Mind Your Language (2)

Ask a group of students (or adults come to that) to draw a parallelogram and see what the most common answer is. When I have done this (with students, educators and parents) by far and away the most common shape is the example A below. I will even have a drawing like this prepared and reveal it to the 'audience' and announce that I read their minds to such an extent that I knew that the longest side of the parallelogram would be horizontal and that it would slope left-to-right.

Rarely will folk draw a parallelogram like B and even more rare will they draw a rectangle or square. This might seem innocent enough but can be a big clue to a huge misconception that often goes unnoticed. This was evident in a Grade 6 class I saw this week. We had given them this question from the 2012 Ontario Junior EQAO test:

 

 

It's a nice question as there are a variety of ways to think about solving this. One student solution was shared with the class and provoked some great discussion:

Some students argued that the shape on the left wasn't a parallelogram. One student argued that it was as parallelograms are shapes that have "...two pairs of parallel sides." Years ago I would have left this statement unchallenged. Now, I jumped at the opportunity it gave and asked if the shape below was a parallelogram:

Probably half the students said yes it was; they understood that a parallelogram is any shape with two pairs of parallel sides. The remainder of the class seemed unsure. We then got in a debate as to whether or not a regular hexagon is a parallelogram(!); some said no as it had three pairs of parallel sides, others said yes as it had at least two pairs of parallel sides.
As we mused how to deal with this, a student asked if she could look up the definition of parallelogram. This she did, and there it was: "A parallelogram is a quadrilateral with two pairs of parallel sides." This was news to a lot of students. However with this new knowledge they were now OK with saying that the rectangle in the solution is also a parallelogram.
On reflection, I now realise that giving insufficient examples and using imprecise language  restricts students' understanding of what a parallelogram is. It would be better for students to construct their own understanding of what a parallelogram is by showing them something like this (from Ontario's MOE's Guide to Effective Instruction Grades 4 to 6: Geometry) and asking them to define 'parallelogram'.



Mind Your Language (1)

Look at this number and say it out loud: 6.125
How did you say it?
'Six point one two five'?
'Six point one hundred and twenty five'?
'Six decimal one two five'?
'Six decimal one twenty-five'?
A few years ago I would have paid no attention to the language that I used to say decimals. In England (where I first taught) I used to say 'six point one two five' so when I first came to Canada it amused me somewhat to hear 'six decimal one two five'.
However, research by Sue Willis (First Steps in Mathematics) made me realise that I wasn't saying decimals properly and that this was not helping students understand decimals.

It is no big secret that decimals are one of the big ideas that really seem to stump some students (and adults). I've taught a lot of tricks to help learners cope with decimals but these were just papering over the cracks: they didn't help students understand the quantity of decimals.
More importantly, I couldn't fathom out what students thought decimals meant. Using a diagnostic provided in First Steps made it a lot easier for me and my colleagues to see what students were thinking. For example, what are students who make this error thinking?:

These students are saying the numbers incorrectly. in example (iii) they are saying 'three point five hundred twenty one, three point six and three point seventy five'. And since 521>75>6 then 3.521 is the biggest number of the three. For example (v) they simply ignore the leading zeroes and say 'four point nine, four point seven, and four point eight'. Fascinating, eh?

Now look how these students often answer the following questions:

I must have had so many students make this mistake without ever realising what they were thinking; instead of correcting their misconception, I tried to give a rule.
Think about how these students get this and then watch this explanation:

 

This is not a one off. I reckon there will be students in every school who make this error.
So how do we correct this? Sometimes by simply saying the number correctly:

"Six and one hundred twenty five thousandths."

This is often enough to students to rethink. In example (v) above, when students say 'four and nine hundredths, four and seven tenths, four and eight thousandths' they often realise 'Hey, seven tenths is much bigger than nine hundredths'. I can convince them of this by modelling these quantities (of which more in a future post).
Skip counting with decimals is also  a really useful activity. Students who make the above error will often continue a pattern that starts 1.2, 1.4, 1.6,  like this: 1.8, 1.10, 1.12, ...
However by saying the pattern as 'one and two tenths, one and four tenths, one and six tenths, one and eight tenths, one and ten tenths...' at this point we often see students realise 'Hey, ten tenths are one whole so it must now be two'. To reinforce this, we can use a calculator: type in 1.2 + 0.2 then keep pressing =,=,= and students will see the count continue. It is a great moment of cognitive dissonance when they see the '2' instead of the expected '1.10'!
These two diagnostics take very little time to do but have helped me so much in correcting students' misconceptions.

A Measurement Mea Culpa

I apologise.
I apologise to any of my previous students who I messed up by saying that the perimeter is "... the distance around the outside of a shape." I meant well, I really did.
But I lied. The perimeter is most certainly not the distance around the outside  of a shape.
I realised this error by looking at students' responses to a seemingly straightforward length question:

This is the type of response we were expecting:

 

 Instead we faced something like this:

Notice that in the left hand path, the student has got 15 in two different ways (the 'outside' path and the 'inside' path) but that this is by luck: the student counted squares that are not involved in the length. Other students made similar mistakes:

There is a definite theme here: counting squares as opposed to line segments.
This response is very telling:

The student has got the right answers (15 and 17) but for the wrong reasons. And if they hadn't have shown how they got the answer (by writing the numbers in the squares) I would have been none the wiser for it. I probably would have thought that they 'got' it. Again, mea culpa.

