Do’s and Don’ts of Primary (Elementary) level Algebra

In my last post I argued that we should be teaching the thinking that becomes algebra from as early an age as possible. But what are those skills? What are the Dos and Don'ts? Many of the don'ts stem from the place of arithmetic thinking in our curriculum. Thinking arithmetically is all about getting a right answer, it's not always about being able to use that right answer to get more right answers in the future, and I think this is at the heart of what follows:

To develop algebraic thinking:

Don't:
  1. Don't use the equals sign as an operator. Many children see the equals sign and think Do something; Work that out; Add those. The equals sign represents balance, equivalence. Children need to learn that in arithmetic to support their algebraic thinking.
  2. Don't represent things with the same initial letter as the problem, like 'a' for apples and 'b' for bananas. All it does is reinforce the misconception that the letter stands for an object or a specific number, rather than a variable.
  3. Don't get tied up in knots about BODMAS (the order that operations are carried out). The context of the given problem will sort that out. It needs to be made explicit when algebraic notation is introduced – you can explain how different calculators work those our sequentially or using an algebraic precedence of operators.
  4. Don't limit thinking about sequence to the next number. See if the children can see the rule or the pattern.

Do:
  1. Teach patterns from an early an age as possible. Here's Marylin Burns fantastic lesson.
  2. Do give children plain paper for them to represent their maths graphically.
  3. Tabulate patterns and sequence so children can move from seeing the 'up-and-down rule' (the sequential generalisation) to the left-to-right rule (the global generalisation).
  4. Follow the previous step by asking 'what's my rule?'
  5. Use empty box problems (e.g. 4+□=11)
  6. Do encourage children to represent the problem, not just solve them. Then the numbers can be changed and children can use the same representation to solve harder problems (perhaps by using a calculator and a spreadsheet).
  7. Do use a trial and improvement approach. This is especially powerful when it can be done using a spreadsheet.
  8. Do use the fantastic free materials that exist free all over the internet. Here's some that help children to find rules and describe patterns that the UK government produced a few years back, stored on the website of Dudley LA.
If there are anymore do's and don'ts, or any that you disagree with, please leave a comment.

At what age should we start teaching algebra?

Like many people, algebra is a slightly painful word. Rows and rows, indeed columns of columns of x's and y's attacked me at secondary school. I didn't really get what they meant, even though I was actually quite good at solving equations.

Now as a primary school teacher I still have a blind spot when it comes to algebra, there's something about it that I don't quite get.

But I've had a revelation today. I think I know what I've not been quite getting all this time.

I've just read a chapter in a wonderful book by Derek Haylock: "Mathematics Explained for Primary Teachers" (4th Edition). I've been able to access the book through the MaST programme I'm on at Edge Hill University – but it was so good that I bought the whole book from Amazon. It starts with a question that illustrates why I don't get question. I don't want to steal Haylock's thunder, so here's a different version of the same concept:

On a school visit, 6 students are can go for every 1 teacher. There are t teachers, s students can make the visit. Describe the relationship between s and t.

The temptation is to say 6s=t. That is exactly what I did in the equivalent problem that Haylock set me. But then, say 30 students make the trip, then according to the equation I just wrote, I need 6*30 teachers. 180 teachers for 30 students? Slightly over-powering! The answer is s=6t

Haylock makes the point that I'm getting confused between 'things' or 'objects' and variables.

In arithmetic, which dominates primary teaching, I use letters as abbreviations – hence 't' for teachers. There's also m for metres, kg, mm, l, and many more. In algebra, letters never represent abbreviations for measurements, they represent variables – they stand for whatever the number you've chosen. An amount that can be changed. It is precisely for this reason that it is unhelpful to use 't' for teachers and 's' for students, because it provides the illusion that you are representing the actual teachers as a tangible thing., rather than the number of them.

I think many of us in teaching younger children think of algebra as a nice extension to do when the children have really got their arithmetic sorted. But I'm seeing now that if we only ever train children to think arithmetically, than we are doing them a disservice. Algebra is a branch off the same mathematical tree that Arithmetic grows on, it is not a branch that nicely extends from Arithmetic. Algebra develops from recognising and playing with patterns, investigating sequences and seeing how things can be represented as bigger or smaller. Many of us teachers, especially in schools were standards are low, look at these lessons and wonder 'how will this help the children's maths?' And by maths we are thinking of arithmetic and doing well in tests (which for 11 year olds are about 50% arithmetic). We are not thinking of developing the children's brains so they can generalise patterns and represent problems.

