What Do Angles Measure?

I've had a lot of fun asking this question both to educators and students recently. Typical replies are; "They measure degrees"; "They measure the size of the vertex/point."; "They measure the distance between the two lines."
The last reply in particular leads to the common misconception that the angle A below is larger than the angle B.

To clarify what angles measure I do a little pirouette and tell people this:

Angles measure turn.

And as with all measures, we shouldn't jump in to teaching about standard units of measuring (degrees) until the students have had experience with non-standard units (e.g. full turns, half turns, right angles etc.)
I used to show students what a right angle is by pointing to the corner of a piece of paper. Now I get them to make their own right angle by doing the simplest Origami as shown below:

A question which I'm often asked is why is a right angle 90 degrees (and not, say, 100 degrees)? Well the answer lies in how many degrees are in a full turn and there will always be some students who know this, especially if they are into skateboarding or snowboarding: 360 degrees. So why 360 degrees? Well the ancient Babylonians were the first folk to consider breaking the full turn into smaller standard parts. They knew that the Earth took 365 days to go around the Sun (long, long before Copernicus) but they also knew that 365 was not exactly a friendly number to work with. they chose 360 instead as they used a base 60 for their numbers. Good job they did otherwise we would be saying that a right angle is 91¼ degrees!
So to get students to really understand the notion that angles measure turn, I have them estimate angles using some cheap-and-cheerful angle measurers as shown:

Here I want students to actually turn the arms of the angle to create the angle. Here is a video of a student using them in a class to see if the angles in a quadrilateral are greater than or less than a right angle.

I find if they have experience estimating angles first, then when they come to measure angles with a protractor, they will not be confused by the two scales that most protractors have.
Finally, to counter the misconception that angles cannot be larger than 360 degrees I might ask students to either use the cardboard angle measurer above or to stand up and turn 180 degrees, then again, then again and ask "How many degrees have you turned now?" This idea of having angles beyond 360 degrees will be important in higher grades when they start learning about periodic functions and unit circles as this site shows.

First Steps in Developing Number Sense

One of my clearest memories of doing maths when I was 6 or 7 was in Mrs. White's class at St. Thomas More R.C. School. We used Cuisenaire rods to develop our understanding of number bonds up to ten. I'm pretty convinced that this laid the foundations for my good number sense and it is an experience I love repeating with young students from Kindergarten, Grade 1 and Grade 2. I start by asking the students to make a 'Staircase' as shown below:

Now this can be quite challenging for some students, but after this has been completed, I then get students to see that each coloured rod represents a quantity between 1 and 10. So I'll ask questions such as "Blue is what number? 5 is what colour". When the students know these they are then ready for the next step: I lay down an orange 10 red and ask:

Put two rods together that match the 10 rod.

This is my 5-year old son doing it for the first time:



When I was in Mrs. White's class, I do remember that before long, I was thinking "8+2 is 10" as I put the brown with the red. And this is what I see kids do now I try it myself with them: they are learning the facts by acting out the operation with the rods. The aim is that after a while they will not need the rods, that they will just 'see' that 7+3 is 10.

So what future knowledge will this connect to?
Well, firstly, it isn't too long before kids will see that 7+3 is the same as 3+7. In other words they will 'discover' the commutative law. I won't call it this though; I'll call it 'Dan's Rule' or 'Samantha's Rule' after whomever first notices it.
The rods also lead nicely into the idea of bar models by which we can represent the concrete (the rods) with a diagram and these in turn lead us to the abstract notation of related facts.

In fact, this development (concrete, diagrammatic, symbolic) is one which we must keep in mind when we are getting our students to develop their number sense. It will give students the opportunity to generalise their number sense into algebraic sense. If a question reads:

In Grade 3 there are 47 girls and 35 boys. How many more girls are there?

Students can represent this with a bar model and then use this to think about how this can be represented with any one of four number sentences:

In this case, some students might choose to do 47-35 but others might think 'What do I have to add to 35 to get 47'. Either way, you get 12.
And all of this thinking can begin in kindergarten with Cuisenaire rods.

Problems With Place Value

Place value is so much more than Base 10 blocks.
This is a common theme has emerged in a number of recent Collaborative Inquiry sessions where we focused on what students know and don't know about place value. A lot of questions from textbooks and worksheets tend be of the type shown below:

The danger with these types of questions is that I have seen students get the right answers but have no firm understanding of quantity whatsoever. They might learn a strategy such as 'The first digit goes in the first space, the second digit goes in the second space and so on...' or 'The digit on the right tells you how many little blocks, the one next to it tells you how many rods, and the one next to that tells you how many flats and so on...' If base 10 blocks are the only representation used then there is a real danger that students will develop misconceptions such as on that was highlighted by Sue Willis in First Steps in Mathematics. Grade 4 students were correctly able to identify that 4 rods and 3 smalls were 43 (below).   

