The Story of Pi
I created this a few years ago and it now adorns our maths corridor. It started by having a Pi number line. There are lots readily available to download. We used one containing the first 1000 digits, I think our corridor was long enough to fit the first 600-700 or so. The idea being that the number of digits discovered relates to a point in history.
I never quite got round to finishing this. If anyone fancies doing some of their own research on the history of pi, and filling in the blanks, then please do so by editing this document. Otherwise, just use what is there – it’s still an interesting story!
It’s a more straightforward operation than adding. I ultimately end up saying “top times top, bottom times bottom”. That works, but why does it work, when it doesn’t work for adding fractions?
This might help a bit… Reinforces the notion that of means times, i.e. when I buy 4 bags of crisps at 30p each, they cost 4 x 30p. Same for fractions, e.g. 1/5 of 2/3
There is so much out there, it is sometimes hard to know where to start. I have compiled a personal Top 10 Maths Teaching Resources page. Following blogs is great as it gives you a drip feed of ideas. It may not feed directly into the next lesson you are about to teach but plants a seed for later. Again, here is my personal Top 10 Maths Blogs. And finally Twitter, which I am very much a novice at (I am @mhorley), so I refer you to Michael Fenton’s post on tips for how to get set up. Also, this post by ICTEvangelist.
With all of these things, there is the risk that you feel deluged with new information. Remember, it’s not like responding to e-mail from your boss! You don’t have to look at anything if you don’t have time. And you really don’t have to spend more than a few minutes a day looking at any of it for it to be really valuable. After a while you develop a sense of want you want to read and what you can skim over. It does take a bit of investment up front to get things like Twitter set up (maybe 30mins – 1hour), but it is really worth it, believe me!
Adding fractions with year 7 today. We’ve spent a bit of time examining equivalent fractions so I felt we were ready. I used an idea that was developed as part of a Lesson Study that I worked on with 2 other teachers a couple of years ago. In the lesson itself we used fraction walls to reinforce the idea of equivalents. After over an hour, we were there, with most being able to find equivalents and successfully add fractions, some no longer using the fraction wall.
So, on to the next lesson. I feel like I have a good sense of how they need to develop this and how I need to scaffold it for them. It’s one of those occasions where I could hunt around online and in text books to find a set of questions that might be OK. Or, if I sit down and think deeply about this, in 10 minutes I have the perfect set. For my class. At this point in their learning.
So here they are. You never know they might be perfect for your class too. But it’s not very likely!
I’m lucky to work in a maths department with many wonderfully collaborative and creative teachers. But something that we have all acknowledged we need to get better at is marking students’ books.
School policy is that we are supposed to mark books every 2 weeks. This is meant to take the form of a marking “dialogue” where we pose questions and set challenges, extension questions, etc. Some staff manage this, but most (including me) don’t. When I do, I feel a faint warm glow that I have done it, but actually on reflection, the students get little benefit from the 2-3 hours it takes me to mark a class set of books. I think about what I could achieve by researching and planning better lessons in that time and I’m convinced this would have a lot more impact on learning.
However, student feedback and formative evaluation is critically important – it’s right at the top of Hattie’s list of effects (here, here and here)
We had a great department meeting today where we discussed some ideas. I will share those here once they are more fully developed (I promise!), but in the meantime, can you share good practice you have seen? Or do you know of any good articles with ideas. In particular, I’m thinking:
What’s the focus of your book marking? Asking questions? setting targets? Providing feedback – what sort of feedback?
Any particular templates that are good? e.g. stickers / slips to stick in books.
What policies work? i.e. they are realistic and are stuck to consistently by teachers.
Please comment below or tweet me @mhorley or e-mail me firstname.lastname@example.org
I’ve been teaching for 3 years, I’ve learnt a lot, but also appreciated that the learning never stops. CPD doesn’t mean going on courses, it means spending a reasonable portion of your time looking at new ways of doing things, never just accepting that you are going to teach things the same this year as you did last year.
This wonderful tome written by a group of fabulous Amercian teachers is something that I think might become one of those sources I go back to again and again.
The basic idea is to replace “Tricks” with a proper teaching of deep understanding. There is a wonderful array of unhelpful mnemonics, cute stories and memorised procedures in here including such gems as “Ball to the Wall” and “Make Mixed Numbers MAD”. For each one, the authors suggest a better approach to instil deep thinking.
On reading this, I was relieved that I don’t use a lot of these bad tricks in my teaching, but there were a few that made me stop and think about my own practice. Including:
- 2.8 BIDMAS (usually PEDMAS in US schools). The first problem is that Divide doesn’t necessarily come before Multiply. The authors suggest an alternative, GEMA, but I’m not sure that this is all that helpful either. I’ve tended to teach this early, in Year 7 and 8, but the problem is that the second item you come in the list, call it Indices, Powers or Exponents, is not something known at this stage. So I just say, ditch the acronym until much later. Just start with saying multiply/divide happens before add/subtract unless we use brackets to indicate otherwise.
- 3.5 Dividing fractions. Whilst I have never said “Ours is not to reason why; just invert and multiply” I must confess that I do use Keep Flip Change (KFC) a lot! So I like the idea of getting common denominators first and then dividing the numerators. The idea being that once students practice a few they will discover the short cut. This takes a confident teacher, though. I can think of some students who would just feel that I have wasted their time showing them a long method and might even think I didn’t know the short-cut!
- 4.5 Pythagoras. Instead of a² + b² = c², leg² + leg² = hypotenuse². Interesting, but I’m looking for a better word than “leg”…
Anyway, that really is just scratching the surface of this wonderful resource. It should just make you stop and think about the fundamental way you explain maths concepts. Because it doesn’t matter how many wonderful resources and activities you plan into your lessons, a significant part of students’ learning will still come from your explanations.
I like to use this as a sort of a crescendo to teaching prime factor decomposition which is itself a very satisfying experience.
Although it sometimes feels a bit procedural it’s a nice way of:
- Practicing times tables
- Getting to know your primes
- Appreciating the commutativity of multiplication.
Anyway, here’s the trick (everyone needs a calculator in front of them)
- Chose any three digit number. Write it down somewhere.
- Type your number into your calculator and divide by 7.
- Hands up if you got an integer answer. Opportunity here for a nice discussion that we might expect 1 in 7 hands to be up at this point.
- Press clear and divide the same number by 11. Repeat again with 13. Right, now they’ve appreciated that not many numbers are divisible exactly by 7, 11 and 13. Time to blow their minds…
- This time type your 3 digit number into your calculator twice so you have a 6-digit number. e.g.
- Divide by 7. Hands up if you have a whole number. Wow, everyone. Now don’t press clear, but divide by 11. And then 13. Wow. Gets you exactly back to your original 3-digit number.
How much you chose to explain this will depend on the ability of the class, but the points are:
- Whatever 3-digit number you chose, the 6-digit number is 1001 times the 3-digit.
- 1001=7x11x13. Weird but true. And this is why it works.
If your students seem to like it, I always ask them to try it out on their family when they get home. I love the idea that I just might have created a discussion about maths around the dinner table – you never know…