So how did this misconception arise?

Probably as a result of the 'count the distance around the outside' rule that has been impressed upon the students. Or maybe students' initial concrete experiences with measuring length involved using, say, square tiles to measure a pencil and so they now think 'squares' are used to measure length. As a result of this, I would certainly get students to use popsicle sticks, string, etc to measure length (i.e. objects that are more blatantly 'length' rather than 'area'). I also now refer to the perimeter of a shape as the sum of all its sides.

So if you have a student who thinks that the perimeter of this shape is 18...



... then show them how highlighting the sides forces us to find the sum of the sides of the rectangle:

Reflecting on Transformations

A few years ago I read a book about the geometer, Donald Coxeter, called King of Infinite Space:The Man Who Saved Geometry. Written by Siobhan Roberts, it is a wonderful insight into the mind of Coxeter who was in love with geometry. What fascinated me about him was how Coxeter, the world's greatest geometer, learned about geometric properties and relationships:

It got me realising that if Coxeter uses manipulatives to learn about geometry, then so should I and so should my students.
When I first started teaching I told students to reflect, rotate and shift shapes by "imagining how they would look after the transformation".
Not brilliant advice any way you look at it.
However, when I started giving tracing paper (or acetate sheets) for students to draw the object and then to find the image, immediately there was greater success. I saw this again in a Grade 6 class this week. We gave the students this question from Ontario's Junior EQAO test of 2010:

Initially, students could sort of make out a reflection, a rotation and a shift (or translation... though I myself don't really like that term because of its ambiguity). However, they had trouble describing  these transformations.
That was until one student walked to the front of the class to help herself to a small sheet of acetate paper that we had surreptitiously placed. And this is how she used this tool to help her tackle the problem:

The tool that she chose suddenly made it so much easier for her to describe the transformations.
Accurately as well.
And when other students saw what she was doing, they immediately wanted to use the acetate too. As one lad said, "It makes my thinking clearer."
                                   *                                       *                                      *
If you want to see engaged students, give them some of M.C. Escher's prints and ask them to find and describe as many transformations as they can. Guaranteed fun.

Why We Need to Listen (1)

I (and pretty much most of my colleagues) have been raised in educational systems where your achievement was measured almost entirely through written products: quizzes, tests, exams. Sometimes these 'written' products were multiple choice tests and involved no writing at all to be evaluated. As teachers, this reliance on a written product is a hard habit to break; it is a habit which I have spent no small amount of time trying to justify to parents and students in the past. 
In the past few years, however, I have been completely deconstructing my original beliefs on how to evaluate students. If truth were to be told, I don't think I was ever truly comfortable with the way I was evaluating students. I often faced situations where I knew that students understood a particular concept but this wouldn't be reflected in what they wrote down. Or maybe I wasn't good enough to interpret what they were writing. Consider this question which I gave a Grade 8 class:
One morning, a cake shop bakes four hundred donuts. They sell two-fifths of these before lunch. They sell one-quarter of the remainder after lunch. How many donuts are left at the end of the day?
Now look at what this student wrote and ask yourself what grade you might give this (If you teach in Ontario, you might want to consider what level you would give this for Communication).

 

Now listen to what the student says when she explains her solution:



I have shown this to hundreds of educators across Ontario and the experience is always the same. Initially most people say that the student's thinking isn't clear. After listening to the explanation almost everyone agrees that the student does indeed explain her thinking clearly.
It stands to reason then that if we are not intentionally incorporating observations and conversations into our evaluations then our evaluation practices are ineffective.

What are the chances?

This past week I was in a couple of different Grade 6 classes doing some probability tasks. Watching the kids tackle these led to some huge insights as to what they actually are thinking when they do probability questions. It got us as teachers thinking if students make mistakes in a probability task, is the misconception to do with the concepts of chance or the concepts of number sense.
But the only way we could get these insights was by listening to the kids' conversations!
So here is the first question we tried, and the first misconception...

To be honest, you can see this student's point! The middle bag has a circle close to the top so that's the one to choose if you want to get a circle... just make sure you pick the top shape (especially if it  feels round!) This got me thinking that the question will be better if I use the same shape with different colours (e.g. just black and white circles). Anyway we addressed this misconception by putting some shapes in a bag and shaking them up and down and asking "Do you know where the shapes are now?" It did the trick!
The more common misconception though showed that the kids were thinking additively and not proportionally.

