In the book Adding it Up: Helping Children Learn Mathematics, the authors discuss the idea of mathematical proficiency – in other words, what it means for anyone to learn mathematics successfully. Mathematical proficiency has five strands:
Intertwined Strands of Proficiency
- Conceptual understanding – comprehension of mathematical concepts, operations, and relations
- Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
- Strategic competence – ability to formulate, represent, and solve mathematical problems
- Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification
- Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy
That’s all well and good, you say, but what does this mean for me as a teacher?
First, what this means is that all of these pieces need to be in place in order for a learner to be successful in mathematics. Second, to me, the most important thing to think about within these five strands is the last one – productive disposition.
Let’s think about that strand for a minute and break it down into its component parts.
1. “Habitual inclination to see mathematics as sensible, useful, and worthwhile…” This is one of the harder parts of this strand. This is the “why the heck am I learning this?” piece to the puzzle. In my post called “When Will I Ever Use This?” , I put up some links from NCTM that give some answers to that question. If students don’t see mathematics as something that makes sense to them, something they can actually use in their daily lives, and something that is worthwhile for them to know, then they will be much less willing to put in the time and effort to learn it. Which brings us to our next point…
2. “…coupled with a belief in diligence…” Diligence. This word is listed on Dictionary.com as “constant and earnest effort to accomplish what is undertaken; persistent exertion of body or mind.” This is that persistence to stay with a problem when it gets difficult, a willingness to get “stuck” and figure a way out of it. As teachers, our first inclination is often to help our students get “unstuck”, and that is probably a huge disservice to them. Students need to feel comfortable with some level of struggle when it comes to problem solving, and we need to feel comfortable with letting them struggle. Without that, the diligence necessary to solve problems and be successful will not be developed. Which now brings us to the last part…
3. “…and one’s own efficacy.” The belief in one’s own ability to do the math and be successful at it. This may be the hardest part, especially with our students that struggle in math. I can’t begin to tell you how many students have said to me over the years, “I can’t do this. I’m not good at math.” Changing that belief alone is both difficult and incredibly important.
If nothing else, if we as teachers can focus on building this strand of mathematical proficiency in our students, we will be helping them on the road to becoming successful, lifelong math learners. What is better than that?