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teach math conceptually

05 Sep The Art of “Doing Math”: Conceptual Instruction

By Dr. Kristy L. Beam, Faculty at University of North Georgia, Former Principal at American School of Valencia, Administrator and Independent Consultant

When I was young, school came easy for me, with very little effort. That changed when I got to high school math. Things started well enough in Algebra, but I barely survived Geometry. Algebra II took a couple of tries to squeak by and Trigonometry might have well have been a foreign language! When I got to college, I had developed a very real “math phobia”.

I was convinced I could not “do math”. Horrified at the prospect of taking a math class, I put them off as long as possible. Finally, with the confidence of a strong, honor student, I registered for College Algebra. I remember, the positive self-talk, convincing myself that I was intelligent and that I could do this. I sat on the front row, class after class. I watched my class dwindle from 30 or 40 students down to less than 20 students, I went home and each night I would practice problem after problem, and I would usually get a few incorrect, and then, the answer would “appear”.

The “Magic” of Math

The best way to describe my experience with math is that it is like a stereogram. If you don’t remember those, they were all the rage in the early 1990’s, about the same time I was struggling to regain my mathematical footing. Stereograms are computer generated and seem to be abstract art. However, if you focus your eyes in a certain way, you are able to see a 3-D image. It is difficult—it can take several tries and can be frustrating, but suddenly, the image appears and it is amazing! This is how “doing math” is for me.

Slowly, I did regain my confidence and I realized that I could very successfully “do math”. I went on to be a top math student in all my degree programs, right through to doctoral level Analysis of Variance, which I took as an independent study course.

“Doing Math”: Active, Engaged Learning

I use the term “doing math” for two reasons:

  1. I have had many students tell me that they “can’t do math”.
  2. I have come to realize that math is not a passive activity. You cannot “sit and get”. Like exercise or sports, you must actively engage in it, in order to become proficient.

I have thought a lot about my experience with math. I think it made me a better teacher and empathic to students that struggled with math.

My experience as an administrator over the last few years has made me think even more deeply about the seemingly global, ongoing “math crisis”. I have been an administrator in two very different schools: one, an economically disadvantaged public school in the United States, and the other, an affluent, international school in Europe. However, data from both schools told a similar story. One piece of data that frequently emerged, revealed students that struggle in the math classroom, many times, had very low reading, not math, scores.

Language-Focused, Conceptual Instruction

A little research showed that, while I had not made the discovery of the century, I was on the right track and quite a bit of recent research surrounding the need to add language-focused conceptual instruction to the math classroom. In a wonderful example, Concepcion Molina, author of The Problem with Math is English, explains that imprecise language that is widespread in math education is even more problematic for English Language Learners.

In this case, his understanding of the question allows him to find the correct answer…with no understanding of the concept. If we try another example, that becomes evident…

Molina (2013) explains that practices in math instruction that limit conceptual understanding while promoting procedural understanding include:

  1. Inattention to language and symbolism (such as the example above)
  2. Teachers’ tendency to use careless vocabulary
  3. The use of shortcuts
  4. Dominant use of naked numbers

Since “my math awakening” and throughout my journey as an educator, I have become increasing more fascinated by the global concerns about mathematics that continue to plague the world of education. I have rarely entered a school that did not report some concern surrounding math instruction.

For years, best practices require that we teach conceptual understanding over procedural fluency, but classroom after classroom still focuses on algorithms and procedural methodology over visualization and deep understanding. Conceptual understanding is critical for all students, and using precise, mathematical language is an important piece of effective instruction. In her recent book, Mathematical Mindsets, Dr. Jo Boaler provides a powerful, data driven argument that there must be a dramatic change in the overall approach to teaching mathematics if we are to end this “misery” and allow students to “realize the joys of learning and understanding math”.

Resources:
  1. Boaler, J. (2016). Mathematical Mindsets, San Francisco, CA: Jossey-Bass.
  2. Molina, C. (2012). The Problem with Math is English, San Francisco, CA: Jossey-Bass.
  3. Molina, C. (2013). The Problem with Math is English Webinar
  4. O’Connell, S. (2005). Now I Get It, Portsmouth, NH: Heinemann.

Kristy Beam has a diverse background in education as a teacher and administrator. After 15 years in a large public school system in the US, she spent the last four as principal of a private, international school in Europe. Currently, she is working in curriculum development and has taken on a greater role with the University of North Georgia, where she has taught in the graduate education program for the last four years.

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