This task was developed because I realised students could not progress in their 'scientific method strategies' without independent analysis skills. Both myself and the maths teacher want the students to acquire a working understanding of proportion and how it's used in maths, sciences and in general. To develop the language and visuals (diagrams etc) for describing proportional relationships. To extend/connect this understanding to fractions, ratios and percentages. To extend/connect and extend this understanding to straight line graphs which go through the origin.

In this task at the final stage, looking at any one variable (parameter) will not work no matter how many variables you check out. They will ultimately need to look at the relationship between 2 variables.

STEP 1: Increasing room for learning...Recognising proportion (3 class periods of 45 mins)

Starting point: SORTING GAME: MATERIALS: Lots of pictures of young/adult animals (dogs, cats, elephants, rhino, birds etc), young/adult people, young/adult human and animal cartoons and young/adult human skeletons. Also: Pencils, rubbers, rulers, plain and graph paper.

  • Students (13) initially working on one large table asked to sort all pictures into groups (emphasis on plural). No other instructions. Sorting/no. of groups is entirely up to them. Each time a set of groups was completed the students would have to explain their choices and legitimacy of their groupings.

Initially they chose 8 groups which appeared to work well. They spoke freely of variables and values and said how there were hundreds of variables if you included all the pictures. However, upon close examination certain members of one group could also be included in other groups, due to the group descriptions. e.g. a cartoon puppy could belong to both 'cartoons' and 'dogs'. Therefore I argued with them about their descriptions. Sometimes the groups could work if they adjusted their descriptions. Idea here is to focus students on making sure their chosen description really does apply.

  • Students allowed to reorganise groupings and refine definitions. Examples are here.

Went through several rotations of this until eventually they were able to confidently recognise/reorganise and define groups that worked. This is an important step so that they understand the point where the task becomes 'impossible' for them.

  • Then I asked for all pictures to be sorted into 2 groups only! (Not impossible yet - I am looking for students to reach 2 groups of 'young' vs 'adult').

This caused some protest, so at this point I divided the class into 2 groups (boys 5 and girls 8 for a change) to allow fresh working realtionships.

Students pretty quickly came up with many groups of 2 e.g. 'celebrities' & 'non-celebrities', 'male' & 'female', 'clothed' (in any way at all) & 'naked',  'birds' & 'mammals' and so on. I kept going around between the groups and checking if they worked (though they were very good at this by now). Then I would say ok now find another criteria that gives only 2 groups (e.g. with the birds v mammals I simply removed the 2 pictures of birds). The students treated this part as a game until they couldn't come up with any more groups.

Just as I thought I'd have to direct the finding of the groups 'adult' & 'young' both groups of students came up with it themselves as in two versions - A. 'kids' and 'adults' and B. 'cute' and 'not cute' which was redefined and 'grown-ups' and 'babies'.

STEP 1 - 2: NOW Take away the fish... (now make the task impossible)

Now they had the idea for 2 groups BUT for the first time they could NOT express/define the criteria by which the pictures were grouped. They could not legitimise/explain their choices in any way or form. I asked HOW DO YOU KNOW this pic (holding pic of baby rhino/puppy in my hand) belongs in this group ('young').

The usual reponse was 'you can see it's a baby or a kid' and variations on that theme. As I said they had become very good at justifying their choices by now so not being able to EXPRESS something which to them was very OBVIOUS was quite uncomfortable. They knew absolutely what their brains were telling them but didn't know either how to communicate it or evaluate it. I chose this quite deliberately because young animals of any species are very obvious indeed but we are totally unaware of how we make this evaluation or even that we are making the evaluation, it's automatic.

The argument progressed to 'well it's instinct' (i.e. recognising it's a subconscious evaluation) and 'we just know because we know from a young age'.

STEP 2: BUILD with the students.

Now ask them to go back to looking at variables and values, which they had been using at the beginning of the exercise. I kept redirecting them back to the pictures if they made comments unrelated to the materials in front of them. I had to ask questions like "well what makes puppies 'cute' ???" "how do you know this is a young rhino/elephant?" "how do you know it's not an adult???". The answers gradually started to come out that the eyes or heads were very large, the paws were large, the legs were short. BUT these ideas by themselves don't work as the pictures of the puppies might be larger than the pictures of the adult dogs (again on purpose to block direct comparison). Some students at this point (in different groups) said that if the photos had all been taken from the same distance with the same camera then direct comparison would be poossible and would show they were right. (This of course is the idea behind a controlled experiment!-save that for later).

