In my years as a math teacher and tutor, I have had many occasions to help with fractions.  Here’s something interesting about nearly every initial conversation I’ve ever had with someone who is having difficulty with learning about fractions for the first time. I’ll usually begin the conversation by asking something like “So you say you’re having a hard time with fractions. What do you know about them so far?” Here’s how it will often proceed from there:

“Nothing. I don’t understand them at all.”

“Okay. But you’ve probably seen them written, right? You have an idea what they look like?”

“Well, yeah, there’s a number on the top and a number on the bottom and a line in between but I don’t understand them.”

“Okay, that’s helpful for me to know. So you can recognize something called a fraction, but you’re confused about how they work or what they are for?”


“There’s one other thing I’m guessing you do already understand about fractions,” I’ll then say. [I get a skeptical look in response, but they’ll usually hear me out.] “You know what half means, right?”

The response I get tends to reflect that offense has been taken. “Well, yeah, obviously. Half is just half of something.”

Indeed. The concept of half is such old news to someone by the time “fractions” get “introduced” in school, that it seems a little insulting to be asked about it as a person of  nine or ten years. One starts to hear about half and halves very early in life, long before any whisper of a thing called math, much less the more specific phenomenon of fractions. The concept of half doesn’t have a chance to feel difficult any more than any other ordinary thing that happens all the time in regular life.

But our use of fractions in everyday parlance is mostly limited to the word half. As a young child you might occasionally hear someone talk about splitting something into thirds or fourths, but you’ll hear about half very often.

Hence the snarly looks I get when I ask about half.  The snarling tends to subside when I protest: “But you told me you don’t understand fractions AT ALL, so I had to ask, didn’t I!?! It’s not my fault that those pesky halves were hiding in your memory disguising themselves as Not Fractions!”

This exaggerated response from me usually gets a smile, and from there we can begin to talk about why the school fractions seem so much more torturous than the easy everyday halves, which truly are so understandable to most people I’ve talked to about them that they don’t even recognize their understanding as understanding. It’s invisible, the way sentence structure is to people who haven’t been taught grammar in their native language.  They can use it, but their understanding is invisible.

Which has always made me wonder if this kind of implicit understanding could be built for other fractional parts quite simply if we just talked more about other fractions, the way we do about halves, such that they’d exist in linguistic experience for young people and thus have the chance of taking conceptual root the way the halves do.



Today is 4/9/16 (in places that put the month first) which means three perfect squares in a row: 22, 32, 42.  Last time that happened was January 4th, seven years ago, and next will not be until 2025, in September on the 16th. After that we have to wait for a new century again because we don’t have enough months to make it work for 16/25/36.

For everyone who lives everywhere else, the first and second dates this century would be 1 April 09 and 4 September 16 instead, and the third one wouldn’t happen for the same reason there won’t be a 16/25/36 on our calendar.  Not enough months to write 9.[16th month].25.

The  moral of the story is that we get one extra perfect square series by naming our days with month first and day second. Not sure it’s worth the confusion created by numbering differently from how everyone else does, but that’s for another time.

Here it is without all my nattering:

1/4/09 (January 4th), 4/9/16 (April 9th), 9/16/25 (September 16th)

everywhere else:

1.4.09 (April 1st), 4.9.16 (September 4th), –.–.– (9th day of 16th month in 25th year)

If you like really big perfect squares, you can smash the days, months, and years together.  I’m going to work on having more capacity for conceiving of these giant ones.  In the meantime, fortunately, someone else posted about that.

words, numbers, and the culture of offering

This is sort of another plug for Ed Zaccaro, but also an observation about language and dignity in the context of math.

I posted a while back about Ed Zaccaro’s Challenge Math series, which I think was written for kids who needed more than the classroom could provide, but also seems to work well with kids with all sorts of different relationships with math.

This week I was surprised when one of the eight year-olds I work with asked specifically if we could work in the Zaccaro book.  I’d done a quick bit of one page with her once before, and she was OK with it, but tends to balk at anything she has to stop and think about, anything that requires reading as the Zaccaro problems do.  This is not because she doesn’t like to read or isn’t a good reader. She is.  But as a math student, she tends to balk at reading. We’d mostly been working on quick little problems I’d chosen for the purpose of letting her see that she is more capable when it comes to math than she has learned to believe.

She leaned right in as soon as I opened the book in front of her.  She remarked on the illustration of the large sweating ants and then started reading the first problem on the page.  She sped through three in a row, understanding exactly what the problems were asking for and doing much of the computation in her head.  This is exactly what she doesn’t do when she’s interacting with her Pearson pages from school (which, as far as I can tell, were written and issued very quickly after the Common Core standards went into effect).  She’s constantly choosing operations that don’t fit the situations in the word problems.

