Methods: Solving Equations

To each his own. All roads lead to Mecca. There’s more than one way to skin a cat. We’ve got plenty of ways to describe people being able to find their own path to the same place, and the path of least resistance to a solution is something massively valued in mathematics. Elegance and efficiency are worthwhile aims, but not all methods support progress to higher levels of maths.

My new role as a school-based tutor means I do lot more close reading of student maths as it happens, and on the whole I’ve been trying to let students use their own methods to cut down confusion. No more ‘whack it in a box’ for anything to do with proportionality and multiplicative reasoning unless the student’s got nowhere to start.

20170216_112744

The Standards Unit box is a go-to for conversions, percentages, etc.

I used to think it doesn’t matter what they do if it ‘makes sense to them’. Teach them 3 ways to do division and let them pick the method that they like, understand or feel most comfortable with.

I’m not convinced anymore. We should be teaching the methods that allow students to access as much of the curriculum as possible.

All teachers have pedagogic opinions on what’s more effective in maths. I love a grid for most things and I find chunking heinous. I think discussing these things with people who don’t agree is one of the most interesting and intellectually stimulating aspects of being a maths teacher. Others disagree. There’s a lot to argue about for a subject that’s often said to be unique in virtue of its definite answers.

So equations.

3x+6=18

The language used varies from teacher to teacher. We could ‘balance’. We could ‘get x on its own’. We could ask ‘what’s 1 x?’.

Some people describe processes I’m convinced come from a random word generator, like ‘float and ping’, ‘crossing the river’ or ‘gouging the onion’. (That last one is a lie)

One of my tutees solves equations like this:

student-method

They start with the unknown, then follow BIDMAS (another thing for another day) until they get to the ‘answer’ of 18. Then they work backwards with inverse operations to find the value of the unknown.

I thought this was brilliant. It was accessible. It linked with something they already knew with the BIDMAS and the student felt confident with it. It feels a bit like dividing fractions with ‘keep, flip, change’, but I’m there to get them through an exam (again, another thing for another day).

Then it dawned on me that this wasn’t going to work for much beyond the most simple one-step equations. This method is not going to support progress in linear equations, never mind beyond.

3=30-3x

Aside from the cacophony of threes, the negative coefficient can become problematic, as does what to do next in my tutee’s method.

330-3x

Much less obvious for the student to unpick.

This BIDMASsy arrows thing could still work if you rewrite the equation from 3=30-3x to 3=-3x+30.

rearrange-3-3x30

But 30-3x being equivalent to -3x+30 is a big leap for lots of kids. I seem to spend my life reminding kids subtraction isn’t commutative, and now I’m saying that thing with a negative symbol can be written either way round?

The only way I approach this is to think of collecting terms, as below, but then this depends on students being able to navigate the links between negatives and subtractions, which for weaker students adds another dimension of difficulty to solving equations instead of fitting in with what they already know.

numbers-as-like-terms

I’m not even going to start on students ending up with (7+4)-(3+6) from BIDMAS

The other way to rewrite the equation would allow the student to crack on with their arrows too:

rearrange-33x30

None of this is wrong per se, as it would enable the student to solve the equation, but learning BIDMASsy arrows for one thing, and then “adding/subtracting unknowns to both sides” is using two different methods for one question. It feels clunky and all over the place. “If it’s negative you have to add to both sides before your start” is only a whisker away from “gouge the onion” in terms of meaning if it doesn’t fit into established understanding.

Students learning methods for every case is ineffective and makes maths more difficult in the long run, as it downplays any flexibility of thought and we end up with hundreds of baseless ‘rules’ with no context.

When I teach solving equations I have post-it notes on a metre stick and do lots of flailing if we don’t apply the same operation and magnitude to each side.

I’m calling it ‘Slapstick for Learning’, and it’s coming to an INSET near you soon.

I want them to be focusing on the truth of the equality instead of what the unknown is. When we first introduce solving equations students can often solve them by inspection.

2x+5=11
“Yeah but it’s 3 though”

I don’t think this is being able to solve an equation though. It isn’t enough. They can solve that equation.

By focusing on the equality,”Is this still true?” becomes the question rather than “what do I do?” or “What’s x?”

4

Is it still true?

7x

Is it still true?

This way, students can practise some substitution, and then when we get to solving inequalities and we need to know if our sign needs changing it’s not another inaccessible idea. It’s all linked.

