The last article, I wrote about the foundation behind the Common Core State Standards (CCSS). In this article, we begin to take the standards apart in order to understand them.
Overarching the whole of the Common Core Framework, there are eight Math Practice Standards. Each Standard represents a way of thinking about mathematics education as a whole, and affect every topic and skill addressed in Mathematics Education. They are:
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Since these Practice Standards lay the foundation of the Common Core, we will look at them individually.
Make sense of problems and persevere in solving them
This phrase seems fairly self-explanatory, but we have to start somewhere! We must start with this standard because it addresses two extremely important concepts: to “make sense” and to “persevere”.
To “make sense of problems” is a key aspect of the Mathnasium Method and the first step to learning. Understanding what a problem is asking is how we are able to select the appropriate technique to solve the problem. If we don’t understand what the question is asking, selecting the correct method is just blind luck. Key questions that we ask our students to help them make sense of a problem are “What do we know here?” and “What are we looking for?” If they’re struggling, scratch paper is your best friend for making sense of the problems. Use it to list the information given, write out the formula they will try, or draw a picture if they are a visual learner. Especially when solving word problems, putting the information in a list will often trigger a student’s prior knowledge of how to solve it. Some of the different ways that Mathnasium utilizes different learning styles can be found here.
Students need to learn how to break down problems into information and questions - what are we given and what are we looking for? This is the true 21st Century skill that is needed in Math, and in all of the STEM driven fields. Dr. Keith Devlin authored an article that highlights some of the reasons that Number Sense is the true skill that students need today. This concentration on making sense of the numbers is what has been called out by many critics as "fuzzy math" and not the precise methodical mathematics that we have been used to. But it is activating critical thinking skills, the one area in which humans will always have the upper hand on machines. Having a true sense of what is happening to the numbers gives students multiple ways to think about a problem, and opens the doors to future possibilities.
To “persevere in solving them” is the second key concept in this standard. One of my favorite phrases to highlight what this means is “I’ve done nothing and I’m all out ideas!" When students don't focus on perservering, this is the trap that they fall into. We need to encourage students to try methods and follow through on what they know. Too often, students have the impression that if they don’t know the answer right away, they aren’t doing it right. We need to encourage them to think through the steps and keep trying. According to Merriam-Webster, to persevere means "to persist in a state, enterprise, or undertaking in spite of counterinfluences, opposition, or discouragement". Mathnasium focuses on providing opportunities for students to persevere by presenting them with challenging problems and making sure that they have the support and time needed to make sense of them. Students need to be able to commit to solving their problems without having their hands held continually. Just like when learning to ride a bike, we need to give support, encouragement, and be there if they fall. But we also need to take a step back and let them try to succeed on their own.
We need to give the students adequate wait time to process the problem. By jumping in with hints or formulas, we interrupt them as they are making the connection for themselves. Even if you find silence awkward, you can still give space for them to connect the solution. The best instructors will find themselves repeating the question, “What are you thinking?” as a way to prod the student’s thinking. We need to recognize that math can be hard sometimes, and that’s OK. This wait time when solving problems allows students to process the situation, select the method, and compute the solution. Students need that space and time to be able to develop fluency with the methods that they are using. Teachers know the value of wait time, but often do not have the freedom to give as much as the students need. At home and at Mathnasium, we have that freedom to wait as our students make connections. Then we can practice with those connections, to cultivate the fluency needed for the classroom. Like Slash, the guitarist from Guns N Roses, said when asked how others can learn to play as fast and well as he does - "You practice it slowly first. When it's right, speed it up."
