I gave my year 8 class a pretest in fractions. The pretest was derived from Idit Harel's "Instructional Software Design Project".
The range of marks for my students was between 20 and 47 out of 67. So, there were significant gaps in the fraction knowledge of all the students.
I then returned the marked test to the students with a copy of the original test questions. Some expressed surprise that I was given them a copy of the test which they had already done. "Do you want us to do it again?", some asked and then answered their own question, "No way!". I explained that I wanted them to look at some of the questions again for discussion, not do them again as a test. I felt that they felt this was a strange request, that they had a task oriented mentality, that you complete a task and then move onto the next task.
I then taught a brief lesson about fractions, explaining that the underlying rationale of the test was transformations between different representations of fractions - transformations between maths like pictures, real life pictures, words and symbols.
I also covered some fraction vocabulary asking for the meanings of numerator, denominator, proper fractions, improper fractions and mixed number. Some of the class members knew the answers to these questions. I invited them out the front to write their answers on the white board and some of the students were happy to do this. Many students enjoy being out the front, writing on the white board.
I then came to the point of the lesson by asking them: "Can you identify for me and somehow articulate a fraction puzzle from the test which you found difficult?"
Once again I gained the impression that I was asking for something strange. One student said that he now understood it all since I had taught a (brief) lesson about fractions. This led to a friendly argument between me and him where I claimed that if he did the test again he would still obtain wrong answers and he insisted that no, he would get them all correct - but no way was he going to do the test again because he had already done it. This went on for some time, a bit like a Monty Python skit about the difference between an argument and a contradiction.
Eventually, after some turmoil and mulling around, the students submitted to my request and began to give me examples of fraction problems from the test which they found hard. One confident student even came out the front and wrote a question she didn't understand on the white board.
But the main thing I noticed from most of the students is that they gave me examples from the end of the test which had the hardest problems, even though many of them had made errors on problems from the easier sections at the beginning and the middle. So they were complying with my request but I think trying to give the impression that they knew more than they actually did know.
I realised then the importance of being able to setup a cross age tutoring scenario. So, that instead of asking what my students know it's far better to ask: "What do you think students in another class would find hard?" Then my students would be more likely to identify a problem in their own zone of proximal development, rather than something that might be too advanced for their current zone at this stage
It's hard work setting up a dialogue with students which involves meta cognition (thinking about their own thinking), meta-conceptions (students thinking about their own knowledge and understanding of concepts) or self evaluation. For many you have to think up ways to trick them into being comfortable with revealing what they don't know. It would be very easy for a teacher to give up on establishing meta dialogues.
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