Teaching Mathematical Practices in the First Days of School

As my first addition to the wealth of knowledge that is the mathtwitterblogosphere, I thought it appropriate that my first blog post should be about the first days of school.  I like to jump into content right on the first day.  More important than the content, though, are the mathematical practices that I teach on the first few weeks of school.  These were taught to me by Ms Leslie Butler, of Millcreek Junior High in Bountiful, UT.  I explicitly taught these practices for the first time at the beginning of school last year. I found that as I repeatedly emphasized these practices to my students, they were solidified in my own brain and became in integral part of my teaching practices throughout the year.

The Answer is Not the Most Important Part:

Phil Daro has a great video online called Against Answer Getting (click on the tab that says ‘Against Answer Getting’ to watch the clip).  One of the things he emphasizes is that in the United States, students and even teachers are in such a rush to get to the answer that we never really get to the mathematics.

To get away from this, I start the first day with one problem – Tiling Pools.  By the end of class, most students are able to efficiently find the ‘answer’ – the number of tiles needed to create a border for a square pool.  And they think they’re done.  Boy are they wrong!   We spend at least a week on this problem and pull every available ounce of mathematics I can out of it.  We draw pictures of pools, we make tables, we make graphs, and we write as many equations as we can possibly think of.  Part of the reason I do this is to emphasize to my students that there is so much more to learn from a problem than just how to find the answer.  It also introduces them to the idea that there are many different roads they could take to get to that answer.  Which leads me to the next mathematical practice.

It’s Important to Develop and Appreciate Different Strategies for Solving a Problem:

By the end of the first day, students will usually know one or two valid strategies for solving the Tiling Pools problem.  Their homework is to go home and develop and illustrate their own strategy.  The next day is devoted to listening to students explain their strategy, discussing when that strategy would be most efficient, and practicing using different strategies.  Through this process, students learn some very deep mathematics and also learn that there really isn’t one “Right Way” do a problem.  Recently I read a tip recommending that math teachers try to avoid saying things like “What is the first step?” to students and replace that with, “What have you tried?” or “Where could you begin?”  This helps students recognize that there isn’t a single valid approach to a problem.

Errors are Opportunities:

Usually as students share their ideas, a few errors will arise.  This is wonderful!  Embrace the errors!  Celebrate them!  In the video Against Answer Getting, Phil Daro explains that often it is easier to get to the mathematics through the errors than through the correct answers.  Errors open a door to discussion.  “Davis made an excellent discovery.  He discovered a strategy that doesn’t work.  Davis, can you explain your strategy to the class?  Can anyone tell me why Davis’s strategy doesn’t work?”  I find that my students often have an easier time explaining why something doesn’t work than why it does.  But in the process they develop a much deeper understanding of the mathematics they are learning.  Also, it is important to highlight errors in order to prevent them from re-occurring.  Daro says that an effective math teacher treats one student’s error as a symptom of the entire class and not just a symptom of the individual student.  This year I’m considering designating a portion of one of my whiteboards as a sort of hall of fame of Marvelous Mistakes.

We Need to Work Together to Build a Climate Where Everyone Feels Comfortable Sharing Their Ideas

Classroom climate is everything.  Without a positive learning environment, students are too afraid of being ridiculed to share their strategies and their errors.  It’s important for students to feel success during the first few days of school.  It’s important for them to get to know each other and also get to know me.  It’s important to have good classroom management strategies and efficient procedures in place. It’s important to let students know that this year they will celebrate and learn from mistakes. And most importantly, students need to know that they are in a safe classroom where they can and will be successful in learning mathematics.

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