Friday, May 10, 2019

A Progression from Long Multiplication to Algebraic Long Division

I’ve not been teaching (much) over the past 5 months. Instead I’ve been doing copious amounts of reading and a heck of a lot of thinking.  In having the opportunity to step back and think deeply about we do, I’ve found myself questioning teaching and thinking more of the nature of mathematics.  Or should I say I’m thinking about mathematics teaching?  To teach, in a general sense, doesn’t quite capture what it means to teach mathematics.  As mathematics teachers we need to have that intimate understanding of how our subject can be arranged to be learned.  We need to take our own understanding and mental schema of this myriad of ideas and be able to transpose this into something which can be shared with pupils. 

At the risk of making a sweeping generalisation, I offer that a problem we have in mathematics teaching in Scotland is that sometimes our thinking is compartmentalised.  We often find methods or ways of teaching which help pupils perform, but which are quite unsustainable for future learning.  One such example is the teaching of ‘change side, change sign’ for solving equations.  This crumble as we progress beyond trivial examples.  

I think we would rather teach in a way which develops conceptual understanding, helps pupils perform and is also sustainable for future learning. 

Our duty is not to teach pupils to be successful in today's lesson.  It is to teach them so that their teachers in subsequent years can build upon what is being taught today! 

This has led me to consider the idea of didactics.  I’ll admit now that my understanding of this is still emerging and I’d be happy to hear from readers who can clarify my thinking. 

Didactic transposition theory is based on an assertion that bodies of knowledge are, with a few exceptions, designed not to be taught but to be used. The didactic transposition of knowledge is the transposition from knowledge regarded as a tool to be put to use to knowledge as something to be taught and learned [Chevallard, 1988, p. 5]. Thus, any modification of knowledge under instructional purposes can be called a didactic transposition. 

The first step in establishing some body of knowledge as teachable knowledge consists in making it into an organized and more or less integrated whole.  This initial process is carried out by curriculum makers, on a national level who prescribe the curriculum content and pathways.  This means that the mathematics to be taught is an interpretation of a subset of the academic body of knowledge.   Textbooks also significantly impact upon teacher’s awareness of what is to be taught.  Textbook writers transform the national curriculum documentation to produce their own interpretations of what is to be taught.  Examination bodies also make interpretations of the body of knowledge to be taught, in order that it can be assessed.

It is primarily from these sources: curriculum documentation, textbooks and examinations that teachers develop a perception of what is to be taught.  What is then actually taught will be further transformed by teachers as a result of their understanding of the subject matter and their pedagogical content knowledge. Finally, what is actually learnt by pupils is highly variable and depends on many complex factors relating to the teacher, the classroom culture and environment and the pupils themselves. 

We can consider the following example to elaborate upon this didactical transposition. 

At university level students encounter the field axioms for the real number system.  One of these axioms is that of distributivity. That is: a(b + c) = ab + ac.  This is an idea which is ubiquitous in its inclusion is school mathematics curricula.  This axiom from the number system appears in the Scottish curriculum through the following statements:

Having explored the distributive law in practical contexts, I can simplify, multiply and evaluate simple algebraic terms involving a bracket.”(Experiences and Outcomes)

and

Expands brackets using the distributive law and simplifies.”  (Benchmarks)

What this means in practice is immediately open to interpretation.  The basis of the idea as being rooted in number is not explicitly mentioned.  A nod in the direction of an underlying concept is made by the words ‘practical contexts’.  However, without further guidance the teacher can only attempt to guess at what the curriculum writers intended.  Is it the case that the axiomatic nature of the distributive law appears to has been reduced to the action “expands brackets”?

In further documentation, for later years of study, we get our answer. The word distributive has been completely omitted. 










How this mathematics fits in the arrangement of the subject as a body of knowledge to be taught is no longer clear.  The origins in the multiplication of numbers, or the connections with area, for instance, are not present.  The idea has again been reduced to “expansion of brackets”. 


One potential didactic transposition may result in this being taught as the following pattern. 

It may not be clear to pupils from this model that the distributive law a(b +c) = ab + ac is still the key idea here.  However, success in carrying out the procedure likely masks this lack of understanding. 

This method may be how the teacher carries out the process, when doing it himself, but one has to consider if it is the correct way to teach the process initially?  The teacher, almost certainly, understands why the process works, but what about the pupil?  Are there opportunities to assimilate this grain of knowledge with previous learning, or does it sit alone as a discrete method to be replicated?

(a + b)(c + d) = a(c + d) + b(c + d).

The idea may, instead, be presented as above.  The connection with previous learning is clearer for pupils to infer, even if it is not explicitly mentioned.  It also clearly connects with future work on factorisation. 





