Monday, August 29, 2016

Elaboration in Math

Porsche, Oriental, Painting, Side View  Elaboration in the English means you go into great detail on something.  It might be adding lots of detail to something but does it apply in the same way to Math?  Lets look first at what elaboration is as a learning technique.

Elaboration is a conscious way of expanding a topic or idea so a student can process it more thoroughly.  One method of elaboration is to take information and turn it into a question.
You might read that warm air rises so you might ask "Why does warm air rise?"  Finding the answer to this question leads you to think about why it might be lighter than colder air.

The materials you know well are called anchor points because it is easy to recall.  The more anchor points you have, in place, the the more meaningful the information becomes and the easier it becomes to remember it.  It helps in the process of learning.  So elaboration helps form a bridge between the new information and previous knowledge. 

This leads us to ask how would we turn the information into questions in Math.  Today I taught x^0 = 1.  Is it possible to ask "Why does x^0 = 1" Yes because of the rules of division of exponents.  X^3/X^3 = 1 becomes X^3-3 = 1 which gives us X^0 = 1. 

What about other topics such as solving one step equations.  One of the standard ways of teaching this topic is to talk about undoing or doing the opposite to solve it.  Can we ask "Why do we do the opposite operation to solve the equation?"  I think we can but the question asked needs to lead to real thought.  For instance when you teach binomial multiplication, you could be asking a couple of different questions to help elaborate the material.

First you might ask "Why do we need to multiply binomials" followed by "Why do we end up with three terms?"  Both of these questions make you acquire a deeper knowledge.  However, research shows that students learn better if they generate the associations rather than having them provided.  So its important for the students to learn to create their own questions rather than relying on the ones the teacher provides.

Suggestions to help students learn methods of elaboration include:
1.  Gaining attention by asking them how the material is relevant to them. This might be seeing how the math is used in real life, or how they might use it in the future.  Most students I know will fall back on the "I need to know it for the test!"

2.  Simulation recall when you ask them to recall something that is similar to the new material.  When looking at binomial multiplication, they might comment that the method you are teaching is similar to the way they learned to multiply regular numbers in elementary school.

3. Present the material by presenting examples and asking them to find the common concept among the examples.  It is possible to ask students how this new concept compares with a previous concept.  One possibility here is using the distributive method for binomial multiplication and asking students to compare it with the distribution property.

4. Error correction where a student explains why the problem was wrong and what is the correct way of doing it.  I ask students to do this for all questions they missed.

Give this some thought and see how this might be used in your classroom.