Do as I DO…

Do as I DO…

I once went to an Inquiry Based Learning conference and attended several lectures about how good teachers employ IBL to teach mathematics. (We learned about IBL through its antithesis….)  IBL is certainly not a process of presenting algorithms.  The heart of the matter is that students make building blocks out of what they already know in order to personalize new knowledge to turn into more steps.

But what it looked like to me is that you give students a bag of apples, some flour, equipment; then show them a pie  (which they are NOT allowed to eat).  Students are set the task of figuring out how to make their own pie.  A priori, we’re assuming students are interested in pie.  And I think teachers are check whether they know how not to chop off a finger with the paring knife.  Or maybe losing a finger would be part of the process….

I didn’t learn to make pie that way.  For years, I watched my mother make pies and helped eat them.  When I was big enough to reach the counter I copied her pie-making steps.  Along the way I learned which steps were essential and which could be varied.  After awhile my pies weren’t copies of my mother’s pies anymore…

Once upon a time my PhD advisor gave me a problem to solve.  He told me to use a certain theorem and pointed out the relevant paper.  For days I banged my head trying to figure out how the results in that paper had any bearing on the problem he’d handed me.

Finally I wrote out a careful description of what the problem and the paper’s result each said and showed how they had nothing to do with each other.  And I hadn’t solved the problem.

He wrote back.  I wasn’t supposed to be using the paper’s result at all.  I was supposed to have noticed that the author’s proof technique presented ideas about how to tackle my problem.

This business of sorting facts from techniques…

My students like it when an algorithm is called an algorithm.  They like to get to practice using algorithms and they learn to recognize the kinds of problems the algorithm may be used to solve.  The good ones eventually get behind the algorithm and make the solutions their own.

Thoughts thought along the road

Thoughts thought along the road

I meant to ask my students yesterday what a number is.

I did that once with a class of College Algebra students who had appointed as spokeswoman a classmate who dared to ask me if I would please explain something “in regular English.”  I told them I would try, but first I needed them to tell me in regular English not what a number does, but what a number is.  They clustered up and went to work; discussing, giggling, yelling; digging in to the chore.  My objective was complete when the reassembled horde reached agreement: “That’s HARD!”

Several years ago, and several years later, I griped, “It is impossible to remediate mathematical deficiencies for university students who don’t understand order or magnitude.”

More and more, university level remedial math is relegated to online drill systems which do a pretty good job of conditioning students to perform logically consistent symbolic manipulations.  Any student with a tolerance for shoving symbols into formal patterns does a good job.  I’ve taught many students who’ve cleared those remedial hurdles, who arrive in my room for their next course which is often “the last math they need.”  These students clamor for detailed algorithms by which to arrive at the right answer: a symbol attained by careful traversal of a necessarily arduous path, in the end a trophy divorced from context.

It is students who pass through that sort of training that tell the world and their children, “I NEVER used algebra again in my life once I graduated.”

Only What They Need to Know

Only What They Need to Know

One of my former students is now teaching middle school.  Last year she wrote to ask me, “Why are Algebra II and PreCalc the same course, essentially? (If they are not, then I need to fix our curriculum!!.)”

“Because,” I explained, “What should be happening is that all the same manipulations the students learned to do as progressively more involved algebra exercises, get reapplied within the context of understanding how elementary functions work.

“The set up is supposed to aim toward a calculus vision of considering the effect that (small) changes in the variable inputs have on the behavior of the outputs.”

I am fiercely convinced that Precalculus should be the arena in which they get used to moving their concern from advanced arithmetic to relationships.  I’d like to be able to focus on number relationships, but calculators have made that much harder to accomplish.

The circular functions provide a splendid forum for introducing number relations which are far removed from the rehash of algebraic manipulation.  So I’ve been hammering at the students to hand-draft graphical images which relate a measured distance around the unit circle to the coordinates of the terminal point.  Now many arrived having already memorized these “special” pairings.  My goal is to cement those pairs to their relation.

A fellow precalculus teacher listened to my ideas, how much time I planned to take on this; what activities I was employing with the students

“Oh,” she replied, “They don’t need all that when they get to calculus!”


On the one hand it is true that their calculus course will revisit the notion of functions — in the same way precalculus revisits algebraic manipulation.  On the other hand, answers to their calculus exercises will come more easily to those who can manipulate symbols quickly and accurately, so the temptation is so focus on rehashing skill drills in the precalculus class.  Any theoretical background will sink into subconscious, at best.

But I don’t want to merely rehash old stuff and I don’t want merely a room full of bored computers.  I want students to take the fundamental ideas and mentally build new things, finishing those computations only as an afterthought.

I want to teach them much more than they’ll ever think they need to know.