Some teachers may instead teach a mnemonic such as foil.  This approach is severely hampered by the difficulty in extending it to the multiplication of a binomial by a trinomial and beyond.  This device might be sufficient for procedural fluency on a very limited range of problems, but it treats the idea an isolated skill to be mastered.  This sort of teaching, without consideration of the progression of the mathematics, can result in various “tricks” being required for different situations encountered.  

Consider the following topics that one might see on Scottish schemes of work:
  • Long multiplication
  • Expand single brackets
  • Expanding pairs of brackets
  • Factorisation
  • Synthetic division
  • Algebraic Long Division

From experience I know that a wide variety of methods are in use for each of these ideas. The problem is that these different methods become to be seen as discrete topics in their own right.  What is often lost, is that each of these is rooted in the distributive law.  It may be applying this law ‘forwards’ or ‘backwards’ but all of these ideas are hinged upon this. 

I've focused on these topics due to some of the work of Mark McCourt who has written an excellent sequence linking these topics using manipulatives and area models.  However, If you want to see that you'll need to come along to one of our sessions!

As an exercise in thinking, I offer the following as a possible more coherent approach.

Algebra from number

Pupils will have experience of the distributive law from primary school.  They will have had some experience of decomposing multiplication problems. 

For example:
5 x 17 = 5 x 10 + 5 x 7

I would seek to capture this and formalise the concept in early secondary.  First, however, I’d put some emphasis on mental agility.  This is invaluable and I’d hope for pupils to be able to fluently do calculations such as the above mentally, using this, and other strategies. 

As we move into Third Level/ KS3 one of our key aims is to begin to capture the relationships we know in more generalised form.  This is done through algebra.  Algebra is the generalisation of number relationships.  It copies the structure of number. Over the past year I have begun to think that the study of algebra can emerge from generalising number work.  As such, algebra is no longer seen as being a discrete topic from number work, but instead, being an expression of the essence of number relationships. Algebra would be studied as topic as it currently is in most schools, but it wouldn’t be disconnected from the number work.  

I have created the following task sequence, originally shared on Starting Pointswhich hopefully illustrates this idea. 








Some people have rightly had concerns about my use of the word “factorisation”.  However, in looking at the above tasks I would hope that study of number structure and the seamless, almost casual, incursion into algebra, is clear.

Long Multiplication

In recent years I’ve found that a majority of pupils arriving at secondary have little fluency in performing long multiplication.  However, in my experience, those who arrive having used a grid method have demonstrated greater levels of success.

Calculations for long multiplication can be constructed using as follows, where we are using the distributive law to express 46 x 32 as (40 + 6)(30 +2).  

For pupils using the standard algorithm, they can become lost when tackling something like 532 x 481.  The idea that the we have a line where we don’t add a zero, a line where we add one zero and a line where we add two zeros, proved to be something that pupils lacked an appreciation of.  Whereas adding the relevant number of rows and columns to the grid, is something almost all have success with. 

However, the rationale for ensuring pupils become fluent with this method isn’t about making long multiplication easier.  It is about looking forwards and building a comprehension of a model which will be sustainable for future learning. 

Another important point to make now is that pupils should see multiple representations in for everything in this blog. Representational fluency is important as it supports conceptual understanding.  The use of algebra tiles and/or Cuisenaire rods to support the development of these ideas and to provide related models is very worthwhile.  Comparing and contrast various procedures can also serve to strengthen understanding.

Careful variety of tasks such as the following helps to develop the reasoning behind this method and draws pupil attention towards the relationships with division and factors.  The example on the bottom is particularly important as this is similar to the sort of problem they will solve (although algebraically) when factorising quadratics. 
  



Expanding Brackets




















At some later point(s) in the curriculum pupils will move towards expanding brackets.  This grid method is extendable here. Unlike an approach such as foil, it is easily extendable.  It is an inherently forward-facing model.  Pupils in S2 or S3 can comfortably multiply together polynomials.  Collecting of like terms is also eased, as those terms of the same power appear in the same diagonal. 

This model is used here, again, not to make performing these processes easier, but as a basis for future learning.  The grid is a model which we can pick up at each stage on this progression.  It helps to unify the ideas. 

Pupils should also have time to practice performing these multiplications without the grids.  Attention should still also be drawn to the important identities such as

(x + a) = x2 + 2ax + a2

Factorisation

The use of algebra tiles can be extremely powerful in opening up the idea of factorisation to pupils.  An area model using the tiles for expansion of brackets can be easily reversed for factorisation.  So too, is this the case for the written grid model. 

For simple common factors the grid method is a very intuitive working backwards problem to be solved.  However, the real power of the model is when we begin to consider quadratics.

