intensive tutoring. They need to learn strategies which work for them. There are programs which do help, but identification of the problem is important. It is not a problem students can overcome merely by “working harder.”

    I believe research in babies has shown the existence of an innate number system. Some people may lack a number sense, just as some people are tone deaf. If you are otherwise intelligent, but can’t see that ten dots are a larger quantity than two dots at a glance, “work harder” isn’t fruitful advice.

  2. Stacy in NJ says:

    It can be taught but must be done at a pace and with a depth that’s appropriate for each individual. Some just need significantly more practice and repetition than others.

  3. cranberry says:

    Interesting article in Nature about dyscalculia:

    If students vary from the norm, working harder at programs aimed at the average student won’t produce the same effect as programs aimed at their particular physical array. So, rather than saying the students need to practice the times tables 90,000 times more than their classmates, with the underlying message being, “you’re stupid,” perhaps some seemingly simpler practice of the skills no one thinks to teach will produce a bigger improvement.

    We don’t think to practice the skill of recognizing 9 is larger than 7, because for most people, that’s too obvious to think it needs to be taught, or practiced. It feels like practicing the concepts Up and Down.

    I looked at the games created by Dr. Butterworth, “Number Sense” and “The Number Catcher.” Links are found at the end of the article.

    • Stacy in NJ says:

      The idea that the concept needs to be broken down into its most basic component parts and taught incrementally is the basis for DI. Zig Engelmann advocated this way back in the ’70s with his book Give Your Child a Superior Mind which is out of print but can be found (for several hundred dollars) at amazon.

    • How would you even begin to teach something like that, though? Something that’s so obvious to everyone else? It boggles my mind to even try to come up with a strategy to show someone who doesn’t know that 9 is larger than 7…

      • cranberry says:

        I guess it’s a real candidate for drill and kill. See Michael Connell’s link in these comments, or the online games Number Sense and Number Catcher, listed at the bottom of the Nature article.

        In some sense, we don’t “know” 9 is larger than 7 as young children–we just see it. This pile of jelly beans is larger than that pile. No one needs to sit down with us, introduce us to the concept of larger and smaller, give us tricks to remember it, and drill us in choosing the larger pile. We do teach children the names and symbols associated with the quantities and relationships they already perceive.

        Just looking at the first games, I’m intrigued. There’s a great deal of doing. Speed is important. Can you select five items and two items to make seven items? Not, can you count to seven? Because you can learn a sequence of words, but not be able to connect seven with 7 with ******* . In some sense, it is mindless. If cats are born with this sense, it’s not a verbal skill.

        If it does work for the (few) students whose number sense is not naturally strong, why not try it? American students spend immense spans of time on totally useless video games. It would be silly to require students who have a good number sense to play the same games in class–and a waste of class time. As an individual method to remediate weakness, though, it would make sense. It could also avoid the stigma of being pulled out of class for tutoring, if the kids could practice at home.

      • Stacy in NJ says:

        It’s sometimes more about vocabulary and not being able to see the distinction between “larger” and “greater” and “taller” and many other terms thrown about by teachers with the assumption the kids understands the precise meaning of the word. By larger does the teacher mean the actual printed number on paper or its value?

        • Cranberry says:

          Stacy in NJ, as I understand it from the articles available online reporting on the research into dyscalculia, it’s not a teacher error, nor a vocabulary problem. It’s more akin to being tone deaf or color blind. Ask the child to tell you which group of dots (or candies, or buttons) is larger (has more candy, etc.), and they can’t automatically point to the larger group. They may be able to painstakingly count–but “normal” children don’t have to painstakingly count items individually to see that a group of 3 is smaller than a group of 6.

          I assume it’s rare.

  4. My kids are 7 and 4, and when they were younger (and still for my 4 year old) I sometimes have them clean up while counting (ie you don’t have to put away all of the cars – you do 5 and I’ll do 5, or you do 15 and I’ll do the rest). When I was small, my mom had me count the blocks for a quilt that she was making. I think that counting things is probably like the difference between singing the alphabet song as music vs being able to point to letters as you sing it, and then knowing the letters independent of the music.

    • Florida resident says:

      Dear Lulu:
      In our family there was a pedagogical tool — to use large number of identical objects, e.g. identical red buttons and identical white buttons.
      The idea was to teach a child to tell, which group contains more buttons: red group, or white group, or their quantities are equal. This can be done without actually counting number in each group, but lining the groups to make one-to-one correspondence of elements of two sets. Either one of the lines becomes ”longer” than the other, or the lines end equally.
      “Counting” is essentially making one-to-one correspondence between elements of actual set (of buttons in this example) and elements of
      “Set of natural numbers: one, two three, … “.