Recalling the words of my PhD advisor, Joe Diestel:

“When you can’t do anything about it, do mathematics.”

I knew that my young math students, first-time voters, would need license to move around and to talk, would be craving reassurance that they weren’t abandoned by dumbfounded adults with no answers, but that most of all they craved what we all want: direction, meaning and purpose.

So I wrote them a stiff, relevant, workable trigonometry review sheet and posted to let them know there would be work to be done on Wednesday.  I strengthened the importance with a warning (No Calculators Or Books Allowed); I offered hope for success (You Will Work With A Study Partner and For The Whole Class Period); and handed the plum (Attendance Points AND Mathematical Achievement Points may be Earned.)

They came.  Exhausted raccoon circles of sleeplessness, wary eyes darting in search of safety, companionship and assurance from each other and finally daring to peek at me to measure what empathy, what hope, what experience and direction, what of any possible help I could offer to them.

They pounced on the worksheets so grateful for order and activity.  They huddled with each other pounding at the exercises and sharing their souls.  Some required hugs; some required my stories; a group demanded to know whom I had supported and what I would do about the result; a few shyly showed me the toys they were carrying for comfort; they demanded mathematical reminders and hints; they argued, roared, laughed and remembered some math together — and most left with a little more self-confidence in their mathematical abilities and their faith in community.

Once upon a time…

Once upon a time…

When I was in grade school I dreaded arithmetic drills.
I didn’t have any idea how to visualize progressions of sixes, say, or how to think about four times eighty-seven.  Memorization drills were on the wane but nothing much was taking their place.
“Hang on,” my mathematically minded mother — who was doing graduate work in Projective Geometry in night school — encouraged me,
“You’ll LIKE algebra.”
And she was really really really right.  I blossomed at the sight of my first “x.”  Then had no trouble proving things in geometry and put together enough ideas to sail through most of Algebra II/Trig.  And then I stopped because I’d already done more than what half the girls did in high school back then.)

Now back in 1964 when I would have taken, as an eighth grader, the entrance exams for the elite high school programs in what came to be called STEM (Science, Technology, Engineering, Mathematics), those programs were closed to girls and had been since the dawn of time.  I never rejected the idea of learning higher mathematics because I was a girl.  Rather, I was utterly ignored with a rejection so entrenched that nobody recognized it as dismissal and rejection.  It was the way things were.

Most women of my generation are still bound by a deeply buried blindness to the thoughtless knee jerk rejection we receive from, then dole out in turn on our sisters; particularly and especially any sisters who exhibit talents and achievements we don’t understand, sisters whose heads and shoulders rise above the rest and attract the potshots, too often from us.

To achieve “gender equality” requires much more than reversing entrance exam rules to admit girls alongside the boys.  It takes deep and careful surgery to remove the cataracts which keep society from seeing women — indeed keep women from accepting women — as smart, skilled, and accomplished.

Tomorrow morning I will vote for Hillary Clinton for President of the United States because she is a principled, smart, skilled, accomplished politician who will do an amazingly good job, whether I like how and what she does or not.

She doesn’t have to like my mathematics either.
The Featured Image is of my collectible Steif donkey who fell off a consignment shelf into my arms a few weeks ago.  Of course I named her Hillary.

“Regular English”

“Regular English”

Once upon a time — long before WiFi and the days of “Ask Siri” — I was egging on a precalculus class to memorize the definition of a mathematical function.

One young lady cried out in anguish, “But can you PLEASE explain that in Regular English!”

I had already given several examples. I knew she had taken notes.  And the rest of the class was clearly relieved that she’d bravely stepped forth with her request….

So I said that I would try.  But first, I said, I wanted them to work out how they would explain to someone, in Regular English, what a number is.  Not so much, I amplified, what a number DOES, but what a number IS.

They worked very hard.  They thought, they argued, they grew frustrated, they broke into gales of laughter, they invented science fiction plots, and they realized they couldn’t do it….

But best of all, they came to understand that a formal definition is really hard to construct, and that once you’ve got one, there’s really no other way to explain things completely.

Students of Mathematics

Students of Mathematics

A professor once noticed that some students think it’s cool to be able to say, “I’m a math major.”

It’s easy, even, to say it truthfully because all you have to do is tell the registrar’s office that you want to be a math major and they tell The System to declare it so.

Then an advisor starts to talk about the classes you have to take and many of them are mathematics classes.  But it’s still okay, because all you have to do is tell The System you want to register for those classes and barring any glitches about “lacking prerequisite classes” or “unpaid tuition,” you are listed as a student in a math class.

Then what happens?

You attend your first class and decide whether to join the ranks of Students of Mathematics.