Feedback should improve the student, not the work

This week I've been thinking about marking, feedback, and corrections. Marking books is one of my least favourite teacher tasks, and it's probably the one that takes up the most of my free time. I also believe it has the least impact on my students' learning, but only because the way I do it is pretty rubbish. And spending hours and hours of your free time doing something that you know is rubbish, makes you feel like rubbish. I decided it was time for a change.

Let me first explain the marking policy in my department, which I think is a pretty good one. Our students in years 7 to 10 have two books: a normal exercise book and an assessment book. They use the exercise book every lesson for writing notes, copying examples, answering questions, and doing homework. We teachers never mark this book, but students are expected to self-mark and correct when we go through answers in class. The assessment books are kept in the cupboard until the end of each chapter (roughly every 2-3 weeks) when students complete a semi-formal (silence, closed-book) assessment in them. We then collect these books in and mark them formatively and provide detailed feedback. The next lesson (ideally) we give these books out and the students have DIRT time where they respond to the feedback and do corrections and maybe follow-up questions. These are the books that we will show to the Ofsted inspectors when they come, and we make sure to impress upon our students the importance of upholding high standards of presentation in these.

This policy means that I only have to spend around three hours per week marking, which from what I've heard is pretty low compared to other schools, and is certainly low compared to teachers from other faculties within my school. I shouldn't look a gift horse in the mouth, and I should just get on with it and not complain. However, my questioner personality means I find it hard to follow policies, even ones that I was a part of creating.

Last weekend I was planning my lessons for the week ahead and I thought about the fact that my year 10s would be doing an assessment on the Tuesday. I knew they would generally struggle with the assessment because the questions were hard (it's a higher tier test and my students have targets of grade 5) and I knew exactly which questions they would be able to do and which they wouldn't, and why that is. That's when I started to question: what actually is the purpose of this assessment? And what will be the purpose of my feedback?

If you ask a load of teachers what the purpose of end-of-topic assessments is, you might get the following responses:

1. so teachers can see how much students understand
2. so teachers can see which bits they've taught well and which bits they haven't
3. to inform the teacher's planning of the next unit
4. so students know how much they understand
5. so students can practise exam-style questions in an exam-style situation

Some of these purposes are good, some of them are bad. Some of them are being fulfilled by my assessments and feedback and some of them aren't. I can immediately eliminate purposes one and two. I already know both of these things, through my use of assessment in lessons throughout the chapter. Purpose three is also irrelevant in my case, because the units are usually disparate and unrelated. Chapter 5 might link to chapter 11, but by the time that comes around I would have forgotten this information anyway. Purpose number four sounds decent, but the assessment I was going to give them doesn't fulfil this purpose, as students could get every question wrong but still have 80% understanding of the unit. They could finish the assessment feeling like a failure when they might be close to getting it. Purpose number five is the only-non problematic purpose on the list, in my opinion. I could also add to that a sixth purpose, one that doesn't often get mentioned, which is to make use of the benefits of the Testing Effect, where being forced to retrieve information improves a student's long-term memory of that information. 

Now that I've decided my purpose of the assessment is actually just to expose students to exam-style questions, what should be the purpose of my feedback?

Some common responses:

1. so students can see how much they understand
2. so students can see areas they need to improve in
3. to let the students know you care about them and build a relationship
4. so the parents can see that you care
5. so the school knows you are doing your job
6. so Ofsted know you are doing your job
7. so students can correct their work

Purpose number one we have already touched on when I said that students can't always gauge their own level of understanding. I could address this in my feedback, by RAGing each sub-topic for them, but what will students do with this information? If Ali is an amber in finding the surface area of a prism, how does knowing he's amber help him? Perhaps in year 11 this would be helpful, as he can add it to his revision timetable. This is also the problem with purpose number two. Yes maybe they need to improve their understanding of surface area, but when will they have the opportunity to do this, and how? I don't think setting them a video to watch at home is really sufficient here. They didn't understand it when you, an expert teacher, explained it first time round and gave examples and independent practice. They're not going to suddenly get it (and crucially, embed it in their long-term memory) when Mr Corbett or the Maths Watch lady explain it to them. Purpose number three is debatable. Surely there are other ways to show you care? Surely planning well-thought out and effective lessons shows students you care about their learning? Surely talking to the students in person is the best way to build a relationship? But I do kind of get it. Students definitely do like it when you write stuff on their work, and I know that I liked it when I was a student. It's almost like when someone "like"s or comments on your social media post. You ticking their work probably gives them a similar feeling of validation that the little heart in Instagram gives them. So fine, I'll accept that one. And then purpose four, whether important or not, is easily fulfilled in the same way. Purposes five and six can piss right off. I spend enough hours in the evenings and on weekends planning high-quality lessons and honing my craft through engaging with books, blogs, podcasts and academic research. Let them try and tell me I'm not doing my job.

And then there's purpose seven: so students can correct their work. This is what my written feedback has always been geared towards. I would circle bits that were wrong, correct little slips, give some scaffolding, write some hints (try factorising!) and then leave a nice highlighted box for them to do their correction in. The more I think about this, the sillier this seems. What is the point in a student correcting their maths work? If it's a small arithmetic mistake or little error (the equivalent of a typo), then correcting that mistake doesn't improve the student's understanding of the topic at all. If it's a conceptual error, then is a correction really appropriate, or would a do-over be better? And how can the student really do that without being re-taught that concept? Your cleverly written hints and scaffolding can allow them to produce the correct answer, but will that have a long-term impact on the child? Even if you have gone so far as to provide a follow-up question for everyone who got question seven wrong, won't it take more than just that to embed the concept that they failed to learn the first time round? And what if they've got five out of the ten questions wrong? Do you really want them to re-learn five concepts or skills in one DIRT session?

Marking students' books to allow them to do corrections individually takes absolutely ages. This careful hinting and scaffolding and follow-up questioning is extremely labour-intensive, especially when it's a challenging assessment. For this reason, many teachers opt to simply tick and cross the students' work, and then spend DIRT time guiding the students through every single question so the students can copy the correct answers down. This is much more efficient but seems kind of like cheating. It also makes you wonder what was the point of taking the books in at all? And if there's not much green pen in the book then how are we fulfilling purposes four and five of the second list?

After thinking about all of this I felt like there was absolutely no point giving this assessment and no point marking it. However, that just felt wrong and also I would get into trouble. So I tried to define for myself some new purposes that I could get on board with. And then I saw a Twitter comment that was linked from a thread that was linked from a thread that was linked from a thread about marking and feedback, and it was a lightbulb moment for me.

"Feedback should improve the student, not the work."

I think this is so powerful. Improving a piece of maths work has no long-term impact. It's kind of satisfying to do and it looks impressive to book-flickers, especially when you've done that thing where you go back and mark the corrections, so there are multiple layers of feedback and it becomes like a written dialogue. But this isn't improving the student. Going through the whole assessment as a class and copying down the correct answers probably has a small impact but it's probably not the best way of achieving the same effect. Going through one question at a time with extra practice in between, or even just re-teaching that sub-topic without referring to the assessment until afterwards may be better. 

However, I do want to do the assessment (because I have to, and because of purpose five from list one) and I do want to give written feedback (because I have to, and because of purposes three and four from list two) and ideally this feedback would do what @EduCaiti suggests and improve my students. But how? 

I did come up with something, but I don't think it's particularly great. I do think it's an improvement on what I'm doing before though. I decided that rather than focusing on improving the work that the students had done in the assessment, I would focus on improving something that would apply to other areas of maths, inside and outside assessments. I came up with the idea of having a specific "feedback focus" - one thing that I would focus on when marking their work. I would give the students specific and actionable feedback about that focus and that focus only (although I would also tick and cross their answers). The feedback focus I chose for this unit was "organising working out in a way that facilitates problem solving". I shared this with my students just before they started the assessment. I took their books in, and I marked for accuracy, and then I wrote a comment at the end, on a stuck-in sheet stating the feedback focus. I wrote things like, "The way you laid out question 4 was really good because you worked out each face separately and wrote these on separate lines. It would have been even more clear though if you had labelled these, e.g. front, back, top etc. This would have helped in question 7, when the top of the shape wasn't required as it's an open box." or "You write out the formula you're using at the start of each question which is great, but if you include the start of the formula too you would have less confusion (e.g. writing circumference = π x d is more helpful than just writing down π x d because then you're less likely to get mixed up with the area)." This might seem like it would take longer than my old style of marking but trust me it was much much faster. When I gave the books back, we spent DIRT time looking at photos I had taken of work that demonstrated good practice related to the feedback focus, then reading the comments I had written, and writing an action step based on my feedback. We didn't bother doing corrections.

Next time my students do an assessment I can remind them to look at their action step from last time and make sure they put it into practice. I can then comment on the new feedback focus as well as whether they have carried out their previous action step or not.

It's not anything amazing but at least I feel better about it. I think that whenever we spend a lot of time doing something we should always make sure to ask ourselves what our purpose is, and consider whether we are actually fulfilling that purpose. And if you can't find a good enough purpose for something, that probably means you shouldn't do it.

Emma
x x x

Rules First, Exceptions Later

I know that we as modern, innovative maths teachers like to say that maths isn't about rules and that it's about relationships and reasoning and all that stuff. But there are some mathematical processes that students have to be able to carry out in a fairly standard way and for the sake of this discussion, we're going to call them rules.

Just so you know what I'm talking about it, here are a few examples of processes that involve rules:
-adding fractions
-finding the area of a sector
-finding the volume of a prism
-solving equations of the form 5(x+3)=7.

When we teach these processes, I think the most obvious way to do it is to start off with easy examples that students can work out without having to use a formal rule, and then extrapolate their reasoning to apply to more complex situations, which is used to develop the rule that is then eventually used. But it's occurred to me that this intuitive approach may actually be inefficient and even detrimental to long term ability to recall these processes.

Let's take adding fractions as our first example. I'm guessing you probably teach this in the same order I've always taught it in:
1) adding fractions that have the same denominators
2) adding fractions with similar denominators (like 5/8 +3/4)
3) adding fractions with denominators that share common factors (like 5/6 + 3/8)
4) adding fractions with denominators that are coprime (like 3/8 + 5/7)

The logic behind this sequencing is obvious: number 1) requires a one-step process, number 2) requires a two- or three-step process, and number 3) requires a three- or four-step process, so this suggests a kind of hierarchy. Number 4) involves applying a rule that you have to use when no other option is available, so it makes sense it goes last.

Students often intuitively understand the rule for number 1), so they can master it within seconds. They are then pretty accepting of the rule for number 2), as it's easy to see that writing 3/4 as 6/8 will allow you to apply the rule from number 1). This then logically leads to a rule for number 3). Number 4) is technically the same rule as number 3) because you are still finding a common multiple, but in practice your approach to these questions is slightly different.

In summary, the rules students need to learn are:
1) Since the denominators are the same, just add the numerators.
2) The denominators are kind of the same, so change one fraction so that it matches the other, and then apply rule 1).
3) The denominators are similar, so identify what the lowest common multiple (or any common multiple) of the two denominators is, and then change both fractions so that the denominators match, and then apply rule 1).
4) Spend a while trying to find a common multiple using your times tables facts, realise you can't think of one, multiply the two numbers together to create a common multiple, then change both fractions to make that denominator. Then apply rule 1).

This is quite a lot of rules to remember just for one process.

Now consider an alternative (still referring to the original four cases):
1) Since the denominators are the same, just add the numerators.
2) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
3) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).
4) Multiply the numerator and denominator of each fraction by the opposite denominator to make the denominators the same. Then apply rule 1).

In this version, there's only two rules to remember. Doesn't that sound so much nicer? The standard rule that works for case 4) works for cases 2) and 3) too, so why not just use this rule for all three?

Now I know a lot of you are feeling very upset about the fact that I am endorsing the idea of students adding 3/4 and 5/8 by making an unnecessarily large denominator of 32. You may also point out that this way would involve a lot more simplifying at the end. But hear me out!

I'm suggesting that we treat fractions with dissimilar denominators as the rule and the fractions with similar denominators as the exceptions. We teach the students the rule, we make sure they're super-duper confident with this process, let them feel really successful in doing this, give them lots of drills in this so that they're well practised and then teach them the exceptions. Say to the class, "of course with question e) there is a slightly easier way to do it, can anyone spot it?" and then discuss this and identify a few more exceptions like this and then do some practice with these. Of course, it's not really an exception because the other rule will still work, but it's exceptional in that there is another method you could use.

