Monday, January 21, 2013


Francis Su is a math professor at Harvey Mudd College* in Claremont, CA. Last week, upon receiving the Haimo Teaching Award of the Mathematics Association of America, he gave a speech to a community of educators that put into words something I have been thinking about for a while:

The Lesson of GRACE:
  •      Your accomplishments are NOT what make you a worthy human being.
  •      You learn this lesson when someone shows you GRACE: good things you didn't earn or deserve, but you're getting them anyway.
You can (and should!) read the entire text of his speech on his blog. Trust me; it is a much better use of the next five minutes than, say, reading this blog. This idea is really powerful and, though it is often relegated to the 'soft' side of teaching, is among the most important things to understand and apply in teaching and in life. Here are some ways in which this idea of grace manifests itself:

1. Dr. Steve Perry, principal of Capital Prep in Hartford, shows us on one of the TV One show Save My Son (check it out here, and keep in mind that I am somewhat obsessed with this show) that the most important step in talking with a young man who is currently on the wrong track is to establish trust by showing him that you value him as a person. For those of us who tend to dive into the issues ("I'm here to help you make better choices!"), it is great to have a model to follow in terms of simply asking what a kid is into. Do you like basketball? What's your favorite team? Do you love your mother? Where do you like to hang out? Why? Dr. Perry asks all of this without judgment, and makes it clear when he is asking kids to open up so that he can help them. Until this point, if a young man doesn't answer in a straightforward manner, Dr. Perry avoids the trap of pushing too hard too soon, and simply takes this as an indication that he has hit upon something that the young man doesn't want to discuss. In the meantime, the young man knows that he is sitting next to somebody who is giving him 100% of his attention. This means a lot especially in Dr. Perry's case: He is a principal (which I've heard can be a bit of a hard job), an author, a correspondent on CNN, a much-sought-after speaker (I heard him speak in Jacksonville) and a father of two. If you are a young person and know who Dr. Perry is, you know how important it is that he is giving you his undivided attention - but that's the catch: The young men on Save My Son don't seem to know who Dr. Perry is, and Dr. Perry himself doesn't make a big deal out of it. Just as his own accomplishments don't define him, he wants the young men he is trying to save that their actions to this point don't have to define them either.

Shawn Jackson, seen here speaking
with Secretary of Education Arne Duncan
and Congresswoman Corrine Brown

2. Shawn Jackson, the Dean of Students at KIPP Impact in Jacksonville (and one of my favorite people anywhere) is a genius when it comes to working with students and parents. Shawn can talk to a kid who is having the worst day of his life, and convince the kid that he is still a great kid with unlimited potential, in a way that gets the kid to reflect on some of the less productive choices he has recently made. Specifically, Shawn has a way of communicating to kids, even at their worst moments, that:

  • your behavior does not define who you are,
  • as a person, you matter to me, and
  • even when you are not at your best, I am happy to spend time with you.

3. Jeffrey Duncan Andrade is a high school teacher in East Oakland (see some of the beautiful and powerful stories at Roses In Concrete) who talks a lot about the need to connect with your students in an authentic way. He advocates living in the neighborhood in which you teach, and finding other ways to express to your students that you do not see yourself as better than them, or above them, in any way. He tells the story of offering a student his Subway footlong sandwich so the student wouldn't have to eat the not-so-tasty school lunch. This is his way of saying, "One of us is going to be hungry, and I'd rather it's not you." This is a segue to showing the child that you are going to be a part of his life forever, and that, as such, you're never going to give up on him. As Professor Su noted above, this favor wasn't earned - it is simply an act of grace.

4. Teachers can also exhibit grace in very small ways: In a lesson our math and science departments watched recently, star math teacher Nicky Berman of Achievement First Bushwick Middle School went through a thorough explanation, involved the class in coming to a shared understanding, and finally ended with, "Did that answer your question, Brandon?" This small addition to what was already a very complete explanation sent a very clear message to Brandon (and, by extension, to the rest of the class): I care about you, as an individual, and I want to make sure that you (not the generalized "every one of the students in this class") understand.

Here are our challenges for this week:
1. If you are a teacher, notice when you show a student grace. If you don't notice yourself doing it, try something.
2. If you're not a teacher, pick somebody who you think is feeling down and show him/her grace. This doesn't have to be fancy.
3. After this week, let us all know how it went. Did you find this natural? Did you struggle with it? What worked? What was the outcome?

Regardless of whether this goes well for you, or it doesn't quite turn out, please know, I - and those who stand with me - will still love you. 

*Fun fact: Harvey Mudd was one of three colleges to which I applied, and the first to accept me. When I opened the acceptance letter, I remember telling my mom, "Well, looks like I'm going to college!" If I had gone to Harvey Mudd, I would have likely had Dr. Su as one of my professors, and I may have learned about grace a little sooner. Then again, I might ride a unicycle. I'm not sure how I feel about that.

Saturday, January 19, 2013

A variety of opinions

What are the official Hawketalke-sanctioned opinions on contentious issues?

No, this is not a collection of my opinions. If you want those, I'm generally not very shy about sharing them - just ask. Or don't, and wait, and I'll probably share anyway.
[Spoiler alert: I share them below.]

This is a post about the value of seeking a variety of opinions, which could avoid a lot of shock and lead to better decision making in general.* However, we should be careful not to take the balancing of opinions too far, particularly in education, when we are responsible for helping young children come to a view of the world that will help them to have incredible lives. There has to come a point when you say, "look, I know there are a lot of opinions about this, but some of them are very dangerous."

Aunt Lila's Question

The other day, my aunt Lila posted on facebook:
Hello friends: I would like to hear from any of you a dispassionate, reasoned argument that will explain to me why a citizen of the United States either needs or should have an assault weapon, armor piercing bullets, or a magazine that holds more than 10 rounds. I am serious about this, and will appreciate no name calling, no sound bites, no sarcasm, but an attempt to help me understand what seems incomprehensible to me. "Because the second amendment says so" will not fulfill this request. Thank you.
This is a well-reasoned request that, absent the disclaimers at the end, could have well ended in a one-sided tirade against advocates of gun rights. Gun rights, along with abortion rights, gay rights, the role of government and, in education, the "reform" movement, are among the most contentious issues right now. When something is contentious, it tends to be extremely polarizing, putting people squarely on one side or another. There just aren't a lot of people who say, "I don't care one way or another whether Americans should be allowed to walk around with assault weapons; whatever the majority decides, I'm OK with it." And just to remove the sense that I am somehow above this and I have perfectly balanced opinions, I'll go ahead and name my opinions on the above: No, yes, hell yes and I'm embarrassed that this is still an issue in 2013, generally more, and in favor but worried about hubris and testing and politics. When I think about these issues, like most people, I don't think about them in terms of my side being equal to the other side; I think there is my side and the wrong side. Those on the wrong side must be some misunderstanding or misinformation, because how could a person reasonably disagree with me? Except the role of government; I get that one.

