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.