### The use of worked examples

We're all familiar, I'm sure, with the use of worked-out examples in mathematics teaching. Worked-out examples are often used to demonstrate problem-solving processes. They generally specify the steps needed to solve a problem in some detail. After working through such examples, students are usually given the same kind of problems to work through on their own. The strategy is generally helpful in teaching students to solve problems that are the same as the examples.

Worked-out examples are also used in small-group settings, either by working on the example together, or by studying the example individually and then getting together to enable those who understood to explain to those who didn't. Explaining something to another person is well-established as an effective method of improving understanding (for the person doing the explaining -- and presumably the person receiving the explanation gets something out of it also!).

### Metacognitive differences between high and low achievers

An interesting study comparing the behavior of high and low achieving students who studied worked-out examples cooperatively found important differences.

High achievers:

- explained things to themselves as they worked through the examples
- tried to construct relationships between the new process and what they already knew
- tended to infer additional information that wasn't directly given

Low achievers on the other hand:

- followed the examples step-by-step without relating it to anything they already knew
- didn't try to construct any broader understanding of the procedure that would enable them to generalize it to new situations

Other studies have since demonstrated that students taught to ask questions that focus on relating new learning to old show greater understanding than students taught to ask different questions, and both do better than students who ask no questions at all.

### Learning to ask the right questions

An instructional method for teaching mathematics that involves training students to ask metacognitive questions has been found to produce significant improvement in students' learning. The method is called IMPROVE -- an acronym for the teaching steps involved:

- Introduce new concepts
- Metacognitive questioning
- Practise
- Review
- Obtain mastery on lower and higher cognitive processes
- Verify
- Enrich

There are four kinds of metacognitive questions the students are taught to ask:

- Comprehension questions (e.g., What is this problem all about?)
- Connection questions (e.g., How is this problem different from/ similar to problems that have already been solved?)
- Strategy questions (e.g., What strategies are appropriate for solving this problem and why?)
- Reflection questions (e.g., does this make sense? why am I stuck?)

A study that compared the effects of using worked-out examples or metacognitive questioning (both in a cooperative setting) found that students given metacognitive training performed significantly better than those who experienced worked-out examples (the participants were 8th grade Israeli students). Lower achievers benefited more from the metacognitive training (not surprising, because presumably the high achievers already used this strategy in the context of the worked-out examples).

### More reading

Here are some papers by the creators of IMPROVE on their studies into the benefits of metacognitive instruction (in PDF format):

http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG8/TG8_Kramarski_cerme3.pdf

http://www.hbcse.tifr.res.in/episteme/allabs/zemira_abs.pdf (no longer available)

http://www.icme-organisers.dk/tsg18/S32MevarechKramarski.pdf (no longer available)