So we tend to dismiss people who disagree with our opinions, and tend to gravitate towards people who agree with us. We engage in dialogue about the issues primarily with people who share our viewpoints, which leads us to inadvertently (and unnecessarily) gathering confirming evidence for what we already thought (see here for my previous post on the dangers of confirming evidence in the classroom)

The Problems

The problem with this is two-fold: It affects us as individuals, and it affects society on the whole.

On the individual level, our opinions become more and more entrenched, and we are less understanding of others. We limit our circle of friends based on what would normally not be an important issue. In conversation with people we don't know well, we condition ourselves to stay away from certain topics, lest we disagree. This limits the scope of what we can discuss, the quality of our relationships, and, ultimately, our own understanding of the issues. We read news stories from outlets that share our views and bring this bias into their reporting, and we avoid the outlets that we disagree with.

For society, the effects are worse. Instead of individuals choosing to spend time with other individuals, now we have groups choosing whether to engage in dialogue with other groups. We accuse politicians of this, but politicians are just a highly visible subset of the population as a whole. We're just not talking to each other and, when we are, we're not really listening to each other.

The solutions:

If you're aiming to balance your thoughts on any particular issue, here's how you can start:

1. Acknowledge that there are other valid opinions out there. You may change your mind at some point, and that's OK. If you don't take this first step, you'll have trouble truly listening to the other side.

2. Ask questions like my aunt Lila's. Actively seek out people who will disagree with you and help you understand their viewpoint. One way I have found to get people's opinions and hear their evidence is to simply tell them that you disagree with them. Most people are happy to oblige with their explanations and evidence, to convince you that they are right, which has the benefit of giving you a window into their way of seeing the world.

3. Expose yourself to multiple sources of news, and understand that what you are seeing is someone's reality. I like to read what the Schoolsmatter blog, the most extreme and scathing critic of education reform, has to say (despite the fact that I disagree with most of it and sometimes have trouble trusting that they are truly interested in what's best for kids). Nonetheless, there are some valid concerns about education reform, and they offer some evidence to refute what I sometimes take to be an absolute truth.

4. On twitter, follow people who disagree with you. Most of the people I follow on twitter are great math teachers, many of whom are very forthcoming about their objections to the education reform movement. I also follow Alfie Kohn, who is about as opposed to standards as you can get, because some of his ideas make a lot of sense any push my thinking about teaching.

The limits of balance

As teachers, we want to present a balanced view of the world, but there are also times when overthinking this step can have a negative impact on students. Many rookie teachers, for example, tend to over-balance their viewpoints when talking to kids, leading to less authority in the classroom and a sense among kids that everything is negotiable. Beyond this, we risk teaching kids that maybe treating others with kindness matters but, hey, that's just my opinion; maybe it doesn't. This moral relativism can spark a lot of deep thought in college philosophy majors, but can easily lead a middle schooler to validate his own self-centered view of the world (says someone who named his blog after himself.) The consequences of this can be disastrous for a classroom full of kids who need to put up with this student, and for the student himself, who can spend several years not necessarily becoming a better person. At some point, you'll encourage them to look for opposing views - just not right now, when you are telling them to stop punching people in the face.

*In a future post, we'll look at how expanding your sources of information and opinions, in addition to encouraging better decision making, can actually lead to more original and creative thought.


Yesterday, while watching my wife earn her white belt at Tae Kwon Do (a precursor to the more modern Rex Kwon Do), my phone buzzed. I had a notification on facebook, from Addie Coulter, so I had to stop to think a bit.

Addie Coulter is my grandmother. She was born in 1919, the year the 19th amendment to the US Constitution gave women the right to vote. When we spoke on the phone last year, she said, "I'm 93 years old, and I wake up every day without any pain." Grammie's mother - my great-grandmother - who I only ever knew as Mom McCrumb, lived to be 102 years old, and went bowling for her hundredth birthday. Our family seems to benefit from some form of the Lake Wobegon effect, where "all the women are strong, all the men are good looking, and all the children are above average." At least the first part.

This post could go a lot of different directions:
a) the advances of technology and what they actually mean in terms of our day-to-day experiences
b) lessons from my grandmother (picking up pennies, opening doors, and authenticity)
c) how not to convince a Coulter woman to do something
d) an homage to my grandparents
...but it will go in none of those directions. My grandmother is using her iPad to connect to me on facebook while my wife breaks a wooden board in half with her bare hands and our kids play with other kids, switching effortlessly between hide-and-seek, make-me-laugh, and Angry Birds.

We live in an incredible, amazing, wonderful, confusing world.

Wednesday, January 16, 2013

Falling Down 2: Teachers Falling Down

Loyal reader "Unknown" posted a great question in response to yesterday's post about pushing our limits in a constant search for improvement:
I can understand how this applies to students - it ties in with the idea of conceptual change, as well as giving children enough "at bats" to fail in order to eventually succeed. However, does the same apply to educators? Are we allowed to employ trial and error techniques to find what works best, even if it comes at the expense of our students' educations?  
Then again, everything I do is trial and error, so this comment is almost irrelevant.
First of all, let me say that this is not at all a trivial question. It gets at the heart of what we do as educators, and at the existential crisis that many of us experience in our first few years of teaching: Am I hurting children? Am I selfishly doing a job that I enjoy (sometimes) to the detriment of the people I aim to serve?

Keep in mind that this is a profession where, every day when you leave, you know you could have done better, and that if you had done better, a child would be better off. You did not achieve what was theoretically possible to achieve with this child today. So, in a sense, you will hurt children every day you're a teacher. This sounds harsh, but if you don't accept it, you'll get upset with yourself every day for not being perfect. In fact, you can extend this same logic to every other part of your life: Unless you're eating the perfect balance of food, you're hurting your health. Unless you exercise just the right amount in just the right way, you're hurting yourself. Unless you spend all of your free time helping society in the most optimal way, you're hurting society. This assumes, of course, that the world would be better off with a perfect person in your place. This is logically true, but I don't think it's a reasonable way to live your life. Instead, I recommend measuring the good you doing versus the good that you are capable of doing in the moment. Nobody can fault you for being in your, say, first full year of teaching, but they can fault you if you're not trying as hard as you can. In other words, for your part of the relay, you don't have to be the fastest person on the team; you're just giving it your all and going for a personal best every day.

