A Closer Look at the Three Part Problem Based Lesson

John Van deWalle, Cathy Fosnot, Marian Small, and Marilyn burns are all key researchers when it comes to mathematics in education. According to these researchers, an ideal math lesson consists of three parts. The HWDSB math facilitation team refers to the three parts as: 1) Getting Started 2) Working On It 3) Reflect and Connect.

Let’s just say we are going to teach a grade 5/6 initial three part problem based lesson on multiplication and the goal of this lesson is simply to see what multiplication strategies students are bringing to the table. The “Getting Started” part of the lesson would involve some sort of activation of students’ prior knowledge related to multiplication (simple problem). During the “Working On It” part of the lesson, the following problem could be presented to the class.

29 students are going on a field trip to a museum. The field trip will cost $20 per student. How much will it cost for 29 students to go on the field trip?

This is an example of an open routed question. There is only one answer but there are multiple strategies to get the answer. Therefore, we ask the students to solve the problem in groups and in more than one way. The first strategy will come naturally for some students however, the second strategy may be more difficult to come up with. Again, the goal of the lesson is to see all the multiplication strategies that the students will use solve the problem. As the students “work on it” we would circulate around the classroom asking questions about students’ strategies, guiding students through the process, and allowing mistakes to occur (these will be addressed during the “reflect and connect”). It is also important to note that not every group needs to be finished before moving on to the “reflect and connect”. Sometimes incomplete solutions provide good starting points for classroom discussion.

The third part of the math lesson is the most important part of the lesson but often the part that gets left out by teachers. It is also considered by many teachers as the most difficult part of the math lesson to facilitate. The “reflect and connect” is when the learning of the math concepts really occurs because the learning comes from the student work. This is the part of the lesson where students are given an opportunity to explain their strategies and solutions and where teachers are given an opportunity to focus on key strategies and concepts by guiding a math discussion through strategic questioning. This math discussion is very important because the conversation is less teacher centred and more student centred. Students ask each other questions about their solutions, make connections between their solutions, and defend their math solutions. The goal of the “reflect and connect” is to create the culture of a math community that allows students to take risks and where mistakes are considered to be opportunities for new learning. Ideally, this is what the “reflect and connect” should and could be like however, it takes time to get there. Students need time to learn how to ask appropriate questions, give constructive feedback, and receive constructive feedback. Teachers need time to learn how to ask probing and guiding questions and look for connections between student work.

There are a few ways to conduct a “reflect and connect”. The following article titled, Communication in the Mathematics Classroom explains three different approaches of math communication that can be implemented during a “reflect and connect”:  1) Gallery Walk, 2) Math Congress, and 3) Bansho.

My next post will focus on the gallery walk and a possible way to enhance math communication via Lino it using the following student solutions:

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Does efficiency = comprehension in mathematics?

Last year, I co-taught in a grade 6 classroom and gave the students the following problem as a part of an initial lesson to begin a unit on division,

“165 parents will be attending a family math night in your school gym. They will sit at tables in groups of 6. How many tables need to be set up?”

We asked the students to work collaboratively to solve the problem in more than one way and to show their math thinking. Below are two solutions from two different groups.

Which group has a better understanding of division? I showed these two solutions to many teachers and at first glance, many chose the group that solved the problem using long division. Why? The main reasons were efficiency and a percieved higher level of thinking. Many teachers viewed long division as a faster strategy and the latter, a time consuming low level strategy. However, is efficiency the ultimate goal in mathematics and does it mean that students that use very efficient standard algorithms to solve problems have a good comprehension of the mathematics?

The standard algorithm for division was invented long before the calculator and was viewed as the most efficient way to divide. However, I don’t think it was viewed as the most efficient way to teach division. Yet, the standard algorithm is frequently the first (and sometimes the only) strategy that is introduced to students when teaching operational sense. The standard algorithm, like many other division strategies can be a great strategy but only if it is understood. For many students, the standard algorithm for division is difficult to understand which forces them to rely on the memorization of the steps without any conceptual understanding of division. If given the opportunity, students are capable of coming up with their own invented strategies for division. More importantly, these are invented strategies that make sense to them which would lead to a better conceptual understanding. Below are the same solutions to the division problem mentioned above but I have included the students’ final statements that answer the question.

The group of students that used a longer and “less efficient” strategy had a better understanding of division than the group that solved using long division. It is crucial that students be given an opportunity to solve problems and on their own in as many ways as they can and allow them to share and dicuss their strategies with their classmates. By doing so, they may be more likely to make connections between their own strategy with others and move towards a conceptual understanding of a more efficient strategy. When you take a step back as a teacher and let the students loose with math and allow them struggle with guidance, you could be pleasantly surprised with their results and the discussion of their results is where you will find that the most teaching and learning occurs.