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.



The volleyball player and entrepreneur Gabrielle Reece has a policy she calls “go first.” (I read about it in Tim Ferriss’ recent book Tools of Titans, a very large collection of information about the work and other habits of a few hundred highly successful people.) Reece says “If I’m checking out at the store, I’ll say hello first. If I’m coming across somebody and make eye contact, I’ll smile first. … [People] are ready, but you have to go first…”

This got my attention because often when I find myself tempted to smile or say hello to people I hesitate, because I imagine that the person I’m encountering doesn’t want to engage. This practice of Reece’s was a good reminder that I don’t actually know whether someone wants to engage or not.

Yesterday I had the chance to try it out. I was at our local community center and a woman walked in by herself. I noticed myself having the thought that she didn’t seem very friendly, as she walked past me. She sat down nearby to change out of her boots into sneakers so she could walk on the indoor track (it had snowed several inches the day before). I remembered about going first, and said “Instant winter out there, isn’t it?” She looked a bit startled that I was talking to her, but only for a split second, and then smiled. “Sure is,” she said. “But I guess we should be glad it didn’t come until the middle of December. Now if only we could be done by mid-February!”

So that was good reinforcement for me for going first. Now if I can just remember that if the next person is not so keen on interacting, it doesn’t mean I should stop trying.


If children kept Frequently Asked Questions pages for their lives, these lists would  consistently feature “How old are you?” and “What grade are you in?”

We could probably do better.


The day was dull with scientific notation.  Sarah didn’t want to be there with me at all.  As I often do when I meet a young person for the first time, I asked her about her interests. She said right away “I hate school. Actually, I love school.  I just hate math.” She didn’t bother to answer my question.  She went right to what she hates, and who could blame her when there it was, math, moving her around like a puppeteer, straight from school to tutoring?

But she was ready to tackle her homework, because at least that could get done while her afternoon ticked away.  She didn’t have a pencil, so I reached to get one from my desk drawer.  I was about to pick up the closest one and then stopped and looked up at her first.  “Which would you prefer,” I asked, “plain old wooden or mechanical?”

She paused before responding. “Mechanical,” she said.  As I handed her the pencil, she took a deeper, slower breath.

The rest of the hour pivoted around this question about the pencil.  It was nothing – a tiny moment – only it wasn’t.  When I thought about it afterward, I realized it was because the general condition of student-hood is mostly free of person-hood, in that students are expected to set aside preference. In fact, preference doesn’t even usually figure in enough to be set aside.

Preference and choice are not the same.  Some schools have lots of choice.  Choice of research topics, choice of Spanish or French, choice of lunch beverage.  Granting preference acknowledges a person’s person-ness in a way that choice doesn’t.  Choice says “The options have already been decided, and it has also been decided that you get to choose from among them.”

To grant preference is not necessarily active.  It’s an acknowledgement that a person has preferences and that those preferences are part of what constitutes their being and person-hood.  It doesn’t matter what kind of pencil a 12 year-old prefers, and she’s generally at liberty to select her own pencil, keep it with her, and no one has to know.  But if there are two kinds of pencil available, and I’m sitting across from a 12 year-old person, I know I’m less likely to ask her preference than I would be if I were sitting across from a 35 year-old person or a 62 year-old person. The relationship between adult and child is usually “I’ll handle this; you just follow my lead/instructions. Here’s a pencil for you to use.”  It’s not usually “Which kind of pencil would you prefer?” And it’s not because there isn’t time to ask.

I didn’t happen upon this because I was keen enough to notice and take an opportunity to treat someone better. I didn’t think to myself “Oh, I should ask what Sarah prefers, and that’ll be a better way to treat her.”  I just noticed that I would behave differently with an adult from how I was about to behave with her. Which I suppose is where we can begin.


I thought a lot today about contempt – what it is exactly and what can be done to dissipate it. It seems to have a big impact on the relationships which contain it and in turn the impact of those relationships on others and on what’s possible for humans attempting collaboration of any kind.

What I think of as contempt is not quite what the dictionary says – a despising or a plain lack of respect – although the definition mentions the words scorn and disdain which hint at what I think of it as. I think of contempt as that steady current of negativity that runs between many pairs or groups of people, flavoring interactions with a note of malice.

