Tag Archives: teaching



NOTE:  The following is my answer to Question 3.5 in Jo Boaler’s online class: How To Learn Math.  “Which ideas from this paragraph struck you? Write 1-2 paragraphs on why you like it. What does it mean for learners of math?”  The original paragraph by Peter Sims can be found here Daring To Stumble on the Road to Recovery (The New York Times, August 6, 2011, paragraph 16.)

The paragraph written by Peter Sims (NY Times) is both succinct and tremendously thought-provoking.  He seems to have captured, in a very brief space, the key elements supporting intellectual creativity.  I think what most resonates with me, however, is the author’s core belief that we must be unafraid.  In short, we cannot conceive new ideas,  create new inventions, or learn new concepts if we are afraid.  To become unafraid, we need to move out of our comfort zone, physically and mentally, and even entertain wild and crazy ideas.  We need to suspend judgment of a situation so we do not limit the possible solutions and thereby stifle our creativity.  Furthermore, we must allow ourselves not only to be wrong but also to be alone in our beliefs – if we believe strongly enough that we are right.   Finally, we must be courageous and not give up when the task is difficult or even overwhelming.

Many students (and adults) fear math, which creates a wall that greatly impedes learning.  Peter Sims’ paragraph lists several specific characteristics that when developed will help students overcome this fear.  By sharing new research, such as we grow new brain synapses when we learn from our mistakes, we may encourage students to become more comfortable with being wrong.  Ironically, being wrong will allow them to learn better.  Also if we create a classroom environment that withholds judgment of all ideas, students may be less inclined to pre-judge their own ideas as unworthy, which will help them to persevere in working toward a solution.  In addition, this positive classroom environment may embolden students to stand up for their solutions, further deepening and enriching the mathematical discussions.  When we fear something, we are often tempted to give up.  If we can help students to be less afraid of making mistakes or being wrong, they may persist when faced with a difficult problem, which will increase their chances of making and learning from their mistakes.

(Photograph from MIcrosoft Office Clip Art, search string “fear.”)

Dinosaur Math

Because I’m home with a broken foot, my daughter happened to overhear part of my latest tutoring session.  What she heard was my student abruptly asking, “Do you know how long ago dinosaurs became extinct?”  (This type of comment is very common during our sessions.)

Later, my daughter, who is 15, told me she was amazed I didn’t immediately shut down the dinosaur conversation.  I explained to her that listening to this student is an important part of teaching him.  Over several months we have built a relationship; we listen to each other.  As a result, I can challenge him to change how he solves problems, or to try something new.  It works.

It was only later I remembered another reason I listen (at least for a while) to these apparently random, off-topic utterances.  It’s because they frequently spring from a math-related source.  Take the dinosaur comment.  He wasn’t asking because he didn’t know how long ago dinosaurs became extinct.  Rather, he wanted to know if I knew because something about it was bugging him.  He wondered:  If 200 years ago scientists discovered that – 65 million years earlier – dinosaurs had become extinct, then why don’t we now say dinosaurs have been extinct for 65 million two hundred years?  (I chose to ignore the when-did-scientists-know-it question and instead focused on the concept of rounding.)

I used a whiteboard and wrote the numbers 65,000,000 and 65,000,100.  (Okay, I finessed the date a little and said, let’s just use 100 years.)  We talked about rounding and the fact that it would take a really long time before it would make sense to say anything other than dinosaurs became extinct 65 million years ago.  He seemed content and I was happy that I had answered what turned out to be a good question.

It was only a day or so later I realized I had missed an opportunity to make the concept more real.

What if, instead, I had said:  How long is 65 million seconds?  Next tutoring session we start with this one!