I’ve been frustrated with the language of many of these problems ever since the materials were issued, but in watching this child with the Zaccaro problems, I realized something new about how they are tripping kids up.  The Pearson word problem language is often ambiguous, which is terribly troublesome in the context of math, but it’s also just plain dull.  For kids who are readers, accustomed to sentences written with care and intention, poorly written problems are not just potentially confusing.  They’re an insult to the sensibilities of the literate child.  The literate person.

I realized that what this student has been doing with her classroom math work is to scan each word problem and attempt to match it with a procedure.  If it looks like the kind of problem for which one should divide, she’ll divide.  If it looks like you have to trace your finger across the table shown and fill in a number according to a pattern, she’ll do that.  No matter that often the problem is asking for something slightly different.  In order to notice that, she’d have to subject herself to the agony of reading the dull text slowly. As a result, her actual ability and number sense is obscured. In a situation in which the reading is in and of itself engaging, she is freed up to think fluidly and flexibly about what there is to do with the numbers.

What the Zaccaro book does for her (that the Pearson book does not) is offer something.  That sounds too simple to matter, but I don’t think it is. An offering concerns itself with presentation and connection, free of attachment to result. It is the fundamental unit of authentic human exchange. It says “Here.  This is for you to have and use if you choose.” It grants dignity, agency, autonomy. In the course of our deep but desperate commitment to educate, we’ve moved away from offering toward mandate and decree, often undermining our ability to pass on what there is of useful knowledge and skill to our young.
The good news is we can resume a culture of offering any time we want.

Zaccaro challenge math

I’ve been trying to figure out what makes Ed Zaccaro’s challenge math books work so well for kids with such different math needs.  I bought the primary level book for one young child who needs more math than he’s getting in school and after working through a few chapters with him I realized the book might be helpful for another student who probably needs less math than she’s getting in school.

Zaccaro’s books have illustrations, and rather silly ones. That could be said of many math texts, but there’s something about the Zaccaro illustrations that endears rather than annoys most of the kids I’ve introduced to them.  I think what might be different is that the illustrations aren’t trying to pretend to be cool, nor are they just there for the sake of imagery.  The content is delivered in conversation between illustrated characters, and often the scenarios are just silly enough to warrant a laugh but not so silly they’re too distracting.  (This does depend on the person of course.)

And then there’s the tone of the explanations.  They’re not dry, but they’re not condescending either. This is a critical element.  It’s a fine line between whimsy and condescension when it comes to writing for children.  Kids can smell condescension.  Something about the Zaccaro tone and illustrations often seems to invite a sort of cheerful eye-rolling perplexedness without distracting too much from the content.

pgcm copy

The other appeal of the books is that at the end of each topic, there are four problem sets.  The problems start at a basic level (always reflecting the content of the topic) and then progress from there.  So depending on the ability of the person working on it, you can stop after the first few basic problems and move on to the next topic, or stop after the first few and come back to the next level later, or jump ahead, or whatever makes sense.  And in my experience, what makes sense for one person on one topic doesn’t necessarily make sense on the next; different ones are more and less difficult depending on all sorts of cognitive and other factors.

If you think this series might be useful for someone you know, I recommend looking for sample pages first; you may find that the appropriate level to start with doesn’t align grade-wise.  This is maybe my only complaint though I don’t see a way around it – we make such a big deal about grade levels that young people are often sensitive about the grade level of the work they’re doing.  The book titles have to give some indication of who they’re written for, and at least these don’t list numbers of grades.  But the primary book can be useful for struggling ten year-olds as well as math-devouring seven year-olds.  So try the sample pages first.

a few things about smallness

The other day I thought maybe it might be time for me to try calculus again.  I didn’t take it in high school the way I might’ve if I’d been born a bit more recently, because there wasn’t a year left for it after 9th grade algebra, 10th grade geometry, etc.  And then I didn’t take it in college because, I think, I was tired.  Or didn’t get along well with the teacher?  I can’t really remember.

But there it is, out there, this un-learned mass with the fancy name. One of the young people I know, with whom I have been doing math for several years because she likes it and doesn’t “need” a tutor but likes having one, is nearing the end of pre-cal.  She and I are really more like math buddies than tutor and tutee.  I help her when she needs help, but a lot of the time we find interesting problems to try to solve together. Now, there she is, about to start calculus, and not only will I not be able to help anymore, she’ll likely be too busy with her calculus to do math with me much anymore.

