We need to tackle the idea of maths anxiety

I have not blogged for years, partly due to fact I have had little worthwhile that has not or could not have been expressed better by others with greater clarity, and partly due to time pressures.

However, I have spent much of my holiday so far catching up on blogs, articles, videos and generally tweets which I had ‘liked’ as a signpost to revisit them at some point.

I have therefore done much reading about the traditional v progressive debate and have been introduced to Greg Ashman’s excellent Filling the Pail blog (https://gregashman.wordpress.com) – should Craig Barton ever get so desperate for guests on his podcast that I find myself on there, this will be number 1 in my ‘Top 3’ section – and through his blog, and others, I have come to the same realisation as many teachers who clearly trained and began their careers at roughly the same time as me – that much of what we were taught and experienced early in our careers was complete nonsense. Brain Gym, learning styles, student led activities – you name it, I did it. I knew it didn’t work – I just assumed it didn’t work for me.

As I stated above, however, I don’t see the point in repeating what others have already said more eloquently than me, so I wanted to take things in a slightly different direction.

As part of my training, either for my initial PGCE OR the maths conversion course I did a few years later, I distinctly remember reading Jo Boaler’s book “Experiencing School Mathematics: Teaching Styles, Sex and Setting”. For those of you who have not done so, let me describe briefly: this is a write up of a research project comparing two schools, one of which exposed its maths students to traditional teaching, and the other of which taught students using projects and themes, where students directed the learning. 

I say I remember reading it; rather, I remember trying to read it. It wasn’t mandatory reading, and I struggled to understand much of the technical jargon around the actual scientific research. But I remember the book and, as a UK teacher who had moved to the US, I was interested in Jo Boaler’s work and her career.

As time has gone by, I have discarded Boaler’s methods, and from my own experience have arrived at the conclusion, borne out by my results, that traditional methods are better, certainly for me. This certainly seems to be backed up by many of the articles I have read, and much of the anecdotal evidence I have seen in the blogs of others. I suppose one caveat is that I am susceptible to living in a bit of an echo chamber – so if you have experienced success with a progressive, reform curriculum, let me know – I have changed my mind once, and will gladly do so again if the evidence persuades me to do so.

One of the bees in Jo Boaler’s bonnet at the moment appears to be the idea of ‘maths anxiety’ and that this is caused, at least in part, by timed tests. As ‘evidence’ for this, she offers this paper – which notes that where students are already anxious about maths, their brains show more activity in areas associated with anxiety, and their accuracy is lower. It shows that students identified as having ‘high maths anxiety’ had a lower difference in their response time for simple and more complex problems than those who had ‘lower maths anxiety’ (i.e. ‘anxious’ students took a similar time to answer both types of question, the implication being that students with ‘lower maths anxiety’ answered the simple questions quickly, and took longer for the more complex problems – which is what I would expect as a maths teacher). What this paper DOES NOT show, at least to my eyes, is any link between the timed nature of the tests and increased maths anxiety (if I have missed this due to inexperience, please let me know and I will retract this). So Boaler is being obfuscatory, you might say.

Skip forward a few days to this: https://twitter.com/joboaler/status/895695882431062016

Here is where I begin to have a REAL problem. From the article linked to in this blog, it becomes crystal clear that Boaler is pushing her progressive, reform agenda by refusing to accept of the benefits of factual memorisation AT ALL. You can find an excellent rebuttal of this article, bit by bit, here (written by @iQuirky_Teacher).

I actually find this post incredibly arrogant in its language and patronising in its tone, but that might just be my perception. Bits of it are laugh out loud funny – or at least they would be, if we weren’t dealing with the education of children (note – we don’t say ‘right honorable [sic] minister’, and nor was Michael Gove the ‘education minister’, but I’ll let that go). The fact that she starts by asserting that not knowing the answer to 7 x 8 is a ‘harmless error’ is worrying. 7 x 8 is a short, indisputable fact. Not only that, but it is a simple mathematical fact. Not as simple as 3 x 5, granted, but simple nonetheless. I don’t see why a man in his 40s should get this wrong at any point, whether being interviewed on radio or not. I would be quite annoyed if any of my year 7s got it wrong. It is a fact to be learned and practiced to the point of automaticity.