I can hear the question being posed. So what? Why should children have to generalise patterns and represent problems?

Well the answer comes down to being able to solve problems with much bigger numbers and larger degree of complexity. I might be able to solve a problem with my arithmetic skills, but if I can represent it I can use a spreadsheet or a scientific calculator to solve it for any number. Likewise I might be able to work out the 15th term of the triangular number sequence, but working out the 77th is a rather harder challenge – I can save loads of time by generalising the pattern, representing it with algebra and calculating from there.

I wonder how many software developers, games designers, app creators and the like can get away with only thinking arithmetically? I don't know anything about how those kinds of jobs work, but I'm sure that some level of algebraic thinking is required for those jobs.

So. An answer to my question: as young as possible. In my next post I'll start to explain how…

Valuing misconceptions on the way to explaining fractions

explaining fractions_0001.wmv
Watch on Posterous

I filmed this about 6 months ago, following an excellent session about fractions on the Mathematics Specialist Teacher Programme. The challenge that we were given, and then I in turn gave to the children, was given a 4-pint bottle of milk that gets 3/5 of a pint drunk each day, how many days does the milk bottle last for? Those of us with a formal background in maths would say:

÷ 3/5
= 4 ÷ 3 x 5
= 4 x 5 ÷ 3
= 20 ÷ 3
= 6 r 2.
So the milk lasts for 6 and a bit days. If we wanted to be really fancy we would say the milk lasts for 6 and 2/3 days. And isn't it more practical to say the milk lasts for 6 days and there's 2/5 of a pint left over? Does our understanding of the algorithms let us say that?

Also can children, who are without the drilled-in knowledge that when you divide a divisor you actually multiply, do this question?

That's what the video explores – and there's some interesting misconceptions on the way.

Learning Creativity in Maths at MaST HEI Day 5

MaST is the Masters level study programme I am on (standing for Mathematics Specialist Teacher). HEI merely stands for Higher Education Day.

Creativity in Maths

The Day begin with a lecture on creativity in maths. It's an interesting idea – creativity – because many teachers have the mental construct that creativity is all about thinking artisticly and creating things of aesthetic value. Derek Haylock went on to talk about about divergence and flexibility – a far different way concept of creativity in maths. One leads to trying to shoe-horn maths into a themed curriculum and doing lots of shape work that becomes artwork, the other leads to open-ended questions, good dialogue and child-centred learning. Here are my tweets:

  • About to hear Dr. Derek Haylock at #MaSThei5. http://derek-haylock.blogspot.com
  • #MaSThei5 creativity is not normally associated with mathematics (confusion between artistic and creativity)
  • #MaSThei5 find 2 numbers with a sum of 9 and a difference of 4? When we have the knowledge, what blocks us accessing it to solve a problem?
  • #masthei5 what are the processes that characterise creative thinking? How do we recognise creative product What kind of people are creative?
  • #masthei5 what conditions foster creative thinking? (all in maths context)
  • #masthei5 Derek Haylock demonstrate that we're all fixed, rigid thinkers by nature. We have to choose to think flexibly.
  • #masthei5. Equal pieces problem – will demonstrate on blog how we're all rigid by nature.
  • #masthei5 flexible thinking is the first step on a creative process in maths. Avoid rigidity an fixation.
  • #masthei5 2 kinds of fixation common in maths that limit creativity: algorithmic and content universe
  • #masthei5 ask children to draw a rectangle. What do most of them do?
  • #masthei5 creativity in maths includes thinking divergently: fluency (many), flexibility (kinds), originality, appropriateness.
  • #Masthei5 appropriateness is easy to define in maths (as opposed to art, writing, etc) so teachers fixate on this one part of divergence
  • #masthei5 how to develop divergent thinking in maths: problems with many solutions; problem-posing; redefinition.
  • #masthei5 redefinition – come up with lots of responses by redefining the elements, eg: what's the same as 16 and 36?
  • #masthei5 redefine by using lots of different ideas to create subsets of a given set of numbers
  • #masthei5 conflict between creativity an accuracy – do we value creativity as much as accuracy in maths?
  • #masthei5 graph of attainment vs. creativity (as Derek Haylock defines it) show 0 children in the high creativity, low attainment sector
  • #masthei5 factors associated with maths creativity include low anxiety, high self-concept, risk-taker, high attainer, being a boy. 
  • #masthei5 creative maths children are also 'broad categorisors'. They are good at identifying the same about numbers+ideas and make links.

