Always True/ Sometimes True/ Never True

I was reminded about how much I love this question today when I took part in a crowdsourcing of questions courtesy of Tracy Zager (@TracyZager) via Twitter and Google docs. I have given this type of questions to students from all grades  (K to 12) and what is great is that it gets them reasoning and proving from a very early age. Consider the following:

Addition makes a number bigger

 

Most students (especially in the primary, junior grades) will say 'Always true' and back it up with examples. Yet there will be some who wonder about what happens when you add zero? Does this make the number bigger? And intermediate students will then begin to reason that adding a negative number actually makes the number smaller. So the answer is 'Sometimes true'. 
Sometimes a question helps them broaden their understanding of math terminology:

Two identical triangles can be put together to make a parallelogram

I know some students will say 'Sometimes true' offering the case where two right-angled isosceles triangles join to make a square (in green below) which, they think, is not a parallelogram. Others might make what they think is a more obvious parallelogram (in blue below).

This gives us a great opportunity to learn why all squares are parallelograms (quadrilaterals with two pairs of parallel sides). This leads into an understanding of why the area of a triangle (½×base×height) is simply half the area of a parallelogram (base×height)

Perhaps my two favourite questions I got today were:

A solid that has a square shadow is a cube

A solid that has a circular shadow is a sphere

 

It immediately got me thinking about other solids that might have these shadows. Or what about if I reverse the order of each statement?:

A cube has a square shadow

A sphere has a circular shadow



So here, for your delight, are some other Always true/Sometimes true/ Never true questions:

• A rectangle is a square
• When you cut a piece off a shape, you reduce its area
• When you cut a piece off a shape, you reduce its perimeter
• Bigger objects are heavier than smaller ones
• The diagonals of a parallelogram are unequal in length
• Multiplication makes numbers bigger
• Division makes numbers smaller
• The sum of four consecutive numbers is a multiple of 4
• The sum of three consecutive numbers is a multiple of 3
• The more you roll a dice, the more likely you are to get a 6.
• The sum of two odd numbers is an odd number
• The product of an even number and an odd number is an odd number.

Next, Near, Far, Any.

"A Maths lesson without the opportunity to generalise is not a Maths lesson."

I love this quote from John Mason as it gets to the heart of what it means to think mathematically. Math is the study of patterns and the power of patterns is that they are predictable. Part of predicting a pattern is generalising a rule which we think we might have spotted. Now this is easier said than done, but doable it is providing we give students opportunities for them to generalise.
Last year, the Ontario Ministry of Education published Paying Attention to Algebraic Reasoning. When I did a word cloud of the text, it was neat to see that the word 'Generalizations' was very prominent.

For me, the biggest learning that I got from this document though was a continuum that has helped me guide students as they learn to generalise:

Next----Near----Far----Any

 

Basically, younger kids will start by saying what the next term in a sequence is. As they get better, they will be able to predict what a near term is (e.g. the tenth term). This will most likely be done by finding the in-between terms. Now we can get students to predict a far term (e.g. the fiftieth, the hundredth, the thousandth term). This will necessitate a move away from concrete into abstract thinking as now it will be cumbersome to find all the in-between terms. Now students are ready to generalise for any term.
So I was in a Grade 6 class recently and gave them this pattern to consider:

I wondered if they could tell me how many sticks and/or dots there would be for any term. They had no trouble giving me the next term and a near term:

At this point, they were pretty clear about describing how to find the number of sticks and noticed that you just add two to this number to get the number of dots. What wasn't clear is what you do to the term number to get the number of dots (i.e. add 3). So I then pushed them to think about the thousandth term:

Note how this couldn't be drawn out but they got the 1001 sticks straight away and soon had also told me that 1003 dots were needed. Had they made the link between the term number and the number of dots? I decided to really push the envelope and asked them to generalise for any (i.e. the nth) term. Straight away they said the number of sticks would be n+1 but they needed a bit more help with the number of dots being n+3. Still, not so bad as it was the start of the unit and with further exposure.
Now I was unsure if it was helpful just drawing the pattern on the board: is it better for students to have the pattern modelled with manipulatives at their desk? I was in another class, a split Grade 6/Grade 7 and we gave each student a set of three cubes that were linked together. We asked the students to put them on the desk and tell us how many square faces they could see. When we all agreed that it was 11 we then asked the Grade 6s to find out how many square faces are visible on a set of 10 cubes (near) and the Grade 7s a set of 50 cubes (far). Notice how we skipped the 'next' stage here. Very quickly they were able to use the cubes to help explain their thinking:

The Grade 6s then tried the Grade 7 question whilst the Grade 7s were asked to make an 'any' prediction. We got three possible answers for this:

n+n+n+2

(n×3)+2

3n+2

(Note: In both these cases, I basically gave the student the patterns but would not always do this: it is vital that students create and test their own patterns.)
It was interesting to see, as my colleague Christine pointed out, that the students who said the first answer may be thinking more additively than the multiplicative thinking of the other two answers. With more experience of these types of tasks, students will develop the skills that are needed to generalise.

(Update to original post):
I spent another lesson with the same class and actually showed them the next-near-far-any continuum. I challenged them to see how far along the continuum they could go for this pattern:

It was great to see that so many students are already using their spatial reasoning to help the generalising as best exemplified by this:

               *             *            *

The Ontario MOE's Paying Attention to Algebraic Reasoning can be found here. 

Why We Need To Listen (2)

So you ask a class of Grade 1/2 students this: "Write a number between 12 and 20." This is what one student replies:

Is this alarming? Er... yes.
Unless you consider what was asked immediately beforehand.

1) Firstly, the brilliant Mr. Stokes had timed how long the class took to get all the materials handed out."Thirty-two seconds: new record!" he said. I asked the students to write this number down.

2) I showed students this and asked them write down the number it represents.

3) Then I asked them to write a number that is bigger than 60 but less than 80.
So for this student, the first three answers were:

At this point, the student then said "Oh! I see the pattern!" But I sort of ignored this and asked the students to write a number between 12 and 20. When I saw the 95 I was a bit alarmed until I asked the student to explain her thinking:

Interesting, eh? The student was so fixated on the pattern that she spotted that she didn't actually hear my question. So I asked her to show me where 95 was on a hundreds chart. She replied, "Right there...oh....wait, I meant to write 15!"
It is so important that we create opportunities so that we can listen to what students are actually thinking (as described in a previous post Why We Need To Listen (1)). Just a quick conversation revealed that the student's answer isn't so alarming as, really, she answered the wrong question.

In fact, I am left with the thought that she is actually pretty good at spotting patterns!

What Does the '=' Sign Mean?

Here is a video of two students in a Grade 9 Academic class talking through their solution to

              -15+x= -x+15.

As you listen to it, can you figure out the misconception?

Despite the otherwise articulate description ("... you need to isolate the variable..." etc.) there is a real fundamental (yet common) algebraic misconception here which is:

when you bring a term to the other side, it switches sign.

I've been wondering when such misconceptions begin. More to the point, I'm wondering if students truly appreciate the meaning of the '=' sign. If we reduce algebra to a bunch of tricks that needs to be memorised ("... when the number jumps over the gate it trips and so it changes sign...") then these might cause these misconceptions.
Some research that Christine Suurtamm presented at the OMCA conference in February also shines light on a common misconception. Students were asked to solve the following problem:

8+4=?+5

This is a breakdown of how different grade responded:

Notice how in each group the most common response is 12. Is this because students translate the '=' sign as 'Now do what you have just read' (in this case 8+4)? If the students have only experienced questions of the type 8+4=? and not, say 8+?=12, then it is no surprise that they make this mistake.
Notice also how that none of  the Grade 5s and 6s (in this study) got the actual answer correct. Is this because their Math experience is now more about following rules and less about understanding the concept? Have these students really developed the concept that the '=' means that the expressions either side of the sign are balanced? It is crucial that they do this. In the book Mathematical Misconceptions by Anne Cockburn and Graham Littler, the point is made that equality is a central but a neglected concept and that there is a strong correlation between the ability to understand the '=' sign and the ability to solve equations.

Even seeing how numbers can be decomposed and recomposed will help with this. For example, asking students to think of how many different ways 12 animals can be split between 2 fields would generate a lot of answers of the type that we can write as 12=8+4 (as opposed to 8+4=12).
Or if I put 5 green marbles and 9 black marbles in one side of a set of scales and 3 red in the other side, students will see that the pans are not balanced. I can ask how many blue marbles do I need to add to make the scales balanced? There are a variety of ways to solve this and when I consolidate the students' thoughts I can tell them that they have really solved this equation

5+9=3+?