However, when asked how many there would be if they were cut into individual pieces, two-thirds of Grade 4 students said “I don’t know, I would have to count them.” 
Similarly, Grade 6-7 students described a large cube as ‘the thousands cube’ but thought that if it was cut up, there would be 600 small cubes (100 each side).  This sort of error explains why some students think that 24+25 is 13 as it is simply 2+4+2+5.

With this in mind, we visited a number of classes to try some different sort of place value and quantity questions. We set out to deliberately bring out any student misconceptions by asking them to order a set of numbers such as:

547, 600 - 3 twenties, 5 tens 7 ones 4 hundreds, 5 hundreds 23 tens, 4 fifties

As the students began to order these, it quickly became clear that many were making errors based on a simplified grasp of standard partitioning. As one teacher said "Boy, we've done way too much of that...".
So in addition to the bog-standard standard partitioning questions, we realised the importance of asking questions like:

• Write 57 in at least three different ways
• 1000 take away 47 tens is the same as how many tens?
• How many 20s in 100? 500? 1000?
• How many 25s in 500? 1000?

This will give students a deeper and more flexible understanding of quantity and place value which will be necessary if they are to have good number sense.

A Mobius Twist

"Why did the chicken cross the Mobius strip? To get to the other..., hey, wait a minute..."
I came across this twist on the Mobius Strip today and it is too good not to share. It is from Martin Gardener whose birthday is today (21st October).
First, cut two strips of paper:

Then tape them in the shape of a cross:

Now tape them so that you create one Mobius strip and one normal loop: 

Now trisect the Mobius strip part and bisect the normal loop part.

But before you do, predict what will happen.

 

I and my colleagues were pleasantly surprised...

So what grade could we do this with? And how can we get it to fit the curriculum?

How are you feeling? Average? Or just mean?

If I ask someone how they are feeling and they reply "Oh, average" it is sometimes very difficult for me not to say "Oh, and what sort of average would that be then? Mean? Median? Or mode?" For the record, I do (mostly) refrain from such a comment but it does get me thinking how misunderstood the idea of average is. For example, if I asked you to work out the average of my (ahem) Math Test marks below:

85, 81, 84, 87, 89

...I would imagine that most people would work out the mean and not the median, and I would be very surprised if anyone would work out the mode (and to be honest, why would you with this set of data?) I would also suspect that for those who work out the mean, a majority would do so by adding up the scores and then dividing by 5. With this set of data though, my first instinct is to look at the numbers and think '85 is in the middle of these, so I wonder if I can adjust the other numbers to get as close to 85 as possible?'
I could do this as follows:
Take 4 from the 89 and add it to the 81 so I now have:

85, 85, 84, 87, 85

Now I can take one from the 87 and add it to the 84 to get:

85, 85, 85, 86, 85

Now I can see the extra one on the 86 can be split nicely between the 5 scores:

85.2, 85.2, 85.2, 85.2, 85.2

So I know my mean score is 85.2 and I can do this quickly (without any need for calculations) because I know that the mean is, in effect, the levelling of the scores. This point is often not understood by students even if they have learned the 'add all the scores and divide by the number of scores' formula.
So how can we get students to think of the mean like this? Well suppose a quick survey was done on the number of siblings that six students have and we get the following data: 2, 3, 1, 4, 1, 1. We could represent this as follows:

 If we see that we have some scores above 2:

...and then level these out:

then we see that the mean number of siblings (as opposed to the number of mean siblings) is 2.
Now I'm not saying that the mean should be worked out like this every single time but I certainly believe that all students should understand that this is what the mean does.
And this is not just a notion that is helpful in elementary schools. Earlier this year I was working on a Calculus problem as part of my Masters in Mathematics teaching at the University of Waterloo. I had to find the mean width of a semi-circle with radius 1. There is a quite amazing formula (below) that can be used to find this out but I didn't need it.

 

I relied on the approach above. Knowing that the semi-circle will have an area of πr²/2 or simply π/2, I realised that to find the mean width I just had to adjust the semi-circle to a rectangle of length 2 with the same area as the semi-circle and from here find out its width.

For good problem solving questions that require students to apply their understanding of the mean, you should look at the University of Waterloo Math Contest site. The one below is from the 2013 Grade 9 Pascal contest.

Math in the Media

Here in Ontario there has been a recent spate of articles in the media regarding the province's standardised test scores in Math (the EQAO results). Most of these have been written by people with no or very little background in mathematics education. Some, in particular this one by a Margaret Wente from The Globe and Mail have been so crass and incoherent that I felt I had to write a response, shown below. I got a reply from an editor who said that he liked and agreed with my arguments but requested that I edit it as it was too long and that I should avoid specifically rebutting the Wente piece (heaven forbid that we offer a counter argument). My edited reply did not get published as apparently the news cycle had moved on.
So here it is, because I cannot stand by and listen to uniformed nonsense any further.

