Viewing posts from the math category

How To Approach Math Learning

Jo Baeler and Pablo Zoido recently published an article in Scientific American Math. (behind a paywall; summary here)

Every three years, the Program for International Student Assessment (PISA) tests hundreds of thousands of 15-year-olds. In the past, the US posted average scores in reading and science, but well below other developed nations in math.

The most recent results for the USA

Since 2012, data collected about how students approach math shows three distinct styles of learning. Some memorize facts; some relate new concepts to old; some self-monitor, evaluating their own understanding and focusing attention on concepts not yet learned.

In every country, the memorizers are the lowest achievers, and the U.S. has high proportions of memorizers. These memorizers are approximately half a year behind those who use relational and self-monitoring strategies. In some countries, pople who use relational and self-monitoring strategies are a whole year ahead of those who just memorize the facts.


American schools routinely present mathematics procedurally, as sets of steps to memorize and apply. Many teachers, faced with long lists of content to cover to satisfy state and federal requirements, worry that students do not have enough time to explore math topics in depth. Others simply teach as they were taught. And few have the opportunity to stay current with what research shows about how kids learn math best: as an open, conceptual, inquiry-based subject.

The foundation all math students need is number sense — essentially a feel for numbers, with the agility to use them flexibly and creatively. A child with number sense might tackle 19 × 9 by first working with “friendlier numbers” — say, 20 × 9 — and then subtracting 9. Students without number sense could arrive at the answer only by using an algorithm. To build number sense, students need the opportunity to approach numbers in different ways, to see and use numbers visually, and to play around with different strategies for combining them. Unfortunately, most elementary classrooms ask students to memorize times tables and other number facts, often under time pressure, which research shows can seed math anxiety. It can actually hinder the development of number sense.

I could not have said it better myself. Behind every math prodigy is a child who has spent lots of time playing with numbers, making friends with them. Today’s schools often spend far too much time with formal testing, leaving too little time for just having fun with numbers.

Some people teach the facts; some teach the procedures; some teach the soul of math. Great mathematicians are like great jazz musicians; they improvise on the fly, searching for a great solution, an inspirational solution.

A Little Calculation

Can I have a few seconds? Researchers have been videotaping classes, taking notes, making an estimate of how much time children are actually receiving instruction, as opposed to walking, listening to announcements, handing in or receiving papers, and so forth; it comes to 90 minutes of actual instruction per school day.

But even that tally is not quite accurate – some of that 90 minutes of instruction is of little value to particular students, since the material is already understood. “Did that already, about three times. Can we move on already?” And for others, the material is incomprehensible.

So, I did a back-of-envelope calculation and came to a figure which could be a little off, but I guesstimate that the average classroom, in one year, wastes (in total, for all 30 or so children), about a gigasecond of their time.

Oh, so sorry. Did you just spit out your coffee? Did I forget the leg-pulling warning? I must advise you to finish swallowing and put your drink or sandwich down. I’ll wait.

A gigasecond is one billion seconds.

A year is about 31.5 million seconds. We can be more exact: 60*60*24*365 = 31,536,000 seconds. Can twiddle this for leap years and leap seconds, but that’s close enough for an approximate  calculation.

Divvy that into a billion, and we have 31.79 years … you see where this is going, don’t you?

31 is roughly the number of students in a class … so, one year wasted per year per child.

Yes, it’s harsh. I sincerely apologize to every hard-working teacher who is trying to do something useful – and yes, some of you are very, very much appreciated from the bottom of my heart, and from many other students who are grateful for those of you who do stand out.

But … why did I have to explain any of this? It’s a simple calculation, and you have had twelve years of instruction in math, which most of you hate. What’s wrong with this picture?

A 30-something friend, with an impressive string of letters in STEM disciplines, shared a thought. He and his friends, some of the brightest people in Southern California, still wake up with nightmares about their K-12 school years.

If it’s that bad for the “good” students, the best and brightest, we might want to try something different. I’d go back to basics: if 90 minutes or less of instruction is all we have to work with, what if those 90 minutes were more efficient? What if, instead of five or ten useful minutes (from the perspective of the child), we find inexpensive (time-wise) methods to find out what is known and unknown for that particular student, and provide only that which is unknown?

Steal Not Thy Child’s Time

My then-wife and I homeschooled two children until they reached ages 12 and 14. I’m also proud and delighted to be a grandfather, that my daughter has been home- or un-schooling her children (seven, aged 1-13 years), and that my son is very engaged with the education of his own.

Unlike most dads, I initiated the discussion of home education. My wife, having a degree in elementary education, balked at the very idea of teaching our own. She had been trained to plan and prepare, to have formal structure, and so forth – a complex task which would daunt any one person.