By manipulating unknowns on both sides at this stage students will be more prepared for unknowns on both sides, or even to solve minefields like:

19=4-3x

In this case, the negative coefficient of x, the negative solution and sometimes even the fact the 19 is on the left side of the equals sign can cause students to stop dead in their tracks and we haven’t even got to equations with unknowns on both sides of the equality yet.

If we know we can do things to that equation that keeps the equality intact, students can now focus on what option would be useful. By framing it this way they can start to have a go at manipulating the equation to see where they get to, and when they might need to go in a different direction, but this too relies on prior knowledge, by being comfortable with algebraic notation and the effects of operations on expressions.

In the first equation, 3x+6=18, I can discuss with students whether or not they would factorise first and divide, when would they add the constant, when wouldn’t they, give me an example of an equation where it would make sense to factorise first, give me an example where it wouldn’t, etc. By approaching teaching equations in this way, students can begin to reflect on what they could do when they get an equation, which is the first step towards what they will do. This is problem solving.

Both fluency in manipulating algebra and the standards unit box diagram allow you to tackle problems across a wide-range of content in algebra and proportionality.

When we talk about students being flexible in their approach to solving equations, I take that to mean their algebraic manipulation. They can preserve the equality of the equation to get to where they need to be and to discern the most efficient way, even at the level of simple two-step equations like 3x+6=18.

That principle of equality becomes essential for solving harder equations and if students are secure with it, solving inequalities and changing the subject becomes an extension of their previous learning rather than ‘another topic’.

In the long run, that’s better for students’ problem solving skills, self-efficacy and resilience. Learning 3 superficial methods and picking one isn’t.

problem-solving

“We’re now going to have a lesson on problem solving.” (Image source)

Questions that increase in difficulty one niggle at a time are great for this, and Dave Taylor‘s website, increasingly difficult questions, is as much a planning tool for honing explanations and concept-building as it is a bank of quality exercises.

I’m sure there’s lots of things I teach in a way that allows students to answer questions and solve problems at the level of challenge they’re at then, but might cause problems later on, particularly at AS/A2 level.

Maths teachers of year 11s up and down the country are getting tactical on what they do between now and exams. When is it alright for a quick and dirty method? I want to say never, but realistically I think sometimes there’s a trade-off for time.

Of course, two days later I taught a paired session and I completely changed my mind on this when it came to expanding brackets, but that too is another thing for another day.

Supercalifragilistic First Time I Taught Quotient

When trying to express oneself it’s frankly quite absurd,
To leaf through lengthy lexicon, to find the perfect word.
A little spontaneity keeps conversation keen,
You need to find, a way to say, precisely what you mean.

Mary Poppins

I’m a bit of a pedant. Irritatingly so. In my defence I’m getting better. I no longer correct apostrophes in text messages from family and friends, or quibble over ‘there is’ being ‘there are’, but in my classroom, be precise.

6+ 1 = 7 + 2 = 9

photo credit: pedro via photopin (license)

Photo credit: pedro via photopin (license)

I correct ‘I done…’ to ‘I did…’ when my classes give explanations, and like a good worker bee I put keywords up on the board. Sometimes I put keywords up on the board. I’m not that consistent with it, as it’s normally an afterthought. It’s ‘best practice’ to highlight keywords with students. It’s ‘literacy across the curriculum’.

It isn’t.

The way I was slapping some maths words on the board at the start of a lesson, never to be seen or heard of again, does nothing to improve literacy. The only students getting to see them are those that get the date and title down in their books pretty sharpish anyway.

I was the Sunday Christian of literacy.

Two of my key stage three classes were learning decimal division last week, and I decided to change my approach. I taught them the terms ‘dividend’, ‘divisor’ and ‘quotient’. Actually taught them.

Capture1

With a bit of questioning on mini whiteboards I got students to identify the dividend, the divisor and the quotient. The example in the attached slides  progress through some question prompts for students. (found here: Dividend, Divisor & Quotient)

Capture2

In the last slide, I first asked students to tell me the quotient and remainder of the division calculation. I then used the same slide and asked them to convert the improper fraction into a mixed number. By changing nothing but the language I used in the question, students explicitly saw the link between fractions and division in a new way, and linked that to finding a mixed number.

I think there’s lots more to build upon with this, to the point where this kind of literacy opens up access to the subject in a much wider way than ‘students can read and understand the questions’.

Knowing a mixed number is the quotient and remainder of a division instead of ‘how many times the bottom goes into the top and write that big and then the rest write over the bottom again next to it’ is difference between knowing maths and knowing rammel.