But what does 'trying as hard as you can' look like? Does this mean pushing yourself to grow, or does it mean serving kids? What is the balance?

Here I will explain why I believe this question addresses the inevitable, give a few suggestions for how to minimize the negative impact of your own learning process, then re-frame the question as a matter of how to maximize student learning over the course of a year.

First of all, let's recognize that, even if we wanted to, we couldn't escape the trial-and-error nature of our jobs. Circumventing the trial-and-error process would require the following elements to be present:
#1 there exists a strong body of knowledge about what good teaching is,
#2 this body of knowledge applies predictably to the group of students you teach, and
#3 acting on this body of knowledge will require no trial and (more importantly) error.

I believe #1 is more true than we generally recognize, and more differentiated by subject matter than we usually discuss. #2 is tricky; every class is unique, and every day is different from the previous day, but I've generally found that good teachers can apply what they know to new classes on new days. #3 is largely a figment of our imaginations. I just don't think there's a way for us to learn that doesn't involve a lot of pushing ourselves what we're comfortably able to do. Isn't that what learning is?

Minimizing the negative impact of your own learning process

But what are some ways to minimize the negative effects of our own inexperience on children? This is a question we already have some helpful answers to:

1. Practice your lessons prior to teaching them. In doing so, you're making mistakes and learning from them without actually affecting students.

2. Perform a controlled experiment: Mitigate the negative effects of something you're working on in a lesson by changing only the one thing you're working on, and keeping everything else about your teaching the same. This helps you to see the effects of what you're working on, and has the added bonus of not allowing the one thing you're working on to ruin the entire class because you kept everything else the same.

3. Hedge your bets: Assign high-quality homework, allow students to read high-quality literature (fiction and nonfiction alike), and keep in mind that your class is likely 45 minutes out of a much longer day. Contrast this with a self-contained classroom teacher just starting out, whose learning process could potentially have a negative impact on students for an entire day at a time. Think of trying out a new recipe; The mahi mahi may turn out incredible, or it may fall flat, but either way you've made your trusty mashed potatoes and corn on the cob, so something positive will come of the experience regardless. As a teacher, you're going to try new things, but you should also keep doing the regular things that are working for you.

Year-long impacts of teachers taking instructional risks

While this began as a discussion of a very practical question, I want to address a more theoretical question: What is the impact of instructional risk-taking on student learning, over the course of a year?

First, let's make the following assumptions:
  • 70% of new ideas are simply not going to work. These are dropped. The other 30% become "strong" instruction add significantly to student learning and are adopted, replacing "safe" instruction.
  • It will take a week of developing a new idea to determine if it is going to work. It takes a month for this idea to become an exemplary practice.
  • Class time will be allocated, in percents, to the following categories, with the assigned number of learning units:
    • Safe: 5 learning units at the beginning of the year
    • Stretch (trial and error): 2 learning units
    • Strong: 10 learning units
Next, I'll introduce you to three teachers:

Teacher A is a rookie teacher who is very risk-averse. She puts herself in situations where she aims to make as few mistakes as possible over the long run. She will still make mistakes, but those mistakes will be spaced out and not catastrophic. Her learning will reflect this; she will not learn very quickly. You have likely met a teacher like this. In this teacher's class, there is no group work because kids might get off task; there is no call and response because kids might be silly. There is plenty of independent reading and responding to questions in writing, because the risk inherent in these activities is relatively low. She stays steady at 95% safe instruction, 5% stretch instruction, throughout the year.

Teacher B is a rookie teacher who likes to break out of her comfort zone a bit. She likes trying new things and is not afraid to make mistakes. She tries out group work, and sometimes it doesn't work. She has kids explore things she has placed around the room, and it doesn't work. She tries a new game and it works. She expects that the probability of the things she tries working is less than 100%, and is not discouraged when something doesn't work out. This teacher begins the year at 80% safe instruction, 20% stretch instruction.

Teacher C* is a total loose cannon. She tries something new every day. She alternates between excited and disappointed. She wants to sing a song to teach today's lesson, but tomorrow she wants students to generate their own knowledge by grappling with difficult concepts and humming to themselves. She starts a blog and wants kids to spend all class writing in the blog once a week. She wants to try full-on science labs tomorrow, despite not having ever taught this way. She tries Total Physical Response the day after she hears about it. She has kids act out plays that they have written about the life cycles of plants. She has kids respond as if they were members of various plant species. Pen pals! Plant pals! Dress up like a dinosaur and roar your answers! In short, she has a lot of ideas and wants to try them all, also realizing that some will work and others will not. Like Teacher B, she is not discouraged by failure. This teacher begins at 100% stretch instruction.

Which teacher's students are going to learn more?

Let's see what happens in the first four months of school:

According to this model, Teacher A's teaching led to 19.85 units of learning (the most), Teacher B's teaching led to 19.4 units of learning, and Teacher C's teaching led to 8 units of learning (the least, by far.)

First, some clarification: Why didn't Teacher C develop any strong practices? Wasn't all that trial and error helpful?

As it turns out, Teacher C has a fatal character flaw which does not allow her to learn from her mistakes. She does not test her variables in isolation, so she doesn't know what is working and what is not. She also lacks the discipline to stay with the same idea for long enough to develop it into a strong practice. So instead of embracing her failures, she quickly moves onto the next new thing. This is very bad. Learn from the cautionary tale of Teacher C. Be the skier who points her shoulders downhill and gives herself permission to fall; do not be the skier who jumps from the ski-lift.

Next, you'll see that Teacher A, the teacher who played it safe, had more net learning in the first four months of the school year than Teachers B and C (though the difference is almost negligible.)

Interestingly, the data from Month 1 tells you which approach is better for students in summer school, for example: Under these assumptions, if you want to serve kids on a one-month basis, don't get too creative with your teaching. Stick with what you know and kids will learn more than if you tinker.

However, most teachers don't have one-month or four-month contracts.They teach for about 9 months:

As we suspected in this highly contrived example, Teacher B's students will learn more. Take a healthy amount of risk, but be thoughtful about the amount of risk you take on at any given time. Your kids will end up learning more as a result.

Plus, I can see two more benefits to teaching in a slightly riskier way:
  1. You are modeling for kids that taking risks is OK. This will help them learn the all-important habit of pushing themselves, even when this is not leading to immediate academic learning.
  2. You are developing yourself as a teacher, which means that you will have a greater effect on the learning of future children. 
...and that can't be all bad, can it? I can't imagine anything better than knowing you will add even more value in the future, and helping kids to internalize one of the most important habits that will serve them for the rest of their lives: how to learn by making mistakes.

