Thursday, 7 April 2011

Scaling in Multiplication and Division: more detail.

This posts expands on the previous one.  Again, any feedback would be most welcome.  Please do take and develop this model and make it your own.  Reference credits would be greatly welcome as would your letting me know if this work is useful to you.
Rebecca Hanson  April 2011

An Exploration of the Use of an Explicit Model for Scaling to Support Skills in Multiplication and Division.

Classes of structures for multiplication
Much attention has been paid in recent years to their being a natural scaling structure for multiplication. 
Confrey and Smith (1995) analyse the limitations of repeated addition and propose that an alternative model which they name ‘splitting’ has an important role to play in supporting students strategies in multiplication.
 “Claim 4: The development of splitting and its connection to ratio crates the basis for what we call the splitting world, whose overall structure and developmental path differs considerably from the counting world that currently dominates the curriculum.”                         Confrey and Smith (1995)
It seems that Confrey and Smith are combining two multiplicative structures into their category of ‘splitting’ here.  The first is the model for multiplication described by Anghileri (1989), namely that when a objects each generate b sub-objects, this is not the same visual image as there being b groups of a (so we have splitting rather than repeated addition).  The second is a model of scaling which involves measures rather than discrete objects and is exemplified by in ratios associated with similar shapes.
Greer (1992) describes Vernaud’s ‘isomorphisms of measure’ and ‘product of measures’.  Similarly Lamon (2007) uses Freudenthal’s classification as ratios as being ‘within (internal) or between (external)’.
Concepts for multiplication which could be considered to be higher level multiplicative activities, such as Vernauld’s multiple proportions the Cartesian product (both described in Greer (1992)) are beyond the scope of this work.

From a formalist to a non-formalist perspective
It is important to draw a distinction between the work described above which seeks to categorise multiplicative structures according to the nature of the situations to which they apply and my work which is interested in the personal structures individuals use to support multiplication and division.
As a teacher I observed my students using a variety of structural models, know facts and algorithms to solve problems.  The algorithms they used would, at some time, have been abstracted from structural models but this was not necessarily done by the student.  Each algorithm they employed may have been passed to them from another person.  While one person may have no ability to link an algorithm to a structural model another student may either consciously or subconsciously link the same algorithm to a generative structure for it.  To confuse matters more whether or not the whole algorithm is structurally supported individuals components of it may be structurally supported in different ways at different times.
So for example if we consider the traditional bus shelter division method.  When I asked my students to explain what it represented I found some were picturing it as being a partitioning (splitting) algorithm while other understood it as being a representation of a quotitioning (chunking) structure and others were unable to link it to any structure.  To confuse matters more, if you focus on an individual computation within the algorithm (e.g. 8 divided by 2), students may resolve this either by splitting 8 into two groups of 4, by grouping 4 twos or by another strategy such as utilising a known result.
Although Vernauld’s work described in Greer (1992) seems to be located within a formalist tradition, Vernauld (1995) clearly makes the case for a  non-formalist analysis of the location of structures for multiplication within the child.
I have often been struck by the intuitive way students swap, seemingly unconsciously, between partitioning and quotitioning models for division.  For example if I ask them to divide 4260 by 2 most will naturally try to split this quantity into two equal parts.  However if I ask them to divide 390 by 130 they are more likely to chunk.  It is natural that students should be taught and encouraged to use both models to support their mathematical thinking.
In multiplication, however, it seems that we only explicitly teach students to use structures based on repeated addition as a model to support thinking.  This PhD will explore how students’ abilities with multiplication and division can be enhanced if they are also explicitly taught to use a scaling model for multiplication.

Why create models?
As a teacher I harnessed very powerful visual models to support students thinking in mathematics.  For example I used the counting of squares in rectangles to support and scaffold thinking on multiplication and higher level concepts such as the expansion of quadratic brackets.  This was a Geoff Giles task which I was taught to use by Geoff Faux at ATM conference.  I also found it in my SMP Interact Text books. 
Another powerful model for thinking which I used extensively as a teacher was the ‘proportion bar’.  Chapman (1986) describes the use of this model as being best practice for the representation of proportional data and it was in this context I first encountered it, but I later found I could use it in a very powerful way to aid the teaching of all mathematics topics relating to proportion when I was given the Fractions ITP (reference to formalise) from the National Strategies to use in my classroom.  I subsequently found the same structure being used in the Freudenthal inspired materials developed at MMU where real life situations are represented by diagrams where which are gradually abstracted to the bar model with which I was already familiar.
Van Den Heuvel-Panhuizen(2003) describes the development of this bar model and also describes the philosophy within which it was created:
“..the philosophy of the curriculum and its development is based on the belief that mathematics, like any other body of knowledge, is the product of human inventiveness and social activities.  This has much in common with RME.  It was Freudenthal’s belief that mathematical structures are not fixed datum, but that they emerge from reality and expand continuously in individual and collective learning processes.”
My experience as a teacher has convinced me that students are much more imaginative and creative with the strategies they employ when their thinking is grounded in robust and flexible models.

