So many students grow up with a fear (or hatred) of mathematics. Mathematical thinking makes up a huge piece of the Common Core standards and this thinking is one way we can help students develop stronger problem solving skills. aimed to solve this by connecting different methods of thinking to the teaching of problem-solving.
The big message is that students must be able to think about different methods for solving mathematical problems that arise in their world. To non-teachers it might look complicated, but these strategies help students visualize mathematical thinking and start building their mental math skills.
Regardless of your feelings about the CCSS, we need students who are not afraid to take risks and try to problem solve in math. Wondering what successful math students look like? Here are a few things we know about these students:
Mathematical thinkers persevere when solving math problems.
Growth mindset is all the rage right now, and for good reason. We know that strong mathematical thinkers persevere, even with challenging problems. They just stick with it and keep trying instead of getting discouraged.
To help build this in our students we need to teach them how to analyze word problems. Students must be able to break down problems and figure out what they are being asked. They need to be able to discard extra information and focus in on just the facts. This has to happen before they even begin to solve.
Praise and encouragement are also essential components to supporting and developing perseverance.
Mathematical thinkers reason & think quantitatively.
These students see the big picture. They use visualization to help them understand the problem they are solving and determine whether their solutions are reasonable.
Drawing pictures can be a first step in this process because it helps build the visual model before the abstract thinking can happen. However, over time problems will become more complex. The hope is to build the strategies that help students determine whether their response makes sense, or is reasonable, without needing a picture every time.
Mathematical thinkers verbalize their thinking.
Mathematical thinkers must be able to talk through how they solved a problem and examine how others solved the problem and critique their reasoning. This is why error analysis is such a huge thing right now.
It used to be enough to memorize a formula and be able to apply it, but to be truly mathematical thinkers, students must be able to carry on a conversation about the strategies applied to solving problems.
Mathematical thinkers model solutions in multiple ways.
From visuals to equations, students must be able to show multiple ways to solve. This allows them to check their work and gives them a number of strategies to fall back on if they get stuck.
This mathematical thinking needs to extend into real-world situations. For example, as students grow, they will be able to use math to plan the logistics for a classroom party…
There are 20 students in the class and everyone wants three cookies. How many cookies do we need?
But Sam can’t eat cookies. Now how many do we need, and what can we provide for Sam?
The need for this type of mathematical thinking continues with increasing difficulty all the way through high school.
They know and use math tools & focus on precision.
Tools can be mechanical tools such as protractors and rulers, or they may be technological tools like calculators and computer applications. The availability of math tools is limitless these days considering the smart phones and other tech that we carry around in our pockets.
Strong mathematical thinkers communicate clearly and precisely. In addition to being able to explain their thinking, students also need to be able to concisely state why they used the operations and tools they did to solve the problem.
Mathematical thinkers learn to see patterns in mathematics.
Young students should be able to see the communicative property of addition, and the relationship between addition and subtraction. They can count the numbers of sides to a shape.
As students grow older, they will see the how the distributive property can be applied to break down more difficult multiplication equations, and they will see how drawing an auxiliary line in a geometric shape can help them solve geometry problems.
By understanding that math follows predictable patterns, they can begin to focus on the deeper levels of problem-solving.
Why this matters…
We are building the foundation of mathematical thinking in our students. The cornerstone of mathematical thinking is problem-solving and reasoning. However, without purposeful, repeated practice many students do not have the opportunity to develop into strong
However, without purposeful, repeated practice many students do not have the opportunity to develop into strong, capable math students. Looking for tools to teach
Want to learn more about how the brain research factors into problem-solving?
Just need more resources for teaching problem-solving?
Daily Problem Solving is a great way to dig deep and help students become strong mathematical thinkers. You can learn more about Daily Problem Solving here.
I hope this article helped spark some ideas for building mathematical thinking. If you’d like more ideas, be sure to follow me on: