I'm going to ask you to take a deep breath and consider your mindset before reading this blog post. I want to ask you to consider why you learn math. What's your goal? Why should you learn math at all? Remember when your teacher told you that you couldn't walk around with a calculator when you are a grown up and that's why you needed to do all the paper and pencil work? Well..that's not exactly true anymore. So, what's the point for learning all this stuff when I can just Google it or use the calculator on my phone?
That question is almost like asking why we need to read a variety of books. I don't plan on being a detective one day, so why should I read Sherlock Holmes? It's about a much bigger picture than just going through the motions.
It's about creating minds that think and problem solve- minds that are curious and reflective. It's about so much more than the content.
As a math teacher, I have to consider my goal for each lesson. I have an objective for my lesson. For example, students will learn how to divide a 3-digit number by a 1-digit number. I know how students will achieve that goal: They will use the standard algorithm. But now, I need to consider the next step. How will I ensure my students understand why the mathematics works?
This is often the step that gets missed. We assume that because students can recall an answer, they understand. We assume that they can apply that thinking to new situations, which again is the goal. Sometimes we even go as far to say those children that do this quickly are gifted or advanced.
According to Jo Boaler in Mathematical Mindsets, there is often this myth that students are gifted or advanced in mathematics, because they are fast with math facts. I'm sure you can picture these students. The student who always finishes early. The student who always knows the answer. The student who may seem bored in class. Without the challenge of the logic involved, these tasks truly can seem below these students' levels, because they are simply replicating a process.
Richard Skemp had a lot of research that showed the power of learning the WHY behind the mathematical process. Skemp introduced the concept of "instrumental understanding" and "relational understanding" in mathematics. Instrumental understanding is when a student can perform mathematical operations or solve problems without necessarily understanding the underlying concepts. This is where you see students who can replicate an answer but not explain why the process worked.
On the other hand, relational understanding is a deeper understanding of the mathematics, where students not only know how to perform operations but also understand the reasons behind them. Skemp argued that explanation, especially verbal and visual explanations, are crucial for students to transition from instrumental to relational understanding. When students explain their mathematical thinking, they are forced to articulate their thought processes, and this process helps them internalize and make sense of the mathematics. It's not just about being able to give an answer but to verbally explain why through the use of visuals. This is where problem-solving and logical thinking truly take place.
But....what about that end goal? How are we teaching them to problem solve if they are just spewing answers with no logical explanation? Let's go back to our example from before:
Objective: Students will learn how to divide a 3-digit number by a 1-digit number.
How: They will use the standard algorithm by exploring its creation through hands-on manipulatives, area models and partial quotients.
Just because a student can tell an answer, doesn't mean they actually understand. Furthermore, an answer doesn't justify why we are learning in the first place. I want my students to think about how things work and are connected in the world around them. I want them to make connections and extend their thinking. They will only do this if they take a moment to change their mindset about what we do in math class and not just about getting an answer. It may even mean that we adjust our mindsets too.
What if we considered math topics as an opportunity to go deeper into the content? What if a student finished a problem and then looked at the problem in another way? What if instead of moving on to something else, the student took the initiative to truly understand why the mathematics works? Sounds awesome, right?
We can do this by creating a climate of growth in our classroom. I've heard teachers say, "You're never done. You have just begun." That totally sums up this idea. When students get an answer, challenge them to take the problem further.
Here are 3 simple strategies:
Ask students to show you another way:
Challenge students to summarize their thinking through writing and visuals.
Encourage students to write their own problems and solve them.
What's so cool about these three ideas is that after using them for awhile, they become second nature to students. Soon, all the students know that when they have an answer they are going to be asked to do it another way or write their own problem. Students begin to take the initiative to do it alone and are now taking tasks further.
As a teacher, it's an awesome feeling. I know I'm preparing them for real life by setting them up with the SKILLS necessary for jobs that haven't even been invented yet. My students know that they are never really done until we are ready to move on. They are extending their thinking beyond the answer, and everyone is getting a chance to try the "extra" activities.
Want to try implementing this in your classroom? Grab these three posters and give it a go! These are great ways to push that opening task in your lesson and encourage students to think beyond just an answer. Leave a comment and let me know how your students enjoy working deeper into a task!
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