*None of these teachers are actual people, of course; any resemblance to actual teachers is pure coincidence.

Monday, January 14, 2013

Falling down

1. A Day at the Ice Rink

Yesterday, we went ice skating, because apparently that's what people do here in the wint'ry North. Sebastian was essentially carried by an adult while he did a high-speed version of a Chandler wobble (which is a real thing.) Barbara fell a little. Maria and I did not fall. A good time was had by all, and this picture serves as a pretty accurate look at how most of the day went:

Let's focus on Barbara's skating experience. She skated for about an hour total.

For the first thirty minutes, Barbara did not fall once. She stayed on the wall, inching her way along, occasionally venturing out a few feet from the wall for a few seconds.

Once she gained enough confidence, she started to spend more and more time away from the wall. She started skating a little faster. And she started falling.

I will argue here that this is exactly the right way to go about learning to skate. And it's precisely why the learning curve is so steep at the beginning of learning to do anything. How can we apply this same principle in the later stages of our development? Simply put, we can accept the first rule of bicycle riding: Riders fall. If you're not falling, you're not really getting better, at least not as quickly as you could be.

2. Put Your Shoulders Down

In Practice Perfect, Doug Lemov, Erica Woolway, and Kate Yezzi describe a skier who realized that she had hit a plateau by not pushing herself to fall. This excerpt from their article says it all:
We know a woman who is a breathtaking skier. She tells an interesting story about her breakthrough moment--and it was just that, a moment--when she started down the road of becoming an expert. It happened on the day she decided to fall. She was getting on the lift at the base of a steep, sunlit ski bowl. She had just come down a twisted, mogul-ridden trail in top form, earning the admiration of a teenager who'd been trailing behind her. At the bottom, amidst words like "stoked" and "killer," the teenager asked, "Do you ever fall?" 
Getting on the lift, she realized that (1) the answer was no, and that (2) if the teenager had been a nephew or a cousin whom she felt invested in developing as a skier, she wouldn't have wanted to admit that to him. Instead she would have pointed out that if you never fall, you aren't pushing yourself and you aren't improving as fast as you could be. Midway up the mountain she realized that she hardly ever fell, perhaps once every eight or ten days on skis, and even then it was usually at tangled moments when she wasn't actually skiing that hard. She realized that if she wasn't falling she probably wasn't pushing herself to learn as hard as she could be. She had gotten lazy because she was so good. 
When she got to the top of the mountain and skied off the chairlift, she knew what she needed to do. She set out to ski hard enough to fall, but she was intentional about how. She knew that there was one thing that she had been working on: pointing her shoulders face down the mountain, no matter how steep. She then set out to execute this skill even if that meant falling. She fell three times that first day. "I could feel myself trying to do exactly the things I was afraid of. I knew if I stuck with it I would conquer my fears." She began skiing without fearing falling. Within a few weeks she was a different skier entirely.

This is powerful for any of us who want to get better at something, and is a great reminder that failure is the only way to know we're pushing ourselves hard enough.

3. Thin Value Bets

There was a time when I played a lot of poker. It's OK, Mom: Nate Silver played a lot, too. In poker, let's say you have a good, but not a great hand, and all of the cards have been dealt. Conventional poker wisdom says that, if you are the last to make a decision, and it has been checked to you, you check behind, because your opponent will always call your bet with a better hand (meaning you threw away money unnecessarily), will sometimes call your bet with a worse hand, and will sometimes fold to your bet with a worse hand. The latter case has no effect, so we'll remove it from the equation. The mathematics of this situation basically ask you to calculate how much money you will make with this bet (referred to as a thin value bet) vs. how much money you will lose when you are called by a better hand.

Most poker players do not make these thin value bets, just like most decent poker players do not try to bluff their opponents excessively. both of these plays are very risky and can potentially cause one to lose a lot of money. But the best players play the game very differently. They are notorious for making more value bets than anybody else, for bluffing in a wider variety of situations, and for being less predictable in general.

All of this comes about because they are not afraid to fail. They are more afraid of leaving value on the table, missing out on what could potentially be an increase in profit over the long run. In short, these world-class players become world-class precisely by preferring to find the limits of their abilities, rather than accepting them.

4. Testing the Ice
(in which this post comes full circle, right back to ice.)

My favorite story to tell students has the same message, but is phrased as a cautionary tale: The only way to know how far you can take something is by taking it too far. In other words, you only know where the line is once you've crossed it.

I was about 10 years old, and my job was to pick up dog poop in our back yard, put it into a plastic bag, and throw it away. We had a pool in the yard, and it had developed thicker ice than I'd ever seen, so naturally I was curious as to how much weight it could hold. I threw a ball onto the ice, and it bounced. I put a little of my own weight, and it still held. Little by little, I put more and more of my weight on it, with the idea that maybe - just maybe - I could walk on the ice. I continued to put weight onto the ice until, in an instant, I fell through, bounced off of the bottom of the pool, and somehow found myself inside. Our cousins were visiting, and everybody else had just taken hot showers, so there was not hot water. I remember my mom heating water over the stove and taking it to the bath so I could warm up. Thank you, Mom :)

I still wonder what happened to that dog poop...

So I learned this important lesson, but it wasn't until recently that I could make sense of the positive corollary, described above. We have to push our limits not only to see what we're capable of, but to ensure that we are pushing ourselves to expand these limits.

Happy falling, everybody.

Sunday, January 13, 2013

Building habits through conversation

Sebastian and I went to Dunkin Donuts this morning. Some families have pancakes every Sunday; some make quesadillas. Some families go to church. We do none of those things, but what we do do (tee hee) is, on occasion, go to Dunkin Donuts. I will usually take one kid (more often than not, it's Barbara, who is awake first) and we'll use the opportunity to chat. Today, my conversation with Sebastian got me thinking about best practices in conversations with kids. Though Sebastian is five years old, I think these guiding principles apply to kids of just about any age:

1. What you talk about is what you care about.
2. Ask for evidence.
3. Encourage divergent thinking.
4. Right is right.

There are many more good ideas out there, and there are shelves of books about this (this one being my favorite), but this is just what came out of this morning's trip to what some people refer to as 'the finest purveyor of caffeinated beverages within two miles of my house.'

I also approach this as a teacher who became a parent, so this is largely based in what I appreciate the most about many of the kids I've taught and what they've been able to add to the school communities I've been able to be a part of.