In every assembly on bullying students are told how powerful the language we use can be, but I rarely take the time to explicitly teach the language of my subject, and then insist on its use in the maths classroom.

Those words are just as powerful, and it’s about time I taught them properly.

Hey Teacher, Leave Them Kids Alone

Induction’s been officially passed, new term INSETs are done and dusted, and it feels like the holidays were an age away already. September’s here, and it’s time to get back into the swing of teaching. The timetable’s somewhat fuller, and there are less names to learn this time round, but it’s still equally nerve-wracking and exciting to be back at school.

When it comes to new year resolutions, mine are to stop over-planning, calming down a bit with marking and then making sure I leave work earlier than last year.

It seems counter-productive to actively try and do less this year, but it’s not. Last year, there were lessons where I was doing more maths than my students. That’s definitely counter-productive.

My new year 9 class were in their third lesson with me last week, (multiplying fractions) and I set them a textbook exercise. They were working in silence, and I gave them the chance to get their teeth into it, to focus. Not having some gre’t lummox faffing about asking you questions and confusing you for 15 minutes whilst you got to get your head down and practised meant that when we stopped, I could really start to question what they had understood.

“Why was question 4 different to question 5?”, “Which question was harder, question 8 or question 14? Why?”, “Write a question you don’t think you could solve this way, and suggest why not.”

If you’ve got no hook to hang your hat on, you’re just chucking questions around.

The breathing space for them allowed them to focus on what they were doing instead of what I was doing. The atmosphere was calm and the students were really proud with what they had achieved, which in terms of building confidence might allow them the security to attempt more involved problems applying their skills.

When it comes to building in routines and getting students to really hone a skill, unsolicited silence as they concentrated on getting through the task was a wonderful surprise.

If students aren’t working on a task long enough to run into a problem, they certainly aren’t going to have the time to dig themselves out of it.

I don’t need to micro-manage my students every second in my classroom and write it down, I don’t need to slave over every single thing my students write in their books either.

This year, I’m going big on routines, and sometimes that routine will involve doing a lot of similar questions quietly.

AfL: Moving Away From A Performance

This year is steaming by. It really doesn’t seem like three months since my last post, but in that time I’ve been focussing on getting on with stuff instead of moaning about it. The pace of learning over this last term has been immense too, from moderating GCSE stats coursework to becoming a proper form tutor (which some days feels like could be a whole part time job in itself!).

AfL is top of mind at the moment, as I’m trying to find AfL techniques that aren’t superficial. I’ve found quite a few times I’ve used things like exit slips in a useless way, which does students no favours and just wastes my time too.

I’ve been using RAG123 from @KevLister lately and found it works really well. I’ve been using it as a 5 second end-of-lesson thought from students. Next lesson, when talking one-to-one, I’ll jot down what I think their effort and understanding was.

Most of the time this is based on the work they’ve shown in their books, allowing me to focus on presentation of their ideas explicitly in a positive way. It makes marking much easier as students have a consistent framework that links to everything they do.

RAG123

I’ve been having a bit of a hoo-hah about dialogic marking too. I get why ‘green-penning’ a response from students helps them to reflect, but I’ve found 5/10 minutes isn’t sufficient for it to be more than a vanity exercise.

Students are going to multiple subjects and writing responses to their feedback in green that are time wasting lip service. Some of the things I was getting back were useless in terms of improvement. “I will try harder next time.” etc. Now this wouldn’t be a problem as such, but I had been giving students personalised tasks on the basis of their work to respond to.

These weren’t read, as often students thought they should just chuck in a comment to appease me. This was taking a ridiculous amount of time to do and is wholly unsustainable, so as an alternative to doing 10/15 minutes every fortnight, I try to get my classes to spend half a lesson every 3/4 weeks on extending and improving their work in the formalised ‘green pen’.

An example of differentiated feedback to be green-penned.

An example of differentiated feedback to be green-penned.

The example above shows tasks in response to a homework. The numbers at the top correspond to the section of the homework students made mistakes on. After a brief discussion of each section, everyone starts from the point indicated on their homework.

It doesn’t sound like that much that often, but my rationale is that students are always responding to feedback. Feedback isn’t an event with its own special pen, but ought to be implicit in every lesson. Extending students’ understanding comes from questioning (be it verbal or through a particular activity) and linking the conceptual ideas in mathematics, not from opening a book and seeing three different colours.

I would much rather do this less often with this degree of detail instead of half as considered twice as often.