Or written another way:

                                        14=3+x

With younger grades, they will rely on their number sense to solve this not some contrived (and partially understood) rule. Activities like this will help students really understand the meaning of '='.
                                          *                               *                                 *
There are some great thoughts and ideas on equality in the Ministry of Ontario's Paying Attention to Algebraic Reasoning document, especially on pages 6 and 7.

Spatial Reasoning

I was at a Ministry-run conference last week working with other Math-folk from all over Ontario and suddenly had an epiphany regarding Spatial Reasoning. To try and understand what Spatial Reasoning is, have a look at this video and predict which of the five squares below the paper will look like when it is unfolded:

I haven't come across a precise definition of what Spatial Reasoning is but in general, most folk agree that it involves visualising, perspective taking, mental transformations, composing and decomposing (shapes, numbers, measurements, data, and algebraic expressions). Nora S. Newcombe in this excellent article says 'Spatial thinking concerns the locations of objects, their shapes, their relations to each other, and the paths they take as they move.' She goes on to show how Spatial Reasoning is not a 'learning style'  but a habit of thinking, that for anyone it can be improved, that whilst there may be gender differences, the important fact is not the causation of these differences but that both genders can still improve their Spatial Reasoning.

My big lightbulb moment was when I realised that Spatial Reasoning is so much more than geometry. Indeed it transcends Math. It is a way of thinking that helps us solve problems in number, measurement, algebra, geometry and data handling; in science (figuring how the atoms in a molecule are arranged); in technology (any work on perspective or visualising how to build a certain structure); in sports (reading the shape of an opponent's defence or making a 'no-look' pass); in the arts (seeing the sculpture in a block of marble or the visualising of dance moves that a choreographer must make; in geography (any map making or map reading activities); in history (visualising what a building must look like from the clues found in an archaeological dig).
This got me wondering, though, how aware are we as teachers of Spatial Reasoning and much time do we spend on it? Working with some colleagues at my table we quickly came up with the following ideas just in Maths:

Primary:

• Use positional language as much as possible
• I'm thinking of a shape that looks like a triangle and a rectangle stuck together. Draw what it might be.
• There is a shape in this bag. Feel it but don't look at it. Now tell me what you think it is.
• Imagine holding a can of soup. How many circles might you see?
• I have a letter. When I turn it upside down, it still looks the same, What might it be? What couldn't it be?
• You measured the table width with your pencil and it was 10 wide. My pencil is twice as big as yours. In your head, imagine me measuring the width. Would I use more, less or the same as your pencil?

Junior:

• I have two rectangles and put them together. What shape could I end up with?
• I cut of the corner of a square. Use a geoboard to show the shape I cut off and the shape that I'm left with.
• I have joined 6 cubes together. From the front, they look like an I. From the side, they look like a T. What do they look like from above?
• Visualising the mean as an evening out of all the scores as shown in this post.
• Visualise two congruent triangles. How could you arrange them to make a parallelogram? How does this help us get a formula for the area of a triangle?

Intermediate:

• Imagine two congruent trapezoids. How could you arrange them to make a parallelogram? How does this help us get a formula for the area of a trapezoid?
• Imagine unfolding a cylinder. What shapes will you see? How would you work out the area of these shapes.
• Imagine two lines. One crosses at (-4, 0) and (0,4). The other crosses at (0,-2)  and (2,0). How would these lines look?
• Imagine completing the square like this

Senior:

• I have a function that has four roots and a range y<4. What could my function be?
• Imagine you slice a cone. What are the different shapes that the cross-section could be?
• Imagine you have three different planes. How many different ways could they intersect?
• How many zeroes are at the end of 125!
• What happens to the secant through two points in a curve as the first point gets closer to the second?

In fact, I will go so far to say that visualising techniques (far more so than rote memory of rules and formulae) are essential in understanding calculus.

After the session, I engaged in a wonderful Twitter dialogue with Malke Rosenfeld who is also learning about the importance of Spatial Reasoning. I highly recommend her blog here.

And as for the solution to the paper folding exercise? Check below!

"Whatever you do to the top,..."