Then some students started to revise the description like this 'the head is bigger for the puppy compared to his body' and many other variations. This idea caught on like wildfire between the groups. Suddenly they could all 'see' it.

STEP 3: TEST their thinking and REFLECT.

So now they were 'finding' many other comparisons of variables that also seemed to work. The size of the paws compared to the body, the width of the head compared to the length of the head.

I asked them to go through all the pictures again one by one and view them all 'through this lens' of comparing one variable with another. They decided that in all cases it allowed identification of the younger ones.

We looked at how simply looking at the size of the eyes didn't work, it had to be the size of the eyes COMPARED to the size of the head.

STEP 3 back to STEP 1: (Take away the fish again).

Ok they have definitely accepted this discovery that the comparison of variables allows them to clearly identify juveniles vs adults. Using the language "X compared to Y shows this one belongs in this group" works and they can complete the task :)

Now I asked them to prove it! In other words they now have a testable statement/a hypothesis. To test it means to make actual measurements and find a way to compare the numbers that allows them to identify the 'juveniles vs adults' again. They can no longer use the language above.

STEP 1 - STEP 2:

Now they have to find a new way to solve the same grouping. We need to learn a new way. This is where we stopped the lesson although some were starting to have the idea of dividing the numbers... We will resume this task tomorrow.



# Alexander Sokol 2011-02-04 08:34
Deirdre, thanks a lot. It's always a pleasure to read your detailed descriptions.
A few things I wanted to specify, though.
1. At Step 2, building the model / hypothesis together with the students, did you ask them to 'fix' their hypothesis in any way? If yes, how? And how many different 'ways of fixing / representation' were there as a result?
2. At Step 3, reflection, there are normally at least two different types of reflection: the one dealing with the actual model / hypothesis and how it worked and the other dealing with the way of developing / testing the hypothesis (a kind of metacognitive level). I more or less imagine how you dealt with the first type but I wonder if you also touched upon the second one? And how did it look like if you did?
Many thanks.
# Deirdre Jennings 2011-02-07 14:59
First for number 1. At step 2 it was very hard to encourage the students to explore possibilities during this activity without giving away the answer i.e. without using words like compare. The idea was for them to develop the language of comparison between 2 variables. They are used to spotting variables and measuring them and so on. They focused quickly on the fact that "children's head are bigger than adults" (but of course they are not). But a compound variable x/y or x*y is an unfamiliar and abstract concept (thought they may have used them in the past without appreciating them as proportions!). They could not at first come up with the language SO I just kept asking questions laced with encouragement e.g. "yes you're on the right track and I know you know what it is, BUT what you are saying/writing doesn't mean that" because it's the concept/language we're after...or just point out "kids heads are not bigger than adults". So after some provocation they came to the point that discussing isolated variables didn't work. Then they started comparing them. They all kind of went for a comparison eyes/head or feet/body or legs/body or head/body. The 'fixing' therefore was making sure the language they used communicated what they wanted to say/show.
Step 3. Students themselves suggested going to the pre-K and measuring them to test the hypothesis - brilliant idea :) we did that today and their reaction was interesting as they realised kids' heads are similar in size to an adult! This made it more concrete too and I don't think some believed it until they measured the small kids.
# Alexander Sokol 2011-02-07 22:41
Did I get you right that:
1. They didn't actually 'put down' the model they came up with as it felt that using it in communication was enough. Right? If so, are you planning to come back to it in some time and see if it was?
I am asking as I am rather sceptical about students remembering things. In my experience, they forget pretty fast and, unless it's 'fixed' somewhere in the written form, it takes a long time to remind them what they already understood at some point.
However, this may be just my 'bad' experience...

Re Step 3. In my view, going to Pre-K and measuring them would be the first kind of reflection, wouldn't it? I still think it'd be important to get them to reflect at the more metacognitive level as well, don't you think?
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