Why create a specific model for scaling?
I have not found a robust model to support thinking about scaling. 
In their description of didactique, Brousseau et al.(2004) it is clear to see that the concept of scaling is deliberately nurtured, but yet again this is situated within a complex, wider ranging methodology.  I was looking for a single, concise, versatile and easily transferable model to support scaling and scaling only.
Montessori resources such as the red rods also lead students to appreciate scaling both through the visual sense of perception of length and through the experience of scaling of weight but their applicability is limited to low integer values.   The scaling of larger quantities is perceived more through weight than length.

What form would a model for scaling take?
The essence of the model is this:
A student is show a line.  They are given a number which is a length for that line (usually 1 at first).  They are then given a second number and they immediately have to show where this end of the line would be were it of length this second number.
______                                                             .
This line is 1      show me 7 (student touches or makes a mark where they guess 7 is – an example of this is shown by the dot above).
One of the first things I would do during my PhD would be to develop interactive whiteboard (or other touch screen device) software which would give instant detailed feedback on the accuracy of the solution shown.
I already have used the structure of this model in different forms.  For example I ask a student to close their eyes and make a jump.  I tell them that jump is 13 and then demand that they jump 11 without giving them time for complex calculations.  I can see that they can perform this task with reasonable accuracy offering some justification for my suspicion that just as Davis and Hunting (1990) have demonstrated that partitioning is a spontaneously emerging model for division, so will scaling be found to be for multiplication.
Trials with this software would seek both to measure and to develop students’ accuracy in rapid scaling.
With their scaled model in their mind they would then be asked to apply that scaled model to their number to be multiplied and to estimate the answer.  So for the picture above if the question was 23 x 7, they would be asked to mentally imagine overlaying a number line where 0 is on the start of the original line and 23 was on the end of if and to estimate on that number line where their dot is.  This is the skill referred to as ‘positioning’ by Van den Hervel-Panhuizen (2008).
If they are not able to confidently rapidly state their answer (as they might easily be able to do had the original answer been, for example 20 x 7) they may start to develop methodologies to support their thinking using this image or they may write down their best estimate and then employ previously established methods using their first result to sense check their second.

The flexibility of this model?
This model does not only scaffold thinking about multiplication of integer values, it can also clearly robustly support thinking about multiplication by non-integer values, multiplication by numbers less than one and it gives routes for supporting thinking about multiplication by fractional values.
This model will also robustly support think about division.
To support thinking about division the same model is constructed but the number line is overlaid with the dividend on the dot and the number at the end of the line is estimated.
Harel and Confrey (1994) describe a commonly held view that there is only one primitive model for multiplication which is repeated addition and that there are two primitive models for division which are quotitioning (chunking or repeated subtraction) and partitioning (splitting).  However Ma’s (1999) description of the wide variety of strategies Chinese teachers employed to solve 1 ¾ ÷1/2 indicates that it is likely that a scaling model is missing not only for multiplication but also from division.  A mental scaling model could have underpinned the decimal strategy demonstrated.  The reciprocity of multiplication and division which was deeply embedded in the thinking of the Chinese teachers is fundamental to this scaling model of multiplication and division. 

Methodology and research design
I anticipate carrying out trial studies to explore the uses of scaling models in practice.  Once I have developed and trialled relevant software and teaching methodologies I anticipate carrying out a series of short studies to explore possible methodologies and contexts within which this model could be used and its potential benefits in each case.
An example of the kind of trials I anticipate carrying out would be that a teacher may teach their class multiplication in their usual way.  We would then split the class by grouping them so that the two sub-sets have equal attainment.  The teacher would then teach one group an extra lesson on multiplication to consolidate their learning while the second group did other work.  I would then train the teacher to teach the second group scaling strategies during their final consolidation lesson.  The progress of students in both groups would be tested at a later date.  If appropriate this strategy would be modified during subsequent trials.
I also anticipate exploring the use of this model with young children.  The overlaid number line would not be used.  Considerable attention would be paid to the exploration of the pedagogical context of young children.  I also anticipate developing strategies for the staged introduction of the overlaid number line.

