We did conditional probability in my IB Higher Level class today, and one of the questions I set them really tripped them up. Not because of the Mathematical content, but rather the context that the question used. Below is the question.
The issues that arose were from a general lack of knowledge about how tennis works. Many of the students did not know that if you miss the first serve, you get a second serve, for example. One of the interesting misconceptions was that if you miss the second serve you get a third serve. Another, that if you got the first serve in, but then lost the point, you got a second serve.
Once I explained the rules of tennis clearly to the whole class, they all completed the question pretty quickly.
This really highlighted to me the importance of knowledge about the context when answering a question. In Why Don't Students Like School, Willingham gives the oft referenced example of children who know a lot about baseball but with low reading ages having better comprehension of a passage about baseball than those children with high reading ages who know little about baseball.
Although I knew this, it was not something I had considered in the realm of Maths teaching before, as I just thought it applicable to subjects with reading comprehension. But clearly reading is a big part of Maths as well.
So what does this mean? Context is important. Or, more specifically, knowledge about the context is important. Lacking knowledge about the context can be a real hindrance to being able to solve a problem, even if the mathematical knowledge is there.
This is an argument for removing all context from the sequence of teaching when initially introducing new content and skills, so that students can focus all their attention on what we want them to learn.
It is also an argument for using lots of questions with contexts after they have mastered the content, and are confident in their abilities to use their new knowledge and skills. This ensures they have seen many contexts, and hopefully, through that and other subjects, they will have knowledge of any contexts that come up later in life (or more likely, in exams).
If I had given that question to a weaker group, it would have thrown them completely, even if they could do the Maths. Fortunately, with a high achieving group, they were secure enough in the Maths to recognise that the problem was the context.
But in future, I am going to be much more aware of the context for questions, and, if necessary, teach the knowledge they need about the context before setting questions, as well as the content knowledge.
UPDATE: I have since read this excellent article from Dylan Wiliam in the IMPACT journal from the Chartered College of teaching, and this quote add nicely to my point.
| "The results of a maths test with a high reading demand are difficult to interpret. We can be reasonably sure that students with high scores can do the mathematics that is tested (and the reading required), but for students with low scores, we cannot be sure whether these are due to the fact that they could not do the mathematics, or whether they could not understand the questions. Such a test would support inferences about mathematical competence for some students (good readers) and not for others (poor readers). The problem with such a test is that the variation in scores between students is partly due to differences in their mathematics achievement (which we want), and partly due to differences in their reading ability (which we do not want)." |
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