Most important topic in maths?

What is the most important topic in maths.  Many, including parents, and (more worryingly government) would have you believe it is 'basic numeracy', which tends to mean 'times tables'.  I agree that a basic awareness of magnitude and the ability to add siubtract or do multiplaications in your head is important, I really don't think that it is the most important thing that maths teachers should be doing with classes.

This is not my idea, by the way, I heard it discussed (briefly) on Radio 4's rather excellent "The Infinite Monkey Cage", about a year ago, but the more I think about it, the more I'm convinced its right.

What we should do more of is probability.  Not just the stuff we do already - words, probability lines, fractions / decimals (do any maths teachers use percentages for probability, and if not, should we if the real world do?).  Not just Venn diagrams and tree diagrams, but actual probability and risk assessment.

Don't panic.  This is not risk assessment from a H and S form point of view, but from a maths point of view!


Why?


Well, every decision we make is based on probabilities and expectations.  Should I buy a lottery ticket? What car finance deal should I go for?  What stocks and shares should I buy?

Everyone needs to be able to properly weigh up the pros and cons of any situation.  We need to teach the appropriate skills to do that. Students need to be able to set up appropriate models (modelling may be the second most important thing we teach!), and interogate those models.  We then need to use our understanding of probabilities (using prior information) to make the 'best' decision.

There's a great example of this earlier on my blog - have a look at http://chrismaths.blogspot.com/2010/05/real-functional-maths.html to see why it's important.

Any other thoughts??

Made up Maths Words

I'm a big fan of creative literacy, generally with a maths focus.  What I'm worried about is whether it may do more harm than good.

Some examples; from recent lessons, to things from a while ago...

Shape names:  I asked a class if they knew the name of a 12 sided polygon. After a sea of blank faces presented itself, I prompted "it begins with D".  Dozenagon was the answer offered.  So, for the rest of the lesson we used Dozenagon alongside the proper name.


Same lesson, outterior angles.  Opposite of interior, in case you were wondering!

I've also used squillimeters and squentimetres for sq mm and sq cm.  This helps avoid the centimetres squared problem. Many students don't appreciate the difference until I tell the (both true) stories of the tech teacher who ordered 10 metres squared of turf for his 10 sq metre garden, and the one (from the telly I'm told) who ordered 5 metres cubed of concrete to be poured on to the floor that required 5 cubic metres.  They certainly know the difference.

I've also verbised words.  Y12/13 in particular use the verbs To Pythagorise and To Trigonometrise to describe what to do to a forces or vector triangle.  Bracketising is something I use as a synonym for factorising.

The question is does it matter?  I am always careful to use the real word too, but I feel it is part of building up a good relationship for learning with pupils, and a few years ago some of a class startd keeping a list of made up words in the back of their books. I tried to get them to create a wiki for it, but the VLE didn't make it easy enough, which is a shame as it would be great to have a full record of words over the year.

All thoughts always gratefully received!

Square of Area 10

This is another one from my PGCE; all those years ago now, but continues to be a favourite little investigation. I take no originality credit! I'm posting in response to @mathsatschool comment on #mathchat that students don't accept that "diamond"s are squares.

I say little, it can be anything from 1 lesson to a week or fortnight's worth.

Resources: Square spotty paper and lots of it.

1. Draw a square of area 1. Easy enough to start. I get peculiar looks at this stage...
2. Draw a sqaure of area 4. Have a look around, you get some 16's at this stage.
3. Draw a square of area 9. They are starting to think either "This is going to be easy" or "He's up to something" depending on how well they know me.
4. Draw a sqaure of ares ... wait for it (they'll be storming ahead with 16 if they're anything like my usual lot).... draw a square of area 10.

At this point you tend to get the stunned silence, followed by some frantic scribbling. Some clever so and so will root 10, and try to do that. "Exactly 10" usually stops them.

It's worth a play, but here's my offering - and indeed I usually have to show them the answer...

I find I need to show them the 'proof' of its area. My usual method is the Big Square around the outside, take away the triangles....

A = 16 - 4 * (0.5 * 3 * 1) = 10.

I find having a notation for labeling squares handy. This one I'd call 3_1. (To move long an edge you go along 3, down 1, etc).

So - what squares can you draw? Area 20? Area 14? Are there any you cannot do?

