Have You Checked Your Work?

"Have you checked your work?"
I reckon this is one of the most common questions that teachers ask their students. And it is usually more of a rhetorical question as we ask it when we know that students have made a glaring error or have come up with an answer that makes no sense.
And usually the students look briefly at the work and say "Yep!" as if to say "Yep, it's still there!"
Checking your work is a vital piece of Polya's four step approach to problem solving. In many ways, I think it should feedback into the first three parts (Understand the Question, Make a Plan, Carry Out the Plan). My sense of things is that this maybe the weakest link for a lot of students.
I have been wondering why this is and now I'm thinking that maybe it is the question itself ("Have you checked your work?") which is causing the problem. Maybe there are better prompts to get the students to reflect on what they have written.
As a case in point, I was in a Grade 9 class recently and they were working in groups on this problem: try it yourself before you read on!!

What was interesting is that many students stopped after they found an answer. They didn't think to consider of there was more than one answer. It was almost as if they stopped out of habit: I have an answer, so now I'll just wait for further instructions.

Biting my tongue, I managed to avoid saying "Have you checked your work?" Instead I asked "Which two sides are equal?" They pointed to the 3x-4 and the 5x-8. I then asked "Are these the only possibilities?" 

Immediately, this had a much better effect than "Have you checked your work?" The students realised that the 3x-4 could be the same as x+6, or the 5x-8 could be the same as the x+6. There were some comments along the lines "They don't look the same!" but they were reminded that they were told that the diagram is not to scale!

But even if they had considered each of the three possible isosceles triangles, and had done the algebra correctly, there was one geometric error that kept on coming up and was not seen by the students: notice the 2cm, 2cm, 8cm triangle:

"How do you know this is an isosceles triangle?"

"Because two sides are the same."

"OK...how do you know it is a triangle?"
"Errr...because it has three sides that are joined together, (said in that 'D'uh' tone!).

"OK...if you can draw me that triangle, then I'll give you twenty dollars!"

Sometimes when we ask students to check their work, they might just check the algebra (which in this case was correct) but not realise that what this leads to is an impossible shape. Which is why I really like this question as it will force students to appreciate the triangle inequality: any two sides of a triangle must add up to more than the third side.
But to get students into this habit of thoroughly seeing if their solutions make total sense, we must provide them with rich questions that force them to look for different cases, or to examine the validity of their solutions.
If, however, we only provide them with questions where there is only one answer (and probably just one way to get this), then they won't get into the habit of effectively reflecting on their work. Thus, we cannot expect them to be complete problem solvers.

                           *                                   *                                    *
The problem above is adapted from the University of Waterloo's CEMC's Problem of the Week. 

Using Pattern Blocks to Reason and Prove

As part of one of our Ministry's Numeracy initiatives, we tried out the following task in a Grade 6 classroom. 

Find the size of each of the angles in the pattern blocks shown. Protractors are not allowed!

Our Minds-On discussion showed that the students had a good grasp of what acute, obtuse and right angles were. At the same time there was some disagreement as to what a polygon was: it was only after a fair bit of using counter-examples to their definitions that they agreed that polygons are closed 2-D figures made up of entirely straight lines (which begs the question, are pattern blocks polygons?). 
When the students began the task proper (and after they asked me a dozen times if I really meant they couldn't use their protractors!) it was interesting to see their strategies.
Some just estimated angles (below). It was a quick fix to show these students that some of their answers were contradictory and that they needed exact answers instead. 

Some recalled that there are 180° in a triangle so this meant that each angle in the equilateral triangle was 60° and progressed from there. Others put compared three equilateral triangles to two right angles (or squares) as shown:

Others benefited from marking the angles they were trying to find actually on the pattern block as this helps them see the amount of 'turn' between the two sides (check out this post for some ideas on how angles measure turn):

As much as possible, I am learning to hang back and not jump in and show students what to do: when I do this, I realise that it is me doing the maths, not them; it is me reasoning and proving, not them. There certainly are some awkward pauses when I do this but this is most likely because the students are mulling over what they could do.
And sure enough, they could solve the problem without my help:

And for fun, they could verify their solutions in more than one way:

So when it comes to problem solving, I really like the philosophy from Singapore's Ministry of Education:

Teach less, learn more.