So I’ve been brushing up on my pre-cal, looking for gaps and rusty spots, and considering  calculus texts.  I’m a Martin Gardner fan, and my library had a copy of Calculus Made Easy, the Silvanus Thompson text with revisions by Gardner, so I tracked it down in the 515s and opened to the table of contents. Thompson’s first chapter is called “To Deliver You from the Preliminary Terrors.” I immediately texted the words to my mom, who is still terrified by most math and went to all sorts of lengths to avoid passing her fear on to her female child.

I didn’t have terrors, per se, but I can’t resist this sort of language.  In this chapter Thompson delivers us from our preliminary terrors by explaining that to learn the two principle symbols d, meaning a little bit of, and ∫, meaning the sum of all the little bits. (He mentions the fancy language as well, but uses the plain to explain.) The second chapter is called On Relative Degrees of Smallness. I will admit that I was so enamored of this title that I had to stop reading for awhile. But then I resumed.  Here’s how the chapter begins:

We shall find that in our processes of calculation we have to deal with small quantities of various degrees of smallness. We shall also have to learn under what circumstances we may consider quantities to be so minute that we may omit them from consideration. Everything depends on relative minuteness.

Here I had to stop reading again and stare out the window.  Who knew this was going to be a philosophy book?

I recently had occasion to look up the word pico.  I have always been partial to words that mean lots of different things and pico is, among other things, the Spanish word for peak (or beak), an island in the Portuguese Azores, an acronym for a method of  locating relevant clinical literature, and a prefix denoting one trillionth. 10−12.  0.000000000001. Very, very little.

The other night I heard about Erika Christakis’ new book The Importance of Being Little.  This book is about very young children, and the way the world view of such people could inform the way we older bigger people interact with them.

This is an awful lot about smallness in one week, I’ve been thinking.  A little hard to ignore.  Also, my father, who is recovering from a massive stroke, cannot remember things I have just told him.  To find the patience and presence to dial back a lifetime of conversational habit in order to sustain an exchange with someone who can cast backward and forward only a few moments is at once an enormous thing and a very very small thing.

I’m not sure how far I’ll get with the calculus, but I can’t help feeling that it’s a terrific thing every time we consider smallness in great depth.


On Saturday we had a visit from four friends – my college roommate, her husband, and their two sons, who are six and three.

The three year-old and I found, in a box of old games, a cribbage board.  He was enamored of the board when he saw the hidden compartment which houses the little pegs.  He also liked the surface, with a track corresponding to each of the three peg colors in the compartment.  He decided that we would each choose a color and then move the pegs around the board according to what we each rolled on a  six-sided die.

He is mostly confident at counting up to six, but has trouble distinguishing between the four and the five on a die.  I was reminded, when I noticed this, of how much math learning will take care of itself in the course of dice games played with young children.

I didn’t have to do any instructing about this four and five thing, though I’d have been willing to if I’d been asked. I just counted out four each time I rolled a four, and counted out five each time I rolled a five.  When he wasn’t sure which he’d rolled, he’d say one or the other, lifting his tone slightly at the end, just enough of a question for me to confirm or correct whichever he’d chose. After a few rounds of this he stopped getting the two confused, and stopped reading the numbers as questions, because he knew he had figured it out.


On Thursday last week, I spoke with the mom of a child who had asked her to find out if someone could help him control his behavior in school.  (I’ll call him Owen.) He has a tendency to interrupt his teacher, to speak up without being called on first, and to have difficulty staying in his seat long enough to finish assignments.  Owen’s nine.

In our first conversation, his mom and I talked about how it’s easy to confuse and confound the symptoms of attention deficit disorder and profound intellectual strength.  The two can look quite similar in the context of a classroom in which children are expected to perform repetitive, stationary, lower-level cognitive tasks.  Owen, his mother told me, has tons of physical energy, and is very curious and interested in learning. He can’t always stand to wait for other students to catch up with where his brain is. He was recently diagnosed with attention deficit disorder/executive function. I told her that what I’d be able to provide for him is support in building strategies for managing his school work and that this work would most likely be successful if combined with intellectual (in the form of logic puzzles, creative problem-solving activities, and other tasks that involve higher-order thinking) that would meet his craving for intellectual engagement.

In our second conversation, after she had spoken with Owen’s teacher, his mom told me that there didn’t seem to be any evidence of the executive function difficulty in the classroom, and what the teacher was more concerned about was impulsivity – the interrupting, the speaking up without being called on, the scooting around the classroom when he was supposed to be in his seat.