I view education from the perspective of the student; I ask not “how to teach,” but how to learn most productively. During 11 years of Catholic school – skipping grade 8 – I often felt that my time was being stolen from me; I wanted something better for my own – not merely in terms of “quantity of stuff learned,” but qualitatively different.

Like the top fifth of entrants, I was already above First Grade level in reading and arithmetic. The most effective way to progress would have been to actually read, to play with new ideas and interesting and challenging math problems. This is precisely what we were not allowed to do.

Instead, we spent 40 minutes or so waiting for a turn to read two lines from a “See Dick Run” book. Far better to use the same time to read interesting books at our own pace – which is what my homeschooled grandchildren do.

With math, we drilled and killed our way through textbooks, page by boring page. This is a horrendous waste of children’s time.

Lately, researchers have been discovering the value of the time spent by that top third or so outside of school. I know what I was doing outside of school – independent reading, playing with math, and otherwise teaching myself. Researchers – such as those at Reading is Fundamental – are often bewildered that children – even at the bottom of the SES rankings – often learn and improve substantially over the course of the summer, with just the slightest bit of assistance – self-chosen books, in this instance.

My wife, the degreed educator, insisted that our son go to a “gifted” first grade class at a nice suburban school. The teacher was respected by our neighbors. We actually had to push to obtain admission; the test administrators balked because his art skills, in their minds, were not quite up to snuff.

Two weeks later, he came home and asked “What is 5-7?”

“What did your teacher say?”

“She says it’s too complicated.”

“What do you think?”

“I know that 7-5 is 2, and 7-7 is zero. So is 5-5, anything minus itself is zero. I think 5-7 must be something else, but I don’t know what it is.”

Anybody who can articulate all that is ready to advance. I briefly explained the idea of negative numbers, using a thermometer diagram to illustrate; turned the diagram on its side, making it a number line, and explained how to think of negative numbers as growing to the left versus the right, and subtraction as moving the opposite direction from addition. He understood immediately. I made sure he learned how to add and subtract all combinations of positive and negative numbers.

A few days later, I come home from work, he’s doodling on his paper, and there’s a number line. This is his own initiative, his own “work.” I asked a few questions, and he’s got the concepts perfectly, in every particular. This took minutes, not days, weeks, or months. I never had to repeat the lesson. It stuck, because it was his question; he was interested.

My wife, observing this, had an epiphany. Our son could learn without complicated textbooks and plans and so forth – a lot more rapidly than at his “gifted” class. We began homeschooling, which continued until our children were 12 and 14 years of age. They had little trouble adapting to formal education. They had some gaps (and what student does not?), but had outstanding mathematical intuition, and rapidly mastered new material.

My wife and I divided our labors; Under no circumstances was she to teach math. I was (and remain) devoted to what I call “organic learning,” and some call “unschooling,” inspired by John Holt. It was my desire to work with the nature of my children, rather than against it. Instead of “begin at page 1,” my children and I had many “random” conversations which where suited to their abilities and interests – sometimes about everyday math, sometimes about more abstract ideas such as binary arithmetic, and we conversed about myriads of non-math topics. We played many games which exercise math skills. Both became skilled mental calculators.

I must stress the role of frequent conversations, often initiated by my children. At age 3, my son asked “What is gwabbity?” (he had overheard the word “gravity” in a conversation). I could have said “gravity is what makes things fall down,” but that seemed too simple, given his state of knowledge at the time. So I went to the whiteboard, writing down the formula F = g \frac {m_1 m_2} {r^2}, and began to talk and draw. In a few minutes, he learned the word “mass,” had a loose idea of its meaning, was reminded that the earth is a really really big ball, was informally introduced to the idea of using vectors to represent force, and a few other ideas. I fudged very little, and built a framework which was close to what he’d want to know as his knowledge of math and physics grew.

To further illustrate the potential, let us move forward about 30 years, to a conversation with my 2nd-generation home-schooled grandson, aged 6.

I asked him to think about adding the integers from 1 to 100. The obvious but slow method: add 1 and 2, add 3, add 4, and so on. 99 additions. 1+2+3+...+100

Or, one could write the numbers down as 1 2 3 … 50, and then write the 2nd half in reverse order:

\begin{tabular}{ r r r r r}    1 & 2 & 3 & ... & 50 \\    +100 & +99 & +98 & +... & +51\\    \end{tabular}

My grandson interjected “each pair adds to 101. There are 50 pairs. 5050.”  That was fast.

Could he generalize? What is the sum of the even numbers, from 2 to 100? He pondered for a few seconds, and replied “2550” – which is correct. This problem stumps most high school students. At age 8 or 9, he tested at the 18th grade equivalent in math. Does he have good math genes? Is he something of a prodigy? Yes – but a prodigy who could race at own his speed, unhindered by a governor.

And that is why we teach our own. We don’t want to hold them back, nor let schools steal their time.