Thursday, 7 April 2011

Developing a scaling primitive in multiplication and division

This post has been prompted by Maria Droujkova's Multiplication Models poster which summaries situations in the world where multiplication happens, showing the great variety of possible situations.

I think it's fair to say, however, that in the individual develops only a small number of structures for multiplication in their brain which they use to support calculation in all of these situations.  There is a point at which they perceive multiplication is going on and their brain starts up its multiplication systems. 

We are consciously aware of some of the mental models for multiplication and division we have but not of others.  For example in division we deliberately teach quotitioning (chunking) and partitioning (splitting/sharing) but even if partitioning is not taught, research shows it develops anyway.  We naturally flip between these two structural methods without being aware of what we are doing.  We also rely on known facts and derived algorithms.

In multiplication we are taught repeated addition (subdivided in different ways for different algorithms), but I suspect most of us also have a scaling structure which supports our thinking.   This is not taught in most schools and even I, as an experienced maths teacher, find it difficult to describe.

I recently worked on a PhD proposal about exploring the consequences of deliberately and explicitly teaching students a scaling model for multiplication (which also applies very easily to division).

You simply draw a line which is 1 (0 is at the beginning of it and 1 is at the end) then ask the student to put a dot where the number we are multiplying BY is.  They are effectively picturing a number line. 

Then they rescale their number line to make the original line (which was 1) the number we are multiplying and estimate the value of the dot on this rescaled number line.

So for example 150 x 7

0_______1                                                                                                .(student might put the dot here)

0_______150                                                                                            .1000? student guess

We only get an estimate for the value of the dot (although students are very likely to start using and developing methods to make their answers more accurate).

My intention was to use this at an intervention with students whereby when they are given a multiplication problem they do this first before calculating the exact answer in whatever way they have already been taught.

Some of you may spot that this challenges not just the way we 'see' multiplication but also the way we see number - bringing out the dual nature of number as being not just about counting but also about scaling.

Anyway I'm not going to be able to take this forward at present so it's here for anyone to take anything they want from it.

I'd absolutely love feedback!

I'll post a more detailed version of the thinking behind it and its place in current academic research on the next post.


  1. Ooh, cool! This is a great visual for what scaling means! I want to try this on a block of graph paper now.

  2. Great idea, I will post a Etoys project that will allow teachers (and kids) to create their own versions of these problems, for you and others to play with and create lessons.

  3. Thank you so much Kathleen and Mr. Steve's Science. Good luck with it and please do post thoughts arising here.

  4. Created post here: with some sample models and two "Step away from the computer activities to give kids experiences with scaling.

  5. One other thing I perhaps should have pulled out from the detailed write-up which follows is that:

    This model lends itself rapidly to multiplication and division by none integers and to rapidly estimating solutions to multiplicatons and divisions with non-integer answers.

    The multiplication by non-integers is obvious.

    To set the model up to do a division (e.g. 1000 divided by 7) you draw the line and the dot for multiplying by 7. Then you put the 1000 on the dot and estimate what goes at the end of the line instead of putting it at the end of the line and estimating what goes on the dot.

    Voila - it's easy to 'see' estimates for any nasty division or multiplication.



  6. I'm writing this from a conference session with Ian Sugerman on subitising in early years number. Subitising is the ability to see a group of a certain number of objects as being that number of objects without actually counting.

    We have been arguing about whether the ability to subitise will be robust if the objects are of different sizes. Given my background described here I suspect not but given my knowledge of human beings I suspect we would rapidly and perhaps even imperceptibly overlay structures to help us cope with the extra complexity of the situation.

    Surely someone must have done this research?

    Many thanks to Ian for a wonderful session.

  7. Of course aspects of subitizing could resemble reading rather than scaling couldn't they? You could learn to 'read' four items in a square as being four just as you learn to read the symbol 4.

  8. Mental models for multiplication are so important for students. Congratulations!

    I’m so impressed with your insistence of children understanding multiplication. If only memorizing correct answers, they are likely to have trouble with all future math.
    I’m retired now but taught for many years. Please see my four models of multiplication that I taught to my advanced first and second graders. The models promote a comprehension of various ways of understanding multiplication. They loved the exercises and did very well. These models could be used for any age involved in learning multiplication.
    I also have models of addition and subtraction, if you'd like them.

  9. Thanks so much for this Peggy. I loved your pictures which really made me think again about the 'real world' structures of multiplication.

    The quality of the illustrations drew me in and I found your picture of combinations great for both illuminating and generalising this structure. Previously I had only seen the line diagram and a picture of the outputs of such a situation separately but I found your way of combining them more illuminating for me as a teacher.

    You have a wonderful blog which makes it possible for you to share your clear thinking. Thank you for sharing it with me.

  10. From Steve Thomas in Natural Maths:


    I just made a tool along the lines described in your blog in Etoys. You can find it here. You will need to download Etoys to use it (its free and open source).

    It's very basic, but could easily be extended to make it simple for teachers and students to create there own problems.

    The scripting is very simple in Etoys (I left the scripts open on the page so you can see them). Email me #### if you would be interested in developing it further.


  11. This looks great Steve but I can't get it to work on this computer. I've downloaded etoys but when I try to start the program it doesn't seem to recognised that etoys is installed. I'll try again later in the week on a different computer.

  12. From David Corking on naturalmath:

    (1) _If_ there are any error messages, it would be really helpful for
    other users if you post them. Etoys is _supposed_ to work fine on
    Windows, Mac, OLPC and Sugar-on-a-Stick (and is available from many
    Linux package managers.)

    (2) In the meantime, you can download Steve's program using the link
    at the bottom of his 'launch' page. Then drag the program and drop it
    onto the Etoys icon.

    > I'll try again later in the week on a
    > different computer.

    If your employer installed management software that stops users
    installing applications, then you will need Etoys-to-Go, and the
    download-drag-and-drop method.

    (Rebecca's note: just posting this before I go off on hols as prompt for the future. Thanks Steve and David)

  13. This is lovely:

  14. and this