*This is the fifth article written by Anne Watson in the 'Dose of Don' series. She posted it on her blog here and I have replicated it word for word. For the background on this series, please see the post Lines and Angles on Square Grids. My thanks go to Anne for giving me permission to share her writing here.*

**Dose of Don 5: Substitution**

John and I hosted a day’s online workshop on ‘Substitution’ recently. Don attended our day workshops regularly and always had something extra, interesting and challenging to offer so we co-opted him onto our team. Hence the workshops we have hosted since May 2020 have always reminded us of him. So I was wondering how Don used substitution in his tasks – whether he made a Big Deal of it or not. We posed the question at our workshop: ‘Is substitution a Thing?’ Our answer in the day was ‘Yes’ as substitution crops up all over mathematics; the purposeful substitution of one expression for another to simplify, gain insight, clarify, test, modify, make new things possible, etc. etc.

Shortly after this Richard
Perring posted a tweet asking the same question but from a different
perspective. His question was about exercises in subbing numbers into algebraic
expressions in early algebra. You know
the kind of thing, the internet is full of them, e.g. ‘*If p = 2; q = -6; r =
10, calculate -pq ^{2}r^{3’}. *The exercise is not about
algebra, it is about calculating with negative numbers once you have understood
the syntax of the symbol system. It changes algebra into numerical answers. By
contrast this task: ‘

*If p = 2; q = -6; r = 10, find at least five different algebraic expressions whose value is 4 using as many of the letters and mathematical signs as you need’*focuses on turning arithmetical understandings into algebraic expressions and launches ‘what if …?’ questions.

So I began to search Don’s
collection of tasks to find places where substitution gives useful mathematical
perspectives and handles.

I didn’t have to look very far: https://donsteward.blogspot.com/2020/04/two-types-of-sum.html

*a*,

*b*, and

*c*(which are generalisations but not very helpful ones) and you have the beginnings of a ‘proof’ for conjectures that arise from doing the calculations. A dance begins between generalisations, structure, examples and relationships that is typical of mathematical exploration and the associated questioning: ‘What can I write instead of ….?’ ‘Can I test that with an example?’ is often about substitutions that are helpful in revealing or expressing structure.

And a later slide of Don’s gives:

This gives a reason for becoming more fluent with such manipulations - ‘doing’ algebra with a purpose. There’s more but you’ll have to go to his website for that while I indicate some other things I found once I had ‘substitution’ in my sights.

I have to force myself to open files with titles like ‘decimal subtraction’ but here goes: https://donsteward.blogspot.com/search/label/decimal%20subtraction.

This does not disappoint; I have
had some great fun showing only the first two lines to people and seeing what
happens. I got myself hooked on wondering about the cyclic nature of what
appears and rewriting numbers as sums of powers of ten, i.e substituting the
separate place values into the ‘answers’.
This is the ‘undoing’ of what is
done in primary school to build up multi-digit numbers from the place value
components and is the basis for many mental methods in Vedic mathematics and
Trachtenberg methods (see Google for these) and an old method that was taught
in schools in the 18^{th} and 19^{th} century called ‘casting
out nines’. Some people immediately substituted ‘nine tenths of…’ for the left
hand sides, which explains something about the answers you get but not (for me)
the cycles. The immediacy of this response was impressive but they were all
people ‘of a certain age’ for whom expressing rational numbers as fractions
retains some of the manipulability that can get lost in decimal notation. So
there are two kinds of substitution at work here, both about equivalent numerical
structures, used to explore and then explain the generic behaviour seen from
the specific examples.

This reminds me of a feature of
substitution that exposes its purposefulness. Firstly, it is a two-way action
in which some things are gained and some are lost: generality/specificity;
approximation/accuracy; manipulability/visualisation etc. This last duo comes from thinking about
modelling phenomena, but also from substitutions that change bases, such as are
done in order to integrate functions.

Don’s website holds many tasks in
which substitution is more explicit than what I have offered so far, see https://donsteward.blogspot.com/search/label/substitution.
Martin Wilson of Harrogate is credited with some of the ideas. There are several of these that can be
explored by trial and adjustment, i.e. purposeful substitution to get a ‘feel’
for what is going on and also raw material for later reflection – why these
numbers? Both
the sets below use structures which we hope will become familiar for learners
but also have extra features to think about.
I could imagine learners being asked to ‘make up some of your own like
these’ and hence writing algebra for themselves, having used substitution to
test their inventions – some two-way number/algebra thinking.

You might be wondering about a
place in the curriculum where the word ‘substitution’ is used explicitly – the
solution of simultaneous equations. In our workshop, and also in some of Don’s
tasks, a powerful use of substition turns out to be the use of equivalent
algebraic expressions, or temporarily equal expressions to simplify the use of
variables. A really simple version of this reasoning is: *if a = b and a = c
then b = c, and a, b and c can be substituted for each other. *Here is a
development of that, where expressions rather than individual letters can be
manipulated to ‘reduce’ the number of variables in a situation (https://donsteward.blogspot.com/2016/03/find-expressions.html
). The task is to express* a*, *b*
and* c* in terms of *n*. Rather than using the language of moving
terms to and fro over the equals sign like deranged chess pieces the language
of logical arithmetical reasoning can be used. For example, in the first set:
‘if b – c = n, then I also know that c = b - n; does that help?’ The standard
question: ‘if I know …. then what else do I know?’ kicks in big time when
transforming algebraic expressions. New expressions for *a* and *b *can
be substituted into the first equation.

This realisation, that so much
mathematics depends on substituting one expression for another when building expressions,
equations, mathematical models and so on, seems to get lost in formulaic
approaches to simultaneous equations. The idea of substituting expressions into
other expressions has recently made more sense to me than the traditional
‘elimination and substitution’ language of methods. For example Don’s ‘where do
the lines meet?’ tasks (https://donsteward.blogspot.com/search/label/simultaneous%20equations)
cry out ‘substitute for y in the second equation’ rather than the ‘rearrange
and match coefficients and subtract the equations’ that appears in some
textbooks.

In my next Dose of Don I shall
return to finding inspiration in his tasks rather than imposing my own
perspective on them.

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