Do Two Minuses Make a Plus? (1)

I've been into a few Grade 7, 8 and 9 classes over the past few weeks and one of the areas of focus has been integers. It's no big secret that integers can be problematic for many students (and, I would add, adults too). Chatting with some students, I heard things like "Two minuses make a plus": is this really true? I saw students use this rule to then say that ¯3+¯3=6. My concern is that students have developed misconceptions because they have skipped straight to the abstract part of the concrete-diagrammatic-abstract part of that continuum. So here are some activities I used to help them through the concrete and diagrammatic phases to allow them to make sense of the abstract:

1) My first goal is to get students to be able to add integers. Like many teachers, I use integer tiles to represent integers concretely:

I tell students that this is the most important concept: if the red tile represents negative 1 and the yellow represents positive 1, then together they make the eponymous 'zero pair'. Anytime they see a zero pair, it represents a value of 0 (even though you see two tiles).
I then ask students to use the tiles to represent quantities such as ¯3 and 5:

Then I ask them to use these two sets to work out ¯3+5. What is useful here is by the actual act of addition (i.e. joining the two groups) we get something like this:

I then give other examples for the students to work:

One of the things I'm aiming to do here is to get students to visualise the solution. For example, in the fourth question above, students (who were using tiles for the first three questions) were able to picture (mentally or with a bar model) 19 yellow tiles pairing with 19 red tiles (giving 19 zero pairs) leaving 4 red tiles, or negative 4. 
After enough of these examples, I ask students to complete this statement:

Adding a negative is the same as ...

Before too long, there is consensus that it is the same as subtracting a positive. This is much better than saying 'a minus and a plus makes a minus'.

2) Once we have established that adding a negative is the same as subtracting a positive, I then like to show how we can use a number line to show this:

Some students really like this way, others less so but I let them choose what method works best for them.

3) Subtracting integers can also be modelled with integer tiles but I've always liked to introduce it in this way. First, I call up three students and give each of them four cards. Two of them have cards numbered 1 to 4, whilst the third has cards numbered ¯4 to ¯1. I then tell a really, really corny joke such as "What do you call a fly with no wings? A walk!" and ask the students to give me a mark for my joke. For example, I might get this:

I feign disappointment at the student who has given me a negative score and the class has a good laugh about this. We then get my total, in this case 4+2+¯3 gives 3. At this point, I make a strong protest against the student who has given me a negative score to such an extent that I demand that this score is removed and I 'escort' them to a corner of the room. Again, laughter all round. I then go back to the remaining students:

I then ask the students what my 'new' score is and they tell me 6. I then say, "We started with 3 and ended with 6. But what happened in between?" The response is usually "You took away the negative 3." I then write down what they have said:

I repeat this for a couple more bad jokes then ask them to look at the results and ask them to complete this sentence:

Subtracting a negative is the same as...

Again, before too long, the consensus is that subtracting a negative is the same as adding a positive. This is mind-blowing for some students and I tell them that I can understand why they might find this counter-intuitive but this is because they have always believed that subtraction makes smaller. However, the activities have shown that subtracting a negative will make a number larger: like it or not, we cannot argue with this beautiful mathematical logic!

It is possible to model this with integer tiles:

Notice that we cannot 'take away' negative 3 because we do not have three red tiles in our model. We can overcome this though, by adding enough zero pairs:

Now we can take away the negative 3:

Some students like this approach, others less so. Either way it reinforces the big idea that subtracting a negative is the same as adding a positive. This is so much better than saying 'two minuses make a plus'. Once again, I follow this up by getting students to try some questions that will wean them off the concrete and getting them to think diagrammatically and abstractly.

Don't Blame the Curriculum

The announcement of Ontario's EQAO results was followed with a predictable tsunami of opinion as to what is wrong with Math in Ontario schools. The media looked to various 'experts' for their thought, many of whom criticised the 'discovery-based' curriculum as being the cause for low math scores. I noticed that many of those making these points are not experienced Ontario teachers with a working knowledge of the Ontario curriculum. I read that the teaching of facts is optional. I read further that the teaching of algorithms is frowned upon. I also read in one tweet that the real culprits were the educational consultants who were forcing bad methods on unsuspecting teachers and students.
Well, I am one of those consultants and have been one for nearly ten years. I have a Math degree and a Masters in Mathematics for Teaching from the University of Waterloo. In these last ten years I have worked in hundreds of classrooms with thousands of students from kindergarten to Grade 12. In doing so I have learned so many things that I wish that I knew when I started teaching in 1990: not that I was doing a bad job back then, but because I would have done a much, much better job.
I have also run various Math nights for parents in our school board, helping them get to grips with their own math phobias as well as giving them strategies to help their children with their Math.
As such, I feel that I have to address some of the myths that have recently been stated.