I wonder how many people read the title of this post and automatically completed it by saying "... you must do to the bottom." It's a phrase I was drilled in when learning about equivalent fractions, a phrase I used myself when I started teaching. It was only after seeing student work like this one that I began to wonder on the wisdom of using such tips:

So when I think of some of the standard procedures for adding, subtracting, multiplying and dividing fractions, I wonder if as teachers we are guilty of rushing in too quickly to computational strategies before students have a solid enough understanding of the quantity of fractions. I read a quote by Jon Allen Paulos that made me ponder on this even more:

"Mathematics is no more computation than typing is literature."

Recently I was at the Ontario Mathematics Coordinators Association's annual conference. The keynote speaker was Christine Suurtamm from the University of Ottawa. Among the many great ideas and activities that she led us through was this one: 

You can see my solutions to the first two questions. What I love about the questions is that I can see how they will expose and challenge many misconceptions that students have about the quantity of fractions. This gives us an opportunity to fix these misconceptions which in turn will put students in a better position to understand any computational procedures they will need to learn.

Of course, we took up the challenge to describe another structure and then have a colleague build it. My challenge was to build a hexagon that is 3/5 yellow, 1/5 green and 1/5 blue; a simple enough question to state but it provoked a lot of thinking. Chad's challenge to me was to build a hexagon that is 1/6 green, 1/2 red and 1/3 blue. After I came up with one answer, I wondered if others were possible and indeed there was. Are there others?



What I really liked about this activity was its openness: there are so many points of entry and it truly is a 'low floor, high ceiling' question.

This was followed by a 'Fractions War' game. If you are unfamiliar with 'War' games, two players have a pack of cards and both turn over one card. The player with the higher card wins. As Sean and I ran through this game, we faced this situation: 



It reminded me of a misconception that I've often seen where students compare the numerators (and see 3>2) then the denominators (and see 12>4) and then conclude that 3/12 must definitely be bigger than 2/4. These students have not had enough hands-on experience to understand the quantity of fractions like those shown above or in a previous post on Fraction Flags .

And if students really do think that 3/12>2/4, how would teaching them to add these two fractions be beneficial for them?





Geoboards in the Car

So I'm driving my daughter to her dance class when she picks up my iPad and asks about the Geoboard app that is on there. I tell her that you can use it to make shapes like the ones she has been learning in class. "Make some trapezoids for example," I tell her. After a while she says "Done! Is this right?". "I can't look now I'm driving!" So we wait until a stop light and then (as she is sitting in the back) she shows me in the rear-view mirror:

"Great. Are they parallelograms too?"
"No... they only have one pair of parallel sides"
"OK. Now make some trapezoids that have a right angle". Now this took longer and a fair amount of "How is that possible?" until a little squeal told me that she got it:

"Right, now make some rhombuses". "Do you mean diamonds?". "No I mean rhombuses!" A little while later she showed me this:

"Hey, what other name can you give that small one in the middle?" A little pause and slight turning of the iPad and then "Oh, a square!!"
"Are all squares rhombuses?"
"Yes! Yes!"
"Are all rhombuses squares?"
"Yes! Wait...NO!!"
"OK. Now it's time to make some pentagons. But make sure that they have at least one right angle"
This is what she made:

I do know some kids who get confused as they think a pentagon will have five 'points' and therefore think that the elastics can only touch five pegs. Thus they won't see these as pentagons as they touch more than five pegs. This is a great opportunity to fine tune what we mean by 'points' and connect it to the number of sides.
Now I know you might be thinking 'Lucky girl, getting to do maths in the car whilst other kids are playing Angry Birds' but it was a neat way to spend 15 minutes. It got me wondering whether or not I prefer this virtual geoboard to the real thing and I think I might be leaning to the virtual side. For a start, the 'elastics' never snap and you never run out of them. Secondly, the vertices look more like they should. For example, look carefully at the 'corners' of this shape below and ask yourself if this really is a rectangle?

That being said, real geoboards are cheaper and I'm sure that some kids will prefer the tactile nature of these as opposed to the virtual geoboard. Either way, geoboards are a great way to get kids really to explore some geometric properties by asking questions such as:

Make a quadrilateral with 3 acute angles.

Make a parallelogram with two right angles.

Are all parallelograms rectangles?

 

For an extra challenge, I sometimes ask to make a shape which I know to be impossible (but the kids don't). This creates huge cognitive dissonance and often gets them reasoning why such a shape is impossible. For example:

Make a triangle with two right angles.

Make a quadrilateral with four acute angles.