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Students Thinking Algebraically

A few days ago I experienced one of the most inspiring moments as an algebra teacher…  I gave my 9th grade students a quiz on solving equations (one-step, two-step, multi-step, literal, etc.).  One of the questions was a word problem that involved buying a season pass ticket to an amusement park versus buying single passes and making multiple visits to the park.  The first part of the question asked students to determine how many trips to the park they would have to make in order for the season pass to be the better deal.  The second part asked the students to write an equation to model the situation.  The third part asked the students to solve the equation.  Most of the students immediately solved the problem by writing and solving an equation.  When they read the second and third parts of the problem they were confused because they had already completed both parts in the beginning.  I was excited!!!!

This is why I was excited…  The students were initially asked to solve the problem using any method (it was an open-ended question).  Most of the students immediately wrote and solved an equation because that was their first thought.  These students were ahead of the test question!  They were already “thinking algebraically” before the question asked them to think algebraically.

After I collected the quizzes the students told me they were confused by the problem and wondered whether they answered it incorrectly.  I told them they answered the question exactly the way they should have.  I told them they were thinking algebraically and that is how they should be thinking.  They were pleased with my response!

The goal of algebra teachers should be to help students think algebraically.  When this happens, students begin to look at problems differently.  They begin to generalize situations and find solutions quickly (and accurately).  Thinking algebraically is a higher level of thinking that most students (and adults) never achieve.  Most of my 9th grade students are already thinking algebraically!  As much as I would like to take full credit for this, I can’t.  Their teachers before me did a phenomenal job and that makes my job easier.

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Bridging the Gap Between Arithmetic and Algebra

Copied from DragoArt.com

Copied from DragoArt.com

I have been teaching college level precalculus for several years.  A running theme of concern has been the lack of preparedness of my students for the course.  The struggling students somehow place into the course, but clearly are not prepared.  My assessment is that the students’ basic algebra skills are weak.  But what happens when a student takes algebra for the first time, but are not prepared?  Why are some students ready for algebra and some students struggle with the basic algebraic concepts covered in Pre-algebra or Algebra 1 courses?  What do you do as a teacher when you are faced with the challenge of bridging the gap between arithmetic and algebra?  How do you incorporate these concepts into your lessons without losing algebra “teaching time?”

This is an issue many Algebra 1 teachers face.  The common concern is that students taking Algebra 1 lack basic arithmetic skills.  But these skills are necessary for success in Algebra 1.  For example, many students struggle with adding fractions.  What happens when those same students have to solve equations with rational expressions?  If they have not mastered adding fractions, they will not be able to solve equations with rational expressions or they will experience difficulty when faced with these problems.

To me the answer is clear… Teach students so that they master basic arithmetic skills before they enter Algebra 1.  This charge is for elementary school teachers.  Here is the reality…  This is not always accomplished.  Elementary school teachers probably have their reasons for why this is not happening, across the board.  In the meantime, students are required to take Algebra 1 with whatever skills they have acquired.  This presents a problem to secondary teachers who have students entering Algebra 1 lacking the basic skills needed to learn and master basic algebra concepts.

How do you bridge that gap as an Algebra 1 teacher?  What does that bridge look like?  How do you help these students without hindering the advancement of the students who were fortunate to have mastered these skills?

These are very valid questions with many valid answers.  What are your thoughts?  What have you done in this situation?