HOME   Curriculum   Fees   Schedules   Testimony   Consultants   Contact Us   Resources

Four Categories of Mathematics Learners 

The expert  remembers her number facts and can efficiently apply procedures like multiplication with regrouping. This fluency and proficiency warms the hearts of those advocating a back to basics policy. At the same time, the expert understands the facts and procedures in two senses: she can provide the ‘official explanation’ as offered by the teacher or the text, for example, ‘Eight times seven gives the same result as seven times eight because multiplication is commutative.’ But this explanation is more than parroted words. She relates it to her own everyday knowledge: ‘If you have an egg carton with 12 eggs, you can turn it to any angle and it is still 12.’ Moreover, if memory fails, she uses backup strategies to produce the needed result. She thinks of mathematics as a meaningful activity requiring thought. And finally, she is aware of her thinking and can describe it to others. She has constructed mathematical knowledge that includes syntheses between various aspects of her personal knowledge and formal, mathematical concepts. True experts may be rare.

The mechanical learner, the master of illusory knowledge, remembers number facts and also employs procedures with efficiency. As a result he scores well on various standard tests and is considered to be a high achiever. But in fact, he understands little. He cannot make sense of much mathematics, and may indeed not believe that doing so would be a good thing. He may parrot the teacher’s explanation, but when pressed does not really understand it, either in purely formal mathematical terms, or with respect to his own everyday knowledge. What’s most important is doing well on the tests. The mechanical learner does the problems in the way he thinks the teacher wants. He tries to outfox the teacher by studying her expression to figure out what answer she expects. This kind of learner is probably very common and may indeed be the ideal for many teachers – but not for those whose goal is sensemaking and critical thinking.

The ordinary student  performs reasonably well but makes mistakes. She remembers many facts and can apply procedures more or less accurately. She is perhaps occasionally sloppy and slothful. But she has a reasonable understanding of the basic formal ideas and can even explain some of them. She can accept that mathematics ought to make sense. She also has many everyday concepts and procedures, but does not always connect them with what is taught in school. Her underlying competence is sound, despite the fact that she often does not exhibit her knowledge to good effect and may perform at a mediocre level. In crude terms we can say that she is smarter than she looks. Perhaps there are many students of this type. They are ripe for good teaching.

The lost mathematical soul  seems to have neither skill nor understanding. He does not remember facts or procedures, and he does not seem to understand what he is doing. Maybe he has various forms of everyday mathematical knowledge, but he does not exhibit them in school, and does not connect them with what he learns there. He does not see mathematics as a meaningful subject: it is rather a major ordeal, a minor torture.

An excerpt from:

Ginsburg, H. P. (2009).  The Challenge of Formative Assessment in Mathematics Education: Children's Minds, Teachers' Minds.  Human Development, 52, 109-128.

HOME   Curriculum   Fees   Schedules   Testimony   Consultants   Contact Us   Resources