The rest of the article makes a number of preposterous arguments that Boaler advances as a way of pushing her own resources, articles and website. Now I don’t have a major problem with her doing this APART from where the points she makes are damaging towards children and teachers. It would be quite easy to dismiss her as someone working in the US, where the system is different and we can ignore her and so on, but this would be a mistake. In an internet age, where I can instantaneously read the blogs and tweets of British academics working in Australia and America, ideas spread quickly. Which means bad and damaging ideas spread quickly.

So without trying to replicate the work of @iQuirky_Teacher, let me discuss a couple of points from Boaler’s article:

  1. the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging” –

No, no, a hundred times no. The memorisation of maths facts is VITAL. Maths facts are like strong, sturdy foundations. Without foundations, a building cannot be constructed. Without maths facts having been memorised, we cannot begin to construct solutions to complex problems.

Ah, but is memorisation of maths facts damaging, though?

No, of course not. Don’t be ridiculous.

2. “[the ideas] that math facts are the essence of mathematics, and, even worse that the fast recall of math facts is what it means to be a strong mathematics student….are wrong and it is critical that we remove them from classrooms, as they play a large role in the production of math anxious and disaffected students.

If maths facts are NOT the essence of mathematics, what is? Granted, maths facts are not exciting, or sexy, but without them students won’t be able to do the fun, exciting and sexy stuff later on – or they will, but it will take them far longer, and they will experience much greater struggle. Maybe this is what Boaler wants – maybe she thinks students SHOULD spend years and years of their lives working on maths, protected in a bubble where they are never tested and they work everything out for themselves. Surely THAT is damaging for them?

And as for playing a large role in the production of maths anxious students – what could possibly make you more anxious than an inability to tackle something because you don’t have the basic skills?

3. “When students focus on memorizing times tables they often memorize facts without number sense, which means they are very limited in what they can do and are prone to making errors –such as the one that led to nationwide ridicule for the British politician.

I don’t know if Boaler is suggesting here that the only possible opposing view to her own is that EVERYTHING is memorised, but I do hope not. 7 x 8 always equals 56. This is what makes it a fact which is so easy to learn, remember, and apply automatically. Once this has been committed to memory to the point of automaticity, then of course any teacher worthy of the name would ensure students are using this fact in a range of contexts. I can assure Jo Boaler than in the time she has been away from the UK, schools have not become rote memorisation factories filled with ‘teachers’ whose sole job it is to ensure students memorise everything mathematical.

4. a) “Some students are not as good at memorizing math facts as others. That is something to be celebrated”

No, it’s not. It’s something to be practiced. Saying ‘I’m not good at memorising things’ is rather similar than saying ‘I’m not good at maths’. It’s damaging and it’s unacceptable. Luckily, you can do something about it.

          b) “many of us would probably assume that those who memorized better were higher achieving or “more intelligent” students.”

Maybe you would. I would assume that those who memorised better were better at memorising, and would use them if I could to help the others.

          c) “Some students will be slower when memorizing but they still have exceptional mathematics potential. Math facts are a very small part of mathematics but unfortunately students who don’t memorize math facts well often come to believe that they can never be successful with math and turn away from the subject.

So what, we test them once, and let them live with those scores forever? Or we encourage them, help them, support them – you know, like good teachers do ANYWAY – and help them see improvements?

