Thinking about Learning,
Learning about Thinking

11. Summing up

I have grouped my thoughts into ten sections, each labeled by an active verb. Here I offer a sentence or two of summary for each verb.

1. Participating. We must keep students actively involved in their own education.
2. Reading. If students don't know how already, we must teach them how. Otherwise they won't be able to cover the syllabus.
3. Writing. Students must write to reveal their thinking. Our colleagues in writing programs can help with this, but we must reinforce their efforts.
4. Calculating. Even if machines can do it all, some calculating is necessary to develop symbol sense -- and it makes students feel good.
5. Assessing. Every learning activity is also an opportunity for assessment. Students need frequent feedback -- formal or informal -- and teachers need frequent updates on their understanding of what their students have learned.
6. Knowing. Students construct their own knowledge. Our job is to make sure they do it right.
7. Differentiating. We must provide routes to success for students who are not like ourselves.
8. Learning. We must pay attention to learning stages and learning styles -- and realize that neither is permanent.
9. Modeling. To understand how our students relate to mathematics, it is useful to recall our own efforts in areas where we never became experts.
10. Experimenting. All experiments succeed. Therefore we should always experiment.

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Last modified: May 17, 1997