I've spent the past two weeks discussing the three different types of math lessons. (You can check out the introduction here and the post on Conceptual Lessons here.) Today, I want to talk about procedural lessons and what they truly look like.
Procedural lessons are probably what most of us remember about math class. Those specific lessons where you learned how to perform an operation were the ones that you followed step-by-step. However, I want us to look at a few components to making procedural lessons go off without a hitch!
First, procedural lessons must come after conceptual understanding. Students have to have a firm foundation to be able to apply the procedural steps. This type of learning may have happened earlier in the chapter or it may have even happened in a previous grade level. If it happened in another grade level, I would firmly recommend doing some kind of review that allows students to explore the concept conceptually for about 5-10 minutes before diving headfirst!
Second, procedural lessons are still going to require the use of manipulatives to help students process what's happening. Remember the research of Jerome Bruner that says students need to be working in two representations at a time while building connections between the two. That means while students are working in the procedure (abstract), they need to make connections to the concrete or pictorial representations. Maybe they draw a model of their thinking or maybe they just use the manipulatives. Either way, continually ask students to explain how the two methods are similar.
Finally, consider how students will engage in the procedure in a meaningful way. Present the task with a rich problem that is based around a real-life scenario. I have seen this best used when students are performing the algorithm and can reference the watermelons or beads that are mentioned in the problem. It's a tangible object for them to count making it real as opposed to some random number presented. The type of problem should be thoughtfully considered and should engage students with interest and relevance.
Once students have understood the process of what's being asked, now the fun begins. Provide students with manipulatives needed and walk away. Yes, walk away. Walk around the room and take observations. Allow students to work together to decide what they will do next. (If doing a homeschool lesson, sit back and allow your student to take the lead. Let him/her tell you what he/she wants to do. Go with it and allow your student to walk through the process.) They already have the conceptual understanding in place. In this situation, they already know the concepts of place value and what it means to add. Your goal is to see how they apply those concepts. Don't forget to challenge them to think of another way when finished. For more information on that, head over to this blog post. For students who may struggle to get started, provide them with a few prompts:
What do you know about the problem?
What would you want to do first?
What are you trying to solve in the problem?
How can you use what you know about _______ to help you solve the problem?
Now, ask students to share their process but with a few things in mind. From looking at your textbook or standards, you know what the expected outcome is for the lesson. In this particular example, I'm wanting students to add using place value with the vertical algorithm. However, this is not the only way. I may have students who decide to use place value chips or to add mentally. I have already seen the types of methods while walking around the classroom. I know what to expect. Some teachers choose to select and sequence moving from basic understanding to more complex, while other teachers just ask students to share their thinking. These thoughts are then recorded on an anchor chart for students to consider and process. (If doing a homeschool lesson, I would challenge my student to share his or her methods and record them on a piece of paper to be referenced throughout the lesson.)
Now ask students to critique each method and decide which methods work. Here are some prompts to consider:
Does this one make sense?
If not, can _____ clarify?
Is there anything you want to change or adjust? (Perfect for if there is a thinking error.)
Which one would be more efficient?
Remember, your goal is to get them to the expected outcome. If no one had done the standard algorithm here, I would then tell them what my friend did to solve the problem. Not what I di, but what a friend of mine did when we were discussing this over dinner. (Because isn't that all teachers do. Ha!) This allows the students to again critique the thinking instead of waiting on me to show them how the adult does it.
Finally, students are ready to practice. I like to have them try two methods when solving the problem. The method that I am aiming for them to master by the end of the lesson/unit is always one that they must do during practice. They can use their own method to check the answer. The beauty in having the chart with methods on it is that students can refer back when working. (I also like to have students record these in a math notebook for their own processing and notes.) It's an additional bonus if student names are on the chart, because I can refer any questions to that particular student. After all, it was his/her's method.
As you can see, these lessons require a lot of steps and processing skills. These lessons tend to go slower. The main problem can take anywhere from 8-20 minutes depending on how involved the students are. Remember that the goal is to go deep into the concept and do fewer problems with understanding. Students will need to continue to practice, but they don't need to do 30 problems to master it.
Next week, we will discuss the Application-Based lessons that wrap concepts and procedures into a nice little package. In the meantime, let me know in the comments your own experiences with procedure lessons. What have you found that helps keep it all organized? How do you push your students to think about it in a different way?