This is where I believe we are getting to the most revealing and damaging part of this whole article, and, by implication, Boaler’s whole philosophy. She seems to have the following thought process:

  1. Timed tests are bad, we shouldn’t do them
  2. This causes maths anxiety for some students
  3. Students who can’t do them are made to feel bad and will want to give up
  4. This will damage those students for the rest of those lives

This makes no sense, but if Jo Boaler wants to bang this drum in her own home, let her. But these ideas are fundamentally dangerous and must be challenged. Here is my take:

  1. We should be doing timed tests, as these promote automaticity. However, we must make them low stakes/no stakes. I personally would recommend low stakes, where scores are recorded, and students are rewarded on the basis of improvement, not simply for getting 100%. We should not expect a class of students to get 100% every time. But we should definitely expect students to improve their scores over time, and we should support them, whether in class or without, to ensure as much as is in our power than this happens.
  2. Despite Boaler’s claims, I am yet to be persuaded that maths anxiety is a thing; furthermore, I do not believe that any such condition, if it does exist, would solely be caused by timed tests. In any case, I refer you to point (1) immediately above.
  3. Who is making these students feel bad? Not any teacher I am aware, nor, I imagine, any teacher worthy of the name. If this is happening, it is happening because of the importance attached to these tests, be that by the teacher, the school, the ‘system’, or the parents.
  4. This is a ridiculous argument, and much as I hate to bring the subject up, seems very ‘fixed mindset’ to me. As teachers, we have a responsibility to convince students that they will improve with hard work. There are any number of examples to draw upon from all walks of life. Andy Murray didn’t quit tennis after years of failing to win a Grand Slam; Thomas Edison didn’t quit after thousands of failed prototypes; JK Rowling didn’t give up after 12 publishing houses turned down Harry Potter. None of this is to say that ALL children can be 100% successful 100% of the time, but it IS to say that it is possible for 100% of children to improve 100% of the time.

And with all of this, let us not forget – we are not talking about timed tests all lesson, every lesson, for every skill. We are talking about short periods of quick recall of basic facts. I am, anyway. Maybe in America they do spend hours on timed tests. But I doubt it. What I do not doubt is that if Boaler is allowed to continue to propogate her ideas about maths anxiety, then maths anxiety will become an issue. It may even become a ‘cool’ badge, the new ‘well I’ve never been good at maths’ or ‘I have a rubbish memory’. This will be incredibly damaging for our current and future students. We have a duty to kill this weed before it takes root in our educational conversations.


A better way of planning – my first attempt

I was blown away by the post by @leadinglearner yesterday (which can be found here – and if you haven’t read it yet, it might be a better use of your time in the short term than continuing reading this). One of the reasons, I think, is that I can see how it links to things I have been trying to do (see my series of posts from last summer) but takes it to a completely higher level – but in a way that is easy to see. I was dead keen to try this out for myself, and so….I did!

Attached below you will find a document for a series of lessons I have designed for my year 7 bottom set. I am sure it can be improved (not least in terms of the SOLO learning intentions), but you have to start somewhere. I do wonder if it is a little repetitive in places, but I do wonder if that is sometimes the nature of the beast in maths, not least at such a low level.

Anyway, have a look, and if you think it can be improved, please let me know! I will continue using this model for my planning of all classes and so may end up sharing more on here, and any feedback will be most useful and helpful. Thanks

New SoW 7.3 – Adding and subtracting!

Half term report (or simply: breathe!)

As I alluded to previously, Christmas 2014 saw me change jobs after 4 and a half years. Having reached the end of 6 hectic weeks, I thought it might be a good time to document my experience, thoughts and fears, not least because hopefully putting my thoughts down in print will allow me to mentally declutter!

