Improving Algebraic Understanding

 I have been at a loss figuring out how to help my students learn and retain their ability to understand and work out Algebraic problems.  One day, they can do it, the next they act like they are reading Greek.

The other night I came across an article focusing on improving algebraic understanding in middle school and high school.

The paper has three major suggestions for ways to improve algebraic understanding and one of the ways mentioned, I've never actually heard but it makes sense.  After reading it, it made so much sense but I also know I have to teach my students how to do it.

The first area to concentrate on is analyzing solved problems that show all the steps.  The idea is they look at the steps and discuss them to find the reasoning and strategies used.  In other words, they are finding connections.  When they find these connections, it makes it easier for them to transfer the information.

Furthermore, the selected problems should apply to the objective of the lesson and some of the examples should include common mistakes so students learn to watch out for them and to look for them in their own work.  This should be done using whole groups, small groups, and independent practice to teach them how to do this.

One big reason for showing solved problems is it allows students to see the the strategy used in total context rather than each step.  When they discuss the reasoning behind the steps, it helps students develop a deeper understanding of the logic involved in solving the problem.  Looking at incorrectly done problems help students learn to think critically.

The second area is to teach students about mathematical structure and algebraic representations by using proper language, teaching them a self reflective method they can use to notice structure as they solve the problems and help them understand that different algebraic representations provide information about the problem.

This process helps students connections between problems, strategies, and representations that at first may seem different yet turn out to be the same.  When they understand the structure of problems, they have time to focus more on the similarities between problems.  In addition, if they understand what makes an algebraic expression an algebraic expression, they will see the connections regardless of how they are presented.

The last area is to work with students to choose from several strategies to solve problems.  It is suggested students make a list of strategies they can use to solve a problem, explain why they chose the strategy, and they need to evaluate a variety of strategies they could use to solve the problem.

Its important to teach students that strategies are different from algorithms.  Algorithms are a set of steps one always follows to get a specific result where as a strategy requires students to make a choice based on the problem and the end goal.  Getting them to move past the algorithm means they can move past memorization to true understanding.

Let me know what you think, I'd love to hear.  Have a great day.