Jemma Sherwood shared this explanation of how the grid can be used to support the development of both procedural fluency and conceptual understanding in factorisation. 



There is a lovely sequence which supports the mathematics behind this by Ashton Coward, on Craig Barton’s Variation Theory Site.


Synthetic Division / Algebraic Long Division

Synthetic division, as taught to most Higher pupils in Scotland is not a good model.  It doesn’t connect well with previous learning.  Further, it is not forward facing as it does not work for more complex divisions.  It is a short term “this will get them the marks” sort of approach.  Since Higher is the final encounter with mathematics for a lot of pupils, I can appreciate this being prevalent.  However, I argue we can actually improve pupil understanding and (for harder questions) pupil performance by using the grid method in conjunction with some other ideas. 


A division such as the following begins by using the grid method in reverse.  We know one of the factors, so this is really easy.  The mathcreation blog details the full process of how this works. 




The beauty of pupils mastering this method instead of synthetic division is that it will work for harder divisions at Advanced Higher such as


The process may be slightly harder to master than synthetic division, although I think this is up for debate, as pupils already possess a significant part of the skill from factorising quadratics.  

Stuart Welsh recently demonstrated a range of Higher questions using this method.  For a question such as the following, I suggest that the dominant method pupils are taught is inordinate. 


For the next question most pupils would again do a full synthetic division, which is completely unnecessary.  An understanding of factors and roots is all that is required.  No division needed.    The division occludes understanding, rather than supporting it. 


Similarly, for the hardest question at Higher, division is unnecessary. 



Conclusions

I offer this progression as a means of showing how all of these very connected ideas, related to distribution, can be unified for pupils.  The model is the vehicle for this.  The model may not be the procedural end goal at any point on this progression, however, it is a worthwhile port of call.  It helps to build connections between previous learning and serves as a basis for understanding later work.  Pupils may still have other algorithms and models for each of the ideas discussed, however, these are complementary to this unifying model. 

I suggest that by having a unifying model which transcends year groups then there is a possible time saving – as we are building from previous learning each time we progress.  The current practice is to see expanding brackets as a distinct method from that required for factorisation, which in turn is distinct from the method of polynomial division.  While we may tell pupils they are connected, telling is not enough.  We need to support their developing schema by having a unified model or models. 

Finally, I return to the axiom of the distributive law.  I have suggested a unified (pictorial?) model for all of the problems discussed.  I would look to unify the algebra too, by at some point expressing things clearly in terms of single brackets.

As usual, I've written this piece to try and make sense of some ideas.  I look forward to hearing replies and comments, both where people are in agreement and where suggestions for modification of my thinking can be made (i.e. where I am wrong!)  



Thursday, September 27, 2018

A Learning Episode: Straight Line


Background
I’ve been introducing the idea of straight line to my third year class recently.  This is a topic where I feel that the “just tell them” approach comes crumbling down.  After having taught this topic a dozen or more times, I’ve come to be of the opinion that direct instruction, no-matter how clear and how explicit will not suffice to develop procedural and conceptual knowledge we’d like pupils to acquire for this topic.  I talked on the recent Craig Barton podcast about pupils (and teachers) requiring the space for sense making.  This topic is a prime example of where this is required.

I think it is worth remembering that this is the first time where the relationship between algebra and geometry comes together.  Until now these have been discrete topics for our pupils.  I’d make the point that Descartes rightly has the coordinate system named after him as it was a big deal for the whole of maths when he came up with it.  For instance, imagine calculus without being able to visualise the functions and their stationary points!  Given that this was quite a breakthrough in the history of mathematics, then we should be prepared to spend more than a couple of hours on this idea with pupils.

I could talk in length about my treatment of the whole topic of straight line; however, I have chosen to focus on a specific learning episode.  To offer some context the class are familiar with the families of lines which are parallel with the axes.  I have done very little work on tables of values, and will come back to this later.  I previously would always start with this, however, I think that once learners have a feel for gradient and y-intercept we can look at patterns in the coordinates with a fresh perspective and tie the topic together.  The class have worked on developing a conceptual understanding of gradient through a slow building up from the idea of vertical over horizontal, counting boxes “for every 1 box along, how many boxes up/down?”, finding gradients by “stepping out” before finally using the standard gradient formula.  The idea of coming at this idea from various perspectives is so that pupils appreciate the affect the numerical value of gradient has, as opposed to simply being the result of some calculation. 

A task I used to support pupils in developing connections between the formula and the concept is the following from John Mason and Anne Watson.  In each case, the pupils have something to relate their answers to, rather than simply churning out meaningless and context less answers.  This task appears simple, but, in my experience, for a novice can be quite enlightening.



