Many people make no distinction between teaching and training. Much of mathematics teaching has slowly evolved into mathematics training. It seems to me that this evolution has occurred in the last 25-30 years.

There is a significant and very complex difference between teaching and training. I will therefore post at least a few essays on the topic. As always my perspective is that of a university or college mathematics professor, but most of what I have to say also has meaning for teaching at the lower grades.

At the most elementary level, one might say teaching instructs the learner about why something is done and training instructs the learner about how something is done.

These simplistic definitions of teaching and training will be enhanced in future posts, but the above definitions will suffice for an understanding of my first examples.

My first illustration is about what happens and what should happen very early in the study of algebra. This multi-blog example should begin to clarify the distinction between teaching and training.

A topic that every student encounters early in algebra is the process of finding the solution of linear equations in one variable of the form ax + b = c. Because most students and many teachers erroneously believe the student’s goal is to “find the answer”, the following instruction is typical

The student is instructed to move b to the other side of the = symbol and change its sign, then divide both sides by a. The student knows this is correct because the teacher says it is correct. THAT IS TRAINING.

I have identified, and will discuss, ten serious flaws built into the above training process.

1) A student with virtually no background knowledge can be trained to perform these steps.

2) No mathematics is involved in the process.

3) The student can become very proficient at these steps without any understanding of the underlying mathematics principles.

4) Nothing in this process can be extended to a different problem situation.

5) Nothing in this process can be generalized.

6) The student has no idea what the process has determined.

7) The student has learned no link to a graph.

8) The student has learned nothing about relation to related inequalities

9) Mastery of these steps leaves the student so limited that changing the numbers from integers to fractions or irrational numbers poses an entirely new problem to the student and training in another process is required.

10) The process is of little (or no) future value to the student.

December 2, 2012 at 9:04 PM |

Good stuff.

December 3, 2012 at 1:23 PM |

This is a test comment.

December 6, 2012 at 11:05 PM |

If you have a reference for young children on the natural assimilation of mathematics in everyday activities, I’d love to explore and test it out with my children and some of our other friends. I need to retrain myself to be a better teacher as I want my children to think, not regurgitate information others deem specific to their chronological age. :p

December 7, 2012 at 7:24 AM |

Kendra;

I know virtually nothing about teaching young children.

All of my work and experience has been with college age adults.

There are, of course, some obvious facts:

Don’t teach them things that are wrong.

Don’t try to use abstractions beyond their grasp.

Teach them basic facts in such a manner that they can extend that knowledge to new situations.

Del