This mom originally called me at the recommendation of the neuropsychologist who evaluated her son.  This evaluator has referred several children to me and I have worked with some of them.  I think it’s expected that I will be able to help children with their executive function difficulties.  It’s true that I have been able to help some children (as well as some adults) with executive function difficulty.  But as often as I help with that (which is only when the person him or herself experiences the executive function difficulty as an impairment), I end up helping with something else that makes a different kind of difference.

Helping children with executive function is usually a matter of helping them comply with environmental demands in which they don’t have much intrinsic investment.  It’s not that they don’t notice or feel the effects of not being able to comply with the demands; it’s that if given the choice, they would rather not be required to.  If you offered them the chance to exist in an environment that didn’t make those demands, and they could exist there without being ostracized for it, they would choose that.

If this is the case with a child, helping with the executive function tends to be a losing battle.  Kids who fit this description tend to undermine their own efforts to improve their ability to stay on task in school, get their homework done, and generally keep track of their assignments.

So executive function is for most children a matter of adaptation.  They’ve been assigned an environment, and in order to succeed there, to be deemed successful according to the measures of the environment, they must alter their tendencies and habits.

Owen’s teacher doesn’t think that he needed help with executive function, but the concerns she does have are also about his adaptive performance.  A typical classroom is not a natural fit for a person with a lot of physical energy, who needs motion and activity in order to keep calm, to think clearly, to feel at ease.

In my second conversation with Owen’s mom, she asked if I might be able to help with the impulsivity. Now that the executive function concern was out of the way, the focus would be staying seated and not interrupting or speaking without being called on by the teacher.

It’s at this point that my job becomes a little weird.  Weird is probably not the right word.  Agonizing is a better word, if you want one that describes my experience of it.  Here’s why: I could do what this mom is asking.  I could help her son get himself to stay in his seat.  I could help him build and use strategies for keeping quiet when he was expected to in the classroom.   And if I did, some things would be better for him than they are now.  He would feel less self-conscious.  He would be embarrassed by his behavior less often.  He might finish more of his work. His parents and teachers would stop worrying about his behavior and about his future.

Is there a cost, though, to staying in your seat when you have the impulse to move?  Or to completing a page of problems you already know how to do rather than thinking about something that intrigues or otherwise engages you?  Or to slowing your own thinking to meet that of a group of age-peers simply because they’re the same age as you are, and thus the only classmates you are entitled to?

In some ways, I wish there weren’t.  I wish I could sit down with a child and coach him or her to adapt to the environment that is the current reality for most children and believe that it would only help.  The push for adaptation, and quick, is tremendous.  I hear so many times a week that children need to be prepared for the way the world is.  They need to learn to do things they don’t want to do, to learn to be told what to do, to find out that life is hard, to get used to not getting their way. I do know that some adaptation is crucial to survival, and not just in the wilderness.  There are ways that the world is, and to be able to navigate them helps.  But to only adapt is not necessarily, in my opinion, to optimize the potential of the human organism.  And sometimes, to become accustomed to adapting without advocating for one’s needs and strengths is to shortchange the species, to deny it the potential for growth and progress.

A few years ago I was working with a fourth grader whose mom was concerned about his math skills.  One week he came into my office and sighed a great, heavy sigh as he settled into a chair across from me.

“What happened?” I asked him.
“I need you to show me the second way of subtracting,” he said.
I asked him to tell me more.
“Well, last week my teacher showed us how to subtract, where you cross out the one number and then add the one to the number next to it…” he looked up at me to confirm that I knew what he was talking about. I nodded, and he continued.  “So then today at math time she wrote another problem like the ones we were doing last week up on the board and started talking.  I thought she was just showing us that same way of subtracting again so… I… stopped… listening.”  He said this quietly.
“It’s OK,” I said.
He sighed. I waited for a moment to see if there was more.
“Is that the end of the story?
“Yeah,” he said. “That’s all.  I mean, I found out later that we were supposed to be learning a second way of subtracting that we have to do for homework and now I don’t know how to do it because I wasn’t listening.”
“OK.  I have a question.”
“Uh-huh,” he said.
“OK if I ask it?”
“You don’t have to answer me unless you want to, but I’m wondering what you were doing with your mind instead of listening to the teacher about the second way of subtracting.”
He took a deep breath.  “Well, my brother and I got a Lego Millenium Falcon for Christmas, and after we built it with the instructions, we took it apart and started building another one with our own design, but we’ve been having trouble getting it to support its own weight.  I had an idea about adding these struts in the back like this (he demonstrated the angle he had in mind) and I was trying to imagine how they could attach.”