This kind of contempt runs quieter but not independent of some of the more visible and widely discussed conditions between humans, but I wonder if it sets the tone for some of the more terrible things we do to each other. It seems to exist not only between the likes of actual warring factions but in families where love is unquestionably present – between spouses and between parents and children, between siblings, between in-laws. What’s interesting to me about it is that because it’s subtle, it’s easier to leave alone, to overlook in our efforts to get along better and improve our relationships with one another.

Anyway, it’s a big beastly thing, contempt, and I’ll be thinking and probably writing about it more soon. But for starters, ever since it came to my attention at a family gathering the other day, I’ve been watching for times when, finding myself wielding it, I could opt to lay it down.


The other day, one of the teenagers I know was talking about one of her teachers and how on certain days he plays music from one of his favorite albums during part of class. She said “He plays it in every class he hosts that day.”

When she’d finished telling the story I asked her about her choice of verb, because of course “host” isn’t what you’d expect in this context, but it hadn’t seemed like an accident. She said “Yeah, I’m not sure why I said it that way, but it seemed like what I meant.”

We had other things to talk about so I didn’t dwell on it. I just wanted to know if she’d chosen the word more or less on purpose. From the sound of it, in that particular teacher’s class, this student feels more like a guest than a… subject, the way many do in many classes.

A small thing, except not at all.


I’m rereading George Dennison’s The Lives of Children and was floored by this passage regarding the role of an adult in the life of a young person:


“…elders of the community rather than heads of organizations…”

More on that here in an old post.


One hundred percent of the parents I know want to be as sure as they can that their children have what they need to get by in the world. There’s a lot of variation in what each parent believes their children need, but that underlying commitment is the same.

Where it can get challenging is in the delivery.  To many parents, it seems as though children don’t want what they and other adults have to offer.  Young people argue with us about how things are, or they refuse to go along with what we suggest, or they seem to ignore our input.

I think about this a lot because these are fairly common complaints among the families I work with.  This week I heard it from one parent who wants her child to be more careful checking her school work.  “I’m trying to get her to care,” the mom told me.  Another parent tried to explain to her son that when the worker at the post office asked him a question and he didn’t answer, it may have been perceived as rude.  “He just argued with me,” she told me. “I know that social situations are not easy for him, so I want to let him know what’s expected.”

This morning I caught myself grappling with a similar situation but sort of in reverse (because I was trying to offer some input to my father, rather than receiving some from him). My dad will be selling his house soon, because he’s no longer able to live alone.  As the day grows nearer when we will get to work at cleaning out Dad’s house, he gets more concerned that we are going to throw away all of his stuff.  He’s been explaining to me that what he plans to do is put it all in boxes and label the boxes carefully, and that he doesn’t need to get rid of anything. I think he actually realizes that he will need very little of what’s there, but I think I understand why he’s talking about it this way. He’s uneasy about not having a house anymore, and uneasy about what his next living situation will be.  And probably also generally uneasy about the later stages of life. Certainly he is uneasy about being dependent on his children.

I also know, though, that he has told us for years that he doesn’t want so much stuff, and would like to live in a much smaller place when he leaves his house, and with much less stuff.  So it’s pretty clear to me that his general commitment (less stuff, more simplicity) is clashing with his nerves about what’s changing.  It happens that I’ve been reading about less stuff and more simplicity because my own house is more cluttered than I would like.  One of the books I’ve read, by the wildly popular tidying guru Marie Kondo, writes about  giving attention to the things you want to keep rather than the ones you are discarding.  This gave me the idea that for my dad, it might be worth thinking through which things he has really missed in past several weeks (when he’s been away from the house).  He has had the unusual opportunity to find out what he does and doesn’t notice the absence of.

As soon as I had the idea to propose this to him, I felt a pang of anxiety.  I imagined that he would probably not want to hear about it – would probably see my suggestion as a ploy to get him to throw things away.  He would probably tell me he’d already sorted through everything and there wasn’t anything left that he didn’t need or want.  I could imagine getting defensive myself and being tempted to try to exclaim things which, while possibly true, would likely just agitate him more.

When I find myself anxious about a conversation like this in which I am planning to have in which I try to offer something to someone else, I get tempted to decide not to have it after all.  It would be so much easier to not talk about the thing.  What I have been trying to remember to do is check with myself in order to find out if the source of my anxiety is that I’m not sure I’m going to be able to get the response that I want.  If I need for the conversation to go a particular way, if I need for the person to respond a particular way, I’m setting myself up for resentment and disappointment.

