Let us begin here by recalling the example of Paul, the fictional teacher introduced in Chapter 4, whose understanding of his two teaching subjects – mathematics and history – contrasts sharply. Where mathematics and its teaching are concerned, Paul is alive with fresh ideas, but he regards history essentially as a body of information – knowledge of events, causes and consequences – for transmission to his students. That transmission, moreover, is guided chiefly by Paul’s familiarity with reliable examination topics, and with the structure of the examination question papers. He is similarly familiar with examination topics in maths, but somehow this remains a background issue rather than a dominant theme in his maths classes. Paul hasn’t made explicit to himself the nature of his understanding of either subject. He readily declares that he finds maths a fascinating subject, and he admits, somewhat reluctantly, that history can be a bit unexciting. He would be offended if it were suggested to him that he had a prejudice against history and a prejudice in favour of maths. Paul insists that he has no preconceived ideas about either subject, but that he just has a natural attraction to maths that he doesn’t have to history. But he also insists that he is no less competent in history than in maths and that his students’ success in examinations bear out this point.