Meandering Maths

Learning journal for MaST

Planning for serendipity: #DoMoreEdu meets level 3 students at my school

I had the fortune of being invited to the #DoMoreEdu event at BETT on Saturday by the team @DellEDU. Whilst I couldn't stay for the full event I was impressed with the way that the discussion, led by Ewan Macintosh and Tom Barrett of Notosh.com, created engagement of its own accord, regardless of the content of the discussion. 

One of the earlier topics in the conversation was about how we use space. Now I have to admit that I'm pretty nonplussed about the issue of 'space' in schools, firstly because I'm of the belief that the relationship between teacher and student is so crucial that the issue of space makes only a tiny fraction of a percentage of difference to education, and secondly because I work in a serviceable 30s built school. It isn't perfect but it works and we get decent results.

What surprised me however was that the discussion engaged me. Motivated me, even.

On to serendipity. It's not the kind of thing that one associates with schools. So much so that I had to ask the guy sitting next to me what it actually means. The top search in Google gives me: The occurrence and development of events by chance in a happy or beneficial way: "a fortunate stroke of serendipity". So it's something about happiness, good luck and benefit. Yet our planning and our timetabling is so tight, so rigorous, so full of targets, that I struggle to see how we can 'plan for serendipity' within the current system.

But maybe the answer was right there before my eyes - it was in the discussion that I was engaged; through the growing relationships around the table that I was motivated. It wasn't the content of the discussion that mattered so much at that time - more the process.

Back to school on Monday and I was preparing my first lesson for a group of level 3 students who are really struggling with maths. Disengaged, with low self belief, they find maths extremely hard. Both their attainment and their progress is below where it should be. I knew all that and realise that I have awful lot of content to teach them if they are to make level 4 by May (when they sit their SATs tests). Could I afford to give up a single lesson just to engage them?

I decided I could and so on Monday we sat around a conference table and held our own #DoMoreEdu. We talked about moments of unhappiness and happiness in maths learning. We talked about how we should organise the space and how we could find more time. The children made suggestions for how and what they should be taught. They resolved to meet online at 6:00pm on Tuesdays for an extra revision timetable. Some of their discussion is recorded here. We also wrote our first blog post about that session. Since then we've had our first real maths lesson, which did actually contain some real content.

It was by chance that I booked an extra night's stay at BETT that allowed my to take advantage of the kind offer from @DellEdu. It was also bad planning when I realised that my train ticket actually said 11:23am, not 1:23pm, meaning that I had to dash off earlier than expected from the session. But that chance and bad planning allowed me more time to reflect on the experience in the light of teaching nine disaffected maths students on Monday morning, and already they have shown a higher level of engagement than I expected. Is that serendipity? Maybe it is. Certainly giving up one lesson of content and direct instruction to gain more motivated students who are willing to participate online in their own time is a win for me.

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Good teaching decreases mathematics anxiety

This weekend, I found myself doing something I've not done before - disagreeing with Professer Derek Haylock. Giving his second lecture to Edge Hill MaST cohort 1, the man who's seminal work "Mathematics Explained for Primary Teachers" has pride of place on my shelf, said some things that didn't quite hang together for me.

His lecture was on the subject of mathematics anxiety - something that most adults have either experienced or can empathise with. His main point was this: if you teach mathematics well, you don't get students who are anxious about maths. As someone tweeted on the day "My God I never thought of that. I hope the person giving this advice is paid a fortune." Given that the audience was a room full of primary maths specialists, or 'maths champions', the advice is more purposeful if given a more negative slant: don't allow bad maths teaching in primary - you'll just get adults who are anxious about maths.

Briefly I will sum up what I thought were his main points and then I'll say where and how I disagreed with him.

  • Many adults experience anxiety in maths when they are afraid to make mistakes in public, or given a mathematical challenge they cannot think clearly to carry it out.
  • These adults can trace their feelings of anxiety back to a single experience usually between the ages of 9-11 at primary school.
  • This experience is always a negative interaction with a teacher - Prof. Haylock quoted adults saying that their teacher had shouted things like "why can't you just get it right?" There was a real emphasis on the negative experience being when maths is thought of as either right or wrong.
  • Many of these adults reported they could only learn maths by learning a rule by rote and couldn't master any conceptual learning.
  • Some of these adults become primary teachers.
  • Teaching styles are to blame for mathematical anxiety - 'traditional methods' create more anxiety; a 'problem-solving / relational approach' creates less anxiety. Quoting from Newsted, he described a traditional approach as one of direct instruction, followed by practice and application, whereas in the 'problem-solving approach' the teacher acted as a facilitator, with the children suggesting their own methods and strategies for solving problems.
Aside from the dangers of telling rooms full of teachers that 'rote learning is always bad' and 'this is the only way to do it', my main disagreement was the way he linked the single negative experience with a given teacher to the traditional teaching method. It doesn't take the room being in rows or table groups for you to have a bad experience with a teacher. Neither does it mean that you if are using a 'problem-solving approach' then teachers can't lose their tempers and make everyone frightened of maths.

In my own experience I've tried both traditional and 'problem solving approaches'.

I would call them using a rigid scaffold and using a negotiated scaffold. In the former, the teacher plots the course through the learning (the scaffold) and takes the students through that course through direct instruction, practice and intervention; in the latter the student and teacher negotiate the path through the learning.

Both approaches work.

In fact this time last year I did an experiment where I did 6 weeks of negotiated scaffolding in maths, then 6 weeks of rigid scaffolding in maths. The children made progress in both periods.