1. What you talk about is what you care about.

Objectively speaking, our kids are adorable. People are always telling our daughter how cute she is, or how much they like her shirt, or that she is dressed so well (obviously not when I dress her.) The downside of all of this attention to her looks is that she comes to believe that appearance matters more than anything else. On the flip side, whenever our kids are talking away, they are often complimented for being 'smart.' I can't affect what people say about our kids, but I do cringe when people say it, so I will usually follow up with a re-framing comment of my own: "They're right, Barbara, you're making a lot of connections!" or "Sebastian, that is a very interesting question." I want our kids to care about thinking, and how they treat people, and just about anything other than how their looks or dress are perceived by others. I want our kids to care about interesting aspects of our world, about other people, and about constantly becoming better in meaningful ways, so this needs to be what we talk about more than, say, their looks.

Barbara's independent reading level was just moved up to an "I" on the Fountas & Pinnell scale; Sebastian's was just moved up to a "G." We talk about all of the reading they did that helped them get there, and we continue to reinforce how great it is that they can now read more interesting books as a result. Instead of calling them smart, we call them hard-working and curious. Those two areas, together, are probably the biggest levers within their control to improve their intelligence anyhow.

2. Ask for Evidence.

We have enough people in this world spouting forth opinions without anything to back up those opinions. I want my kids to grow up to be the kinds of people who think about the world scientifically, who can think about the world and formulate opinions based on what they observe, who look to prove positives and think about the world probabilistically. Basically, if Fox News is still around when my kids are older, I want my kids to laugh at it.

Besides, focusing on math for a second, most higher math is predicated on the idea of rigorous proof, the beginnings of which can be discovered in simply asking how we know something must be true. For now, I'm accepting any anecdotal evidence, but there will come a time and a situation that calls for a little more logic, and I hope I'll be able to take advantage of this situation when it comes.

This morning, at Dunkin Donuts:
Sebastian: "I know what that says! It says "PICK...UP...HERE." [He speaks very loudly.]
Me: "What do you think happens over there?"
Sebastian: "I don't know..."
Me: "How could you find out?"

Sebastian goes over to the counter, snoops around a little bit, and comes back to me.

S: "Maybe it means people work over there."
M: "Why do you think that?"
S: "That's because I see people working behind the thing."
M: "What thing?"
S: "That thing over there."
M: "The counter?"
S: "Behind the counter."
M: "Let's wait and see."

We paid for our fifteen dollars worth of beverages and sandwiches and went over to where the sign was. I asked, "Why are we coming to this area?"

S: I don't know.*
M: Look at the sign; what do you think we're supposed to do here?
S: I don't know.
M: What does the sign say?
S: Pick up here.
M: So why did we come to this area?
M: Let's see if that's true. How will you know if you're right?
S: If they bring the food, that means we are right! [Notice how he spreads his prediction over both of us, cleverly hedging his bets.]

I then picked him up and we laughed about my interpretation of the sign. He said, "that's not what it means, but thank you for holding me, Daddy!"

Such a sweet kid.

3. Encourage divergent thinking.

This one goes by a lot of names.Some people talk about avoiding yes/no questions; others refer to open-ended questions. I say we should just try to capitalize on the fact that kids aren't tied to "what makes sense" - let them dream a little, and encourage them to, as Robert Kennedy said, "dream things that never were and ask, 'why not?'" So once you have an answer, ask for another answer. Ask a lot of questions about why things are the way they are, and don't stop at one answer.

Barbara: [points to a snowblower] What's that for?
Me: What do you think it's for?
B: I don't know.
M: Well, what could you use that for?
B: Maybe you could use it to dry your hair.
M: How would that work?
B: The hot air comes out of that part.
M: OK, what else could you use this for?
B: Maybe you can put food down that tube so animals can't get it.
M: What else?
B: Maybe you can blow into it and it makes noise?

...a few weeks later, we saw one working, and Barbara was excited to see what this machine actually does, which is pretty impressive to 6-( and 32-)year-olds who have never seen a machine shoot a stream of snow 20 feet in the air.

The key here is to help kids become creative, divergent thinkers. Dan Pink writes about the need for this in A Whole New Mind, and makes the case that this is a necessary way to develop into the kind of person who will be more employable in the years to come. Pragmatism aside, I like that it makes the world more interesting.

4. Right is right.

Via xkcd:

Grandpa, what was it like in the Before time? "It was hell. People went around saying glass was a slow-flowing liquid. You folks these days don't know how good you have it."

Doug Lemov has coined the term 'Right Is Right', which distinguishes between something that is partially right and something that is 100% right. Nowhere does this apply more than in how kids use words to communicate, though it is also a helpful way to deal with not-quite apologies and almost-following of directions like "please turn off the TV quickly."

At Dunkin Donuts, the counter is called a counter. This word is not pronounced "supposibly." Winter is not the time when we are farthest away from the sun (at least here in New Haven.) The topping on the donuts is not called sparkles, but "sprinkles." In our house, we get do-overs when we don't say what we mean the first time. We make a lot of mistakes, and we learn from them. The key is to get it right, and walk away having gotten it right at least once. Though this may run counter to the idea of encouraging divergent thinking, I believe that they go hand-in-hand quite nicely. In order to understand the context of your creative ideas, you have to have a solid understanding of how the world actually works right now. In order to communicate your ideas, you need to use words that mean roughly the same thing to you as they do to your audience.

Anyway, this is important not only because of the power of getting things 100% right, but because it reinforces the all-important growth mindset. After all, nothing shows you that you can learn from mistakes quite like learning from mistakes all the time.

What about you? What do you think are the most important other things to keep in mind when talking with kids? What are your favorite ways to help your kids turn into the people you hope they become? What else do you try to avoid?

*If you have gotten this far, you are a loyal reader. What are your thoughts on quotation marks vs. no quotation marks? I can't locate my Chicago Manual of Blogging Style.

Thursday, January 10, 2013

Sharing the peanut butter sandwich

The other day, a great blogger named Christopher Danielson, who thinks about units and place value on a level that I hope to reach some day, posed an intriguing question:

I like this question because it asks us to get at the heart of how kids (and adults untainted by math instruction) conceptualize numbers. I responded that I thought 1 1/2 made more sense because 3 sixths would require an implicit cutting of all of the pieces of what I interpreted to be the quesadilla (because, really, who splits a cookie into thirds? Who shares cookies at all?) into six pieces, in order to share them equally. I applied what I thought Barbara and Sebastian would do, which is to take one piece each, then split the remaining piece in half.

