Mathematical Practices

I have been reading articles, blog, tweets, and other mediums on math education.  They all don’t necessarily align, but there are some common themes.  Essentially there are two camps of advocates in the field of math education.  The first believes the mastery of skills, often referred to as a “back to the basics” curriculum, should be the focus.  The second believes that the conceptual understanding of mathematics should be the focus of a strong math curriculum.  When one reads these statements independently, it is hard to disagree with either one.  Both of these ideals need to be addressed.  One can not have a strong conceptual understanding without having mastered some skills. Thus, one should consider Dr. Keith Devlin’s perspective on a focus of producing innovative mathematical thinkers.  The last four paragraphs of one of his blog entries really speaks to what all educators who are responsible for teaching students mathematics should think about. 

Traditionally, a mathematician had to acquire mastery of a wide range of mathematical techniques, and be able to work alone for long periods, deeply focused on a specific mathematical problem. Doubtless there will continue to be native-born Americans who are attracted to that activity, and our education system should support them. We definitely need such individuals. But our future lies elsewhere, in producing people who fall into my second category: what I propose to call the innovative mathematical thinkers.

This new breed of individuals (actually, it's not new, it's just that no one has shone a spotlight on them before) will need to have, above all else, a good conceptual understanding of mathematics, its power and scope -- when and how it can be applied -- and its limitations. They will also have to have a solid mastery of a few very basic mathematical skills, but they do not have to be stellar. A far more important requirement is that they can work well in teams, often cross-disciplinary teams, they can see things in new ways, they can quickly come up to speed on a new technique that seems to be required, and they are very good at adapting old methods to new situations.

Arguably the worst way to educate such individuals is to force them through a traditional mathematics curriculum, with students working alone through a linear sequence of discrete mathematical topics. To produce the twenty-first century, innovative mathematical thinker, you need project-based, group learning in which teams of students are presented with realistic problems that require mathematical and other kinds of thinking for their solution.

Of course, you still need a curriculum, in the sense of a list of topics that students need to master at some point or other. But it should be a short list, and should not be used as a list to proceed through topic by topic, as is current practice in the US. There needs to be a shift in STEM education from (topic-based) instruction (hashtags #traditional and #back-to-basics) to guided-discovery and project-based learning (#reform, #inquiry-based-learning). The primary focus needs to be not on what people know, but on how they think.

In the end, I think there needs to be a balance in the classroom.  Think about the essentials skills and knowledge students need to have and how does one seamlessly integrate that with conceptual understanding.  At the heart of the matter are the eight Mathematical Practices from the Common Core State Standards for mathematics: 


1. Reason abstractly and quantitatively.
2. Construct viable arguments and critique the reasoning of others.
3. Model with mathematics.
4. Use appropriate tools strategically.
5. Attend to precision.
6. Look for and make use of structure.
7. Look for and express regularity in repeated reasoning.

By considering these practices, what skills are essential, and rich mathematical problems grounded in real-world application when lesson planning, our future students will be innovative mathematical thinkers.