- Explain it, highlighting the key points in special boxes or formats.
- Illustrate it, giving 2-3 example problems/solutions that explore the concept
- Provide a list of problems to gain proficiency, which students generally do at home. Textbooks usually provide answers (not solutions) to half of the homework problems so students can check their work.
Modern textbooks supplement this with video clips of great teachers explaining the concepts and working 2-3 examples for students to watch. A few textbooks even provide step-by-step hints to guide the student through some of the homework problems - these are very helpful.
This traditional approach is effective and will probably remain the core teaching approach for a long time.
The second way
Students can use this approach to augment the traditional way. They still need someone explaining each concept (in person, on video, or thorough words in the textbook). And they still need someone walking them through 2-3 solutions. But instead of going right to the homework problems, the next step is to give students many problems with solutions already worked out. I'm not saying all students need this, but it is very useful for some. Of course, students still need the practice of working quite a few problems all on their own - but not as many if they carefully study provided solutions to a wide range of problems. This second way exposes students to a more and wider variety of problems in the same study time as the traditional method - or it reduces their study time.
It's definitely a very useful study strategy for someone who once knew the material but needs to get back up-to-speed quickly, such as a student preparing for a final exam or an ACT/SAT test. It's also a great strategy for tutors/coaches to refresh their skills before meeting with a student. And the tutor/coach can use such solutions in their time together. In a very short time this strategy helps a tutor/coach refresh their skills.
But where can you find a large supply of nicely organized solutions to problems?
This is where JDS Mathematics comes in. This is a free website providing full solutions to a wide range of mathematics problems: arithmetic, algebra, coordinate geometry, calculus and statistics. It automatically generates as many unique problem examples as you will ever want, all organized by problem type and difficulty level. (There's also an Android version.)
An example of this second way
One Tuesday night I was looking for a way for 2 pre-algebra students to experience working together to learn how to multiply 2 polynomials together. It was a new concept to both of them. I used Google Docs to write a step-by-step learning exercise incorporating the JDS Mathematics website. The two students sat together in front of a computer while I sat back and watched - helping, as needed. They opened up the Google Doc and began following the instructions. As they worked their way through the Google Doc they discovered the link to JDS Mathematics, which provided great example problems/solutions in a good progression to grasp the topics. The general thrust of the teaching was to learn the general case of how to multiply 2 polynomials together, and then to understand the FOIL method for the special case where the polynomials are binomials. To see the Google Doc, go to JDS Math enrichment exercize.
By now you have noticed the picture of Dr. Purohit. He is the developer of JDS Mathematics. I met him on Google+ and have interacted with him as he developed his software. Development is ongoing.
All the best,
Tim
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