Whats great is that you can check their answers really quickly. You'll either see the answer really instantly, or you'll do the algebra with square x_y, and go "Oh - yeah".

Enjoy. Good for Shape / Area work, and for Algebra.

Helping students less.

I don't like helping students with their maths.

That's probably controversial (and probably not true either), so I'd better qualify it.

I don't like helping student's with their maths, until they have made an effort to help themselves.

That's better. The question is how to best help them do that! It's all about independent learning and learning to learn. Here is one way I've developed in my classroom, that is now all around my department, and starting to spread, as I've spoken with at least one other Maths Subject leader about it...

Maths Posters. Seems simple enough. I've downloaded and printed the "Maths Facts" posters from here. I printed them onto different colour card - green for shape, yellow for data, blue for algebra and pink for number, and had them laminated.

I did have them around the back of my room, but my LA advisor suggested moving them to the front.

Now, whenever a student first asks for help, all I need to say (much of the time) is "there's a poster about that", and they go to find it. If they're still stuck then of course I'll help.

It's getting to the stage where the question is not "Which one is the mean again?" but becoming "Where is the poster about the mean again?"

Yes, I let them get up during a test to look - after all tests are formative as well as summative. I also get them to write down on their test paper which posters they needed to look at. This gives them a starting point for their revision for their exams.

Rocket Science - no.
Simple and effective - yes.BTW -apologies for the fuzzy photo; I want a better phone...

Factor caterpillars

1. Choose a start number. Find the factors of the number..
e.g. 16. Factors are 1,2,4,8,16
2. Add up the PROPER Factors of the number.
e.g. 1+2+4+8=15.

Repeat for 15.
Factors are 1,3,5,15.
1+3+5=9.

Factors of 9 are 1,3,9
1+3=4

Factors of 4 are 1,2,4
1+2=3

Factors of 3 are 1,3 (3 is prime as it has two factors).

1=1 So stop here.

If you write each number in your chain on a circle of card, you can make lovely caterpillars.

Good questions. Whixh numbers under 100 make the longest chain.
Which numbers are boring! (Primes)
Why are square numbers interesting here?
What is special about powers of 2 to start?
What is really special about 6 and 28!
Try 220!

FULL PAGE LINK


Here are some pictures...
One of them includes what happens when you start at 30 ...

Difference of 4

This is one of my favoruite early lessons, and really lends itself to mixed ability Y7. I'm posting it in response to the mathchat discussion tonight.

Start with a 2x2 grid, and put numbers in the spaces...

Having done this, drawe a few arrows, and do some times tables...

Followed by some adding...











Then, simply - the difference between 46 and 64 is 18.

Questions you can ask...

"Can you choose numbers that give you a difference of 4"

"What happens if you use the same numbers, in a different order"

"What happens if you add 1 to every number"

"Are there any differences you cannot make"

"what happens if you put numbers in in a pattern - 2,4,6,8, etc"

"Is there more than 1 way to make a difference of 4?"

....

It is a lot of fun (and I take no original credit for the lesson. Picked it up on my PGCE many moons ago (OK - 1995/96!)

I feel a bit let down by the technology.

I always feel a high tech solution should be better (more efficient, slicker, make me look better) than a low tech one. With that in mind when planning to collect heights of Y9 to do some serious stats with, I decided to set up a motion sensor in the top of the doorway, and measure the distance to the floor.

Then as students arrive, they stand in the doorway (no heels please), and the sensor measures the distance to the top of their heads. They type that in to a Google Form (along with Age, gender, and some other bits), and then the spreadsheet subtracts one from the other to give their height.

This should have been fool proof. Yeah, right.

Sensor in doorway. Check.

It measures 2.04 m to the ground. Check with tape measure. Check.

I stand underneath, subtract reading from 2.04m to get my height. It tells me I'm 186 cm - 6'1" (according to Wolfram|Alpha).

The problem? I'm 6'3". Double check against my known kids heights. It puts them down 2 inches (and Pippa was measured by school nurse recently). Check with the missus - it puts her short too, by an inch and a half (and she checked at the gym recently).

I can't work out how it could possibly be wrong, but it is.

So my lesson plan goes back to having a metre ruler on the wall, which upsets me more than is reasonable! At least I'm still using GoogleForms to collect the data. That's progress at least.