This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network. He started off by running through the classic 1089 “trick” which I have written about before here.
These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject. If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.
I’d used this before but had never given it the time that it deserves, nor had I extended it:
- Try it with 4 digits, 5 digits, etc.
- Go back to three digits and try it with a, b, c. In other words prove it algebraically.
- Now try in a different base, e.g. Base 7!
I actually got a bit obsessed with this when I got home. I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…
My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!
I’ve been thinking a lot recently about “formal methods”, e.g. Column Addition or Long Multiplication.
As secondary maths teachers we don’t pay much attention to these. I assess their ability to carry out the procedure and have sometimes attempted to teach long multiplication explicitly. But more often than not it is assumed that these methods have been learnt at primary school have been pretty well-practised and are therefore secure.
We have been doing a lot of number work with Year 7 so far this term including exploring in some detail the Laws of Arithmetic.
Here is an example of the type of question we have been looking at:
127 + 54 + 73
Most of my students’ first instinct with a question like this is to draw up a nice column addition and solve it. Would you agree that most students do this?
I want them to look at the structure of the numbers first. To realise that addition is commutative and that it’s much easier if we add the 127 to the 73 first because that’s a number bond to 200.
I am having 2 issues with this approach:
- My students are not convinced that this is a valid exercise. Basically they think it is contrived (which, of course all exercises are). They think I have chosen numbers to make this work, it will only work for those certain numbers and basically they feel like I have tricked them.
- Some students perceive the message that the formal methods they spent hours practising at primary school are now not the way we do things at secondary school. I have said that this is not the case, they are not the devil’s work, and that those methods still have a place. But we need to have smarter ways to work. The fact that we have spent many lessons not solving calculations using formal methods leads me to believe that they are confused about what I want them to do.
Fundamentally I want my students to rely less on the authority of the teacher and rely more on their own understanding. I want them to see the structure in everything they do, to be certain that what they have done is valid because the mathematics tells them it is so, not because I tell them that they are right.
How should we bridge the gap between informal methods which require (and develop) depth of understanding and formal methods which are efficient and accurate?
I have an idea using manipulatives. I am not sure if it will work, but if I get chance to try it with the class, I will write about it again later.
In the meantime, does this resonate with anyone else? How do you bridge the gap? I’d love to read your comments.
One of those classic investigations that gets forgotten about all too easily. So much scope for generalising at different levels.
The fact that all odd numbers can be expressed as sum of two consecutive numbers is probably the first thing that will be established. But why is this the case? And can students express this as a generalisation, first in the form of concise words and then algebraically?
The beauty of this is that there are then many other layers of things to discover, right down to a generalisation explaining which numbers cannot be expressed as a sum of consecutive numbers. And maybe even a proof.
This nRich page gives away some of the answers.
Thanks to Alan Parr for reminding me about it with this excellent blog post:
The All I Can Throwers – Sessions with Den and Jenna. #1 – Consecutive Numbers
There is a range of “take turns” dice games like this one on nRich.
I see this activity as a way of practicing key skills (in this case column addition) but in a much deeper way than repeated practice as you are working backwards to achieve a result. I would think of this as adding a significant degree of difficulty over simply doing a page of sums; it would be something you would only chose to do once the basic process of column addition is reasonably secure. However, because it is engaging (i.e. competitive) students are more likely to stick with it.
To avoid the need to provide students with 10-dice an interactive dice could be displayed on the board. nRich have this handy spinners tool here and there are lots of other options online.
All students would then have the same set of numbers and it would be a competition to get closest to 1000. You might need to note down the numbers as you go to prevent cheating!
Again, of course, the value in the activity comes from the discussion, both in pairs and in whole class. I like the idea of saying that the target is 1000 but actually rewarding good discussion and reasoning rather than just closest answer (i.e 5 points for a new “noticing”, 5 points for the closest answer)
Another way is to determine all the required random numbers at the outset and the students can fill in the grid with full knowledge of their options. Less luck involved and so probably less fun!
I would do one game whole-class, where students are playing individually. Then a second game where students are to work together in pairs competing against another pair so they can compare strategies once they have a degree of familiarilty with the problem and get some good discussion going on strategy.
These only cover adding and subtracting negative number, i.e. they can be used before going onto multiplying and dividing negative number.
They can be printed (here is the pdf), cut out and then stuck on A3 paper under the three headings with examples and counter-examples to explain why they have been put under that heading.
I should point out that two statements on here are deliberately vague, i.e. “two negatives make a positive” and “a positive and negative make a negative”. This is often how students remember them and this can lead to problems down the line (e.g. the misconception that -3-2=5). My idea with these is that the end up in the “Sometimes” column but ultimately we dismiss them and not being very useful.
I made this a while ago for supporting Year 7 students on directed number (i.e. positive and negative numbers). I think there is something more intuitive about a vertical number line – if you are adding you go up, subtracting you go down. Having said this, I have always had a horizontal number line across the top of my board!
If the number line is stuck on the inside back cover of the exercise book, it is always there whatever page the child is working on. It can then be folded safely away whether using large or small format books.
Doing some work in primary this week, I realised that the same idea could be useful for supporting younger children learning the essential skills of counting back and counting on when doing addition and subtraction of positive numbers. So I have made another version just with positive numbers.
The end points of these lines are arbitrary of course. I have deliberately gone for something a bit random to start a discussion, “Sir, why does it stop at 44?”. But if that offends your preference for order in life, then feel free to adjust it on the spreadsheet that I used to create the pdfs in the first place.
Vertical number line to stick in books.xls
Vertical number line to stick in books zero to 44.pdf
Vertical number line to stick in books minus22 to plus22.pdf
So, I learnt something today from a kid in Year 3. He gave a perfect explanation of a column addition that made me stop and think. How would you explain this?
“4 plus 1 is 5, so put a 5 in the ones column. Then twenty plus ninety is one hundred and ten, so we put a 1 in the hundreds column and a 1 in the tens column. Then one hundred plus three hundred is four hundred so put a 4 in the hundreds column”
This has got me thinking about where else I can reinforce place value when discussing procedures.