Do Two Minuses Make a Plus? (2)

For me, getting students to understand what happens when we multiply integers depends on them:
1) Understanding what happens when we add integers
2) Understanding the commutative property and
3) Extending patterns.
I start by getting them to model ¯3 + ¯3 + ¯3 + ¯3 concretely:

 I ask how else could this be written and the usual replies are 4ׯ3=¯12 or ¯3×4=¯12

Next, I ask them to model ¯4 + ¯4 +¯4 concretely and typically I get something like this:

although some students will just the use the first array. Either way when I ask how this could be written, the usual replies are 3ׯ4= ¯12 or ¯4×3=¯12. After a few more examples of this type I then ask the students to complete this statement:

Multiplying a positive and a negative gives...

Here I want the students to see that because of commutativity, it doesn't matter if we multiply a negative by a positive, or a positive by a negative: the product will always be negative. I deal with division in a similar way as well as connecting to fact families or related facts.

Multiplying two negatives is a bit more tricky to represent concretely though (at least in my opinion). I can tell the students that multiplying by ¯1 is equivalent to flipping the two-colour counters:

but this of course begs the question: Why?

So I prefer using patterns to make sense of multiplying two negatives. Firstly, I ask them to continue this pattern which will confirm the negative multiplied by a positive gives a negative result:

Next, I ask them to extend and complete this pattern:

As the students notice that the product of each row increases by 2 each time (i.e. from ¯6 to ¯4 to ¯2 to 0) then the next two rows must be 2 and 4 respectively. In other words, a negative multiplied by a negative gives a positive.

This is mindblowing for some students: they think that if you have a negative number and multiply it by another negative number, then the product will be somehow even more negative! So I make the point of assuring them that it's OK to be perplexed by this as this was how I felt learning this result. But, like it or not, we cannot escape with the beautiful mathematical logic shown above. So negative multiplied by negative will give a positive. This is much better than saying 'two minuses make a plus'.

Some Thoughts on Terminology and Notation

In the UK, I remember there being an emphasis on careful use of terminology with integers. For example, for the expression:

¯3–¯5

I say "negative 3 subtract negative 5". I notice that here in Canada, the tendency is to say "minus 3 minus minus 5". I often see it written as (–3)–(–5). To me this cumbersome and could lead to some misconceptions. So when writing expressions like these, I make a point of using superscript symbols when I went to indicate negative.

Also, I give a little cheer whenever I hear a weather reporter say something like, "Today's low will be negative 7."

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!