Let me give you another example: finding the area of a sector. I've always started out by finding the area of a semicircle, because students know straight away that this involves halving the area of the whole circle. They'll also get how to do quarter circles straight away. Then I would throw in a 120 degree angle or a 60 degree angle, and with a bit of thought they'd be able to tell me that you need to divide the area by 3 or 6. And then I'll put up a 24 degree angle and watch them struggle. They might eventually do 360 divided by 24 to get 15 and then tell me we need to divide by 15. And then finally I'll go to a 37 degree angle and they wouldn't be able to do it. At that point I would tell them, well the fraction that's shaded is 37 out of 360, so we do blah blah blah.

Now consider an alternative: start off with the 37 degree angle. Ask the class what fraction of the circle is shaded. Eventually they'll get that it's 37 out of 360. Then simply show them the correct calculation. And that's it. No faff. If they come across a question with a 24 degree angle, they can still do the same calculation, there's absolutely no need to work out that you can divide the circle area by 15. And even if you have a semicircle, the same calculation will work, and it won't exactly take much longer that halving (especially if it's a calculator question). Feel free to teach semicircles and quarter circles and even third circles as exceptions afterwards, as a little bonus shortcut that students will grasp easily. But I don't really see how starting with that will help students understand the process of finding the area of the 37 degree sector.

Another example: finding the volume of a prism. Isn't it weird that we learn how to find the volume of a cuboid separately from finding the volume of a prism, despite a cuboid being a prism and the prism rule working perfectly well for a cuboid? Why not just teach how to find the volume of a prism, and throw in cuboids, triangular prisms, trapezoidal prisms and compound prisms altogether (assuming they are all fluent in finding the area of rectangles, triangles, trapezia and compound shapes)? I would love to add cylinders to that list too but I found out embarrassingly recently that a cylinder is not, as I always believed it to be, a prism. Who knew?

One final example: solving equations that look like 4(x+2) = 8. Now there are two obvious methods to solving this equation. You could divide both sides by 4 or you could expand the bracket first. Dividing by 4 is a nicer method in this case and gets you the answer faster. However, what about the equation 4(x+2)=7? In this case, it would be unwise to divide by 4 first, so you should really expand the bracket. But there are no examples where expanding the bracket first doesn't work, so why don't we just teach our students to always expand first? (I have a feeling a lot of you do teach them this actually!) There are a lot more numbers that don't divide nicely by 4 than numbers that do (although Hilbert might disagree...) so most of the time the dividing method won't work, but the expanding method always will. It's nice to show the dividing shortcut as a little aside, once the students are really secure in the process of solving other equations with brackets. But I believe fluency in the rule should take precedence over the ability to identify exceptions.

I hope I've managed to explain my point well enough in this post. It would have been much easier if I could scribble stuff onto a whiteboard at times! Maybe that's the push I need to finally start the NotJustSums YouTube channel...

Oh and I should mention that this post was inspired by things I heard in Craig Barton's podcast which is excellent. My favourite episodes have been his interviews with Dani Quinn and with Tom Francome, and I strongly recommend that you check them out.

Emma x x x

I Was Wrong

That's right, I'm still here!

It's been a long time since my last confession.

I've been wanting to get back into blog writing for a while now but for some reason no particular topic really seemed interesting enough to write about (and let's face it, I've set the bar pretty low with some of my previous topics - "is zero a factor of zero?", anyone?) However, this week I've been inspired to crack open the ol' blog and do something with my spare time that doesn't involve my Nintendo Switch.

The reason for this sudden burst of inspiration? Reading Craig Barton's wonderful book, How I Wish I'd Taught Maths. If you've not heard of Mr Barton, you're missing out. He's a UK Maths teacher whose resources website has got me through many a Tuesday afternoon double lesson. His book is part memoir and part how-to guide, with lots of lovely academic references to back up his claims. I'd definitely recommend getting yourself a copy.

