Wednesday, March 20, 2019

Scaling Up

One of the biggest hurdles to mathematical understanding is moving out of additive thinking into multiplicative thinking. There are many reasons why students get stuck in an additive phase so what can we as teachers do to move them into a multiplicative phase?
I worked with a Grade 9 Applied teacher recently who noticed from her diagnostic tasks that many of her students could not think multiplicatively. As they were about to begin some work on ratios this was going to be a problem. We decided to adopt a concrete-diagrammatic-symbolic approach to move students on from additive thinking. 
We began with a simple problem:

The weights of two dogs as puppies and fully grown are shown:

Which dog grew more?

Without exception, the students said that they grew by the same amount (i.e. 6 kg). They were looking at how much weight had been ADDED.
So we then asked them, is there another way of thinking about this. After a bit, one student noticed that the first dog had DOUBLED in weight whilst the second dog had not increased by the same rate.

This was the platform we needed to build on.

I was clear with them: we need to learn how to compare things not just by addition but also by multiplication. I told the that we were going to do some activities that would help them how to see things in terms of multiplication and not just addition, and that this would make them better mathematicians. 
I also told them that we were going to do this in three steps: concretely, then diagrammatically, then symbolically. 

Each student was then given a set of cuisenaire rods.
I told them to find two orange ones and put them end-to-end. "If one of these is 10, how much will two be?" "20!" came the instant reply.
I then told them to put a yellow rod directly below the two orange ones (and showed this using the mathies.ca Relational Rods tool). I then asked them to estimate how many rods would be need to match the two orange rods. 

After they made some suggestions, I asked them to find out and then tell me how much a yellow rod was worth: they were able to tell me that it was 5.
I then asked them to write a number sentence for what they had just done. Over half wrote 5+5+5+5=20 so I then asked them to write a number sentence without using an addition sign. This nudged them toward multiplication and they wrote 5x4=20.

Again, I was clear with them: this is the goal of today's lesson...to think multiplicatively.
Next, I asked the students to do this again but this time with the purple rod. Seeing the students carefully lining up the rods to make sure they were equal to the two orange rods (and the four yellow rods) made me realise that maybe this is the experience that they had missed out on: the actual concrete act of creating equality using equal groups.
They wrote 4x5=20 without any prompting. One student then noticed something: "I can write it another way without using addition. If you split the rods up again you are dividing the 20 so you can write them using divisions!"

This led to related facts:
4×5=20
5×4=20
20÷5=4
20÷4=5

Next, I told them that as they were grasping this so well, it was time to scale up: now we need to use larger numbers and that these would be better modelled with diagrams. So I asked them to write a set of related facts for this diagram:
From this alone, they were able to write:
20×6=120
6×20=120
120÷6=20
120÷20=6
No-one wrote 20+100=120. We were seeing the students shift away from additive thinking.
Curious I wrote down my favourite math fact on the board:
37×3=111
and told them that we were about to scale up again. I asked them to complete the set of related facts which they were able to do even though they had not learned the 37-times table!

We then split them into visibly random groups and gave them a problem to try:
Two people do some decorating. Ann worked for 2 hours, Bill worked for one hour. Together they were paid $30. How much should each person get?
As the groups worked on this, it was clear that they realised that it would be unfair for the people to be paid the same amount. Most groups got the sense that Ann should get paid twice as much as Bill and used different ways to come up with an answer. We summarised their their thinking by using a bar model approach:
This allowed them to see the 'three-ness' of this problem and allowed them to see that each hour block is equivalent to $30÷3 or $10. We then challenged them with the following set of problems and encouraged them to use bar models to show their thinking.
It was pleasing to see many of them successfully use the bar models to solve the problems (though I wish I took more pictures of their work).

When I asked how they felt about this concrete-diagrammatic-symbolic approach at the end of the lesson, the students told me that it really helped them. 

Sometimes it takes just a well-timed nudge to move students on.

Thursday, February 7, 2019

Polygon Angle Sums: Develop, Don't Give.

A common way to get students to see that the sum of three angles is 180° is to get them to tear the three angles and rearrange them to create a straight line. Rather than giving the angle sum formula for any polygon, I wondered if I could use this approach would work with other polygons so as to develop the formula instead. I tried this with a couple of grade 9 Applied classes.
I visibly random grouped the students in to threes and gave each group a different paper quadrilateral. It didn't take long for the to rearrange the angle to form a complete turn and for them to tell me that quadrilaterals have an angle sum of 360°.
Moving on to pentagons, I wanted to make sure that the angle would rearrange clearly into one and half complete turns. I figured that the best way to do this was to give pentagons that had two right angles like this one:
This allowed the students to rearrange like so:
...and then tell me that pentagons have an angle sum of 540°