As it turned out, I was indeed setting myself up for this in this case.  I was nervous because I wanted the conversation to go something like this:

Me: Dad, I was thinking, what if you…
Dad: Oh, OK.  That sounds much better than the way I’ve been thinking about it.  I’m going to make a list right now of things I think I really want and need to keep, and then we should just get rid of the rest.

I’m exaggerating for the sake of illustration.  That would have been my ideal outcome. I also wouldn’t have minded something like the following as an indication of success:

Me: Dad, I was thinking, what if you…
Dad: Yeah, I guess that makes sense.  I’ll think about it.

What I was worried about was that it might go more like this:

Me: Dad, I was thinking, what if you…
Dad (interrupting me somewhere in the course of it): I’m not going to just get rid of things because it’s more convenient for you.  I can fit most of what’s there at the house into… (and on like that with no acknowledgment of the content of what I’ve said)

If it went that way, it would feel like a failure, like I hadn’t done a good job proposing what I was proposing, or I’d chosen the wrong moment to bring it up, or he’d been a jerk about it, and it would have been a waste of time.  A failure.

But all of that is about my interpretation of his response.  If I were right that each of these outcomes meant whatever I thought it meant (success or failure), then indeed it would be a failure if I got the third outcome.  On the other hand, I could be wrong.  I have had enough conversations like this one – in which I suggest something and am met with a snarly response only to later discover that my suggestion was taken to heart in some way (likely not exactly the way I’d intended, but still taken) – to know that the way a person responds to something in the moment it’s offered is not a reliable predictor of how they will use the information or content of the conversation once it is over.  Especially if the person doing the offering (in this case me) is doing the predicting!  It’s easy to be attached to how a person responds to an offering and to let it prevent you from doing it at all, or make you wish you hadn’t bothered.  But if you can get good at checking to make sure that it’s the content of the offering that you really care about, and not whether or not you get a good grateful gracious response, then it gets less scary to approach people with sensitive suggestions or recommendations.

I talked about this with the mom who had tried to explain to her son how his silence at the post office may have been received.  She and I were able to find a few examples of situations in which she’d talked with her son about his interactions with others and responded really defensively and angrily, but then later showed in some way that he’d heard her and used what she’d suggested in navigating a subsequent situation. His response in the moment wasn’t necessarily an indication of whether or not he’d heard or appreciated or taken to heart what she’d said. If she could manage her own expectations going into the conversation, she could continue to offer him information and suggestion that he might want and be able to use.

We put together a two-pronged strategy for situations in which she wanted to offer advice or counsel to her son.  The first prong was that she would, before making a suggestion about something, remind herself that how he reacted in the moment would likely depend on many factors other than simply the content of what she said.  He may respond angrily or in a grouchy manner, giving the impression that he was rejecting the content of her suggestion when other factors might be driving the response entirely or in part.  (Maybe  the suggestion took him by surprise, or he was especially tired or still upset about something else that happened earlier in the day.)  If she felt that it would still be worth it, in case the suggestion planted itself for him in some way regardless of how he reacted, then she would proceed.

She also decided to set aside time with him for such suggestions and wait for the go-ahead from him to offer them, rather than bringing them up in a way that might feel sudden to him.  We talked about how sometimes it can feel blindsiding and invoke defensiveness to be confronted with something about your own behavior or choices when you don’t realize it’s about to happen, and it makes it much more difficult to take it in even if it’s information you’d really like to have.  So she planned to set aside time when things were relatively calm and peaceful between her and her son, and let him know that she had some feedback for him if he wanted it.  If he was up for receiving it then, they’d talk about it, and if he wasn’t, she’d offer again another time.

I decided on a similar plan for future conversations with my dad, (and other family members and adults!). With adults I think we’re more likely to ask for the go-ahead before offering advice or other input, so that part is likely easier adult-to-adult than adult-to-child, but the piece about releasing attachment to a particular response and recognizing the limits of our ability to know how something has been received by another person is no different between adults than between adult and child.

This inquiry reminded me of Adele Faber and Elaine Mazlish’s timeless book on talking with kids. Here’s a link.