I asked Barbara and Sebastian this question, using a sheet of paper I'd cut into thirds, and presented it like this:

I'm going to take this paper and cut it into three equal-sized pieces, and then you and I are each going to take the same amount of paper. This is called 1/3, this is called 1/3, and this is called 1/3.

Barbara gave herself two of the pieces and gave me one. She was done.

I made a face and told her this wasn't fair. She switched so I had 2/3 and she had 1/3. I made another face and she said, "or, we could rip this one." She ripped the third piece, so now we each had one of the original 1/3 pieces and half of the remaining 1/3 piece. I asked her how much she had and she said I have one piece, and a smaller piece.

I asked her what the smaller piece was called and she was stumped. "A little piece." I reported that this was the answer our kids gave, committing the sin of assuming understanding based on the most advanced student's response.

I tried the same thing with Sebastian today. Though there are some highlights, my main takeaway was that I wanted to have a conversation about math, making it relevant by using the fact that he'd just asked me for a peanut butter sandwich, but he really just wanted a peanut butter sandwich. When talking math with kids, timing is everything:

Some thoughts:

1. Both of our kids first thought that the best way to share was to take 2/3 for themselves and leave me 1/3. This is either an indictment or a triumph of our parenting style.

2. This moment with Sebastian, feeling silly and hungry, reminded me that often, when we think we're talking math with our kids, they are really more into the surface structure of the task. The underlying work with number sense, mathematical problem solving, and struggle to name fractional amounts take up no more than 25% of Sebastian's working memory, the rest of it filled with 5-year-old versions of ideas like "why can't you just make your own damn sandwich?" and "now can I eat it?"

3. After thinking about this for the last two days, and trying this with both Barbara and Sebastian, I am struck by the ease with which each of them decided to split the remaining piece, and the lack of vocabulary to describe the precise amount that each piece represented. I was also struck that, by essentially telling them that each of the three pieces was 1/3 of the whole, I was naming a new unit for them, giving a name to what they otherwise would call a "piece." I don't know enough about early childhood development to describe this precisely, but they are at a stage where they count things. I could have called each third a whatsit, and they would have described the final amount the same way: A whatsit and a smaller piece. The term 1/3 means nothing as of right now, but the concept of splitting things into equal-sized parts makes a lot of sense. Not in some abstract sense, but in a very real, rip-the-bread kind of way.

4. Another take-away for me was that the reason we wouldn't accept either kid's answer beyond a certain age is that, at least in math classes, we expect every part of an amount to be expressed in terms of a single unit. If the unit is the whole sandwich (or quesadilla, or piece of paper), we want to know how much of it we have. If the unit is a third, we want to know how many thirds. We don't want to know how many thirds and how many other pieces, nor do we ask for the number of thirds and fourths. Egyptians didn't write fractions this way back in the day, expressing fractions as the sum of unit fractions, but I'm not taking this as evidence that they actually perceived fractions that way. I have a hard time believing that the pharaohs really conceived of Sebastian's original partition (2/3) as 1/3 + 1/4 + 1/12, meaning "OK, let's take one of these original pieces, mark it for later, put the original back together and cut into four pieces, take one of those and mark it, then put the original back together and cut into twelve pieces, mark it, and take the marked pieces. No, wait; that's not even. I don't know how they viewed fractions conceptually, but I know that our current representation differs from how both of my kids (small sample size, grew up in the same home, but still) view fractions, and I imagine this concept/notation gap is not unique to our household or culture.

The same-unit idea is a new one for me, though. It got me to thinking that we don't say things like "the pencil was five inches and three centimeters long", but we can say, "the table was three feet, six inches tall" because inches and feet share a common unit (the unit either being the inch, decomposed from feet, or the foot, composed from inches.) For what it's worth, I think "three and a half feet" very clearly refers to the foot as the unit, whereas "three feet, six inches" uses the inch as the basic unit. But at some point, kids realize that an answer to "how much of the quesadilla do you have?" should not be answered with "one piece of this size, and one piece of this unrelated size."

5. Perhaps most importantly, I need to spend more time talking math with my kids. Not because they need to get ahead, or because of increasing global competition, but because it lets the people I love the most fall in love with the subject I love the most, and makes us all think in the process.

Tuesday, January 8, 2013

The Rigor Revolution, Revisited

First of all, I recently learned that I have twice increased the readership of this blog over the last two days. Most recently it was doubled, which was nice, but the day before it became INFINITELY more popular than it had been. Sorry; you just can't do any better than that. So to all (both) of my dear readers: Thank you for reading.

This article from Popular Science details the story of Taylor Wilson, a young genius (or, in Workaholics parlance, a "human genius") who managed to achieve nuclear fusion at age 14. My favorite quote from the article is the last line of this paragraph:
The Davidson Academy is a subsidized public school for the nation’s smartest and most motivated students, those who score in the top 99.9th percentile on standardized tests. The school, which allows students to pursue advanced research at the adjacent University of Nevada–Reno, was founded in 2006 by software entrepreneurs Janice and Robert Davidson. Since then, the Davidsons have championed the idea that the most underserved students in the country are those at the top.
First of all, I know I'm looking for confirming evidence, not taking my own advice from this last post. That notwithstanding, I'm increasingly convinced I am on the right track. Before laying out my best arguments, let me first provide a little background into what I see as the heart of the issue and a proposed solution:

We currently expend the vast majority of our resources (time, money, and energy) pursuing the ends of teaching as many of our kids as possible something that will further their education (AKA "teaching to the middle") and teaching the kids who need the most help as much as possible, with the hope that they will eventually join the middle and be able to learn alongside them at a similar pace (AKA "bringing up the bottom.") The goal of this is to provide as many kids as possible as good an education as we can muster, which is noble, but which I believe ignores some of what we have learned about teaching and learning over the last hundred years. I honestly think we stick to this model both because we care about test scores (and test scores are measured by 'percent proficient' or, at best, a mean level of performance) and because we are good people with good hearts, and we got into education because we believe we can make a difference. Nothing tells you you're making a difference like helping kids who couldn't read learn to master difficult texts, or taking a kid from 3x4=No idea to creative problem solving and factoring polynomials. Plus, if we really want to put kids on a better life path, we owe it to them to bring them up to speed as quickly as possible, right?

All true.

I just don't believe this is really the most worthwhile goal to pursue. Yes, it's noble, and yes, it feels like the right thing to do, but it's not what I think we're all really trying to do.