In a similar way, I then gave each group of students a different, hexagon, heptagon,  or octagon whose angles could be torn off and easily rearranged into full and half turns such as these below:

This nudged the students into quickly rearranging the angles:

As the students were finding the angle sums, I recorded their results using Desmos so that we could all see what was happening:

I could now ask the class "What do you notice? What do you wonder?"
They quickly noticed that the angle sum was increasing by 180° each time the number of sides increased by one. Some students wondered if this was something to do with the angle sum of a triangle.
So, I then sent them in their groups to the vertical whiteboards and asked them (one polygon at a time) do choose a single vertex, and from there, draw as many diagonals as possible to any other vertex. 
As this decomposes the polygon into smaller triangles (each of which has an angle sum of 180°) they could then confirm their earlier results.
I now challenged them to predict the angle sum of a dodecagon. As they were now recording their data in a table, it made it easier for them to spot and extend this pattern:

I followed this by asking them to tell me the angle sum of a 102-sided shape and then to generalise for any shape. They were able to see that each polygon with n sides could be split into (n-2) triangles and so the angle sum is (n-2)×180°. 
This was pleasing as we had not only used the concrete-diagrammatic-symbolic continuum but also the next-near-far-any continuum for patterns. The students seemed pretty chuffed that they had 'discovered' this rule themselves. I finished the lesson by giving them these questions to consolidate what they had just learned.




Monday, January 21, 2019

A Tale of Two Questions Part 2: Junior

Continuing on from my last post, one of the things I like to do when analysing EQAO results is to look at how students responded to individual questions. Good questions that expose students' misconceptions can give us pointers as to what we can do to fine-tune our teaching.
Two questions that I found particularly interest in last year's Junior EQAO were these:

Again, when I show this to teachers, principals and parents, I ask them to predict what percentage of students they think got these correct. For the first question, bearing in mind that primary students struggled with a similar equality question, typically people reckon that about 50% of students got this correct.
The actual amount provincially is 68% and the breakdown for each response is shown below.

To an extent, this is encouraging: It would appear that students have a better understanding of the meaning of the '=' sign than their primary counterparts and chatting to a small number of junior students would at least anecdotally confirm this. From a number sense point of view, it would appear that many of them are comfortable enough to work with both a subtraction that might involve regrouping and a multiplication that requires one of the more commonly mistaken facts (i.e. 8×7=56). A very small minority (those who answered 11) appear to have added 72 and 16 instead of finding the difference. And there is another group who have answered 56. Now this maybe because they have misconceptions with the '=' sign and simply answered 72–16. But I have also seen both students and adults do something like this:
...and for whatever reason selected the 56 option in the answer key. When I saw one parent do this at one of our Math Café for parents, I asked them to tell me the value of m. They correctly told me 7. I then asked them, what answer they selected. Again they answered 7, so I asked them to look at what they had actually written. When they saw they had selected 56, they were stunned and said "How on earth did I put that?".
It made me realise that just because somebody puts the wrong answer down on a multiple choice test, doesn't mean to say that they don't understand what to do.

The second question was the answered correctly by the lowest number of junior students (34%). The actual breakdown is shown below:


It is classified as a measurement question but my sense is that the errors that are being made here are not to do with how many millilitres are in a litre. The mistakes are made because students (and adults) need to decide which operations to use. They have deficits in their operational understanding. If students are told what operation to use (e.g. calculate 45+99, work out 2496÷24) then this is computational understanding which, though useful, does not prepare them to answer the milk question. If students experience a diet of nothing but computational questions then they will always have difficulties when they have to decide what operations to use. Doing a hundred long division questions will not teach students what division is and when it should be used.

But also, if students are just given one-step word problems where they have to decide on what single operation to use, then this will not be enough to prepare them for the question above: two of the answers above are the result of students most likely selecting a single operation (6×4=24, and 250÷4=62.5≈63).

Perhaps a strategy that I would avoid at all costs is the keyword approach. Here students are taught a list of words and their accompanying operation. One of the problems with this approach is that sometimes these keywords are not present in the question (are there any keywords in the example above?). Additionally, I have seen students read a question, write down two numbers involved then an operation suggested by the keyword regardless of whether or not this makes sense:

In terms of how to develop operational understanding, my go to is always the concrete-diagrammatic-symbolic approach. It was interesting to see how adults tackled this question when I gave it to them. Some are surprised that this is on the junior EQAO as they think (rightly or not) that it is too difficult. There were more than a few who struggled to figure out what operations to use until they tried drawing a diagram: then the operations required became clearer. This is one of the reasons why I like bar models (as I have blogged before here and here.)
As operations are in essence actions, any strategy that gets students thinking about the actions involved in the question will tend to be a good one.