The recent article by Margaret Wente contained so many unsubstantiated claims and vagaries that I feel compelled to write in response. I have a Math degree, an Honours Specialist in Math and I am currently studying for my Masters in Mathematics Teaching at the University of Waterloo (whose Math Department is world-renowned). I am also a Math Consultant for the Durham CDSB and as such I work with students and teachers from Kindergarten to Grade 12. I have a love of Math and a love of teaching and in my role I see what methods are successful and what methods are not.

The first issue I have is the emphasis on Math being nothing more than merely the memorizing of facts, formulae and standard algorithms. Any mathematician will tell you that this is not what Math is about. Math is the study of patterns: sometimes this is patterns in numbers; sometimes it is patterns in shapes or patterns in data. The famous British Mathematician G.H. Hardy wrote “A mathematician, like a painter or poet, is a maker of patterns. If his (or her) patterns are more permanent than theirs, it is because they are made with ideas.” Facts, formulae and standard algorithms are but a tiny part of Math and if taught improperly do nothing to improve students’ number sense. In fact there is plenty of research (e.g. Kamii (2000), Fosnot and Dolk (2001), Cockburn and Littler (2008) to name but a few) that standard algorithms (if taught before the students are ready for them) can actually undo their number sense. I tell my students and teachers that good mathematicians look at the numbers first before they decide what to do. For example, look at these questions:

1) $4.99×5

2) 800-481

3) 37+38

4) 2368+9417

If the only way that you can answer these is to use a standard algorithm then you do not have good number sense. I want my students to look at the first question and think ‘$25 take away 5 cents’ as opposed to reaching for pencil and paper to write out the multiplication. I have seen Grade 4 students efficiently do the second questionin their head without any need to use the algorithm. Instead they use good number sense (‘800 take 400 gives 400, then take the 80 to get 320 then take the 1 to get 319’). I give the third question to parents at Math Nights that I run and very few use the algorithm to get the answer. Some do 40+40-5, others do 37+40 to get 77 then take 2, others double 38 to get 76 then take 1. These parents are not ‘baffled and confused’ by the methods they use but they do agree that it makes no sense to say to a child that they have to use the standard algorithm to get the answer. I actually did the fourth question (one that was in the article) in my head working from left to right far more quickly than anyone could do by writing out the standard algorithm even though Ms. Wente states that the algorithm is efficient. This is a common myth about algorithms, that they are quicker and more efficient. There may be times when this is the case but these times are surprisingly rare. Really, would anyone suggest that the best way to do 1000-995 is to use the standard algorithm? Or to work out how much change you would get from $10 if your Tim Horton’s bill is $7.85?

Ms. Wente states that the curriculum has ‘downgraded arithmetic to near-invisibility’ and that children will ‘not understand fractions, will not learn to multiply or divide two-digit numbers on their own’. It is a pity that she did not take the time to do a bit of research and actually look at the curriculum. If she had looked in the Ontario curriculum for example she will have noticed that fractions begin being taught in Grade 2 and are a huge part of the Number Sense and Numeration strand in each grade thereafter, including learning standard algorithms for adding, subtracting, multiplying and dividing fractions. If Ms. Wente had used any basic journalistic skills, she would have noticed that in the Ontario curriculum students will learn to multiply and divide two-digit numbers in Grade 4. It is worth pointing out here that the standard algorithms for multiplying and dividing are not the only ways of doing so. There are other pencil and paper methods which we show students and parents which work just as well and are just as efficient (just Google ‘array multiplication’ and ‘Elizabethan multiplication’ to see these in action). When I have shown these methods to parents at Math Nights they often comment how much more clear these are compared to the standard algorithms.

There is also a myth in that if a student can do the standard algorithms then they are good mathematicians. This is not the case. Computational understanding is not the same as operational understanding. I know students whose parents have spent a lot of money on certain well-known Math clinics so that they can do long division but, when these students are given a word problem, they do not know what to do: they are not sure if they should add, subtract, multiply or divide the numbers; or which numbers they should work with; or the order in which they should work with these numbers. When students are taught the algorithms by rote this operational sense is something that is not developed; it is the Achilles heel of certain well-advertised after school Math programs. When I hear people emphasise the importance of getting students to do these standard algorithms by rote I must point out the research by Northcote and McIntosh (1995) which shows that most adults do not use written algorithms in their daily lives.

