Sense of Number versus Number sense

I know I have talked and written about this extensively over the past 12 years but it seems in Ontario this age-old argument is coming back with a vengeance.

Now some may say that my title is misleading that there really is no difference but over the past couple of years I have really thought hard about this very topic. Over these years I have seen students come into my classroom with amazing fact recall. In fact, I have never seen a community that has spent countless hours practising math facts and algorithms. But I have also seen over this time I have also seen students have a lack of what those facts truly mean.

Our new government is calling for a “back to basics” and let’s get rid of “discovery math” but all of this is rhetoric by politicians who don’t understand how students development a sense of number. I see it in my own daughter. Izzy has trouble in math for numerous reasons, one of which is a low working memory but another is her understanding of place value. For Izzy, she is reworking on understanding unitization, grouping, and counting by fives and tens. At the moment she is playing Duck Duck Moose and app that helps build those foundational skills.

But back to my classroom experience. Many of my students struggled with the very same problems that Izzy has. They struggle to understand part-whole relationships in number, they struggle to understand estimation skills and most importantly they struggle with understanding reasonability of number. And now I think back to the thinking of our Governments call. They fear that this generation cannot do the math, they fear they don’t know how to make quick change but the reality is they have no sense of number or how our number system works.

When I think about how students learn I think about Dr. Alex Lawson’s work on developmental learning. This is also a great article that she has written: http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/ww_modelling_proficiency.pdf .

I don’t have many answers right now but I do know that learning of mathematics needs to be a balanced approach. Math needs skills but they need to be developed properly.

As students are developing skills trying ideas like Which one Doesn’t Belong, Building spatial sense, Estimation 180 or even Math Before Bed. All of these are amazing activities to help your students build a sense of number. Combine these activities with rich problems and then purposeful practise and now we are building students up to be successful mathematicians.

So as we plan and think about our math curriculum over the summer I encourage you to think about how we are making our students think about numbers along with teaching number sense. A final thought as well, as math teachers continue to teach as I know you teach. You are strong individuals with sound pedagogy. Show students what mathematics is about, how it is a beautiful language, how it is comprised of beautiful and elegant thinking. That it isn’t just rules and magic tricks. Please do not get discouraged by politicians or other professions that have no idea how children learn. In the end, be true to yourself no matter what.

Author: MrSoClassroom

I am a grade school Teacher, promoting creativity and exploration in all of my students. My classroom is always in a state of Inquiry.

7 thoughts on “Sense of Number versus Number sense”

  1. A couple if questions. Why do the parents feel such a need to compensate for work the school should be doing? What problem are they seeing? And this is not about your school as thus pattern is found across the province. There seems to be some implication that the parents are getting it wrong. I am sure it is unintended, but it is implied. Most , if not all the issues you are mentioning are the responsibility of the education system, and yet that is not mentioned. The major role of parents is review and yet it seems to have become teaching/paying tutor. I can’t understand why parents were even mentioned. What would these students be like if the parents, who sound very conscientious, hadn’t overcompensated. Where is the ed system failing? I don’t know what grade this is, but some of these issues are fairly typical. Students have different levels of understanding. If you see gaps fill them. That is the job. Send home a bit of review if necessary. Hint from a long long time teacher. Take PD wirh a big grain of salt. I think the issues outlined here fall almost exclusively in the lap of the education system and are far bigger than number sense.

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    1. Hi Teresa, Thanks for the comments. I do think that the education system needs some changes, maybe not the changes you see but I feel that we need more focus on a sense of number. As for parents, I mention it as a parent and seeing the lack of skill in my daughter. I feel that she is in a classroom of worksheets and DI and not really learning. But as a parent it is my job to make sure my children learn the best that the can in order to be successful. Now why you may see more parents feeling that they need supplement then I am not too sure. There are many factors I know one is it is hard to teach your own child especially when you both are very similar. But there are others too.

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  2. I am a middle school math teacher. I used JUMP Math in my classroom which provides a very good presentation on ratios and part/whole, as well as estimation skills, multiplication and division skills and other areas in which students may have deficits. The best way to build up number sense is to have them work problems, without calculators. Long division is useful in that it requires students to make estimates of what multiplier will yield a number less than the partial divisor.

    My 7th grade class had had a teacher in 6th grade that relied on discovery techniques, math talks (ala Jo Boaler) with the emphasis on “understanding” not “doing”. The students did poorly, and were traumatized by their feelings of inadequacy. Parents complained and tried to compensate at home. Thus, I chose to use JUMP math with this class to make up the deficits they had endured and build up their confidence. It helped a great deal, and by doing the problems with explicit and whole class instruction, their comprehension improved.

    As far as the view that students “do” math but don’t “know” math, that is missing the point. If a concept is part and parcel to a procedure, it is not difficult for students to understand that concept. But some concepts are not so close, as in the case of the invert and multiply rule for fractional division. In such instance, provide examples of whole numbers divided by fractions so they can see the relationship of invert and multiply. As far as fraction divided by fraction, that comes about with more tools–in algebra they have enough tools to be able to see algebraically why it works for all fractions. Some will get it; some won’t. The important concepts with fractional division are what does fractional division represent, and what types of problems are solved using fractional division.

    The focus on “understanding” and particularly the notion that understanding MUST come before a procedure is taught is damaging as I saw with the students I had, and with many parents of such students who confided in me. Interestingly, when parents hire tutors or enroll their kids in learning centers, they obtain the kind of instruction that has been missing in schools for years.

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    1. Hi Barry, thanks for your comment and sorry for the late reply. I have been on vacation and not getting these alerts as I thought I was. I think JUMP has many good qualities but also many shortcomings. I find that with any program even JUMP you need to have a solid understanding of mathematics. Even Mighton would suggest that when he wrote and did JUMP with students he had this solid understanding. When reading his supplementary books Mighton talks about a landscape of learning or continuum that he has in his head and how he breaks down the numbers and scaffolds the learning. Unfortunately, this is not always done with JUMP. In the end, I don’t really care what program you use or how you teach that is not the purpose of the blog. My purpose was to talk about the inefficiencies that I see in students learning and the bad rhetoric that I feel is being created. It is not just the so-called “discovery” versus the so-called “back to basics” but about seeing mathematics for what it is. We need to do understanding and procedure at the same time. This is why I find it interesting that you saw a deficiency in students learning with math talks as I normally get the opposite with my classroom students and with the kids that come into my classroom. However, that is another argument. Math is a balance and there is no one approach to teaching nor should there be. Each student is different and each student needs a certain way of learning. I never used calculators in my classroom but I also don’t just teach procedures. You said it is easy for students to get but it is not. The division algorithm is mathematically incorrect and has many flaws with it. Yes it helps to see the relationships but it is fundamentally incorrect (i.e. 360/6 you would say 6 can’t go into 3 but that is not a 3 nor worth 3 ones but 300 which in that case it can it is 50!) This is my problem with the long division, along with some other algorithms we take for granted. I also don’t like the comment about some will get it, some won’t. The problem is that the job of a teacher is to teach. I am hoping you are not implying that if students in your classroom don’t get it well I guess so be it but I don’t think that is what you imply.
      Thanks again for the convo, happy to continue.

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