So, what is the difference between struggling in mathematics and productive struggle?
The shifts in the common core require that the students do the majority of the thinking and the work - that they are the ones productively struggling with the problem solving, questioning, thinking, and explaining.
It really comes down to conceptual understanding. Conceptual understanding is how we make meaning of what we are learning - it's the mental models and images that we create to help us understanding what we are doing.
Prior to common core, learning math was largely about about procedures. It was up to us to make our own meaning and find ways to understand the concepts behind the procedures if we did not immediately "get it." Twenty years ago, we didn't provide students with manipulatives to make meaning - we gave them procedures and expected them to figure it out. As a result, many students continued to feel like they didn't "get" math.
What is the lesson here for teachers of common core mathematics?
Think about how long it took you to make meaning of mathematical concepts. Now think about how long you give kids to really make meaning of mathematical concepts. If you give students two days with manipulatives, have you really just substituted different procedures and students still don't have enough time to make meaning?
Here's an example. (https://www.engageny.org/sites/default/files/resource/attachments/g4-m1-full-module.pdf)
Lucy plays an online math game. She scored 100,000 more points on Level 2 than on Level 3. If she
scored 349,867 points on Level 2, what was her score on Level 3? Use pictures, words, or numbers to
explain your thinking.
Struggling would mean that you would give this problem to students and let them try to figure it out., Productive struggle means that you woud read through this problem with your students, giving them an opportunity to discuss what strategies they might use to solve the problem and what information they see as critical to understanding what the problem is asking.
In order to do teach in this way, you have to plan differently. First of all, you need to do the math. You have to understand what students are going to do when they first approach this problem. You have to plan for an ideal student response and also plan for misconceptions that students are going to have. If you don't do the math, you are merely teaching to procedures and not to conceptual understanding.
Secondly, you need to plan for students to productively struggle. Again, that means that they have to have some information to use before they begin to problem solve. In this problem, you might talk about the numbers that are important in the problem, and you might even brainstorm strategies that students could use to solve the problem, but in order to ensure that students would be doing the thinking, you would not set up a tape diagram for them. Productive struggle means that students have to have an idea about the work they are going to engage in and they would be able to explain the reasons that they chose the approach that they did. Do you provide students with manipulatives or do you have them get their own? Do you encourage them to use their white board? How do you help guide students toward potential strategies that will help them make meaning without telling them how to think?
Students also need time to do the math. Too often, this time is cut significantly short because teachers have not taught students to productively struggle. You have to have an idea of what you expect to see while kids are working in order to plan effective questions, redirections, or next steps. Your job during this phase is really to facilitate thinking. Students have to have a starting point and they have to be able to determine which strategies are most efficient. As you gather information from students, you have to be able to really listen to them in order to understand their thinking and plan for next steps in moving them forward. Students should be encouraged to work together if it helps to facilitate their thinking - think about what you would do if you had to solve a problem. Chances are, you would at least talk it through with someone before you decided what steps made the msot sense to you.
The real impact comes in the student discourse - where students explain their thinking whether they have a right or a wrong answer. The confident practitioner will embrace the questions that help move other students toward understanding during this part and will not get nervous about students modeling incorrect answers or divergent thinking. If student discourse sounds just like students filling in the blanks in your classroom, it is not getting to real understanding and it is certainly not moving students toward proficiency. Students should be able to think through their understanding and simply snot encouraged to find a friend if it takes them longer than 15 seconds to answer or explain their thinking. Classrooms that really understand that every child makes meaning about mathematics in their own way are classrooms that will celebrate divergent thinking, will build on other students' responses, and will ultimately see the greatest growth in mathematical understanding.
I have come to look at mathematics instruction so differently than when I was a student. Our children are ready, willing, and absolutely able to master the mathematical concepts and skills that are a part of our common core curriculum. Teachers must look at teaching as more than simply covering the lesson and must do the math in order to be able to facilitate conceptual understanding for students and build on their questions in order to secure foundational skills. Our students must have more time with manipulatives and making meaning in order to build conceptual understanding.
Making math matter in classrooms requires teachers that believe in the power of making meaning, building conceptual understanding with tools and manipulatives, and the ability of our children to conceptualize how different parts of mathematics align and connect. Adults must engage in productive struggle with that kind of planning in order to create clasrooms where students can productively struggle with concepts in mathematics.
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