Today
The approach I’m trying out this year is based upon some excellent work by my colleague @mpcopland.  I have not yet talked explicitly about y = mx + c.  My experience is that this becomes some abstract notation which bears no correspondence to the notions we are trying to convey.  I am happy to delay the introduction of that until the next lesson. 

I used Desmos to illustrate each of the following lines.  I think using the family of lines with gradient 2 draws more attention to the role of the coefficient of the x.  



I asked what was happening.  An interesting reply from several pupils was “the lines are moving sideways”.  This response is notable as I’ve had pupils come out with this on previous occasions when I’ve been using dynamic software such as Desmos or Autograph.  I wonder if I restricted the length of the lines that the vertical movement would be more obvious.  It took more probing and specific questioning “what about where the line is crossing the y-axis?” before the idea of vertical movement became apparent.  

The demonstration with an invariant gradient drew the attention to the y-intercept.  I then explored the lines with gradient of -2 and pupils were able to answer out with successful predictions of how the lines would appear.  

By showing a few more examples and cold calling it seemed to be the case that many pupils appreciated that, for instance y = 3x + 2 is a line which has the following properties:  A.  For every 1 box along it goes up by 3.  B.  It cuts the y –axis at 3.  I was feeling quite confident at this point – this was na├»ve, as no child will have fully internalised these ideas through simply observing or being told.  The opportunity to attend to the ideas for oneself and to play with the mathematics is essential. 

The follow up task for learners is shown below, focused on drawing parallel lines with varying y-intercepts.  Again this idea of varying just the y-intercept is important, to keep this as the main focus.  You’ll notice how this pupil appears to have a limited appreciation of the infinite length of the lines.  I’ll probe this more tomorrow. 



The next pupil, was one of a few who produced the following.  It’s at moments like this where, as a teacher, there are two options: become incredibly frustrated or instead appreciate the mistake and use it as the basis for more conversation.


I shared this with the class and asked them to discuss in their pairs what might be right and what might be wrong about this attempt.  Some rich conversation ensued and when I brought the class back together there were a sea of hands eagerly wanting to tell me how to rectify the situation.  Interestingly, while this mistake appeared in some pupils work for the first set of lines, nobody repeated it for the second set.  I’d said very little, the knowledge, the appreciation of the ideas and the sense of the topic was already in the room.  I find it quite important for learners to be able to explain their ideas to each other.  I like to take advantage of this to address misconceptions.  I conjecture that pupils may be used to hearing my voice and therefore switch off quicker, but are more inclined to listen to a friend and ask demanding questions to clarify their understanding.  So rather than me telling the class the mistake and possibly having a positive outcome, a whole host of other positives may have had opportunity to arise in terms of confidence and deepening of understanding, by learners talking to each other. 

Exploring the dimensions of the problem

During class discussion a couple pupils posed my favourite type of question.  “What would happen if?”  They each suggested equations, both of which are shown below.  One of them imagined that the line might not be a line at all.  I put the equations into Desmos.  We looked at their gradient and y-intercept and then I demonstrated some simple rearrangement.  This was a nice opportunity to explain that if we have the line in the form y =  that we can easily read the gradient and y-intercept.  This, I feel, will set us up nicely for the tomorrow’s lesson on formal y = mx + c.



Taking things further
Having spent a few periods on gradient and then today on the y-intercept, I wondered if learners would be able to generate the lines represented by the equations below.  Tables of values, as I previously said, have not been emphasised yet.  If pupils were to succeed with this task then, I think, it would be from a good understanding of line equations.  Richard Feynman has a lovely quote “what I cannot create, I do not understand”. Being able to generate the diagram from the algebra is fundamentally important here.  I think that a much higher level of understanding is required to do it without a table of values than is required with one, thus my avoidance for now. 



The following is one of many successful attempts.

  

I then asked pupils to draw a line of their own choice.  None of them were trying to vary from the format of y=mx+c now as, I think, they are beginning to see why this representation is useful.  A number of interesting lines were drawn correctly, such as y = 0.5x + 0.25. 

Conclusions

The last time I wrote about a learning episode like this some people commented along the lines of “why bother?”   In this case, I think it is very much worth the effort to use a range of tasks, teaching approaches and representations to develop that fuller understanding.  Straight line is a topic that can be taught at a very superficial level.  “Here are two points, find the gradient using this formula, come up with the equation of the line using a point and the gradient in this formula.”  However, what is the lasting legacy of this?  A very limited instrumental understanding.   Another critique of a recent post was that “pupils aren’t interested in why”.  I disagree entirely.  If we offer the material at the appropriate time and in the appropriate manner, ask the right questions and, more importantly, leave space for the learners to make sense of things, ask their own questions and generate their own examples then engagement (whatever that is) can be extremely high.