Delving a bit deeper into the Newstead report I see that the traditional approach includes: "The teacher decides what is right or wrong and intervenes in the case of mistakes. Later word 
sums may be used as application of methods. Social norms are more static and involve more discipline, rewards and teacher authority." Now to me that's not traditional teaching. Traditional teaching is where direct instruction is followed by practice, yes, but then appropriate intervention from the teacher. And so now it leaves me thinking that Haylock, quoting Newstead isn't comparing 'Problem Solving' with 'Traditional', but is comparing 'Problem Solving' with 'Bad Teaching'.

I'll go on to say that Haylock is right by saying that for a student to have one-to-one negative interactions with an authority figure such as a teacher will cause anxiety, in any subject. The teacher that chooses 'traditional teaching methods' but can avoid the negative interactions can still teach a class without causing anxiety amongst the students. And a teacher that attempts to be a 'facilitator' but then loses their temper when the students don't choose a method they were anticipating will also cause anxiety. It's not about the style, or dare I even say it the teaching, it's about the teacher themselves.

Good teachers reduce anxiety.

Good for the fractions learning; bad for the coffee mug

Fraction_mug

Sometimes children hear the word 'fractions' and they turn off.

I saw it on Wednesday when I started my lesson on comparing and ordering fractions. I had barely uttered the words when I saw a few heads drop. A few children joined in when I asked them what they knew about fractions - one knew the word 'third'; someone else knew 'part'; yet another one knew they have something to do with division. But quite a few heads with dropped.

So while the keen had their hands up, and others were looking to avoid eye contact, I slid an empty coffee mug into an empty plastic bag. Then, for security, whilst the conversation continued, I placed the first plastic bag into a second one.

Then I smacked it against the wall. Really hard.

All the children looked - some jumped.

I proceeded to pull pieces out of the bag and estimate how much of the mug each piece had been, from the large chunks (1/3 or 1/5) to the tiny chips that were only 1/1000 or maybe even smaller.

The children were engaged and by the end of the lesson all of them had made some progress about ordering and comparing fractions. Even the special needs group children who, according to their data, struggle to order numbers 1-100.

As a bonus, we even specified that the bottom of the fraction was called the denominator and the top number the numerator - I love it when children learn proper maths words, although it was amusing to hear one child call the top number the nominator and the bottom number the dominator.

So, if you're stuck with teaching fractions - break something. At least you'll stop the heads from dropping...

The unconsciously incompetent

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What happens when you build something on shaky foundations? It falls down.

Knowledge is a bit like that. Concept is built on prior concept giving rise to skills that can be practised and applied.

I was reminded about this in a maths lecture today at Edge Hill University on the subject of 'errors and misconceptions'. We were challenged to draw a picture or diagram of errors and misconceptions and how they relate to each other. I tried to draw a picture of a house standing on shaky foundations with each stone representing a concept - some of which were solid, some broken and some completely absent. Whatever your picture might be, the fact is that as maths teachers we can easily see the errors, but it's a bit more difficult to identiy the misconceptions.

An error could be a simple calculation mistake arising from being hasty or not checking the answer. Or it could point to an underlying misconception - a broken foundation - which could lead to many repeated errors.

I remember speaking with a friend when, both in our twenties, I realised he could not subtract even small numbers accurately. He would always be one out. For example 24-9 would be 16. This was because when counting back, he would always start by counting the initial number as 1, rather than 0 - if only he'd been shown number lines at primary school! I'm not quite sure how he got through 'A' levels and a degree - but he did. Anyway, after that conversation his misconception was fixed - the foundation was more sturdy than before.

So as teachers how can we correct misconceptions?

Firstly we need to avoid causing misconceptions ourselves. Have you ever heard a teacher say: "Five take away seven: you can't do that!" or ""Fives into three can't go!"? Have you ever said that yourself? O have you ever drawn triangles, squares or rectangles with all their bases horizontal and parallel with the bottom of the page? The lecturer today called this being 'unconciously incompetent' - a condition we should try to avoid. We need to be deliberate and precise in our language, using words that will enable future concepts to be easily built on what we have taught. In his report of 2008, Williams wrote: "It is often suggested that 'mathematics itself is a language' but it must not be overlooked that only by constructive dialogue in the medium of the English language in the classroom can logic and reasoning be fully developed". So Mathematics teachers need also to be masters of english so that they don't unintentionally teach misconceptions.

Secondly spot the misconception. If an the same error is repeatedly occuring in a student's work, that's a glaring clue. However some misconceptions are harder to find and one-to-one conversations with children - that 'holy grail' of 'Quality First Teaching' which all teachers aspire to where they can spend some time in purposeful dialogue with each child in their care at least once a week.

Thirdly use concrete examples, or models and images. Misconceptions are often because students don't get the abstract form of maths. They need to have to take it back to concrete examples - counters, teddy bears, things, whatever, or at at least use powerful models and images such as the number line. It's amazing how many uses teachers can find for chocolate when illustrating some mathematically...

I'd love to have examples of others experiences of errors and misconceptions - please comment below.

Photo: Broken Brick by Ternus on Flickr

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...

Running out of ideas to increase skills in gifted children

Giga_stars

The picture is what we use as the pinnacle of our '99' club test. This was a test that we introduced a few years ago to help increase children's instant recall and mental mathematics skills. Unfortunately, 99 club wasn't big enough, so introduced a 'superstars club'. Then that was completed so we made up a 'mega-stars' club. Even that wasn't enough, so I designed a killer test - Gigastars club, thinking no self-respecting 10 or 11 year old would pass that. Even my brother-in-law who's a red-hot software designer type could only do half of it in the required 10 minutes.

But then last year, one plucky 10-year old did the test. In only ten-minutes. We allow 3 wrong to get the certificate. She got 2 wrong. What a star. What a 'giga-star' in fact!

But now, what do we do now? She's still got a year with us. Helpful suggestions about where to go next with our skills development would be most appreciated.

Valuing misconceptions on the way to explaining fractions

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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:

4 ÷ 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)