In terms of AfL I lean heavily on questioning, as it’s one of my strengths as a practitioner, so I’ve been running with it, which is fine, but it’s been a while since I’ve used mini whiteboards.

I normally fully rate a mini whiteboard.

The only problem is, when I’m looking over at mini whiteboards, I quickly forget who’s got a misconception and what specific aspect it involves. Spending time finding or writing questions meant to expose misconceptions and then training students in the use of mini whiteboards is all wasted if that information’s not used to address them effectively.

I’ve seen a few tweets mentioning Plickers and didn’t really think I’d be right to use it, as our school has recently clamped down on students using mobile devices, so having the teacher walk around scanning stuff might seem a bit hypocritical.

But what an amazing tool! If I could not only take this information to inform future planning, but have it to refer back to later I can address misconceptions properly, instead of the ‘look at me doing AfL, aren’t I dead good’ trap I’ve fell into before (traffic lights anyone?)

As for the questions themselves, I can’t get enough of Diagnostic Questions. Students want to know why their answer wasn’t correct, which is incredibly powerful to start discussions. By combining these two amazing resources, I’m hoping to see some real improvement in the way I tackle misconceptions.

I’m going to give Plickers a go next week in conjunction with Diagnostic Questions, and hopefully this will enable my day-to-day feedback improve for individual students.

If you’ve any strategies you use regularly, please let me know in the comments below. I’m always on the lookout to snaffle ideas from all and sundry!

Overreacting To Feedback

“Sir, when you look out at us as a whole class, do you just see, like, loads of Bs you want to to turn into As?”

This question caught me unaware and I wondered where this had come from. Was this a response to a school-wide push on stretching and challenging that might have caught some of our year 10s by surprise? Was this the way I was coming across with my zealous discussions of what we think the grade of this question is?

Straight away I’d fallen into that trap of taking an offhand student comment personally. A common error. Whilst everything I do in the classroom has an effect on the students, this is not the same as saying everything the students do in the classroom is directly caused by me.

As a logician, that’s an embarrassing mistake to make. Then again, as a person, with all the emotions, messiness and ‘off days’ that you’d expect, it’s understandable.

I answered the question honestly.

“When I look out I don’t see grades. I’m looking for who’s paying attention, who’s trying to hide that they don’t understand and who’s trying to get away with chewing gum on the sly.”

In my attempt to be a ninja and pick up on as much that’s going on in the classroom as I can, I see people. People with all the emotions, messiness and ‘off days’ that you’d expect.

The conversation continued. This student was a bit worried that now they’re in year 10, everything they do is graded, which might explain why they feel more like letters than people. Whilst this isn’t directly aimed at me in the way I first thought, my insistence on using grades as motivation and outlining expectations was certainly contributing.

This set me thinking during break duty, which by the way, is a right treat. An entire 20 minutes where I’m not able to run around or feel like I should be working. I am working. Just by standing still. Winner.

What is it about my lessons then that brought this question up? I thought about how I have been clamouring on about the grades for quite a few lessons now, when it clicked. In the hectic first month of term, I’d let a bad habit creep back in.

During my training, I’d get feedback and then go overboard with it in the following weeks. Regardless of the observation feedback, the same four steps kept repeating.

  1. Have an observation where something’s mentioned I can improve on.
  2. Do it to death.
  3. Realise that there’s a point where it isn’t productive.
  4. Back off and find a bit of balance.

This is what was happening here. My first observation of the year last week went really well, and one of the main pieces of feedback was that I should let students know what grade the work is at more often.

  1. What was that? Levels and grades on questions you say?
  2. Put grades on every learning objective and ask students what level you think a pot noodle is compared to a posh restaurant. Apply this to algebra. (“Your work’s a kebab. It does the job, but it’s not that nutritious or good for you long-term”)*
  3. Put (Grade F) on the first lesson of a unit for year 11, face a mutiny and total work refusal.
  4. Realise that there’s a time and a place for slapping letters all over lessons.

*A bit of an exaggeration, but you get the gist.

The main difference between this example and my training was that I had only stopped putting an abundance of grades on year 11’s activities. Whilst reacting and reflecting on the needs of individual classes is great, I hadn’t really considered the impact on the classes that didn’t react quite so robustly.

In that one offhand question as I was helping the student next to him, this particular pupil helped me realised where I can hone my practice with that little bit more subtlety, and also gave me another perspective on the student experience.

We know that high quality questioning from teachers to students helps them improve, but then this week I realised I’d forgotten it works the other way around too.