Wednesday, January 16, 2019

A Tale of Two Questions Part 1: Primary

As part of our analysis of the EQAO data form our schools, I like to move way beyond a simple percentage figure that tells how many students are at or above provincial standard. In terms of looking for data that will have an impact on the way math is taught, I find it more useful to look at how students responded to individual questions. Good questions can often expose students' misconceptions which in might in turn give us pointers as to how adapt our teaching.
Two questions which I found particularly interesting in the Primary Math EQAO were these:



Both of these questions are on the section of the paper where calculators are not allowed.
I have shared these two questions with teachers, principals, and superintendents as well as parents at the first of our Math Cafés for Parents. I ask them to do the questions and then tell me which one they think was the most difficult to do. Some think the first one is more difficult because it has large numbers, whilst others think that the second one requires a bit more thinking even though the numbers involved are small.
I then ask them to estimate what percentage of students they think got the first answer correct. Bearing in mind that my audiences have heard and read a lot about how students are not taught the basics (whatever they are) and that they are not required to learn facts (which is definitely not true) I get a range of answers from 30% to 70%. 

I then tell them that provincially 88% of our students got this correct.

This is surprising for the people I show it to and I can get them to agree that this shows us that students, in general, can add two 3-digit numbers without a calculator. Now I don't know how many of these students did this mentally, or how many needed to write the numbers in columns to use a standard algorithm, or how many used an empty number line, or whether we would have got different answers if some regrouping was required. But I do know that for this question, the vast majority of students met the expectation, contrary to how adults expected them to perform. I then ask the question: do you think that this question shows that the vast majority of our students understand addition. Most people say yes.

The second question is more interesting. Again, I ask the question: what percentage of students do you think got this question correct. Typically I get a range of answers from 20% to 60%. 

The actual provincial results are that 43% of students got this correct.

I find this statistic incredibly informative. Firstly, this is not a 'gotcha' question. It is explicitly stated in the curriculum and a version has been used on many previous EQAO tests. In fact, previous EQAO tests would have had one answer as 9 but this was not an option on this year's test as too many students were choosing it! We actually gave this question as is to all the current grade 3s in our Board and 2% of them actually added '9' as a fifth option, drew a circle alongside and helpfully shaded this in! The actual provincial breakdown for this question is as follows:
Notice that almost as many students selected the incorrect answer '12' as the correct answer. These students have just added all the numbers that they see.


But why do so few get this correct? It isn't anything to do with not knowing their addition facts as we would have seen lower marks in the first question. Instead, I think that this is due to our students not understanding what the equals sign means (as I have written about before here).

For all the talk of 'going back to basics' and memorising facts, we are in danger of being blind to some extremely important mathematical concepts, in this case, the idea of equality. We can get students to develop a solid understanding of equality through the Concrete-Diagrammatic-Symbolic approach (as referred to in an earlier post here). For example, we can begin to develop a concrete understanding through models such as this: 



However, if our students experience nothing but a diet of 2 + 3 = , 9 – 6 = , 32 + 89 =  questions, then they will not develop a true understanding of equality thus putting their future math learning in jeopardy.

Monday, December 3, 2018

Nets of Solids

One of my favourite things to teach is nets. The way I teach it is very different to the way I was taught it. Basically, I remember having to look at something like this:
and then say which of these were nets of a cube. As my spatial reasoning was net well-developed, I had trouble with this type of question. When I began teaching, I would 'teach' nets by getting students to cut out and glue something like this:

Whilst this was an improvement on how I learnt, it was all a bit messy and time-consuming. The way I teach now is much better as I get the students to use polydrons. These are two-dimensional shapes that click together to form nets that can be folded together to form a solid.
I visited two Grade 5 classes recently and, having explained what a net was, challenged the students to find as many nets of a cube as they could. The students were randomly grouped into threes and, to help them record their results, I provided one sheet of grid paper to each group.
Each group got stuck into the challenge immediately. The wonderful thing about polydrons is that it is that it is so quick for students to either prove that their net works:
 Or to disprove:

 After about ten minutes, I always get this question from students: are these two nets the same:
It is a perfect opportunity to stop the class and get their views: some say they are the same, others say that they are different. I can then tell them that to a mathematician, they are the same as they are congruent: they are exactly the same size and shape. I can even show this to the students by reflecting or rotating the nets so that they coincide.
Sooner or later, the students ask me how many nets there are so I tell them that there are eleven. If any student find this net, I make sure that they know that it is my favourite net:

 And if they find this one, I will tell them that this is the one net that I still can't believe folds to make a cube:

At the end of class, we then summarised our results: success! we had found all eleven.