Part of our mission (my favorite part, by the way) states that we aim to "serve as the next generation of leaders in our communities." Though I definitely believe that leadership happens at all levels, and that even mundane tasks offer us all a chance to lead by example, this is not what is meant by our mission statement. We aim to help our kids become the people who will change the world for the better. In science, we want to influence the Taylor Wilsons of the world. In politics, we aim for incredibly bright, thoughtful, caring people at the top, like Barack Obama (one of the benefits of a limited readership is that I can post this and you will both know what I mean...and neither of you will yell or break your monitor.) Consider the extent that the Obamas and Wilsons of the world can make real change happen. Now consider the change that is going to be created by people who nobly put their nose to the grindstone and managed to earn an Associate's Degree before working at a job that allows them to provide for their families. Both are doing a lot of good, but only one of these groups can truly be considered to be the next generation of leadership.

We don't even have to wait that long for the positive effects of better teaching our kids at the top. They can continue to learn at an increased rate, once they are provided some direction and the freedom to keep learning. They can help illuminate concepts that other kids may not understand as easily. They can bring expertise in a single area to help other kids make connections that are currently absent from our classroom discourse.

I'm not talking about creating another Davidson Academy. And I'm definitely not talking about pulling support away from the kids who are the furthest behind. What I'm talking about is changing the ratios a bit. The way I see it, two things have to happen at the same time. The first is that we need to start teaching in a more constructivist way, and in a way that allows our kids to all learn from each other and benefit from the collective wisdom in the room. As long as the teacher is the keeper of knowledge, we're not going to benefit as much as we could (at least not for a long time) from a small group of kids suddenly knowing a lot more things.

If we group our kids into Rafe Esquith's three groups (Kid As are the 20% at the top, Kid Bs are the 60% in the middle, and Kid Cs are the 20% at the bottom), I'd say we currently allocate about 10% of our resources to Kid As, 50% to Kid Bs, and 40% to Kid Cs (or, in my case, 10% to A, 30% to B, 50% to C, and 10% to blogging about it.) I think we have it backwards, and I think our Return On Investment (ROI) for energy spent on Kid As is much more than our ROI spent on Kid Cs, both in the short term (through their contributions to the learning of Kid Bs and Kid Cs) and in the long term (through their contributions to society.) In fact, I think we can lose no net learning from Kid Bs by fixing the formula so we're still allocating 20% to Kid Cs, but now 40% to Kid Bs and 40% to Kid As.

Can you imagine if our top kids actually got twice as much attention as our kids at the bottom? I don't mean the kind of attention they already get more of, in terms of getting not-so-cold called for tough questions in class that we know they can supply the correct answers to. I mean actually designing tasks that will push them forward academically...planning separate lessons for our high flyers, creating projects tailored to their interests, allowing more time for individual 'tutoring' to push their thoughts on certain issues and give them feedback, which they are more likely to take and run with anyway.

I'm not the first one to think of this. This idea is often championed by misguided teachers who just want kids who are super compliant, get all the right answers, and require little-to-no teaching in order to reach whatever minimum bar of proficiency has been set by the state. These are the teachers who just wish all their kids were like little Jamie, smart girl that she is. These people should probably not be teachers.

But this idea is also at the heart of what makes the Success Academies across New York so impressive. I don't mean impressive like OMG LOOK AT THEIR TEST SCORES (though they're super high); I mean impressive as in THIS IS THE SCHOOL EVERY PARENT WISHES THEIR KID COULD GO TO. When I visited Success Academy Bronx 2 in December 2012 (last month), the principal told us that she had to talk her teachers (most of whom, like her, have backgrounds in special education) out of spending most of their time with the most struggling scholars. Instead, she had them push the kids at the top. Once those kids were set and learning faster than ever before, the teachers had much more time to work with the kids who struggled the most - now unburdened with the idea that they were stagnating the kids at the top so that they could help those who needed the most help.

This is my real hope. I don't know that it's sustainable to spend half of our energy on the 20 percent at the top. But what if we did this for 2-4 weeks? Could we get kids at the top learning, and set up to keep learning for another 2-4 weeks while we intensively helped the kids at the bottom? What would we lose if we tried and failed? What would we gain if we tried and succeeded?

To me, this seems like a risk worth taking. Nassim Nicholas Taleb describes this situation as antifragile, and  posits that we have much more to gain than we do to lose from taking this risk. I personally don't think we have much to lose from trying this out, but we do have a lot to lose if we try this without thinking it through: We could potentially lose the permission to try again.

If you have any ideas about how we could go about designing this experiment, or how you could see us allocating our resources differently for a short burst of time, I'd love to read about them in the comments.

Sunday, January 6, 2013

Checking for Misunderstanding

You are teaching young kids about shapes. You explain what a triangle is, draw some triangles, then explain what rectangles are and draw some rectangles. You then decide, because you are a good teacher, that you should check for understanding. You draw a triangle and ask, "what is this called?" and you even follow up with "how do you know?"Or, you have recently gone to a wicked awesome PD session about rigor, and instead you say, "who can come and draw a triangle on the board?" - because you want kids to APPLY what they've learned.

In either case, kids call the triangle by name, tell you it has three sides, and/or draw a three-sided figure that you would call a triangle. So - good news - they got it!

The issue is that you checked to see if they understood what you had said, instead of checking to see if they DIDN'T understand what you said. You have fallen victim to your own confirmation bias, which leads to you seek data to confirm what you already believe (in this case, that your kids have learned what a triangle is.) 

Don't feel bad, though; this happens all the time. We look for evidence to confirm what we believe all the time, so much that it affects our perception. In a classic study, linguists played tapes of people from northern Michigan and then asked Michiganders what they heard. The researchers told the control group that the tapes were of speakers from Michigan. The experimental group, however, was told that the speakers were Canadian. Shockingly, the groups actually heard the speakers differently. The control group clearly heard "out" where the experimental group heard "oot." The researchers even reported the experimental group mocking the speakers for the thickness of their 'Canadian' accent. If our hearing is affected by what we expect, it should come as no surprise that our professional judgment as teachers can also be negatively affected by our expectations.

My solution is to reframe checks for understanding as checks for misunderstanding. This is what many great teachers already do instinctively, but I think this framing is helpful for those of us that don't always do this naturally. 

In the triangle example above, think of a way to probe to see where kids have misunderstood you. Perhaps they think triangles and rectangles are just two different names for all closed figures. Perhaps rectangles have more than three sides? Perhaps triangles have three sides but need not be closed? 

To address these possible misconceptions, I could imagine having all kids draw something that is almost a triangle but not quite, and to explain what would need to change for this to be a rectangle. I could also imagine giving kids an impossible task: Draw something that is both a rectangle and triangle at the same time.