Myth 1) Teaching of Facts is Optional
No it isn't. It is clearly in the curriculum.

Myth 2) Teaching of Formal Algorithms is not Allowed.
No, they are there in the curriculum too.

Myth 3) The Ontario Curriculum is Discovery-Based.
No it is not. Read the front matter and it will say that direct instruction is part of good teaching. This (and the need for students to be fluent in their math facts) has been emphasised in many different sessions that the Ministry has run that I have attended.
There is a great word that is used throughout the curricula though: 
Develop
This is a much more powerful word than 'Give'. Sometimes, a student led activity will develop the formula, sometimes a teacher-led activity will be used.

Now I am sure that there are some people who want to argue that developing formula is 'discovery-based' and thus a waste of time, that we should just give the formulas to the students instead. This attitude (whilst bordering on elitist) is also easily disproved: when I have developed formulas with students and parents and then ask them 'Would you have preferred it if I just gave you the formula and told you not to worry about why it works, just memorise it?', they always reply 'No!'. 

Myth 4) Low scores are a result of the discovery-based curriculum
Even my grade 9 students know that correlation does not mean causation. Yet this is the conclusion I have seen many folk jumping to. Yet none of these people can back up this claim unless they have gone into the classrooms to see how math is being taught there. Even if a curriculum is 'discovery-based' (whatever that means) that certainly does not mean that every classroom will be discovery-based.

Now this doesn't mean to say that we don't need to improve Math teaching in Ontario: of course we do. We can always get better. As educators and parents, we need to actively seek out the most effective ways for teaching Math. Countries that tend to do well on international Math tests such as PISA and TIMSS have math curricula that emphasise both the conceptual and the procedural aspects of learning Math. This is backed up by research which maintains that these are bidirectional: it is not necessarily so that we need to learn all the facts and rules before we can learn to solve problems. Likewise, it is not necessarily true that all of our facts and rules are learned after we have solved some real-world problems.
And, of course, it is very important that students are given good opportunities to practice what they have learned. But what constitutes good practice? Again, research points strongly toward spaced practice. We need to think about how we can incorporate this into our schools.

Having worked in Ontario schools, I know that there are some brilliant Ontario teachers who are getting great results (EQAO and otherwise) with their math students. These are the people we need to look to when searching for answers on the best methods of teaching and learning Math. And when we do, we will see that there is a lot of common ground in the methods that they use to deliver the curriculum effectively.

So as an experienced, qualified Math teacher I will say this: it is not about 'back-to-basics' and it is not about 'discovery-based' Math. It is about balance. The Ontario curriculum (whilst it might need some fine-tuning) allows for this.

My Top Ten Favourite Math Games and Puzzles

I'm a firm believer that Math games and puzzles offer an excellent opportunity for practising Math. I sometimes wonder if kids these days are at a disadvantage because they don't play board games to the same extent as we did growing up. As an example, when playing Monopoly, if I landed on Northumberland Avenue and want to buy it for £160 then I need to use number sense to figure how to do this (three fifties and a ten?, a hundred and three twenties?, two hundreds and get two twenties back?) Today, there is a version of Monopoly where all transactions are done via swiping a credit card!

I often present at Math Nights in various schools throughout our Board and one of the things I do is show parents games and puzzles that I play with my own kids. In doing so, I also let them know how these games help a learner develop mathematically.

So here are my top ten favourite games and puzzles:

10) Swish. Fantastic for developing spatial reasoning. This pack of transparent cards has a variety of markings on. The goal is to pair up cards but this will require reflections, rotations and translations.

9) Darts Having a dartboard in my bedroom, and playing countless games with my brothers did wonders for my mental arithmetic. Still one of my proudest achievements is when I checked out on a 170 (treble 20, treble 20, bull). Admittedly, might not be the best thing to have in a classroom for a variety of health and safety reasons...