      Using these identical objects, you can teach “rectangular numbers”, i.e. “composite numbers” as opposed to “prime numbers”.
      Then you teach “square numbers”, k*k, putting a square with side “k” of these identical objects,
      Then you teach “triangular numbers”; triangle with side “k” contains
      k*(k+1)/2 elements.
      Then you discuss “area of rectangle”, Area=(a*b). Then you discuss “geometrically” the formula
      (a+b)*(a+b)={i.e. square of (a+b)}=(a*a +2a*b +b*b}.
      As a mental exercise you can chat about “cubic numbers”, “pyramidal numbers”, etc. With my kids I (and latter my son with his two kids) used little cubes (bought at “Jo-Ann” store). We managed to get cube 5*5*5=125. It was fun.

      Back to the topic. Yes, I have met some children born with poor sense of numbers.
      Best regards, your F.r.

      • Florida resident says:

        Clarification. “Triangle” above means
        “equilateral triangle” with side “k”.

      • I didn’t mention it in the previous post, but my older child seems to grasp numbers without needing a lot of concrete examples. He did arithmetic very early, and was still in preschool when he figured out negative numbers. A friend taught him the idea of perfect squares and square roots over Christmas dinner when he was 4. I don’t know if the early practice caused this or if it’s just the way that he is.

        My younger child (who just turned 4) is doing fine – she counts, understands more and less, corrected people who were guessing her age by telling them ‘younger’, and can do simple arithmetic if she uses her fingers. Practice with us and a brother who teaches her math for fun doesn’t hurt, but when I look at the 2 of them, it’s obvious that one has a great number sense and the other has a way with words and music.
        I tend to think that most people can be taught competence, but that certain things will be easier for some people than others because they’re just naturally better at it. It doesn’t have to be deterministic, though. When I was young, my parents were told that I’d be successful if I avoided math or science. They never told me until I was grown…and I have a PhD in genetics and teach biology. :-)

  5. Florida resident says:

    “– Describe shapes: The ellipse is round like a circle but flatter.”

    Here is the definiton of ellipse by military teacher:
    “Ellipse is a circle, inscribed into a square 3 by 4″.
    Translation of this joke
    {actually ellipse is not a circle;
    there is no such thing as square 3 by 4}
    into scientific language:
    Affine transformation of a plane with a square and an inscribed circle, both drawn on that plane, transfroms
    square into rectangle (not necesserily 3 by 4) and at the same time transforms the said circle into ellipse.

  6. Much like literacy, numeracy is something that takes a ton of exposure. The kinds of things the researchers suggest you do are common sense to the kinds of parents who automatically provide a word-rich environment. That means there are plenty of parents who don’t do those things and their children are at automatic disadvantage. I think we’ve drilled literacy into everyone for years. There are plenty of companies who are finding Americans’ lack of numeracy makes hiring them difficult.

    While I realize that there are some people with a math processing disability, it’s not very common. Instead, difficulties are usually from a lack of developed number sense (note that I said developed, not innate) and a learned anxiety about math (as in a parent who tells a kid that getting a bad grade in math is OK because they weren’t very good at math, either, as though it’s something you’re born with). Math is a skill, just like reading, playing an instrument, or learning a sport. While some people will find it easier than others, age-appropriate practice can make most people fluent. Children with a true disability need to be identified earlier than they are now.

    I used to teach preschool and now I teach middle school math and science, so I’ve done math instruction from both ends. It’s MUCH easier to teach number sense to little kids than it is to big kids. Same goes for reading.

    It’s all about repeated exposure.

  7. Number sense is definitely an important foundation for children’s success in school and in life. But this insight isn’t new – there is a mountain of research on number sense extending back decades – including the work of Clements and Sarama at University of Buffalo, Nancy Jordan at University of Delaware, and the work of Griffin, Case, and Siegler in the early 1990′s on RightStart. In fact, we have known for over 20 years how to teach number sense to any individual child – we just haven’t been able to teach it at scale to *every* child. Coincidentally, this was the subject of a recent video:

  8. Richard Aubrey says:

    Unless an absolute neurological blockage is involved, any capacity can be increased or improved–not to infinity, of course–by practice.
    My granddaughter is pretty bright, five and a half years old. Even her pesky grandparents might ask her a simple arithmetic question, and she gets a few moments of flash card time every so often, and her parents use counting in various conversations.
    Some kids aren’t that lucky.