In any case, these tasks seem more likely to get incorrect answers, which is really what you want. You shouldn't ask questions to confirm that every kid "gets it"; instead, you should ask questions to try to draw out misconceptions or misapplications of what has been learned.

Let's try another math example: You have taught the definition of prime numbers (common definition: a number that has exactly two factors) and have given examples. 

Checks for understanding: So is 17 a prime number? Great; what about 18?

Checks for misunderstanding: Is 9 a prime number? (students could reasonably think 3 and 9 are the factors of 9.) Given that 1x5, 2.5x2, and 0.2x25 all yield a product of 5, what kind of number is 5, prime or composite?

Note that this is different from (though sometimes concurrent to) the concept of higher-order thinking or task complexity. The key, however, isn't necessarily to make things more complicated, but to give students a chance to misapply what they (reasonably) believe to be true.

I would love to collect more examples of checks for misunderstanding, from any discipline. Leave me some goodies in the comments if you have a chance!

Path Dependence

Q: Why are we so scared to change our default model for math instruction from direct instruction to something a lot more constructive?

A: Path dependence.

Wikipedia defines path dependence as follows:

Path dependence explains how the set of decisions one faces for any given circumstance is limited by the decisions one has made in the past, even though past circumstances may no longer be relevant.

Path dependence explains why computer manufacturers still use the QWERTY keyboard, and explains why the software we use dictates so much of our lives.

Path dependence is rampant in our schools - our discipline systems themselves include several path dependencies...but that's not what interests me the most. What I find most fascinating right now is our generally consistent approach to math education.

Here's the genesis of math education at No Excuses charter schools, as I see it:

The Setup: For years, math education had become increasingly 'progressive', asking kids to think about concepts deeply. They grappled with difficult problems. However, the thought that direct instruction would stifle their thinking or creativity stopped them from being directly taught some key math concepts or skills. Parents were frustrated: How could it be that their 4th grader didn't know how to add fluently?

Some teachers ignored the new wave of math instruction and just taught kids the way they knew best. One of the best such educators was Ms. Harriett Ball. She taught Dave Levin, who, together with Mike Feinberg, started KIPP. KIPP immediately saw incredible gains in math scores on the Texas state test. Dave went to NYC, where he started a school that got similarly high results.

This style of teaching was very heavy on the basics.I remember from the early-ish KIPP days that we used to share chants and songs and tricks, all with the goal of increasing our kids' procedural fluency. There was also an emphasis on "critical thinking", and though we spent a lot of time on it, it basically boiled down to a lot of word problems. We tried to make them relevant, though many ended up as pseudocontext with our own kids' names added to show them we cared about them.

(Aside: I still think most schools I have seen outside of No Excuses schools generally under-emphasize procedural fluency; it is certainly a key to understanding math down the road, as it frees up working memory to think at higher levels and make connections with new material.)

The gist is this: Our schools have distinguished themselves through the use of some form of direct instruction to fill gaps that were not being filled by other schools. Parents wanted our schools because they knew we would teach the basics. We have earned some accolades based on this style of teaching. It has served us well.

Zoom forward to 2013, and here's where we are:
Students at No Excuses schools still score relatively high on math tests. Though there is a very low sample size, I think it's safe to say that this is clearly the case more in states with easier math tests than in those states that moved toward a more rigorous assessment early on. At our school in Florida, we were not prepared for the rigor of the FCAT. Schools in Tennessee struggled when the test became more rigorous. Minneapolis struggled. KIPP Lynn seems to be a positive outlier here - their math scores are still good, partly because they have consistently focused on high-quality instruction.

The Common Core is coming. Rigor is going up across the board.

So we're trying to change up our instruction to meet these demands. It's great, and it's definitely best for kids. After all, how many people actually still believe a direct-instruction approach to math is really the best way to teach? Not many.

The problem is that we're going through some discomfort. We're afraid of what happens when our kids don't master the basics. We're afraid we'll end up like the schools our parents were running away from so many years ago. We're uncomfortable, because we've never taught like this before. A lot of this thinking is path dependent - because we began in states that emphasized low-level fluency, because we've developed curricula in this way, it feels very odd to depart from it.

But we shouldn't let this stop us from doing what we think is right. There are great examples out there - probably MORE great examples than there ever were of direct instruction lessons. We have Dan Meyer, Fawn Nguyen, and Kate Nowak. We have a whole community of teachers that has been thinking about this longer than we have. They tweet, they blog, and they share. We have tasks that are created to bring out the standards in Illustrative Math, and, as always, we have Marilyn Burns, Marcy Cook, Magdalene Lampert, and Deborah Ball.

This journey may not be easy - there will be a lot of bumps, and we'll all feel we're not quiet as good at our jobs as we once felt we were - but it is necessary. In the end, I look forward to the day where kids love math for math's sake, where the joy in our room reflects the joy of our discipline, and where we can bring the small pieces we've done very well and embed them in a rich, thought-provoking curriculum.

Saturday, January 5, 2013

Talking math with kids

One of my favorite things to do is talk math with my own kids. Today, Barbara asks me how old I'll be when she is 10. I love this question and ask her what she thinks.

She has no idea, so I ask, "how old are you now?" 6.
"How old am I now?" I don't remember.
"I'm 32."

She is silent for a while, then says 41. I ask her how she got 41 and she says, "I counted." I ask how and she gives me a vague answer. I wish I could go back in time and probe more here - this is a good -like situation, and I wish I could have asked for direct evidence of her thinking.

Instead, I say, well it's not right, so let's try a different way. She wants to get it right. I say again, "How old am I right now?...and how old are you right now?"

Me: "How old will you be next year?"
Barbara: "7."
Me: "And how old will I be next year?"
Barbara: "33."
Me: "And what will happen after that?"

...and then I close my mouth and wait. Of all the things in this conversation, this was by far the most effective thing I did - I just waited. I also think the "what will happen" question is the right one, as opposed to "what about the next year?" or "how old will we be next?"

After a long pause, Barbara starts..."I will turn 8..."

Still waiting. I'm pretty sure she'll get there.

"...and you will turn 34."

Got it; she's on the way.

"...and then I'll be 9, and you'll be 35..."



Me: "How did you get that?"
Barbara: "I counted."
Me: "Tell me how you counted."
Barbara then walks me through her start process, starting with "this year I'm 6 and you're 32, so next year..." and ending with "then I'll be 10 and you'll be 36."

Then we were interrupted, so there was no chance to extend this or apply the same reasoning least not immediately.

@trianglemancsd I am not, but I thought - despite the initial hiccup - this one went fairly well.