Mr Barton (why does it feel wrong to use his first name, like he's my teacher or something?) talks in his introductory chapters about how his opinion of what good maths teaching looks like has changed dramatically over recent years, and I've experienced something similar. When I was in my PGCE training year back in 2010/11, there was a belief held by many cutting edge maths teachers that maths lessons should be all about exploring, discovering, and open-ended problem solving. The suggestion that a traditional "chalk and talk" lesson which was teacher-led and full of repetitive exercises may be effective was met with haughty derision by us (wise beyond our years) trainees and may have even elicited an "OK boomer" type eye roll. We knew that the way forward was rich tasks from NRICH and a hands-on activity involving spaghetti and marshmallows. We quoted Piaget and Skemp as if they were old friends and bragged to our colleagues "oh I never use textbooks". I think we were rebelling against our own education, which might have involved those SMP booklet thingies or OHP slides covered in hundreds of near-identical quadratic equations. And the fact that we were really rather successful under that system was irrelevant; we knew that we could do better, and better must mean different. So clutching a copy of Jo Boaler's latest with our paper-cut (bloody Tarsias!) fingers, we marched into our classrooms determined to teach our students nothing and yet have them learn everything and when things didn't quite go the way we planned (although the marshmallow came unstuck from the ceiling eventually), we blamed the Primary school teaches for raising our students to have fixed mindsets and no independent learning skills. We pulled off Outstanding (capital O) lessons and wowed our observers whilst managing to cover remarkably little content. Those were the glory years, the halcyon days of group work and bits of string and "what do you notice?". But somewhere along the road I think we all started to realise that not only was this approach not anywhere near as much fun as we thought it would be, it also just didn't seem to work.

I'm in my ninth year of teaching now and my pedagogical beliefs have definitely changed. Here is a summary of how:

I used to think:
You couldn't tell students anything explicitly, you had to let them discover it for themselves, otherwise it wouldn't be meaningful.

Now I think:
Mathematical concepts don't become meaningful when you discover them, they become meaningful when you use them to solve a problem. Think about the circle theorems. Discovering the "angle at the centre is twice the angle at the circumference subtended by the same chord" rule by drawing a lot of random lines inside circles is very unlikely to happen without significant scaffolding (is it really discovering if you were given a guided tour?) and students may well discover something completely different (or "discover" something that's not even true!) along the way. And would they even know they've discovered it when they discover it? Would they even see its value? How about instead, we give students a circle with some angles in it and ask them to find the missing angle. They will try to apply the rules they already know, but will eventually get stuck. We can then say, "hmmm, you need a rule that's going to help you solve this easily. Luckily, I have just the thing!" and then the circle theorem saves the day and makes a difficult looking question dead easy. Much more meaningful, in my opinion.

I used to think:
You must show students the proof of rules or theorems before you let them use them. This includes basic rules like the laws of indices, up to more complex ones like differentiation.

Now I think:
Proofs should only ever be shown after students are fully comfortable with using and applying the rule or theorem. When I was shown proofs of Pythagoras' theorem as an adult, I thought they were amazing and interesting and beautiful, and I thought, why didn't my year 9 teacher show me this? So then I decided I would always show my students these proofs when introducing Pythagoras. It turns out, the only reason I found these proofs so fascinating is because I had already been using Pythagoras for eight years and was pretty intellectually invested in it. Why should students who have never heard of or used the theorem to solve any problems care about how the theorem is derived or why it works? I'm even starting to question my approach to teaching differentiation: I usually demonstrate graphically using a concrete first principles approach to find the derivatives of x² and x³. Along the way I generally lose the interest of 50% of my class, and the other 50% finish the lesson believing that differentiation is the most difficult topic EVER and they should drop A Level Maths before it's too late. Wouldn't it be better to teach them the quick little "lower the power" rule, get them used to it, and then show them where it all comes from? And then the sudden appearance of the 3x² out of nowhere may be met with a chorus of satisfied "ahhhhh"s rather than a sea of bewildered faces.

I used to think:
Slower methods are better than quick ones if it means that students understand how and why the method works.

Now I think:
The best method is whichever one has the lowest cognitive burden, as this frees up valuable thinking space for other aspects of the question at hand. Problem solving questions can sometimes involve five different mathematical processes to be carried out, wrapped in a "real life" situation that involves unpicking. The problem solving aspect of the question takes up a considerable amount of brain power, so being able to carry out the mathematical processes efficiently is really beneficial. I would say, however, that for students with certain SEND needs, sometimes the memorability of the method is more important than the cognitive burden, which means that slower methods can sometimes be favourable. But for students with a decent long term memory, the quickest method is usually the best.

I could probably list another ten beliefs that have changed, but I've written an awful lot today and I want to save some material for future posts. Stay tuned!

Have your beliefs about teaching changed since your training year? Let me know in the comments.

Emma x x x