Pedegogical Context
This research aims to be explicitly aware of its pedagogical context.  The original genesis of my perceptions of the variety of the structures students employed in multiplication and division came from the specific methodologies I employed as a teacher which rigorously required students to draw or visualise and explain their working.  Students were taught that correct answers which were derived from mathematical tricks they did not understand would gain less credit than approximations which could be robustly defended according to structural models.
In Ma (1999) Schulman emphasises the importance of understanding the pedagogical contexts in which the diverse and robust models of calculation demonstrated by the Chinese teachers existed.  Evidence to support this claim in the book includes the results of Ma’s testing of teacher’s ability to describe situations which could be described by the calculations they are attempting.   The Chinese teachers were able to describe practical contexts for the calculations they were trying to solve while the US teachers could not, indicating that the Chinese teachers calculations strategies were connected to structural models while the US teachers were using ‘inherited’ algorithms.  It is my intention that my work will remain attentive to these wider pedagogical issues.

This PhD research will build on a very substantial amount of work I have already done both in practice as a teacher, through online discussions and in reading relevant literature.  It’s genesis is in my realisation that the way in which I am now defining and visualizing the primitive structures of multiplication and division seems to be innovative yet robust.  
I have decided not to take this work further at present but would like to take this opportunity to thank Jeremy Hodgen (King's London) for his support it developing it this far.

Anghileri, J. (1989). An Investigation of Young Children's Understanding of Multiplication. Educational Studies in Mathematics, 20(4), 367-385.
Brousseau, G., Brousseau, N., & Warfield, V. M. (2004). Rationals and decimals as required in the school curriculum: Part 1: Rationals as measurement. The Journal of Mathematical Behavior, 23(1), 1-20.
Confrey, J., & Smith, E. (1995). Splitting, Covariation, and Their Role in the Development of Exponential Functions. Journal for Research in Mathematics Education, 26(1), 66-86.
Chapman, M. (1986). Plain Figures. HMSO: London
Davis, G. & Hunting, R. (1990). Spontaneous Partitioning: Pre-Schoolers and Discrete Items. Education Studies in Mathematics 21: 367-374
Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grows (E..), Handbook of research on mathematics teaching and learning (pp. 276-295). New York: Macmillan
Harel, G., & Confrey. (1994). The Development of Multiplicative Reasoning in the Learning of Mathematics(pp. 41-60). Albany: State University of New York Press
Lamon, S. J. (2007). Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for Research. In F. K. J. Lester (Ed.),Second handbook of mathematics teaching and learning (pp. 629-667). Greenwich, CT: Information Age Publishing.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah, New Jersey: Lawrence Erlbaum Associates.
Van Den Heuvel-Panhuizen, M. (2003). The Didactical Use of Models in Realistic Mathematics Education: An Example from a Longitudinal Trajectory on Percentage. Educational Studies in Mathematics, 54(1), 9-35.
Van Den Heuvel-Panhuizen, M (2008). Children Learn Mathematics.  A Learning-Teaching Trajectory with Intermediate Attainment Targets for Calculation with Whole Numbers in Primary Schoools. Sense Publishers, Rotterdam Taipei.
Vernauld, G. (1994). Multiplicative Conceptual Field: What and Why? In Harel, G., & Confrey. (Eds). The Development of Multiplicative Reasoning in the Learning of Mathematics(pp. 41-60). Albany: State University of New York Press
Fractions ITP reference to formalize.  Accessed 23 Feb 2011 []


  1. This part of your post struck me:

    >I ask a student to close their eyes and make a jump. I tell them that jump is 13 and then demand that they jump 11 without giving them time for complex calculations. I can see that they can perform this task with reasonable accuracy ...

    If someone can jump 11/13ths of their first jump (without instruction), that means we have a pretty clear understanding of what multiplication of fractions means, before doing it in the math classroom. Did you see exercises like this helping students to understand the math on paper version?

  2. Rather than saying that we have a pretty clear understanding of what multiplication is Sue, I see it more as meaning that we have many relevant experiences and structures which could be connected to enhance our understanding (and currently often aren't).

    With regards to improving students understanding of written calculations.... I think that depends on the student (what methods they already have) and the particular written calculation. A student who has a strong 'repeated addition' understanding of a particular algorithm is unlikely to change their view of that method.

    But it's always worth adding more strings to your bow to enhance your mental strategies. I think we underestimate the importance of versatility and of having a variety of robust mental structures to support mathematical thinking.

    Please do probe further if you'd like to.