Algebra – frogphilp.com

Teaching computing to a blank page

At my school, I’m on a journey of learning how to both lead and teach computing. I wrote about a planning framework previously. These next 2 posts are about lessons.

In some ways it’s easy teaching computing to children who have had no prior experience. Children at my school, whilst they are strong in IT and digital literacy, have had minimal experience of what used to be called the ‘control’ strand of the ICT curriculum, and is now called ‘computer science’. They are very much a blank page.

I am aware of the damage that can be done to blank pages. When teachers-who-know-a-little misteach, it makes teachers-who-know-a-lot despair. A criticism of much primary science by secondary science teachers is that children often do the fun stuff without really understanding it at primary, so that by the time they’re ready to do the fun stuff and really understand it at secondary, the students dismiss it because they’ve ‘done that lesson before’. Obviously without Bunsen burners. We don’t have Bunsen burners in primary schools.

Another example is algebra. @oldandrewuk was telling me recently how he would prefer it if no algebra teaching was done at primary, because it would make his job teaching algebra in secondary maths so much easier. Non-specialist maths teachers can’t help but teach misconceptions with a complex area such as algebra and thus it would be better to leave it to the specialists.

I’m aware that computing may be similar and I would be interested to know what secondary colleagues think about the computing teaching going in primary schools – do they expect to have to correct children’s misconceptions? Would it be easier to start from a secondary school blank page? Or is some knowledge a good thing?

Either way, I’ve taught three hours of computing today to a class in my school who were very much ‘a blank page’ and I’d be interested for people to pick apart my teaching and consider what is helpful and unhelpful to their long term progression as computer scientists. I’ll write about my lesson in my next post.

Image source: http://upload.wikimedia.org/wikipedia/commons/7/70/Bunsen_Burner_(PSF).jpg

A greater stretch in mathematics? If only.

I read the letter from Michael Gove to Tim Oates about how the new National Curriculum Review would affect the Programmes of Study within Primary Schools with a great deal of anticipation, and then a growing feeling of disappointment. There are various areas that disappoint me, but the area of maths teaching is perhaps the greatest. I read:

In mathematics there will be additional stretch, with much more challenging content than in the current National Curriculum. We will expect pupils to be more proficient on arithmetic, including knowing number bonds to 20 by Year 2 and times tables up to 12 x 12 by the end of Year 4. The development of written methods – including long multiplication and division – will be given greater emphasis, and pupils will be taught more challenging content using fractions, decimals and negative numbers so that they have a more secure foundation for secondary school.

Minor Disappointments

Let me break this down then. Here are some of the minor points that disappoint me:

“Knowing number bonds to 20 by Year 2“ – this is already the case. How can it be more stretching to maintain an expectation – surely the bar should be raised somehow.
“Times tables up to 12×12 by the end of year“ – this is a slight rise in expectation as children are currently expected to know up to 10×10, but in my experience it is not the knowing of extra times tables facts that hold back children in the latter part of primary schools, it is the lack of knowledge of corresponding division facts (which happens to be part of the current national curriculum).
Greater emphasis on written methods (like long multiplication and division) – this sounds good, but it’s already in the current programme of study. Just saying something more doesn’t make it more rigorous, nor does it necessarily raise expectations. In addition, I’m all for children learning skills in school such as the skill of performing long division, but I had been under the impression that the new curriculum would be more knowledge based.
“Pupils will be taught more challenging content using fractions, decimals and negative numbers.“ Again, these are all in the current programme of study for children from the age of about 7. Whether children are taught these are up to their teachers and their schools.

So when I read the phrase “much more challenging content“, and put it next to the above examples of challenging content, I’m struggling to see the giant leaps in standards that Michael Gove would be hoping for when his new National Curriculum is implemented.

A medium-sized disappointment

A greater disappointment is to see the phrase “more proficient in arithmetic” without any reference to algebra. As I have written before, children start learning about algebra from a very young age when they start investigating concepts such as larger and smaller. It is the concepts involved in algebra, often linked with precise language teaching, that I think are under-taught or mis-taught at primary level. Teachers shy away from this vocabulary-driven area because it doesn’t feel like maths to them – there aren’t numbers and operations written in children’s books – it doesn’t look as good as arithmetic. When children leave primary school I think they are often under-prepared for algebra – it is in this area that I was hoping for higher expectations within our new National Curriculum.

If you’re good at arithmetic you can go to the shops. Or maybe become an accountant. If you’re good at algebra you can become a rocket scientist. Not that education should just be about gaining a decent job – Gove himself says he wants “a love of education for its own sake” – but I have a feeling that for many algebra isn’t the elegant thing of beauty that I perceive it as, but is a rather lumpy, ugly thing, ringed with fire and tears.

A major disappointment

Aside from my algebra foibles, there is a further disappointment that I think will have a wider implication on maths teaching: teacher subject knowledge. No matter how detailed a Programme of Study or a National Curriculum might be, neither address this problem: we have many teachers within the primary sector who do not have sufficient subject knowledge to teach mathematics effectively. Many primary teachers only have a grade C at GCSE and have had to repeat their mathematics test required by teaching training in order to pass their course.

I have no problem with teachers teaching mathematics concepts that they’re not to sure about, so long as they know what to do when they’re stuck. There should be an expert teacher within each primary school – the maths co-ordinator or similar – who can share their knowledge and expertise when others don’t know the next steps. Too often less-skilled maths teachers don’t seek help from their more experienced colleagues, but struggle with the text of the National Curriculum and any scheme books that support it. Not knowing what to do, they either miss the tricky areas or teach it badly – characterised by repeating themselves more loudly and more slowly, a bit like the traditional Englishman abroad.

It is not a new Curriculum that is going to improve standards in mathematics. We will stretch primary mathematics only by increasing subject knowledge within our teachers.

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:

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.
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.
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.
Don't limit thinking about sequence to the next number. See if the children can see the rule or the pattern.

Do:

Teach patterns from an early an age as possible. Here's Marylin Burns fantastic lesson.
Do give children plain paper for them to represent their maths graphically.
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).
Follow the previous step by asking 'what's my rule?'
Use empty box problems (e.g. 4+□=11)
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).
Do use a trial and improvement approach. This is especially powerful when it can be done using a spreadsheet.
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…