Generalizations in Mathematics

We find a lot of generalizations in math but most students have difficulty in looking at generalizations and applying them to different situations.  For instance, multiplying numbers and multiplying binomials are the same process so if you know how to do one, you can do the other but most students see them as totally different.

By definition, generalizations are patterns that are always true. It is taking ones observations to create conjectures based on those observations, so they can be applied to the same type of situations.  An example of this might be that when two odd numbers are added, the resulting number will always be even.  This is a generalization because it is true for all cases.

Many math teachers do not take time to identify generalizations when teaching them.  Generalizations can involve either processes or products.  There are three components to generalization - expansive, reconstructive, and disjunctive.

Expansive refers to expanding the existing information of the student without changing what they already know.  This means the new information is close to what they already know.  In addition, if this information is close, students often reconstruct their cognitive knowledge.  Disjunctive is when a student uses disconnected pieces of information put together to provide the generalized knowledge to solve new problems.

The process most students go through begins with specific problems to formulate conjectures, which are then symbolized before arriving at generalizations.  Generalizations help students jump from what they already know to incorporating new material into their understanding by providing a link.

The thing about generalizations in math is they should start in Kindergarten so by the time they get to middle school and high school, they are accustomed to working with generalizations.  Many times as we teach general formulas such as (a + b)^2 = a^2 + 2ab + b^2, students have difficulty connecting this with a problem such as (x + 3)^2.  In addition they don't always recognize x^2 + 3x + 9 as factoring to (x + 3)^2.  The formula is the generalization while the (x+3)^2 is the specific application.

So as part of the mathematical programs, we need to spend time helping students learn generalizations either through direct instruction or through the use of manipulatives so they can then apply the generalizations to specific situations.

One way to help students learn to make generalizations is to use open-ended problems and tasks.  Once students have made a generalizations, they need to answer the questions "Does it always work?"  and "How do I know it works?"

One example of an open ended task is to provide students with 12 addition and subtraction problems.  Some of the subtraction problems might be 12 - 5 or 7 - 11 while the addition problems might be 12 + 3 or 7 + 0.  Have them group the expressions according to their own rules.  When students are done, write down the different ways the groupings occurred and see if students can then create their own generalizations, ending with the two questions.

Another one might be to come up with a variety of shapes that all have the same areas and have students divide them into groups.  Repeat the write the criteria they used on the board and see if they can come up generalizations.

So look for ways to encourage students to generalize so they can learn the material better.  Let me know what you think, I'd love to hear.  Have a great day.