  • Change is difficult at any time, but especially part way through the year – I guess this is obvious, but even now I still don’t think I am fully ‘there’. I was so comfortable at my old school – too comfortable, in fact, which was one of my reasons for leaving – and the first week in particular was a huge shock to the system. In fact, it became a simple case of survival at times. But a rough plan of what I was doing, and supportive colleagues who helped me bed in quickly, got me through to the weekend, where I could regroup.
  • Schools which appear quite similar can be very different in practice – both my old and new schools serve predominantly white, working class intakes from ex-mining communities. But that is pretty much where the similarities end. The ethos of my new school, the behaviour management systems, and the aspirations are all very different (in a good way). I know this was a great move for me. I can’t wait for that day when I finally feel like I’ve arrived and am fully comfortable in the post.
  • Expectations on all levels need raising  – both in terms of behaviour and in work ethic, it’s clear standards for some of my students were a little low. My year 8 class admitted earlier in the week that the previous teacher was soft on them in terms of the BM system in place – which all staff are required to follow religiously. I feel like I’ve made some progress here, although there is still a way to go.
  • I have a large/scary amount of control over my class AND my environment – the freedom I have is something I have found difficult to cope with at times, and has led to me having crises of confidence on many occasions. We have no scheme of work at all, so I decide what to teach and for how long. This can be quite a daunting prospect, although I also recognise the huge opportunity it provides. Similarly, the fact I now have my own room is a great opportunity for me to impress my values and expectations on students – it had been my aim to get a display put up this week, but the replacement of windows and the associated removal of asbestos prevents this from happening.
  • I just need to focus on the process and not worry (too much) about the outcomes: like George Michael, I gotta have faith – in my running, I’m all about the process – running consistently, through all weathers and good and bad runs alike, will deliver the outcomes I want. Similarly, I need to be confident in my methods and strategies of teaching, and know that if I am producing the goods day in, day out, students will make the desired progress.

Although half term hasn’t really started yet, I’m looking forward greatly to the second half of the spring term. I feel in a much happier place, and feel this is the time to start kicking on and making big inroads on student progress. I look forward to catching up with old blogs to inspire me as ever!

A potential epiphany….

“Jaws isn’t about a shark, and Tinker Tailor’s not about spying” – Mark Kermode (paraphrased)

This quote from one of my favourite podcasts struck me as I began planning for next week this morning. I was creating my objectives/SO THATs/success criteria, when I began to wonder whether SO THATs were actually pretty indistinguishable from success criteria. A quick glimpse at few of my ‘go to’ blogs in this area (here by Dan Brinton, and here and here by Zoe Elder) convinced me that there was sufficient difference to continue treating them as separate parts. So I began planning. But two different lessons, for two different groups, brought about the same observation in mind – “it’s not about that, it’s about something else!”

Example 1: Year 11 – WAL about estimating the mean SO THAT we can achieve full marks on estimating the mean questions.

Success criteria: a) we can calculate an estimate for the mean from a set of grouped data; b) we can calculate proportions from a set of grouped data

Example 2: Year 7 – WAL about area and perimeter SO THAT we can calculate area and perimeter of squares and rectangles.

Success criteria: a) we can calculate area and perimeter by counting squares; b) we can find area and perimeter of shapes not drawn on squared paper; c) we can explain the formulae for area and perimeter

Now you are probably ahead of me already reading that, and know what I’m about to say, but I can honestly say this is a bit of a Eureka moment for me which will change my planning from this day forward. I was using Zoe’s approach of the WAL as the ‘what’ of learning and the SO THAT as the ‘why’ of learning, with the success criteria being a ‘how will I know I have learned it?’ check. But a brief glimpse of my plans for year 11 would suggest the lesson isn’t about estimating the mean, it’s about grouped data, with estimating the mean providing a context for that learning. It’s what I believe Dan Brinton writes about in his blog (citing Shirley Clarke) which I have linked above. Similarly, if we look at my plans for the year 7 lesson, I’m not sure that is a lesson about area and perimeter; rather, I think it’s a lesson about formulae using the context of area and perimeter. And actually, that’s not what I want that lesson to be about.

So I made refinements. My year 11 lesson became:

WAL about grouped data SO THAT we can accurately answer questions on estimating the mean and cumulative frequency.

Success criteria: a) we can estimate the mean from a set of grouped data; b) we can draw an accurate cumulative frequency diagram and derive quartiles and the median from it; c) we can calculate proportions satisfying a condition from both types of representations of data

and my year 7 lesson became:

WAL about shapes SO THAT we can find the area and perimeter of squares and rectangles.