Essentially what this student (and some others) were saying was 'Since the first bag has the least number of squares, you are less likely to pick a square... therefore you are more likely to pick a circle.' Conversely, there were other students who picked the third bag as it had more circles. The interesting thing was that when we asked the students to write the probability of getting a circle they were good at this: most were able to say 2/5, 6/15, and 10/25. So they might have understood the chance of getting a circle but they misunderstood the concept of equivalence. We challenged these ideas by representing the fractions using a virtual manipulative:

It was clear that students will still need to see concrete representations of fractions for them to move from additive thinkers to proportional reasoners.
We then stumbled across another misconception; we asked students what will happen if they flip a coin. They wrote their answers on post-it notes which allowed us to quickly display the results:

 

Here, the majority of students don't clearly understand chance. Some students thought that if you begin with the heads 'up' then it will land heads 'up'! Of course, we tested these ideas straight away.

What was also apparent was that if students only ever experience situations in which all the possible outcomes are equally likely then they might that in every situation, every possible situation is equally likely. Here is the second question we tried (in a different class):

 



 

We know that some students have the misconception that as there are three outcomes (i.e. you could pick a green, a yellow, or a blue) each one is equally likely. Again, a quick experiment usually is enough to challenge such notions. Or, in this case, say (as my colleague Chad did) "How about we play a game; every time a yellow is drawn, you give me $10 and every time a blue is drawn, I give you $10." They don't usually want to put their money where their mouths are!

Students began by representing their ideas with tiles as shown:



They could even write the probabilities correctly:

 However they had the misconception that another yellow was being added, the probability was increasing. In other words they couldn't see that the original 4/8 probability was the same as the final 5/10 probability. We asked them to think of how they could use the tiles to convince us that in fact 5/10 was larger than 4/8 (rather than us show them that in fact they are equivalent). Light bulbs began to go on when they used representations such as these:

We also showed equivalence using the virtual manipulatives above for further proof (always a good thing).

 

So for students to experience success in probability we concluded that they must have the opportunity to use represent it in a variety of ways (e.g. spinners, tiles, number lines etc.) The nlvm site has a superb applet for spinners by the way in which you can design your own spinner and have it spun for up to a thousand spins and have the results displayed in a live bar graph.. Here is one such bar graph which begs the question 'What did the spinner look like?'

But we also wondered if students would also improve their understanding of probability by playing board games, card games, and dice games like Yahtzee. Maybe this could be the perfect homework assignment!

 

 



"We're Into the Third and Final Quarter..."

So once said a sports commentator on British TV. Yogi Berra (the famous New York Yankees catcher) once was asked if he preferred his pizza cut into four pieces or eight. He replied, four as eight pieces would be too many! Fractional misconceptions are everywhere as seen in this advert:

The chart above actually gets my vote for The Most Pointless Legend Ever (look closely at the bottom left-hand corner).
As I and my colleagues bounced ideas off each other as to why fractions are so problematic, it became clear that what worried us most was that students might not have a concrete grasp of the basics of fractions. That maybe when we teach fractions we rush too quickly to procedures and algorithms (and end up saying things like "'reduce' this fraction" or "there's no need to wonder why, just invert and multiply").
So our line of inquiry going into a Grade 2/3 class was based on these two wonderings:
Do the students really understand fractional names?
What is the best way of getting students to work with these fractional names?

We showed the following screen to students and asked them to discuss in pairs where the labels should go:

 

Good class dialogue led to some consensus:





But when they reached the last label there were more than a few puzzled looks:

Some students wanted to put the fourths label on the last circle simply because it was the only one without a label. Others argued that the last circle was in fact split into fourths, in fact there were three fourths:

This puzzled me at first until I realised that what the student saw was three fours and not three fourths.
So to address this we tried to connect it to the language of fifths, sixths, eights and tenths. By getting them to see the pattern that five fifths make a whole as do six sixths, eight eighths and ten tenths, then you must need four fourths to make a whole. They then realised that the fourths label would go on the same circle as the quarters label. Of course the language of thirds and halves (why halves and not twoths?) doesn't make this linguistic pattern easier to spot: sometimes the English language gets in the of learning Math. All of this certainly challenged any notions that we might have had that these students all knew how to label fractions; to assume otherwise will lead to a host of problems. 

We then gave the students this Yogi Berra-inspired problem:
Would you  prefer five half-pizzas or nine quarter-pizzas?
We were able to give three pairs of students iPads and they used an app called Virtual Manipulatives. It was noticeable how much easier this made it for these students to model and communicate their thinking:

Other students relying on pencil and paper sometimes struggled to see how to compare the two quantities and some were still stuck on the idea that nine pieces must be better than five pieces:

There was also one pair who drew five pizzas split in halves and nine pizzas split in quarters: they had misinterpreted what five half-pizzas and nine quarter-pizzas meant. I wonder how many other students make this error?

All of us left realising that students need a wealth of concrete experience with fractions, much more than they are probably currently getting. We also agreed that this experience shouldn't just be confined to a two-week unit; it is something that should be experienced across (and even beyond) the Math curriculum throughout the year.

Fraction immersion, as it where.

A word of caution though: learning about fractions involves so much more than the 'pizza' fractions seen above. Here fractions are being used to describe an area, but fractions can also be used to describe a set (in a class of 13 girls and 15 boys, what fraction are girls?) and fractions can be used as a number (what number is halfway between three-quarters and two?) Students must  experience fractions in each of these situations.