8) Pentago A twist on Connect 4. Players take turns placing black and white marbles on a playing board. After each go, they can turn one of the quadrants 90°. The goal is to get 5 in a row. Easy to learn, not so easy to master!

7) Tantrix Can be played solo with others. The basic idea is to place hexagonal tiles together so that you form a continuous loop of one colour. The puzzles increase in complexity as you add more tiles. Great for spatial reasoning and developing persistence.

6) Pass the Pigs Great for practicing mental arithmetic with numbers up to 100. Throw two pigs and how they land will determine the number of points you get. Now decide if you should (piggy) bank these points or continue rolling. However, if the pigs fall in a certain way, you lose all your points.

5) Kanoodle Another spatial reasoning puzzle that is available in three different versions. In the Kanoodle Genius version, you have to arrange the seven pieces into either a hexagon or a tetrahedron. A puzzle booklet provides you with starting positions for some of the pieces: it's up to you to work out where the others go. I have seen students spend their whole indoor recess on these puzzles.

4) City of Zombies A cooperative game in which you either all win or all lose. Zombies are attacking you and the only way to destroy them is through Math! Combine the numbers on three dice using a variety of operations to kill as many zombies as you can. Great for mental arithmetic especially with junior students. Only downside is that I've not seen it being sold in the usual toy shops in Canada so you'll have to have it shipped from the U.K.

3) Shut the Box I have played this so many times with my own children and it has done wonders for their mental arithmetic and their ability to decompose numbers. Throw two dice, add them and then knock down the tiles (1 to 9) that add to that total. So if you throw a 1 and a 5, you could knock down the 6, or the 1 and 5, or the 2 and 4, or the 1, 2 and 3. Continue throwing the dice and knocking down totals until you get to a point where you cannot knock down any combination of tiles to match the dice total. The tiles you have left are your total. Now it's the turn of the other player who needs to get a lower final score than yours. When I play this with my kids, we usually play first to three games wins. Once, I started by throwing a double 1 (so knocked down the 2 tile) then threw another double one which meant I ended up with the worst possible score!

2) Farkle A dice game with some similarities to Yahtzee. The rules (found here) might look a bit confusing to begin with but are easily understood once you start playing. Really good for mental arithmetic involving larger numbers (that you might find in junior grades) and also strategising. 

1) Cribbage Controversial? Maybe I've been influenced by happy memories of playing this in various pubs and in tents whilst wild camping in the Scottish highlands. But for me, cribbage is bursting with Math: decomposing numbers, mental math, combinations, probability. And there is something very satisfying about pegging your score.

Now I know that many of you are probably thinking something along the lines of "How can you leave out Yahtzee?" or "No Blokus? What's wrong with you!" Even as I write this, I'm debating whether I should have included Pandemic, a cooperative board game in which either all players win (by discovering four cures) or all players die from a pandemic. 

Also, I recently bought Prime Climb which is a great for practicing mental arithmetic as well as learning (in a game situation) about prime and composite numbers.

And I'm sure some of you will even point to the Game of Life as being a board game where players will have to use their math skills. So for all of you who disagree with my top ten, let me know what yours is!

Measuring Up

I was in a grade 6 class yesterday and a couple of quick questions really made me understand why students might have some measurement misconceptions: they see something like length as a count and not a measure. 
This was the first question:

We were expecting some answers of 15cm but in actual fact, there were very few of these. The most common answer was 9cm. They weren't measuring, they were counting. But they started the count "1" at the '7' mark, then "2" at the 8 mark and so on. Now, both myself and the class teacher were pretty sure that if we gave the students a proper ruler, they would have been more successful, but is this because they are actually measuring or merely reading a number? 
The second question revealed the extent of this misconception though:

The majority of the students circled the correct answer, c. We then asked the students to explain how they got their answer. Some gave this:

It was the same mistake seen in the first question! These students were counting dots, not measuring length. It was clear that they knew how to calculate area of the two shapes, but were using the wrong values.
Others gave this answer:

These students were again counting to get the length/base and width/height, but this time were counting the dots inside the shape to do so!
It is perhaps not surprising why students should have this misconception. All their measurement experience from primary years is based on using a count to get a measure. So they have developed this misconception that measurement is discrete not continuous.
So how do we change this? Well, we started by using the response to the first question to create a contradiction:

Starting at the tail of the caterpillar, I asked who thought it was 9cm. I then gradually moved my finger towards the head, and then asked how long the head was. They said 2cm which I wrote above the head. I then put my marker on the 7cm mark and told them that I was about to draw a line and that they had to tell me when I reached 1cm. (By drawing the line, I was hoping that they would see the length as a continuous measure and not a discrete count). Sure enough, they told me to stop when I had drawn 1cm and I recorded this on the diagram. I then compared the 1cm to the 2cm (see above) and asked "Can it be both?" At this point, faced with the contradiction, the students had to agree that their 2cm answer (and hence their 9cm one) was wrong. By then drawing further line segments, they could see that the actual length was 8cm. We immediately followed this with some practice whereby they used a broken ruler to measure the perimeter of various shapes and they did this with great success.
What I learned from this is that when I see students use a ruler to get the right numeric value, I assume that they have measured this. Now I know that they might have counted. These quick questions helped us uncover and confront this potentially problematic misconception.

Return of the Math Vines

In a previous post I shared some Math Vines that I have created with my colleagues Chad Richard and Dan Allen. Here are some others we have recently made. The first one which develops the area of a trapezoid formula has proved to be very popular (and, we think, is much more beautiful than our original attempt which is below it).


We do like using Vines to show why formulas work. The next one was actually inspired by brother-in-law, Stuart Sudlow, who is tech teacher for TDSB.



Vines can also be used to develop a concrete understanding of things such as fractional equivalence.

Or Vines could be used to prove things through spatial reasoning:


Or reveal hidden mathematical gems:

Finally, look at this Vine and ask yourself where the missing square went:

Yohaku: A Different Type of Number Puzzle

**Update: I have now created a website with these yohaku puzzles: http://www.yohaku.ca/

Like many folk, I'm a big fan of number puzzles. I always look forward to the Kakuro in the Toronto Star every Sunday: sometimes this will take me 5 minutes to solve, sometimes 5 days. I also love KenKens and have been seeing these used in classrooms as a great way of practicing number bonds and problem solving at the same time (always a win-win situation in my view). Sudokus are also OK but I won't go out of my way to work on one of these. I recently came across Multiplication Squares on the nrich maths site. This one particular problem struck me as having lots of potential of allowing students to practice and to problem solve both with additive and multiplicative thinking. I quickly created a few of these and, as I often do, I tried these with my own children.
I started with a 2 by 2 multiplicative one for my daughter:

The idea is to fill in the cells so the top two multiply to give 40, the bottom two multiply to give 14, the two left multiply to give 56 and the two right multiply to give 10. As she did this, I could see her use multiplication and division to fill in the cells. She also had to think about which pairs of numbers multiplied to give 40: in my opinion this is a far richer question than the more closed 4×10. I then gave her a 3 by 3 one: 

This definitely caused her to think more yet all the time she was using her number bonds: what three numbers multiply to give 42? To give 70? She had to stick at it for a while before she got it, trying a couple of possibilities in the process: much like I do when working on a KenKen or Kakuro.
At this point, my son wanted a piece of the action, so I gave him a 2 by 2 additive thinking square:

When I created this, I had a 2 in the top left hand cell: he started with 5. And that's when it suddenly hit me: there are an infinite number of solutions to these! I began to love these even more.
My son, seeing his sister work on the 3 by 3 multiplication, then asked for one for himself. I duly obliged:

As he persevered on this, it was clear that there was lots of lovely decomposition and recomposition going on. Then this happened:

I tried these puzzles in a grade 3/4 class for the first time yesterday. We began with a 2 by 2 additive puzzle which they absolutely gobbled up. When they finished, it was really powerful to say, "Compare your solution to your neighbour's" and then see them get excited when they had done it in different ways. 
I then wanted to see if they were ready for a 2 by 2 multiplicative puzzle so gave them this:

Now this is a class which has very recently begun learning about multiplication through the use of arrays and been recording their findings:

Without a doubt, they had to think more about this puzzle but the fact that they could think of 12 in different pairs of factors helped them.

      *           *          *

As a teacher, it takes hardly any time at all to create these and I can structure them such that if I want to focus on a particular fact, I can include that as a possible solution. I can also add an extra level by including some restrictions to the solution. For example, my son gave me this one to try:

 "I already know the answer," he told me. I looked at it and said, "Is it all fives?"
"Yes, but your not allowed to use fives!"
So this got me thinking. I can create a whole new set of puzzles with restrictions. These will require more perseverance to solve (you're welcome). For example:

Call me bias, but I really like these and as I haven't seen this type of puzzle before, I get to choose what to call them. 