Writing Assignments

Course Tutor, Mary McAteer gave us some top tips and hints to help us successfully write our first piece of level 7 writing.

  • #masthei5 Mary McAteer reminds us to demonstrate an understanding of ethical issues in essay and PLL
  • #masthei5 warns us against over use of Excel as a presentational tool for simple data

Place Value

Ian Sugarman definitely had the graveyard shift on the day. The last session after a big lunch on a 6 day week – on a Saturday when most would be out shopping, or slobbing in front of the TV – can't have been an easy lecture. And when the subject is the dry area of place value, it's always going to be a tricky one. The biggest thing I got out of this lecture is the warning against the indiscriminate use of number lines and the value of teacher column methods for securing place value when ordering decimals.

  • Context for place value #masthei5 getting things 10 times out can be at best expensive; at worst lethal…
  • #masthei5 misconceptions of place value after the decimal point are rife between ages of 7-11. Half-learned rules and over-generalisations
  • #masthei5 when pupils are given opportunities to explain their thinking, they often spot their own flaws.
  • #masthei5 to get place value it's helpful to sort and justify before ordering
  • #masthei5 talks about left-justifying decimals when I think it's helpful to justify by the decimal point
  • #masthei5 to get x10 relationship it's helpful to use pictures or Dienes apparatus to visualise place value
  • #masthei5 recommends http://nlvm.usu.edu – university of Utah website for good models and images.
  • Great activities advertised at #masthei5 at http://numbergym.co.uk (but not free)
  • At #masthei5 Ian Sugarman talks about standard algorithms can be a sledgehammer to crack a nut in questions like 81-78.
  • #masthei5 numberlines vs standard algorithms vs necessity of getting place value = conflicting interests
  • #masthei5 British children have been referred to as 'pathological splitters', as they partition numbers in both addition and subtraction.
  • #masthei5 Ian Sugarman advocates empty number lines, but not as another rote-learned method. Draw from 0 and emphasize progression.
  • #masthei5 maths in Holland always starts with a real setting, whereas in UK we start with pure maths.
  • #masthei5 can use 'same difference' method as alternative to empty number line for examples such as 83-37 (86-40 is much easier)  

Does ‘mean’ mean ‘mean’, ‘mean’ or are you being mean?

Ambiguities

In researching for my assignment (about whether social media can enhance maths learning), I've come across lots of interesting stuff about language. I'm bemused with myself that I'm saddened at the fact that the mere 2500 words I've got to write my assignment won't be enough to include many of the interesting things that I've found out. Interesting questions more than anything – because it's a whole world of learning. But it's also another essay to be written at a different time.

Suffice to say, at this point: maths teachers – you are amazing.

Why are we amazing?

Because we can explain to our children when leaves means equals.

And when vulgar isn't rude.

We know that multiply can make things smaller; that even isn't always smooth; divide can mean share and group; differentiation is more than just good practice for including all children in learning.

And we can explain that mean means mean (as in average), as opposed to nasty (although it might feel like it), or intend.

Maths. It's all about the words, don't you know.

What is Maths?

This is my definition of maths.

Maths is elegant. It is graceful and swift.

However my favourite is by my MaST colleague Matthew Cham who worked with his children on the definition. Together they came up with:

Mathematics –
a spoonful of numbers,
a sprinkling of symbols,
200g of quantity,
100g of measurement,
All mixed together and moulded into the 
shape of your choice!

MaST stands for Mathematics Specialist Teacher. It's a Master's level study designed to help us primary school teachers be a bit more clued up on the big picture of maths. We all had to come up with our definitions of maths, but they're all hidden away on the universities Blackboard system.

So did the exercise with my children on this wallwisher.

Please respond to this post with your own definition of maths or post on the Wallwisher (although as I write, wallwisher does seem to be misbehaving a little at the moment – bizarrely I find it works best on IE8)