 I revisited one of these classes the next day to follow up with this challenge: Find as many nets as you can of a triangular-based prism. It was fascinating to see how well the students were using their spatial reasoning to discover these nets. Again, some worked:
 And some didn't:
 Again, the students recorded their results on grid paper, taking a bit more care to draw the triangular faces:
 When students found this net, I made sure that they knew that it was my favourite:
 It was another successful lesson and at the end, I consolidated by showing all the solutions that the students found and showed them how a mathematician might classify these:
Having done all this investigation, students will now be in a better position to look at the first picture at the top of this post and use their spatial reasoning to decide which of these are nets.

Wednesday, May 23, 2018

Thinking Outside the 'Box'

I'm a big fan of the array or 'box' method for multiplication (as I blogged earlier here.)  A twitter chat with Britnny Schjolin last week raised this troubling point however:
I know that many of my colleagues are also impressed by it even though they, like me, might not have see it when they were students. I have also worked with colleagues who have openly stated that the 'box' method is not the proper way to show your work or that they don't like it so they won't show this to their students. I'm not sure how widespread such attitudes are but I honestly feel that we are doing our students a huge disservice by not showing them such a powerful representation that allows for so many different connections to be made. If this means that we as teachers should learn something new, then so be it: as educators we must always be prepared to learn new things.

To show how useful I have found this, here is how I recently solved a problem that I came across by using arrays or the 'box' method. 


Prove that the product of four consecutive numbers is always one less than a perfect square.

I started pretty conventionally by trying to generalise the product of four consecutive numbers:


Well, I don't fancy working out that product, but I know if I rewrite the four consecutive numbers like so:
Now I can rearrange to make use of the difference of squares to make something a little more delightful:
 A quick array is drawn to help me work out this product:
Since I have to prove that the product is one less than a perfect square then I need to consider this:
I am good at factoring quadratics by inspection but not so good with quartics! However, I now decide to draw a square array to help me factor by working out the components of each side. The first part solves itself:
To get the 2a³ term, I need to split this symmetrically across the square and think what the next component must be. This makes things very clear:
 This also helps me get the middle product:
 Now to get the -a² term, I need to have a -a² in each of the top right and bottom left cells:
This immediately gives me the last component from which I can write the term as a perfect square:

Thus the product of four consecutive numbers is always one less than a perfect square.

Now, before I learned about the array or 'box' method, I would have chugged through with the algebra and probably would have eventually reached the same conclusion. However, now I can attack and solve such problems in a fraction of the time and with more clarity. This is why we must teach this method:


It is an incredible mathematical tool.

It is not a new idea either. Recently, for fun, I have decided to work my way through Silvanus P. Thompson's classic Calculus Made Easy and I came across this:

Array models were being used back in 1910!

Tuesday, May 15, 2018

Open Middle Fraction Problems

I have often shared the Open Middle website as a source of good thinking problems with the teachers that I work with. This week I created a couple of my own and used them with a grade 6 class who have been working on equivalence in fractions, decimals and percentages. The first problem (given using verbal instructions) I gave was this: 

Use any of the digits 0 to 6 once only to satisfy the statement below.
We used visibly random groups to get the students into threes and gave each group seven tiles to work with. As we began to walk around the room, a few students asked 'What do I have to do?' Taking the lead from Peter Liljedahl's work, I smiled and told them to find out from a friend.
We could see some students getting stuck or making mistakes but resisted the urge to jump in and show them how to get a solution. 

We let them think their way out of it.


Sure enough, we soon heard some 'A-ha's around the room as students realised that the first space had to be a zero, and as they figured out the decimal equivalent of a fraction:
Once they had found one solution, I asked them to find me another. Some used two-fourths as the fraction, others used one-fifth or two-fifths:

One group bent the rules a little bit:

Another group tried something similar but put 2.5 instead of 0.25. I simply pointed to this and asked 'Is this more than one or less than one?' and walked away. When I came back, they had corrected it.
Another group tried this:
When I asked about this, they knew that two-sixths was one-third but they also thought that its decimal equivalent was exactly 0.3. As they knew how to use a calculator to change a fraction into a decimal, I asked them to do this for one-third. This made it easier for me to convince them that 0.3 is different to 0.333333333... 

As we had fifteen minutes left, I gave a variation of a question that I tweeted last week that proved popular:

Any thoughts we might have had about this question being tricky for them were soon dispelled:


As we circulated, we could hear the students justify their solutions and, if we were unsure, simply asked a question like "Can you convince me that two-sixths is less than five-eighths?"
One student asked if it was OK to use a fraction whose numerator was larger than its denominator. This led into a nice discussion about improper fractions and how these are all larger than one:
We could see a couple of groups who found solutions quite quickly so we gave them an added challenge by removing the '1' tile. They relished the added level of difficulty:

All in all, it was a really pleasing lesson that allowed the students to show us their problem solving skills as well as allow us as teachers to assess their understanding of fractional equivalence.