Success criteria: a) we can find area/perimeter by counting; b) we can find area/perimeter of shapes not drawn on squared paper; c) we can explain how to find area/perimeter for a shape where we don’t know the measurements

I think, particularly with the year 7 lesson, there is still an implicit tendency towards formulae at the end, but the lesson is now much more clearly focused towards area and perimeter. What are we learning about? Shapes. Why? So we can find the area and perimeter of squares and rectangles. How will we know we have been successful?….. and so on.

Having thought closely about this area this morning, I think there will be occasions where the WAL, SO THAT and success criteria may need to be closely linked, particularly towards the knowledge acquisition end of things. But equally, most of the time it will be appropriate to redraft the plans, even before the content of the lesson in considered.

What do you think? Have I got the wrong end of the stick, or does it seem like I’m on the right track? I’d love any feedback you may have, either here in the comments section or on Twitter via @Still_Improving. Thanks for reading!

Day 2 reflections – Don’t worry about a thing….

….’cos every (little) thing is gonna be alright!’

Well, day 2 seemed to begin with a bit of internal contrived panic – I had a last minute revision session with my year 11s despite not having seen them yet – but once that had passed, things seemed to become clearer in my mind. I ran around less, people came to find me less, and I taught a few lessons, too!

It just takes a bit of getting used to. All of it. Having my own room again (I still keep thinking somebody’s going to come and take it off me any time now!), split lunches, and the quiet on the corridors at break and lunch (it’s out of bounds for students). I love the PD system the school has – instantaneous and with no follow up or paperwork required! As a result, my lessons need tightening up a bit, but I’ll get there soon enough.

There are still of course SOME issues, but it is still early days. I have noticed I am slouching at the board, and in fact I’m doing too much at the board full stop. A balance that needs redressing. Overall, though, today was a much happier day than yesterday, and I suspect tomorrow will be even calmer still. My books are neatly filed, my seating plans written up neatly and printed out, my desk is tidy. My inbox is virtually empty. I’m feeling much more organised, and once again my new colleagues have been amazingly supportive. And I’m even smiling, more so than I ever did at my last school (again, new year, new habit!)

And best of all, it’s nearly Friday…..!

Day 1 reflections

New year, new term…..NEW JOB?!?!?

Finally, after much waiting, I took up my new post. I felt I’d had a decent break, I slept well and woke at the time I’d planned. I was in school early, eager to get cracking….until stuff got in the way. Like not being able to get in the building (note to self: if the school has a keypad entry, find out the code BEFORE the first day), not being able to connect to the school wifi, finding out my year 9 class have 6 lessons a week, not 5, and therefore one lesson a week with another member of staff, who teaches them on a Monday, and………

Yes, it was busy. No, I still don’t have a clue what I’m doing. But yes, it WILL get better. I am sure of it. Not least because I have my own room (although I haven’t really even begun to think about what I’m going to do with it), lots of TAs to help me, a very understanding HoD, and a corridor full of colleagues willing to listen, and help.

I’ll be fine. It’s just going to take a while to get up to speed.

I can’t wait for that day….!

Looped polygons (odd number sides)

So much of my Sunday was occupied with the looped pentagon problem. Having solved that, I was intrigued to see how the problem would play out for other polygons. I have spent most of my time since focusing on odd-sided polygons (because I think even-sided ones are relatively straightforward, although I need to check) and thought I might share what I had come up with so far.

Firstly, I have to say this has been a bit of an eye-opening experience for me. Not having a solution, or any real help, has added to the intrigue. I’ve extended the problem, I’ve made connections and generalisations, and I’ve refined my approach all the time until now, where I feel I have a method to share. It’s the kind of thing I’d love to spend time on with students….if only!

The general problem is like this: a polygon is marked, with circles of 1cm radius placed around its vertices. A piece of string in looped around each vertex and pulled tight so there is no sag. The challenge is to calculate the length of string needed. In the original pentagon problem, the distance from vertex a (at the top) to vertex c (bottom left) is given as 5cm. I will refer to it simply as l.