Ms. Wente also states that ‘For years, Math professors at our leading universities have been telling elementary and high-educators that their methods don’t work’. Well, I went to a four-day Math Teachers conference at the University of Waterloo in August and this certainly wasn’t the message. On the contrary, there was an emphasis on getting students prepared by making sure that they are good problem solvers, flexible thinkers and not reliant and facts and formulae that they don’t understand. Not once did any of the professors say ‘Make sure that they know their long division’. The ‘discovery’ method which was alluded to by Ms. Wente is in fact the method that Mathematicians use all the time. The great geometer Donald Coxeter said“The best path to learning is hands-on… leading to the thrilling intellectual buzz of a self-made discovery”. Karl Freidrich Gauss (arguably the greatest mathematician ever) said“It is not the knowledge but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment.” When it comes to getting advice about how to best teach Math, I will defer to Coxeter and Gauss and not to Ms. Wente.

Should students learn their basic number facts and multiplication tables? Absolutely. But a better question is ‘How should they learn their facts?’. Rote memorization is not good enough. It leads to what is referred to as pattern interference whereby rules, formulae and facts get confused and the learner is unable to apply these in problem solving situations. Understanding why rules work is crucial. For example, knowing your nine times table is all very well but will not help you efficiently figure out 9×37. Knowing that nine times a number is the same as ten times a number minus that number is far more powerful and will let you figure out that 9×37 is just 370-37 or 333.

Should teachers have a greater comfort level with teaching Math? Absolutely but a better question is ‘How should teachers get a greater comfort level?’ Simply teaching from a textbook, giving out worksheets is not the answer. The faculties of education must revisit how much time they allocate in their programs to Math (often a poor distant cousin to Language). If one looks at countries that perform well in international Math tests prepare their teachers there is a common theme: Collaborative Inquiry. Teachers participate regularly in lesson study, watching students tackle problems and then engage in professional dialogue to learn from what they have seen. It is a process which Ontario’s Ministry of Education has been using and the teachers whom I work with have said is the best professional development they have had.

Should we improve the standard of Math education in this country? Absolutely. Why would you want to lower it? But responding to the cries of ‘back to basics’ (whatever that means) from people who have no background in Math Education is not wise. Instead, look to the countries we want to compete with and see how they approach Math. For example, in Singapore, Finland and Japan the curriculum is not dominated by rote learning; it is dominated by problem-based learning balanced with opportunities for students to practice what they have learned. I would respectfully suggest that the next time the Globe and Mail wants to make a meaningful contribution to improving Math education they don’t give platforms to people with no background in the area and that instead they do some proper journalism and seek the educators who have consistently gotten their students to love Math and be successful at it, no matter what grade they teach. These are the people who will be much better placed to inform your readers of what good Math looks like and what good Math learning and teaching looks like.

Why We Need Puzzles in Maths

Now that September has come again, I realise that I have spent a fair amount of my summer 'off' doing maths. In July I finished off a Calculus course that was part of the Masters in Mathematics Teaching I'm doing at the University of Waterloo. I was also at Waterloo for a four-day Summer Conference for teachers which was well worth it. In between, I took Jo Boaler's excellent How to Learn Math online course and also started reading Alex Bellos's fascinating book Alex's Adventures in Numberland as part of an online Maths bookclub http://mathsbookclub.wordpress.com/.
More importantly, my daughter (who is 9) completed her first Sudoku on her own. Now it wasn't a Sudoku with numbers (it had Disney characters!) but it still required a fair amount of puzzling and struggling on her part.
And I liked it that she struggled.

And so did she.

 

It got me thinking about how we should use puzzles on a regular basis with our students. Now I'm not as fond of Sudoku as I am of some other puzzles like Kakuro and KenKens; not only do these require problem solving and resilience but they also require good number sense.
I first came across Kakuros in the Toronto Star:



The idea is to fill this in using the numbers 1 to 9 that add to give the values shown in the row to the left or the column above with the restriction being that you cannot use a number more than once in any sum. So with a little thought, I can fill in the following as a start:

What I also do is look for certain rows that have triangle numbers (e.g. 15=1+2+3+4+5) or what I call 'frustrum numbers' (e.g. 24=7+8+9) as these will give me some clues as to what numbers must be used.

 

With a bit of puzzling (OK, sometimes a lot) the numbers topple like dominoes.

KenKens are a bit like Sudoku in that they are based on a Latin Square (i.e. a grid in which no row or column can have the same symbol appear more than once). The wonderful website http://www.kenken.com/ allows you to vary the size and level of difficulty of the puzzle but you basically start with a square like this which needs to be filled with the numbers 1 to 4: 

The single squares can be filled straight away: 

Now I can work on some of the blocks:  

This allows me to complete the bottom row, then the third column, then complete the third row and the second column: 



 

For more of a challange, all four operations can be used with bigger squares: 

 

 

 

And if you are teaching complex numbers, how about a complex KenKen (courtesy of The College Mathematics Journal):

 

 Part of being a good problem solver is tenacity, the ability to stick-at-it. It strikes me that this is a quality that is positively encouraged through puzzles such at these.

As such, why should they not be a regular part of any Maths classroom?