So I have decided to call these puzzles: YOHAKU.

Catchy, eh?

I have created a website with these on: http://www.yohaku.ca/ Who knows, maybe people will like doing these and Yohaku will become really popular. Maybe one day they will be as big as Sudoku and KenKen! I'd love to know what you think.

Measurement to the Fore

I am sometimes concerned that our students' experience of Measurement consists primarily of memorising formulas and conversions at the expense of the the actual skills of estimating and measuring objects. It is all very well if a student can tell me how many millimetres are in three kilometres, but if they cannot measure the length of a floor accurately, then they have not learned about Measurement. 
My experience has been that as students use metre sticks, measuring jugs, and sets of scales, they develop such good measurement sense that they can see that since one metre is 100cm, then they will also understand that 5m is 500cm or that 725cm is 7.25m. In other words, they will use their measurement sense to make conversions. Those who rely solely on the staircase method for conversions will not develop their measurement sense so easily. 
As such, I like giving students activities which actually gets them measuring. It allows me to see if the are using the tools available to accurately measure objects. In two classes this week, I began the following task:

Design a container to hold twelve golf balls.

We gave each pair of students a single golf ball and told them that they could choose to use any further tools they might need from around the classroom. Right from the get go, all students were engaged in the task. They quickly began thinking about how they should arrange the golf balls. Some initially decided to design a bucket (but quickly abandoned this idea!). Others looked at a 12 by 1 arrangement; or a 2 by 6 arrangement; or a 3 by 4 arrangement. A couple of groups even thought about a 2 by 2 by 3 box. As they began to measuring the golf ball we noticed a few misconceptions:

1) Measuring from the edge of the ruler and not the zero mark.
I reckon most teachers have seen some of their students make this mistake or a variant of it (e.g. measuring from the 'one' mark). One way to correct this is explicit teaching: demonstrate to the student that measuring this way leads to inaccuracies and have them move the object (or the ruler) so that they measure from the zero mark.

2) Measuring the circumference and not the diameter

Maybe some students 'see' this attribute (circumference) more readily than the diameter. Having measured this, one group multiplied by 12 to get 158 cm. We asked them how they would use this value to design their container. When they seemed unsure, we showed them a metre stick to help them realise that this measurement was greater than a metre. At this point, they began to realise that they were in a bunker so we prompted them with 'Is there a better measurement to use?'

3) Spheres are difficult to measure!

We could have given the students a similar task with objects that are much easier to measure (e.g. design a box that will hold 12 juice boxes) but that being said, I think it was great that they were challenged; it brought out some creative approaches. For students who really struggled, I put a ruler either side of the golf ball and they measured the distance between these two rulers.



4) Only thinking two-dimensionally
This was fairly common and we dealt with it using this prompt: 'OK, you are phoning me, the manufacturer of these containers. Tell me all the information I need to know to make your container accurately.' This got the students to think about the height of the container.



The next step for both of these classes is to take their plans and actually make the boxes out of card. However, before this happens, the students need to know the old adage, 'Measure twice, cut once'. Most of them measured the diameter as 4cm when really it is about 43mm. They need to factor this exact measurement into their design (and consider if they need to add a little extra) before they actually begin construction.
Once all these boxes are constructed (again, this will involve a lot of hands-on measurement), they can argue which of the boxes is best for the job. I already know one student who is convinced that the 1 by 12 box will be best as it will fit in nicely alongside the clubs in the golf bag: it will be interesting to hear the other students' thoughts on this!

       *       *      *

I once gave students the task of designing a pasta box that would hold 1000 cubic centimetres. The choice boiled down to a 10 by 10 by 10cm cube or a 10 by 5 by 20cm rectangular prism. Many students made the argument that the cube was the better box as it had less surface area and therefore would be cheaper to manufacture. One student though argued against the cube as it was difficult to hold and therefore pouring the past into a pot of boiling water would be an issue. Another student argued that the cube had a smaller 'front' than the rectangular prism (100 compared to 200 square centimetres) so it would not be as visible on the shelf of a grocery store which would impact sales. 
It goes to show that sometimes the mathematically best solution is not the best solution practically.