Learning how to ride a bicycle requires a bicycle, some basic knowledge of how to operate it, a safe place, and the willingness to try. On the first attempt, you lose balance or fall over. You know when you’ve fallen over. You don’t need someone to tell you that you lost your balance. Being on the ground prompts you to wonder why you are down there and not on the bike. If you are not too embarrassed or frustrated and still have some hope that you will be able to stay upright on the next attempt, you pick yourself and the bike back up and try again.
This is the only way to learn to ride a bicycle. No amount of listening to famous cyclists, note taking, reading, or watching videos of the Tour de France can replace the act of getting on the bike and trying to ride it. It is only through active learning that complex skills, behaviors, and values are gained.
Does this seem to be what is happening when learning math? What is the bicycle? How do learners know they’ve fallen off or are losing balance and are therefore in need of reflection before trying again? How do they know what to reflect on? How do they manage their frustration or embarrassment and quickly go back for another attempt?
The answers to those questions are what is missing from learning math. Math students need to be able to attempt math in a way that makes building their expertise in it as efficient of a process as riding a bicycle. How can my students go between attempting math, realizing they are doing it incorrectly (in the beginning or middle of a problem – not just at the end – and without someone else telling them), reflecting on what they might have done wrong in a meaningful way, and then trying again? All with no significant delays and not too much frustration so that they stop trying.
This efficient learning process does not exist in math. As a result, math gets reduced to rules, steps, and formulas students try to remember and follow instead of understand. The problems educators tend to give students consist of recognizable cues that students react to in the way they remember being told how to. Exposure to word problems is limited. Failure, even if it is productive, is uncomfortable and avoided at all times. Everyone goes home thinking math was learned.
Yet, if we look at the Common Core State Standards for Mathematics, there is a section titled Standards for Mathematical Practice which lists areas of expertise math teachers should strive to develop in their students. This section comes first in the document before the mathematical content standards. The Mathematical Practices listed in the Standard 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.
What this means is that students should engage with mathematical content by using the Practices listed above. Teachers should facilitate this by connecting the mathematical practices to the content. Importance is placed not only on gaining expertise in the content but gaining expertise in the Practices, as well.
Essentially, the Standards are calling for an end to long daily lectures, repetitious worksheets, and rote learning. Students are intended to interact with math in a way that does not involve remembering steps and rules and reacting to only the cues they have been told to react to. The Standards call for a much more complex way to interact with math. At a high level, this is a fantastic change in the lens through which learning math is viewed. However, these Practices are highly complex and are difficult to learn and therefore difficult to teach.
Building expertise in the skills and abilities listed in the Standards requires active learning. Therefore, someone needs to invent the bicycle to ride for learning math. The student must be enticed to try and ride it. The teacher will only ever be a facilitator in this environment – knowing when to remove the training wheels, hold the back of the seat, and when to let it go. The teachers, administrators, schools, and communities must provide whatever is needed to keep the student in the learning process by teaching young people how to handle the shame and frustration of numerous scraped knees and bruised egos.
When learning math today, students are not limited by their ability. They are severely limited by the inefficiency of the process of learning math. It is under the Active section on this website where I intend to create ways to make this process more efficient.