Algebra - 8ways, literacy, and unmasking our demons

I had some exciting ideas about things to do in our first look at algebra with my lovely year 7s. In the process of asking people for input, I got a bunch of better ideas from them (although my original ones will probably still appear).

For the first lesson, I took the inspiration from 8ways and we had a 'yarn up', pretty much as described here. The lesson got off to a good start down the comfortable-chat road with a discussion of the results from last term's feedback survey, the holiday homework, my new hair colour etc. I put up the new topic title: "Patterns, Algebra and Equations" and immediately there were groans and protests. With our scrap paper in front of us, I started asking questions. "What do you know about algebra?" "Have you done any before?" "When did you first learn about it?"

They knew some good stuff. It involves letters. It is where letters represent numbers. Usually you have to find out how the letters relate to the numbers. Most of them hadn't done any before, they said. Some had.

They had some fears, which came out when talking about their first meetings with algebra. "I first heard about it when my stepsister was saying how hard it was." "I looked over my sister's shoulder and couldn't understand anything on the page."

"So you hate algebra because other people told you they hate it?" They laughed and admitted it.

Someone said something about shapes representing numbers as well, instead of letters. Suddenly others realised they had done that too, and they didn't realise that was algebra, and that wasn't so hard. I said that I thought letters were easier to write when the algebra got more complicated. "It gets harder?!" Groans again. I explained that is wasn't hard as long as you learn one step at a time and that soon they would be able to do stuff that looks completely confusing now.

Then I used some ideas that were generously shared on the Maang group, exploring literacy and linking pronumerals to other "pros" they might know (pros and cons, pro-labour or pro-liberal, and pronouns) and discussing why pronumerals are used. We talked about the different types of symbols that can be used in place of numbers. I was taught how to pronounce xi, and admitted that at uni I referred to it as "squoogle". Other Greek letters were discussed in some depth. "It's definitely psi, it looks like the ninja weapon." "Oh yeah, like Raphael uses." "Who's Raphael?" Kids.

It took the right group of kids, but this was one of the most positive, enjoyable lessons I've had. It was a good way to start the topic, a good way to spend a Friday afternoon lesson, and a good way to get all the feelings and fears out in the open.

Tarsia

I love Tarsia. I have been using it as part of my year 7 Weekend Homework Menu quite successfully and as classwork or revision with other groups. It always takes longer than I expect for them to finish the puzzle and some kids just don't seem to have any experience with puzzles and take a while to understand what they have to do.

We also used them in our transition-to-high-school Maths Olympiads, with giant puzzles on coloured card to make it more interesting. I've sometimes done one large copy when I work with a class, and had some students solve that one and blu-tac it to the wall. (As well as their smaller copies, as you can see. They seem to be very proud when they finish them!)

Refraction

Refraction is an awesome game. Starts out as a fairly simple puzzle with simple fraction concepts but gets up to the concept of adding fractions by converting them to the same denominator, so it is pretty impressive! And loads of fun and addictive.

I've finished, but I still have a few coins and cards to win :)

Maths Taboo

I know someone else has done this, but I couldn't find it at the time I needed it, so I made my own. At first it was geared towards School Certificate content only, as revision for my lovely year 10s last year, then I added stage 5.2 and 5.3 content for further revision later.

If you haven't played Taboo, the idea is to give clues to get your team members to guess the top word, without using any of the words below (the 'taboo' words).

The file is here, it requires editing to make sure the tables aren't split across pages and so forth. And I recommend adding more, as I plan to in future.

AngleJack and other games with a Deck of Angles

Ok so bear with me here, I might rant. I got a bit excited about this one.

I found a piece of paper on which I had scribbled the idea of a game of making shapes out of angle cards, to practise angle sums of triangle and quadrilaterals (there is a post still to come about another game on this topic I made earlier). I made a test deck and started to play with the idea.

Basically the deck is angles in increments of 10 up to 170 degrees. There are more of the ones from 10 to 90 to provide more balance in the games. My first version of the deck only had the numbers, except for right angles which had a diagram only, but I'll add diagrams to them all (Firstlyl to remind them that they are combining angles, not just numbers, and secondly in the hope of subconsciously familiarising them with size of angles to help them visualise and estimate in future). The games focus on combining angles into right angles, straight lines, revolutions, complementary and supplementary pairs, triangles, and quadrilaterals.

Here are some ideas:

• Go fish with complementary or supplementary pairs. Seemed quite successful in the playtest with year 10 at the end of term. The kids these days seem to really like go fish. Low on strategy, but easy to explain and get going.
• Snap with complements and supplements. Never as good because a difference in skill swings the game a lot but always popular with some groups.
• AngleJack (or Revolution). I was inspired to create Blackjack-style games by this one for negative numbers and also the fact that some students seem to know this game idea quite well. Instead of 21, the aim is to get as close to a revolution as possible (optional - without going over). Can be adapted to getting as close to a straight line as possible but you need to be allowed to go over (or remove the cards above 90), otherwise you can go bust as soon as you start. This was our favourite at home (I have the best, most tolerant boyfriend!)
• Texas Hold 'em Angles - still some fine details to work out, including rankings and the tricky "betting" thing. But basically two cards in hand, five face up, from which the players have to make the best "hand" they can by combing the cards. So far the ranking of combinations is quadrilateral, triangle, revolution, straight line, supplementary, complementary. (Even though a supplementary pair should be less common than a straight line made of any number of angles, there is more thinking involved in the latter, so I ranked it higher). I think to split within the levels, the largest unique angle wins. I did contemplate introducing suits or colours of some kind but I think it's unneccessary. Not sure what the suits would have been but I do like the idea of someone anouncing "quadrilateral of abacuses" or something.
• My original one was a Numero-ish game where each player has a hand of 5 and the table has 5 cards face up. On your turn, you combine cards from your hand with one from the table to make a triangle or quadrilateral to put into your pile. Replenish hand and table. Most cards in pile at the end wins (therefore quadrilaterals are worth more, which is fair, they require more thought). Could also be allowed to make supplementary and complementary pairs to keep the game going.
• At a lower level, the cards could also be all spread out and players take turns to make and take a quadrilateral or triangle (or whatever else mentioned above).
• We also pondered the idea of a Scrabble-like board where you play your triangle or quadrilateral onto one already on the board, using one of their angles. This will take some more thinking about! (And probably the kids would think we had gone insane)

Heaps of thanks to Michael and Bennie for developing these ideas with me.

Some thoughts about games

Further to my previous post about up-skilling my students, I'm dwelling on one of my favourite things - games.

I love maths games and I try to use them in class, but I do find they meet with mixed success, and it often isn't the maths that is the problem. I began by thinking "Kids these days don't know how to play games" and feeling sad about that. But sometimes I go to a friends place for dinner or a party and a board game comes out, and sometimes there is an adult who doesn't seem to really "get" how to play the game. So I suppose it isn't generational, it's just that not everyone plays games and I don't normally associate with that sort of person.

I think I should persist though, because using maths in a game seems like another level to me. Once the student has a good grasp of the concept, playing a game about it can work on using the knowledge/skill strategically, reasoning about the knowledge, and gaining more fluency at applying the skill. Which potentially also provides differentiation, (although I might be reaching here) as one student makes the most obvious move while another considers multiple options and chooses the best one for the situation.

Over the years I have played a lot of games in class, that I have discovered or invented (mostly invented). I spent a lot of my early part-time years throwing myself into making cards for cool games I came up with and then they didn't work that well or I got bored of them. The legacy I will leave behind me in the teaching world is a large pile of neglected hand-made games. But I will be sharing my more successful creations here in the hope someone else might enjoy them. Stay tuned!

What kind of students do I want to pass on?

I've been thinking a lot about what it is I am actually trying to achieve as a maths teacher. With good students it's easy to not question the standard form of maths education, but with middle and lower-achieving classes I question the relevance of a lot of content.

I've been slowly reading through some of the blogs I found in my first over-excited blog weekend, and after reading all of exzuberant and infinigons, I've thought a lot about quality teaching, assessment, problem-solving skills and engaging, relevant learning activities. Reading this post today made me think about what the key, transferable, whole-of-life skills are that I should be trying to impart. If I taught at the kind of school where many kids went to university I might not care as much about this, but probably 50% of our students won't do maths in year 11 and 12, and only a handful will be studying any mathematics after school. So what the hell are we doing? And why?

This is a big problem and I have a short attention span but I kind of continued the thought on a slight tangent.

Today I was also making up a new maths game (as you do), and I though about how most of my students aren't great at playing games that require much strategy. And in fact a lot of the "cool", engaging activities we want to do, anything that requires independent thought, critical thinking, deep understanding, dealing with open-ended problems, etc. etc. seems to stump our kids.

So I thought, that's something to start thinking about. This type of thinking, dealing with these types of problems and activities, these are all just skills too. They need to practise them. We need to be persistent, and we need to maybe scaffold learning how to learn that way?

I have the top year 7 class. They are pretty good at maths so far (most of them) and from here I have to try to turn them into hard-working, keen, interested, engaged, skilled maths learners. And maybe that means I need them to be:

• used to learning games, playing games, strategising and coming up with variations to games
• used to solving problems
• used to working on projects
• used to working in groups
• used to reasoning about and communicating concepts
• used to tackling open-ended problems

as much as, or even instead of:

• doing lots of drill questions from textbooks or worksheets 

Which means I need to make sure I'm doing these things. I've made a start on games and have some group projects about communicating concepts planned, but need lots more problem-solving especially.

Asking scary questions

So I took Nordin Zuber's advice and decided to get student feedback from my year 7s about my teaching this term. I followed his template and plan pretty closely, although unfortunately I can't give feedback until next term, because this was the last lesson. I felt so good about the process and the results when I compiled them that I took an even scarier step: I did the same thing with my year 9 class (cue horror movie music and distant scream). I was pleasantly surprised.

What I was afraid of:

• Obviously, getting totally slammed and told I was a bad teacher
• The top year 7 kids telling me the work was too easy
• All of year 9 saying the work was too hard
• Lots of "I hated it" responses

What actually happened:

• Most students seemed pretty pleased with how things were going (relative to their achievement level - the high achieving year 7s were more positive than the low-achieving year 9s)
• The difficulty level seemed well-balanced (although year 9 did just get pretty decent test results back which has improved their outlook)
• No one said they hated it (One year 7 kid was very surprised that I had given that as an option!)
• The year 7s weren't as keen on the Weekend Homework Menu as I was. They did say they liked it and liked having choice, but I expected a bit more excitement about getting to choose and getting to play games! Apparently not.
• Year 9 find each other very annoying! And hate seating plans (fair enough).
• I didn't get any mind-blowing surprises, but I hope they will be more forthcoming with their comments if the process is repeated, both out of comfort and familiarity and because repetition will make them feel that their input is genuinely wanted and will be acted on.

Here are my difficulty graphs (year 7 at top, year 9 below):

I was pleased to see a fairly close distribution, and nicely balanced in year 7. I had been working hard to keep things easy for year 9, and kept feeling that it still wasn't easy enough. I think the graph suggests I need to keep working on it, but it isn't too far off!

Another funny thing is that it increases my feelings of goodwill towards my classes. It's good for me to stop seeing them in terms of behaviour and focus on how they are learning and feeling as individuals.