Patterns are Predictable

I made the point in previous posts (The Study of Patterns and Next-Near-Far-Any ) that algebraic reasoning is based on our ability to notice patterns and generalise from them. As our students become more proficient at spotting patterns we must get them to realise that patterns are predictable: if you have spotted the pattern rule correctly, then you will be able to predict any term with 100% accuracy. 
This differs from the predictions we make in, say, handling data, where our predictions (even though they might be well thought out) aren’t guaranteed to hold. 
The Next-Near-Far-Any continuum is also useful in helping our students make predictions. First get them to predict what might come next. Then predict a ‘near’ term such as the tenth (this might be done continuing the pattern one at a time). From junior grades onwards, students might then be asked to predict a ‘far’ term such as the hundredth. As it is now impractical to continue the pattern one at a time, students will now begin to generalise patterns between the position number and the term number. This is more easily done when we use visual patterns. From intermediate grades onwards, students might be asked to predict any term: this is true generalising. In other words, for position number n, what is the term number in terms of n?
A great source of pattern prompts can be found at Fawn Nguyen's Visual Patterns site. These type of questions are 'Here are the first three terms: figure out what happens next.' They can be relatively straightforward with colour being used as a clue to how the pattern grows:

to more complex such as this one which asks 'How many visible spots in this view are there?' (The first term has 33 visible dots):

I love these types of questions but I like to balance them questions where students are given the chance to make their own patterns: I am always blown away by how creative they can be. Last week I was in a Grade 5/6 class and we asked the students to make their own patterns. They then had to look at someone else's pattern and predict the tenth and hundredth terms. Here is a selection of what we saw:

Notice how some of these combine growing patterns with repeating patterns. For example, to predict the hundredth term of the first pattern above, you would need to notice that the vertical line increases by one each time and that the colours rotate in a cycle of four. Since every fourth term is a blue vertical line with a single green square, we can predict that the hundredth term will be 100 vertical blue squares attached to a single green square. It was great seeing the students make and justify their predictions.
A variation on this is where you give a single term and ask the students to fill in the pattern from this single piece of information. This one (from Marian Small) is my favourite and I've used it with students from grades 4 to 11.

This is the fifth term. Make the first four terms. 
Students always come up with a variety of patterns: some increasing; some decreasing; some linear; some non-linear. I like to plot these patterns on a single graph and ask students to explain why these plots cross at the same point.



And if you need further evidence of the creativity of students, a colleague gave the dice problem above to his computing class and before long, one of his students had written the following program (in Turing, I believe) to solve the problem. It is, I hope you agree, a thing of beauty.

function dice (term : int) : int
    var dotnum : int := 0
    dotnum += 10 %Tops of yellow and red dice *
    dotnum += term * 15 %Sides of yellow and red dice *
    dotnum += 5 * term ** 2 %Tops of white dice *
    dotnum += 3 * term ** 2 %Sides of white dice *
    dotnum += 6 * term ** 2 %Sides of white dice
    dotnum -= 6 * term %Covered sides of white dice
    result dotnum
end dice

Bar Models 2

Following on from my last post, here are a couple of further examples of bar models being used.
The first is in a Grade 6 class and we wanted to specifically give them a multiplicative comparison problem. These problems tend to be more difficult for students compared to equal groups problems; I think this may be because students have a hard time visualising what is happening. Bar models can certainly help here.
Our question was:

Mr. Jacobs eats some candies. Mrs. Delaney eats four times as many candies as him. Together, they eat a total of 260 candies. How many candies did each of them each?

A typical response is shown below.

By using bars to represent how many candies Mr. Jacobs has and how many candies Mrs. Delaney has, the calculations needed to be performed are clearer to see: the combined five bars must represent the total 260 candies. Thus one bar must be 260÷5=52.

In another example (this time from a Grade 2/3 class) the student uses a bar model to visualise this part-part-whole problem:

A school has 83 water bottles. If 29 are filled, how many are not filled?

A typical response is shown below:

What I like about this, is that the student drew a bar model first and used this to help decide what number sentence could be written. Once this was done, the student decided to use an empty number line to calculate the difference. So whilst this might be thought of as a subtraction problem (when the unknown isolated, you get 83-29) it can be solved by addition. The beautiful thing about bar models is that they allow students to visualise both ways.
I'm not sure if students will become experts at using bar models to solve problems unless we as teachers model them effectively and consistently. In many ways a whole school approach is necessary. In primary, teachers should model how to use bar models to solve additive thinking problems. In junior grades, teachers need to model how to use bar models to solve multiplicative thinking problems. In intermediate grades, teachers need to model how to use bar models to solve proportional reasoning problems.

The pay off will be massive.

Bar Models 1

For me, one of the most important continua in learning Maths is the following:

Concrete-Diagrammatic-Symbolic

Too often, I feel that Math teaching rushes to the symbolic representation of the problem and downplays the importance of the diagrammatic and concrete representations. Which is strange really: if I get stuck on a problem, one of the first things I do, is draw a picture to help me better understand what is going on. And if this doesn't help, then I'll get some hands-on materials to help me figure out what to do.

