Learning Mathematics

Erma Anderson’s professional development work with our faculty this week left me with three reflections about the learning of mathematics and the Common Core.

Number Sense

Let us start with a quick math quiz.  Quickly answer the following question relying only on your sense of numbers (i.e. do not calculate the exact value):

If you are “one billion seconds “old, then you have lived for approximately 31 years.  How long have you lived if you are “one million seconds” old?

Many people find the answer to be shocking, highlighting some of the challenges we face associated with number sense.  When we speak of budgets, populations, and exponential growth in the billions, do we really have a sense of what the numbers mean?

To answer the question above, “one million seconds” equates to 11 days, as compared to a billion seconds equating to 31 years.  Extending this example, “one trillion seconds” is approximately equal to 32,000 years!  These numbers hopefully put the concept of a trillion dollar debt into a different perspective.

Students usually do not develop a strong sense of numbers by blindly following algorithmic procedures or memorizing formulas without developing a deeper sense of the numbers they work with during their studies.  While algorithmic procedures the memorization of formulas have their place in mathematics, it is only through a deeper conceptual knowledge of mathematics that an enduring understanding of mathematics will be achieved. If more people learned mathematics this way, perhaps there would be less of a negative stigma highlighted by adults and students and their learning of mathematics.

Ways of Knowing

To achieve enduring understandings, mathematics must be taught in a manner that requires students to interact with concepts in a variety of manners, including visual, contextual, algorithmic, procedural, and theoretical, among others. The common theme, however, is that the learning must ensure a strong conceptual understanding.  By way of example, if you ask adults to state the quadratic formula, few of us will be able to do so correctly.  However, if these same adults understood where the quadratic formula comes from and how to derive it from first principles, then the likelihood of being able to correctly state the quadratic formula is very high.  We must move from memorization, as the focus, to strong conceptual understandings.

Teaching of Mathematics

During a conversation with Erma Anderson, we discovered that we both experienced a similar “learning” moment at the start of our respective teaching careers. We both majored in mathematics at university and graduated as mathematicians before choosing a career in education.  While teaching our first calculus classes, we, like so many other teachers, came to a stark realization. While we could always “do” mathematics very well, our deeper conceptual understanding of the subject was questioned, for the first time in our careers, through the challenge of teaching the conceptual understanding of calculus.  Whether teachers admit it or not, most educators go through a similar experience as it is one thing to be able to “do” mathematics but quite another to be able to explain your understanding of these same concepts. Our responsibility as teachers is to continuously seek ways to better understand our subjects while also finding ways to effectively work with students so that they develop their own deep and meaningful conceptual understandings.

This is an exciting time at EAB as our teachers are dedicating a significant amount of time and energy towards the ongoing development of a strong mathematics curriculum and, in parallel, the ongoing development of our collective teaching practices.

Featured image: cc licensed ( BY NC SA 2.0 ) flickr photo by Tom Magliery: http://www.flickr.com/photos/mag3737/6266477735/