How do I plan for problem solving?

Almost a month ago, I wrote a post "Is direct instruction a better approach to teaching math?" that got a lot of attention (relatively speaking). My post was in response to an article which used one poorly constructed (my opinion) study to suggest that problem-solving or inquiry-based lessons were less effective than lecture-style instruction when it comes to standardized-test results. What seemed to get the most attention/ire was a comment by the article's author, Paul E. Peterson. 

"I, too, like those problem-solving classes. They require less preparation and are easier to teach."

This might be true for a tenured university professor who: (1) enjoys academic freedom; (2) has no "accountability" to a standardized, national test; and (3) does not believe in the problem-solving lesson as an instructional approach. But for the rest of us, a problem-solving lesson requires a great deal of effort. I want to share my process of preparation for such a lesson in this post.

Because I am also a university professor who enjoys those first two perks, I want to focus on a fraction lesson that I planned and taught a few years ago at a local elementary school. When the fifth grade teacher contacted me for help, she was very specific about the content that I needed to addressed in the unit. In Michigan, the driving force for most K-8 teachers is the Grade Level Content Expectations (GLCE) and for my series of lessons I needed to get at this standard: 

N.FL.05.14: Add and subtract fraction with unlike denominators of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 100, using the common denominator that is the product of the denominators of the 2 fractions.

My first step in planning was to gather data about my learners. Through a series of emails, I found out that this school grouped fifth graders by ability for math and that this teacher taught the lowest group. This included several learners with special needs supported by a special education teacher. While I am not a fan of ability grouping, this was not my fight. I was just grateful for the information.

Mathematics understanding is about experience not ability. It was up to me to plan a problem-solving lesson that offered learners an experience that would support their development of a relational understanding of fraction computation. Fortunately, I was familiar with an excellent resource that provides just such an experience. Planning a problem-solving lesson is not about developing activities from scratch (but if you have time and training, then this can work). Still, the planning does require effort in identifying appropriate resources and structuring them in such a way that they support learning.

Cathy Campbell wrote about this excellent resource on her blog. In particular, she discusses the clock model used for adding and subtracting fractions, which can be found in Minilesson for Operations with Fractions, Decimals, and Percents. Cathy does an great job describing this resource, so there is no reason for me to say much more except that I find its use of context and connections to prior successes very supportive for learners.

I am also fortunate to have the professional development packet that goes along with the series. This packet includes videos of teachers modeling some of the lessons. Before planning my lesson, I watched Joel teach the clock model, and it gave me some ideas of how to organize the lesson. In particular, it showed that he introduced the model in a whole-group setting.

Finally, I was ready to write out the plan. I decided to use a slight modification of a lesson planning framework Debbie Miller shared at the Michigan Reading Association Conference in 2008 and described in her excellent book, Teaching with Intention. You can view my plan here. As you can see, it is quite detailed, yet I do not consider it a script. I am a firm believer in Jon Stewart's approach to planning, "Creativity comes from limits not freedom ... When you have a structure, then you can improvise off of it..." (I still wish I had remembered to share that quote during my TEDx Talk). This detailed plan allowed me to make necessary adjustments as I taught the lesson, but that is for another time.

I hope this makes the point that planning for problem-solving is not easy. "Where's the problem-solving?" you ask. Let's compare the plan with the National Council of Teachers of Mathematics Process Standard for Problem Solving:

• Build new mathematical knowledge through problem solving;
• Solve problems that arise in mathematics and in other contexts;
• Apply and adapt a variety of appropriate strategies to solve a problem; and
• Monitor and reflect on the process of mathematical problem solving.

Please let me know in the comments if any of these are unclear in my planning.