When solving problems, probably the most common question that students ask their math teachers is "What do I have to do?" Often they want us to tell them what they have to add, subtract, multiply, divide (or any combination thereof). Over the years, I have become more convinced that this is because students haven't developed a schema of what the operations look like in real-life. Instead they are reduced to looking for keywords that might (or might not) be in the question and that might (or might not) actually mean what they are supposed to mean.
This is why I have been recently singing the praises of bar models to the educators I work with. These are sometimes referred to as 'Singapore Bar Models' due to their extensive use by students in that country to solve problems. However, I saw bar models when I began teaching in 1990; they were used to illustrate how to visualise percentages in the SMP Red series texts. And, I suppose, my first concrete experience of bar models as a student, was when I worked with Cuisenaire rods as shown in this earlier post.
For those unfamiliar with bar models, see how this Grade 7 student used one for the very first time in solving the following problem:

A crowd of 2400 go to see the local hockey team play but as they are doing so poorly, three-quarters of the fans leave at the end of the first period. A further third of the remaining fans leave at the end of the second period. How many fans watch the third period?

What I think is great about this solution is that by drawing the bars, the student can see what operations need to be done (and then does these in his head). If he was struggling before about what operations to use, now he can see them and, if necessary, he can write these symbolically.
Much as I love bar models, I don't think it is a method that students will come up with themselves independently. It will require a lot of co-ordinated effort and consistent modelling from teachers all the way from Kindergarten up. I will post some ideas on how we can develop students' efficiency in using bar models in future posts, but in the meantime, it is worth checking out these summaries here and here.

Developing Formulas (3)

A few years ago, I was in a room of 100 or so educators who were asked to draw a picture to represent the Pythagorean Rule. Maybe 5 or 6 people were actually successful. I would say half didn't even know the Pythagorean Rule. Others drew a triangle (sometime right-angled, sometimes not) a bit like this:

But this doesn't illustrate what the Pythagorean Rule is, namely that the sum of the areas of the squares on the two shorter sides are identical to the area of the square on the longer side:

At the time I made the point that if we don't understand this, then we don't truly understand the Pythagorean Rule. I reckon that this is the result of teaching the Pythagorean Rule purely from an algebraic point of view. I was once shocked to see a textbook from the UK that told students to memorise these three formulas for the Pythagorean Rule:

a²+b²=c²

c²-a²=b²

c²-b²=a²

But this is not how the rule was discovered. It was discovered geometrically long before algebra was invented. Here in snowy, frigid Ontario, the curriculum has got it right: in Grade 8 the focus is on a geometric understanding which is followed up in Grade 9 by connecting this to an algebraic understanding. If this is the way it was discovered, then it should be the way we develop it with our students.
So let's not just give students the abstract formulae.
Instead, have them try some decomposition activities like the following (notice how one of the triangles is not right-angled: I always like to offer a counter-example!)

This can be followed by this video which has been doing the rounds on social media (I don't know who first created it so apologies for not giving this person credit):

Now this isn't a proof as such but merely one example. Using a dynamic geometry software such as Geometer's Sketchpad or Geogebra

helps us generalise that it doesn't matter what the original right-angled triangle looks like, the sum of the two smaller squares adds up to the area of the larger square. This is the point where we should bring in algebra to PROVE that it works for all right-angled triangles

But what happens if we draw other shapes on the sides of the triangle such as semi-circles?

It appears to hold true for semi-circles too. Can we prove this algebraically? Spoiler alert:  Yes we can, and it is a lovely wee proof too.
If we get our students to develop a geometric understanding of the Pythagorean Rule, they will be in a better position to use the algebraic representation to tackle problems like the one above or the one below (which is in my Top 10 favourite math problems of all time).

Developing Formulas (2)

"I deeply worry about a curriculum that pushes students to results and not let mathematics be the organic conversation it deserves to be..." James Tanton

When I began teaching, I (like many others) simply gave students the formulas, and some worked examples, and expected this knowledge to stick. 
It did not work as well as I wanted to. 
When I started showing why the formulas worked, students were far more likely to recall and use the correct formula. When studying area, we would learn about rectangles, then triangles, then parallelograms, then trapezoids (or, if you will, trapeziums (or, if you will, trapezia!)) before getting stuck into circles. Recently, I have been wondering if a more logical order would be rectangles, then parallelograms, then triangles, then trapezoids. Fundamental to all of this is learning why the area of a rectangle is length times width and the best way to get students to develop this idea is to consider arrays (as touched upon in this earlier post ) How I then get the students to develop the formulas for parallelograms, triangles, and trapezoids can be seen below. I must point out, that I do not do all of this in one lesson!

Usually at this point, I am pretty confident that most students will now understand why the formulas work. Part of a balanced Math program must involve putting this knowledge to practice. This practice should involve a good balance of closed questions (the standard text book ones where a diagram is given with different measurements given) as well as open questions e.g. a trapezoid has an area between 60 cm² and 70cm², what could its dimensions be? The question below came from Anne Yeager and I have used the question with many grade 7 and grade 8 classes. It has always generated a lot of different solutions as well as great thinking and discussion amongst the students as they decide which formulas to use and when.

James Tanton (whose quote appears at the top) provides some fantastic resources for Math teachers. In particular, I love his curriculum videos and his Mathematical Essays. Do check out his site here.