Malcolm Swan Day

Recently mathematics education lost one of its leading thinkers, Professor Malcolm Swan. The impact that Professor Swan had on developing mathematics teaching and mathematics teachers cannot be overstated, and also cannot be adequately described in words. This post is not an obituary, I didn't ever have the pleasure of meeting Professor Swan, but despite that I have been massively influenced by his resources and the development materials he has published, primarily for me in the Standards Unit (or Improving Learning in Maths).

The purpose of this post is to highlight an opportunity to celebrate the life and work of this great Maths educator. Professor Swan's funeral is on Tuesday 23rd May, and so we are calling on Maths teachers to use Malcolm's materials in as many lessons as possible, and tweet pictures and examples using the #malcolmswanday

For those people who may not realise what we have to thank Malcolm Swan for, his materials include:

• the aforementioned Standards Unit, which can be found on mrbartonmaths website here.
• the Mathematics Assessment Project materials, which have their own website here
• The 'How risky is life?' Bowland Maths project, which can be found here
• The Language of Function and Graphs - a fantastic book, which the Shell centre have kindly provided photocopiable masters on their site here

The posts and images tweeted on the day will be collated and given to his family as a tribute from maths teachers across the country to this inspirational hero of maths education.

Approaches to teaching simultaneous equations

My esteemed colleague Mark Horley (@mhorley) wrote an excellent blog recently about the balance between the need for understanding when teaching simultaneous equations balanced against ensuring procedures are straightforward enough to support pupils ability to follow (read it here). Reading his reflections led me to reflect on my own approach to simultaneous equations, as well as others I have previously seen, and one that occurred to me literally as I was thinking about them. This blog is designed to act as a summary and chart my journey through the teaching of this topic.

Elimination: This is probably the first method I used, and is definitely the sort of approach I was taught at school. Very much a process driven method, I can't remember understanding much about the algebra beyond the idea that I was trying to get rid of one variable so that I could find the other. I find that the subtraction often causes problems (which is partly why Mark's idea of multiplying by -2 instead of 2 is very interesting) and of course the method doesn't generalise well to non-linear equations. I can see this being a popular approach for those people teaching simultaneous equations in Foundation tier.

                    Substitution: Another one from school,

                    this was the alternative I was taught to 

                    elimination, which was mainly because it

                    was necessary to solve non-linear 

                    simultaneous equations. I can't remember

                    it being the method of choice for myself 

                    or any of my classmates, and that is 

                    certainly borne out with my experience of

                    using it with any other than the highest 

                    attaining pupils.

       Comparison: Similar to elimination, but for me less 

       process driven and more focused on understanding the

       relationship between the two different equations. This 

       removes the difficulty around dealing with subtracting 

       negatives, and allows for the exploration of which

       comparisons are useful and which aren't, so it is a little

       less 'all or nothing' than the process drive elimination

       approach. It also copes nicely with having variables with

       coefficients that are the additive inverse of each other, for 

       example in the pair of equations above if instead of the

       approach outlined we multiply the second equation by 3 

       and get:

       4x - 3y = 9     and       6x + 3y = 21

       then the comparison would be "the left hand sides have a 

       total of 10x, and the right hand sides have a total of 30, so

       10x = 30."

       This is the approach I used when recapping simultaneous 

       equations with my pupils in Year 11 and they certainly 

       took to it a lot better than the elimination or substitution 

       that had used with them the previous year.

                                              

                                               Transformation: This approach is the

                                               one I have very recently considered, but

                                               not yet tried. The general idea is that you 

                                               isolate one of the variables, and then look

                                               at how you can transform that variable in

                                               one of the equations into the other. The

                                               same transformation applied to the other

                                               side of the equation then gives a solvable 

                                               equation. Although the equation may be 

                                               slightly harder to solve at first, I do believe

                                               this approach has merit. I would suggest 

                                               that this approach develops pupils'

                                               appreciation of the algebra and the

                                               relationships between the different 

                                               equations in a similar way to the

                                               comparison approach above. I can also see 

                                               this approach working for non-linear

                                               equations, like the one below:

           

etc...

I will almost certainly give this approach a try when I next teach simultaneous equations - when I do I will try and blog the results!         

Methods of Last Resort 4 - Comparing/Adding/Subtracting Fractions

Working with fractions is notoriously something that teachers complain about when it comes to pupils' understanding and ability to manipulate. As a result it often seems to me that working with fractions is a place where even the best maths teachers can often fall back into what Skemp would call 'instrumental understanding'; pupils mechanically following procedures rather than applying any understanding of the relationships between the different parts of the process or between the question and the result.

This was brought to mind for me recently when I saw the question below mixed into a group of questions about comparing fractions:

From the rest of the questions listed it was quite clear that the intention would be that pupils write the second fraction as a fraction of 30 so that the comparison between the numerators would yield clearly that the first fractions is bigger than the second. Which of course is completely apparent because the first is more than ½ and the second less than ½. Any halfway competent mathematician wouldn't even bother equating the denominators, and this is the sort of thing I would want to highlight to pupils in order to try and develop their relational understanding.

The process of finding common denominators for comparing, adding and subtracting fractions is one that can easily become automatic for pupils, and I would argue that if pupils are to really understand fractions then they need to be able to take a more discriminatory approach. The following are all examples of questions that pupils could tackle without finding common denominators:

I would argue that the first and second points are more easily done by converting to decimals than fractions (which people may or may not agree with), and that the last one certainly doesn't require a common denominator; the first is greater than ½ whilst the second is equal to ½.

So if you are truly committed to developing pupils' relational understanding of fractions then the next time you look at the sorts of comparisons or calculations that often benefit from converting into equivalent fractions with common denominators, it might be worth throwing in some examples and questions of calculations where this is a method of last resort.