I’ve asked kids this question, about what they were thinking about when the classroom environment demanded they be thinking about something else, attending to something chosen by the adult in charge, and it’s not always a complicated engineering problem as it was this time for this child, but it’s rarely frivolous. I suppose that frivolity is a matter of opinion, so I should add that it’s rarely frivolous in my opinion.  A wise friend of mine told me once that his primary reason for considering the possibility of not sending his children to school as soon as they turned five was that it was important to him that they have access to their own thoughts – that they have a chance to figure out what they were interested in, what mattered to them, and he thought that if they were hurried off to school, where they would much of the time be told what to think about and when, that they might not have the chance to get acquainted with the contents of their own minds and imaginations.

This fourth grader, once he thought he was all set with subtraction, turned his attention to a fairly complicated problem of engineering.

From a very young age, humans go looking for good uses to put their minds to.

From a very young age, humans go looking for good uses to put their minds to. There’s an extent to which we sometimes undermine and interrupt that process in the name of adaptation and presumed survival.  We imagine that if we just force a kid like this one to comply with our curricular plans for him, that later, once he’s been taught (and re-taught) everything we consider basic, he can turn his attention to contribution and participation.

There are a few problems with this model.  One is that the curiosities of a young person don’t necessarily hang in there in the face of being ignored, suppressed, or otherwise dishonored.  A child who is fascinated by technical problems as a five and six and seven year-old but is not free to explore such problems and is told that the most important thing is to do well in his schoolwork, his spelling and his multiplication, will often not remain loyal to his earlier curiosities.  (In fact, a child who does remain loyal is often considered a behavior problem, or difficult, or resistant, at risk for failure as a student.)

Also, we know that the brain is most elastic when it’s young. The flexibility and openness to new ways of thinking is not confined to language learning. If we spend all that elasticity on predetermined and prescribed content, we leave little room for the thoughts and unique creative potentials of the individual.  (And I don’t mean creative in an artistic sense, though I include the artistic – I mean creative in its basic sense, making things up, thinking of things that haven’t been thought of or haven’t been thought of in a particular way.)

This is all to say, I suppose, in many many paragraphs, that I think there’s a balance to be struck, to be sought, between adaptation and actualization.  Between supporting an individual in finding his or her way among the obstacles erected around him over time and supporting him or her in exploring and realizing the fullest possible extent of her capacities and curiosities. A balance which, if struck or even aspired to, could save a lot of frustration for a lot of students (and teachers) and offer all of us better access to the extent of potential contained in our human selves.

the sorter is broken

I asked a high-school aged friend if she’s thought about what she might want to study in college (she’d already told me she wants to go) or what she might want to do after college.  She hesitated, and then said that she really likes science but she’s terrible at memorizing things, so probably she’ll do something with history instead. She said she’s especially interested in environmental science and issues related to climate change.

You could think of this as a teaching or curriculum problem, but I don’t think it really is.  I think it’s a sorting problem.  Or a problem with how we’ve taught ourselves to imagine we should be sorted when it comes to ability and suited-ness to particular tasks and professions.  And also a problem with how we think about what our brains are best used for.

This young person thinks that her difficulty with memorizing will exclude her (or should exclude her) from a career in science.  She has classmates who memorize easily, and they are the ones with high scores on tests and in courses. So on paper, on transcripts, if it’s good grades that tell us about someone’s aptitude for a particular area of study, we can see that her quick-memorizing classmates are the ones destined for careers in science.  Yes, many a good teacher will tell you that if you don’t emphasize the memorization you can show something different with how you grade, but for most students in most schools, information recall is a big factor in grade determination and a teacher who makes it otherwise is swimming upstream and trying to pull her students along with her against a strong current.

What would we have to do to make it otherwise?  First we’d have to decide whether we believe that a scientist must be able to keep on hand a multitude of data.  Must one?  I’d guess not, at a time when it’s possible to put a handheld device with endless data in the hand of any professional anything.  Wouldn’t it make sense for the first qualifying characteristic for a career in science (or any participation in science) be an interest in participating, and then perhaps the second an interest in and capacity for problem solving and analytical thought?  Science once required extensive memorization, but it doesn’t any more, and we’ve got big enough problems, and many enough problems, that the more solvers we can get on them the better, it seems to me.  To exclude the ones who can’t memorize stuff as well as some others can memorize stuff seems unwise.  Not to mention that the good memorizers might be put to better use elsewhere especially if they’re not interested in the careers and occupations that their memorizing might qualify them for and point them in the direction of.