Category: Teaching

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/

 

The Learning Process

Last week, I was invited by a group of EAB students to participate in a formal debate over whether or not cell phones should be allowed in classrooms. While listening to the students’ arguments for and against the proposition, I reflected, as I often do, on the process of learning and what constitutes effective instruction. While the issues associated with the use of technology in education are complex, it was the process the students were embracing that was of most interest. Preparing a rational and researched defense of their argument that will pass the scrutiny of their peers is no easy feat. The fact that there is no clearly defined response to the cell phone proposition is emblematic of the issues we often face as adults; meaningful and important problems are rarely categorized as simple, binary black and white options, but rather reside in the awkward grey areas.

The critical thinking element of the learning process is arguably one of the most important skills students can develop, especially when navigating the challenges associated with establishing informed opinions about issues that fall into the thorny grey areas. It was, therefore, reassuring to hear an EAB alumna, speaking at today’s high school assembly about her transition to university, highlight this very point through the following statement: “EAB prepared me for the expectations of an UnB teacher: proactivity and critical thinking.”

Critical thinking is a focus area that is prioritized at EAB. While it is often easier to deal with categorical problems that can easily be pigeonholed, a more essential learning process involves a commitment to thinking deeply about complex issues and forming an educated and defensible opinion. However, this deeper form of learning is only achieved through a higher degree of commitment, perseverance, patience, and deferred gratification.

The question of deferred gratification was examined in a Stanford University experiment conducted more than 40 years ago. Young children were left alone in a room with one marshmallow sitting on plate in front of them. The children were told that if they did not eat the marshmallow while the adult was not present, then the child would receive a second marshmallow when the adult returned. Of the 600 children who participated in the experiment, approximately one third were able to delay gratification by not eating the marshmallow, who were then rewarded with a second marshmallow. While it appeared to be a simple experiment, the consequences associated with the ability to delay gratification were significant. Over the next 30 years, the researchers followed the students as they matured into adulthood and discovered a very strong correlation such that the students who waited had ended up with better grades and higher SAT scores, were healthier and in better physical shape, enjoyed more successful relationships, and achieved greater professional success. The implications related to the ability to delay gratification have had a significant influence on education and learning. [More information about the “marshmallow experiment” can be found via the following links: Businessweek, New Yorker, TED]

Returning to the cell phone debate, it was intriguing to hear one of the students comment on her inability to control a tendency to check her cell phone every few minutes for messages, which the group concurred represented a distraction that everyone in the room also found to be a challenge. The concept of delayed gratification was a prominent feature with the use of cell phones in this class, as it also seems to be with many of us.

The cell phone debate ended with both student groups presenting compelling arguments for and against the use of cell phones in the classroom. Beyond the debate, however, it was the instructional practice that I found to be the most intriguing aspect of the class. Students were developing the skills to delay gratification through the process of learning how to think critically about a complex issue, rather than blindly accepting a simple “yes” or “no” response, and publicly articulating their refined thoughts in a passionate and articulate manner. It was learning at its best.

Featured image: cc licensed ( BY NC ND 2.0 ) flickr photo by Octavio Solórzano: http://www.flickr.com/photos/octaviosolorzano/5010443243/