Stage 4: Disciplinary Thinking

Is mathematical thinking natural?

Many math teachers struggle to teach disciplinary thinking in their classrooms. The list of procedures that students must “master” by the end of each math course seems overwhelmingly long. There is no time for students to “play” with math, explore its beauty, or think deeply about any one idea in the course when a new topic must be covered every day.

The result? Teachers frantically teach mathematical procedures — factoring, FOIL, finding the area of a triangle — without a chance to make them meaningful. Students grow frustrated, and many develop the attitude that math is just not for them. This presents teachers with another giant problem: how can you teach mathematical thinking, which involves persevering to reason through complex problems, when students are afraid of the discipline?

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Interestingly enough, research shows that even though students may be afraid of math in the classroom, thinking mathematically comes quite naturally in many other aspects of life. Take this excerpt from How Students Learn: Mathematics in the Classroom as an example:

Both children and adults engage in mathematical problem solving, developing untrained strategies to do so successfully when formal experiences are not provided. For example, it was found that Brazilian street children could perform mathematics when making sales in the street, but were unable to answer similar problems presented in a school context. Likewise, a study of housewives in California uncovered an ability to solve mathematical problems when comparison shopping, even when the women could not solve some problems presented abstractly in a classroom that required the same mathematics…And men who successfully handicapped horse races could not apply the same skill to securities in the stock market.

It seems that some mathematical thinking and quantitative reasoning come naturally as we solve real problems in the real world. So what’s happening in math class? Why is mathematical thinking so hard to teach if people can do it outside of school?

According to the National Academy of the Sciences, “mathematics instruction often overrides students’ reasoning processes, replacing them with a set of rules and procedures that disconnects problem solving from meaning making.” Somehow, we are working against students’ natural abilities to do math and reason through difficult problems!

So, as we seek to overcome the challenges of teaching disciplinary thinking in math, how can we tap into students’ natural mathematical reasoning instead of “overriding” it?

Let us know what you think!

4 thoughts on “Is mathematical thinking natural?”

  1. A lot of what we do is work to explore the mathematical concepts thoroughly. Our students spend at least 1/2 of the time in between assessments exploring and playing with a concept. the more concrete aspect of problem solving and procedural thinking is introduced after the concept is explored. Often, students are able to think about this procedural thinking and they propose how to solve using the concept. This is a time-consuming way of teaching, there’s not “blowing through material” because we have to teach to a test. It requires faith in the system and faith in what our students are capable of understanding.

  2. It is very true that most math teaching works against deep learning and understanding. We need to support the learning process and support students’ own thinking in order to create the experience of being competent. Only confident students think for themselves. And, sadly enough, when we follow the thinking of someone else we will never create the same competence as we do while thinking things through with our own understanding. The longer this cycle keeps spinning, the harder it is to break. Self-determination theory provides evidence about how intrinsic motivation improves the learning quality, and my 3C tools show teachers how to teach with intrinsic motivation in mind.

  3. Or just teach your kids math skills in the real world instead of a classroom; i.e. homeschool / unschool them, then all math will be meaningful.

What do you think?

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