A couple days ago, I wrote about doing some fill-in teaching at Our Ladies of Perpetual Properness. That post concerned a crappy incident involving the Year 11 class. This one is about the Year 9 class, which I adored. All the classes were enjoyable, give or take the interference of a Deputy twat, but the Year 9 classes I actively looked forward to. The girls were charming: polite and very friendly, happy and hard-working; they wanted to learn.
The class was not overly strong, the more advanced kids having been siphoned off into extension and acceleration classes. Still the girls were smart enough, and with some definite sparks. They worked hard, asked good questions, listened to the answers and learned. They were probably Methods-bound at most but there was plenty of opportunity to teach them some mathematics and it was fun and rewarding to do so.
The mathematics to be taught was unthrilling, or worse, but it was ok. The Cambridge text was awful and there were calculators, but not too much. OLPP was wise enough to delay the CAS perversion until later years. Mostly we covered the basics of trig – flagpoles to the nearest centimetre and other burning issues – and “measurement”-geometry. The measurement-geometry was annoyingly formulaic as this stuff always is:
“So, dear girls, this is the formula for the area of a fucking kite.”
OK, I didn’t say “fucking”, but I rolled my eyes in a “fucking” motion. I was caretaker for the regular teacher and so did not take too much liberty, but I tried to indicate, gently, that area is nothing but rectangles and triangles (which are in turn, of course, half-rectangles), with similar messages for perimeter and volume. I tried to get across that the geometry they had to learn was about three ideas rather than a dozen formulas.
It all went well enough, but then the common test was coming up. This was the big test, covering a bunch of topics, including topics that the girls had studied before I arrived. I can’t remember, but I think they had revision sheets to practise for this. In any case, one of the more studious girls, let’s call her Lucy, spoke up in class, asking for help on an earlier topic:
“Please, Sir, could you go through expansion for us?”
“Expansion? You mean like expanding (2x + 3) times (x + 4)?”
“Yes.”
“Sure, let’s go through it. But isn’t that easy for you guys?”
“But we don’t understand it, Sir.”
“Um, ok. Let’s go through a couple.”
I wrote the expression on the board:
“Ok, so we’re going to end up with some x squareds and some x’s and a number, the constant term, right?”
(Understanding nods.)
“Right. So,”
“Great. So how many x-squareds will we get?”
(A second’s pause and then many replies:)
“Two.”
“Yep. Two:”
“OK, and what about the number at the end, the constant term?”
(Two second’s pause and then many replies:)
“Twelve.”
“Right. Three times four is twelve:”
“OK, and how about the x’s? How many x’s will we get?”
(Five second’s pause and then a few tentative replies:)
“Eleven?”
“Exactly. Eleven:”
“Right? The two x times four gives us eight x’s, and the three times one x gives us three x’s, and eight plus three is eleven. Make sense?”
(Mostly understanding nods, with some uncertainty.)
“That’s ok if this is a little quick. We’ll practise a couple. Getting the x’s is more work, and you have to be careful, but after a few you’ll be fine.”
Then Lucy piped up:
“But Sir, we’re not allowed to do that.”
“What?”
“We’re not allowed to write it that way.”
“Why not?”
“Because we have to show our working.”
“……………………?”
“……………………!”
“AARRGGHH!!”
I may have been in caretaker mode, but I had my limits.
The class then explained to me what the reader has probably already figured out: the girls were expected to expand step by step, 2x times (x + 4) and so forth. It was pointless and mad, and I told the girls so. And I didn’t have to tell them because they already knew. They knew that they had failed to understand a simple concept because they had been forced to write out unnecessary details in a ritualistic manner.
It is difficult to convey how bad it is to force kids to do this. Sure, if a kid has weak arithmetic and is struggling with expansion then you have to ease her into it, include more detail. But if the teacher’s end goal, even with a weak kid, is to write this every-step crap every time then the teacher’s end goal is plain wrong. Have the kid doodle “3 x 4 = 12” and the like along the way, as she needs it, but do not, ever, teach expansion as if it is required to be presented as an extended essay. You write what you need. More than that is painfully slow, tedious, obfuscating and it is hugely prone to error, creating further delay and tedium and obfuscation.
Anyway, the girls and I were quickly sorted. We practised a few, together and then individually, and they were fine with it. There was one final question from Lucy:
“What should we write on the test? What should we do for working?”
“Just doodle ‘3 x 4 = 12’ on the side or something. That’ll do.”
Lucy was happy, her classmates were happy, I was happy.
I recall reading a story by a mathematician who later won the Fields medal. He was told at school by the physics teacher that you should show your working in every examination problem. If you don’t, the best you will get is 30% of the value of the question. In the next examination, the student gave only answers and no working. He scored 30%.
Proving what?
“Show your working” is a dud phrase.
What you have written about has been a weeping sore for ions. Common sense does not rule.
We should be asking kids to write down their “thinking/calculations/results”, as makes sense.
These are the things that we are assessing, and rewarding with marks.
But what makes sense?
Different things in different situations.
Which is probably why this has been a weeping sore for ions.
Depending on the situation, what we want to see could well be different – just a result, a calculation and a result, a short phrase that indicates a thought was had that led to …
Part of the teaching is to be sure students know what makes sense, the expectations during the learning process and when being assessed.
But how do people learn what makes sense? Who says?
If they were to write what is in textbook examples …
(Kids should also know that writing down more, if it helps you to get to the correct result, makes sense!)
Yes, of course this a long-standing and systemic issue, and it is getting worse. The less substance, the more one is reduced to assessing form, VCE being the stellar example. But I thought this was a very telling instance.
My suspicion is that less well trained maths teachers fall back on doing it one way, rather than as I always say – if it’s mathematically correct and you’ve shown enough for me to understand it, then it’s full marks. How much is enough then is of course a judgement call and if you haven’t got enough training, that judgement may be hard to make.
Of course. If you’re not comfortable assessing meaning then you assess the form only.
When we were initially learning that, we used the acronym FOIL (first terms, outer terms, inner terms, last terms). So for that equation the only line of working out required would be 2x²+8x+3x+12=2x²+11x+12
That working out somewhat makes sense in my opinion, since it’s not too long, it’s not confusing, and it makes the arithmetic easier (I still use it if the co-efficient of x is large and hard to solve without working out)
On the other hand, expanding it to 2x(x+4)+3(x+4) would be complete madness, and would only dumb down kids’ understanding of how to expand equations
Thanks, Jay. Maybe the previous teacher was expecting FOIL, I don’t know. But it doesn’t matter. Demanding FOIL, or any working whatsoever, is absurd. If the kid can do expansion reliably in her head there is zero argument, in fact negative argument, for encouraging or demanding anything else.
The way you teach something does not necessarily tell you how to assess whether the kid has learned it. Also, although I’ver never thought hard about it, I think I’m of the opinion that FOIL sucks. A future blog post …
I sort of agree. At the same time, doing it in your head can come back to bite you later on. For example, I think a kid using FOIL would have an advantage in the question `(15x+74)(23x+35)` (that is an exaggeration, but my point is that less working out makes it harder to solve more complicated equations).
Also, I think you’d definitely need to use FOIL to solve (ax+b)(cx+d), for example
I’m curious, in your opinion should VCAA (and other exam organisations) award full marks for correct answers that show no working out, on CAS-free exams (whether working out should be required on CAS exams is a whole other discussion)
Hi Jay. Of course if the numbers are sufficiently hairy then you’re not going to work it out in your head. That’s really evidence of nothing.
I don’t want to give detailed thoughts on how to grade high stakes exams, because I really have no clue, even the ones I’ve graded. But:
*) the current VCAA system, of awarding a mark for each individual fart, is madness.
*) In general you should get marks for demonstrating that you can compute something or that you understand/can-prove something, and partial marks for semi-successful demonstrations. I think that principle is all you need. Which means if you are required to compute an answer of some polynomial or something, and you correctly do it in your head, then writing down that answer should be sufficient. (But, you risk not getting partial marks if you provide only an incorrect answer.)
A general principle of grading (and teaching): first, and above all, be human.
That principle sweeps away a lot of pointless questions.
Hello Marty,
Do you think kids need to show working for a question like “find the surface area of this cylinder to 2 decimal places”? If it’s worth 3 marks maybe some working is required, but you could easily just do it in one line on the calculator. Every teacher I’ve talked to reckons they need to write the formula, substitute the values in, then write the answer down for full marks. I can see what the teachers mean (if they get the answer wrong they get 0, but they get some marks for working out), but I’ve been giving full marks for just the answer and they’ve been saying it’s not the right thing to do. I honestly don’t know what to think and it’s driving me crazy. This is for year 9, if that makes any difference. Thanks.
This reminds me of the joke about the guy asking for directions:
“Excuse me, could you tell me how to get to X?”
“Well, I wouldn’t start here.”
How should one grade that calculator question? Well, I wouldn’t start with the calculator.
But, thank you, Teacher. It is a very good question and deserves a long answer. I have such a blog post planned, on my “short” list, so I’ll keep it (relatively) brief here. Also, please keep in mind that any intelligent answer will get you in trouble with your colleagues. So, read on at your peril.
The primary purpose of grading is to determine whether students have learned the desired information and/or skills. In your Y9 context there is also an important and perverting secondary purpose, which we’ll come to, but let’s begin with with the primary purpose. For that primary purpose, the only question is,
Has (or to what to what extent has) the student demonstrated the desired knowledge and understanding?
That’s it. Anything else is secondary, if not entirely superfluous. An immediate consequence of this principle is:
If a kid’s answer could only have been plausibly arrived at with the appropriate knowledge and skills then the kid’s answer should be awarded appropriately full marks, whether or not the kid has included all or any working to that end.
So, in your context, if the kid cannot plausibly get the surface area other than by plugging the correct values into the correct formula then the teacher can be sure beyond reasonable doubt that the kid knew what he was doing and the question should be graded accordingly.
To me, this principle and its consequence is entirely obvious. It is also used in some contexts, and is known as “implied grading”. I think it may be standard in IB. In Victoria, however, which has elevated grading to an anal retentive and fetishistic art form, implied grading is largely unheard of and, when proposed, is considered an affront to God.
Having said that, there is an obvious danger to the student whose working remains in his head: it is difficult or impossible to award an unsupported and incorrect answer any partial credit. So, the kid takes a risk, and it is reasonable and necessary to warn kids of that risk. Still, the demonstration of understanding is the demonstration of understanding, and the prudence of showing some working does not reasonably imply that it be mandatory.
But, there is then the secondary purpose of grading, at least at Year 9: prepare the kids for the vicious morons who will grade them in VCE. I think this argument is way oversold, and that there is still time in Year 9 and Year 10 to teach proper mathematics rather than the pointless, ugly as dogs balls CAS-soaked bullshit that VCAA expects. But the argument obviously has weight.
More soon, I hope.