This blog seems to be turning into a Dear Abby column, but what the hell. If readers keep posing questions then we’ll keep posting them, to see what answers arise. You gotta keep the customer satisfied.
This latest question comes from frequent commenter, JJ, who has begun doing extension work with Year 6-ish primary school students:
I’m keen to keep it to broadening rather than accelerating. … Not doing things they will cover later rules out most of algebra (though I might make rare exceptions such as converting recurring decimals to fractions). …
I’m interested in two things – developing a framework for the approach (ie a sense of the bigger picture and areas to go to – apart from ensuring the basics +-*/, decimals, fractions, % are fully covered and automatic) and also sources of good materials.
I have never taught Primary, but I would never not show a student something that they are capable of doing just because it comes up in the program at a later year.
Showing a student is not the same as designing a program for a class.
Hi Marty. This is a great post; thanks. And a great opportunity to stress the importance of inverse processes in the teaching of Mathematics
So (i) I can add two numbers. Two numbers add up to 15. What are they? Can I include decimals or fractions?
(ii) I can find the area and perimeter of a rectangle. Design a rectangle with a specific area and /or perimeter.
(iii) What about right angled triangles? Design one with a certain area.
The second point I would like to stress is the ability to become more comfortable with talking about numbers say from 1 to 100. Can you talk for 1 minute about the number 24? You need to use the words factor, prime. Why is 24 special? Plenty of scope for imagination! Being able to remember important numbers is critical to move forward in Maths. As Tony Gardiner pointed out, so many kids don’t know there’s only 1 prime number in the 90’s. Make a real effort to remember those 25 prime numbers less than 100. Spot those prime numbers in the decimal expansion of pi and use this as a way to help kids remember important numbers and interesting sequences of ‘random’ numbers. Build up kids confidence in their knowledge of numbers.
Finally, get your students to actually DRAW those rectangles, WRITE out those numbers and activate their ability to remember important numerical facts.
Dear JJ
Here is a lesson
that I used with a Year 5 student. It involves add ups and take aways and ends with a pointer to negative numbers. It seemed to be a success, albeit with a sample of 1 very good student. I’m still keen to see if something similar would benefit a whole class.
When I was a young child I went to some extension event or something with a bunch of other kids. They gave us the Seven Bridges of Konigsberg problem and told us to find a path. A kid a couple years older than me immediately said it was impossible, but the teacher leading the exercise said that it was possible. So, we spent a bunch of time trying to find a solution, drawing out attempted solutions on paper. After some time, the teacher called our attention and told us that it was, in fact, impossible. Of course, everyone objected. To which the teacher said something along the lines of “in mathematics, you can’t just say that it’s impossible, you have to prove why!” I can’t remember whether or not the teacher gave a proof, but if he did it was surely inappropriate for the level of the class (these were children maybe at the oldest 12 or 13.)
Choosing to “broaden rather than accelerate” is in my opinion a bad decision. I can appreciate the Seven Bridges of Konigsberg problem now, but at the time it would have been better if they just introduced quadratics or something instead.
I don’t see Konigsberg as an example of broadening rather than non sequituring.
I think it is so hard to pitch extension activities because you don’t know if the students are really keen on mathematics especially or just used to being good at everything and wanting further confirmation of that. If it’s the first case, they will want serious challenge; and if it’s the second one, then you have to be careful because they might get upset as soon as they find something difficult for the first time in their lives. So, it really would depend on the students and I’d be inclined to choose fairly flexible activities until I felt like I knew them.
One time I taught a combined class for Year 7 (I think it was everyone who didn’t go to a sports day or something like that) and needed to fill in time, so I chose this as a starting point and it was nice enough: https://mathpickle.com/project/rainbow-squares/ It was good in being flexible according to what the students could handle, and they actually seemed to appreciate the weird story too.
Thanks very much, wst.
I agree that the distribution of Good Maths Kids one sees in such extension classes is pretty bipolar. Of course, if you have them on a weekly basis or whatever then you can quickly figure out who is who. I am not particularly fussed, however, about upsetting those who are “used to being good at everything”. Not that I’d be actively seeking them to bring them down a peg, and the more unsure and sensitive the kid the more I’ll handle them with care. But the general over-protection of egos, the general unwillingness to have kids feel even momentary discomfort, is a serious problem with primary schooling.
The rainbow squares activity looks interesting and fun, but I’m not sure I get it. It is definitely a fun way to practise solid playing with numbers and arithmetic. But is one expecting more than that? Are there nice general solution methods and/or proofs of unsolvability that one is expecting the kids (with possible assistance) to work out?
The rainbow squares task really was mostly just for the arithmetic practice. At the time, I was looking for ways to get students to the point where they could recognise which small whole numbers were perfect squares without too much effort. (I was shocked at how many Year 7 students couldn’t do that already, and in hindsight I can’t remember why I decided it was important. I was maybe drowning in the amount of basic arithmetic that I’d taken for granted and which it turned out students didn’t know.)
I didn’t see any general arguments or proofs of unsolvability (there are some general results mentioned in the article linked above), but there were piecemeal arguments for why you couldn’t solve certain examples and things like that. I could get a sense of students’ tendencies from their attempts to make one. And it was interesting enough to keep students going for an hour. So, it’s not amazing, but I think it was an okay starting point.
Thanks, wst, and fair enough.
Thanks very much for your suggestions folks!