The main issue I have found so far, and which I may well go back and revisit at some point, is deciding the order in which the string is looped. Once you move to a heptagon there are options. In all cases I present below, I have opted to go for the furthest vertex from the top vertex, which I have called the bottom left. The string then loops back up to the vertex immediately to the right of the top vertex, and so on.

I had, you will recall, noticed that l was parallel to the string – it must be so, as extending the string would create a tangent to the centre of the circle. My approach therefore involved a lot of angle theory. Beginning with the equilateral triangle, I split the interior angle in half to give 30 degrees, as this allowed me, by deduction, to calculate the angle from due North to the string. This done, and the two added together to give 150 degrees, I subtracted the tangential right angle to leave an angle of 60 degrees from due North to the tangent. Drawing a right angled triangle showed me that the angle from the centre of the circle to the tangent was 30 degrees from East, meaning that on both sides of the circle, the string was wrapped around an extra 30 degrees before moving towards the next triangle.

This gives a total contact with the circle of 180 + 30 + 30 = 240 degrees, or 2/3 of the circle’s circumference. The LENGTH of string needed can be calculated by 2/3 x 2 x pi for one circle, and then multiplied by 3 for all 3 circles. This gives a total of 4 pi for the circles, plus 3l for the straight bits, for a total of 3l + 4 pi. Knowing that the pentagon gave a solution of 5l + 6 pi for the length of string, I began to conjecture that for any s sided polygon, the length of string needed was sl + (s +1) pi.

I next tried a nonagon, being a shape with the number of sides being a factor of 360. And this was where I got stuck for a while. I guessed the answer was 9l + 10 pi, but this was based on me guessing the angle from due North to the bottom left circle. I eventually realised that the answer may lie inside the polygon – in the centre, in fact. By taking a line segment from the centre of the polygon to each vertex, I had the angles there, ready to go. So for the nonagon, each vertex was a movement of 40 degrees (360/9). Using angles on parallel lines, I could see that the angle created by drawing an isosceles triangle from the points due North, in the centre of the polygon and the bottom left vertex, gave me the angle I needed to subtract from 180 degrees. In this case that angle is 10 degrees, meaning the tangent/due North angle was 170. This, to cut things a little short, means that 10 degrees was the angle beyond 180 where the string was touching the circle on each side. Here, this resulted in 180 + (2 x 10) = 200 degrees, a total of 5/9 of the circle’s circumference. Again, the total curved contact can be found by calculating 5/9 x 2 x pi x 9, which equals 10 pi. So the total length is 9l + 10 pi.

Having found a shortcut, I then tried the heptagon. This was particularly brave given my lack of a scientific calculator, but I got there in the end. The large angle in the isosceles triangle I needed was 3/7 x 360, so the small angle worked out at 12 6/7 degrees. This was the angle beyond 180…..etc etc….and I eventually arrived at a length of 7l + 8 pi.

I had spotted another pattern here, and stopped to investigate. The proportion of the circle which was touching the string was 2/3, 3/5, 4/7, and 5/9, and can be expressed as (s-1)/s for a s-sided polygon. What I also found fascinating was angle from due North vertex to bottom left vertex as a proportion of 360: 1/3, 2/5, 3/7, 4/9…clearly this will gradually get closer to 1/2 without ever reaching it, and can be expressed as 1/2(s – 1) / s for an s sided shape. And if you add this angle to the sector touched by the string, you get 1 every time. Pretty neat, huh?

I realised now that this allows me to quickly work out the information I need without having to worry too about the angles – so for a 15-sided shape, we take 7/15 (168 degrees) as the largest angle in the isosceles triangle, meaning the angle beyond 180 degrees will be 4/15 (6 degrees) on each side, or 8/15 (12 degrees) in total. This will give total string contact of 15l + 16 pi.

With odd sided polygons, I think the next step is to investigate whether a different (but still repetitive) looping arrangement changes the total  amount of string needed. I’m not sure at the moment, but I do know I have some strategies which will hopefully allow me to find the answer pretty quickly!