The Same Amount of Tall

My 5-year old daughter, Daphne, was looking out the car window when she laid this question on me:

“I’ve been wondering something. If there’s a whole of a tree, and half of a tree next to it, what do you call it? A whole half?”

At the next light, I turned on my phone and recorded our conversation. What follows is a word-for-word transcript. It’s one of my favorite math discussions ever, because I learned so much about her thinking.

“First of all, tell me about the half of a tree. How do you know it’s a half?”

“It’s a stump.”

“It’s a stump?”

“No. I mixed up the word. It’s a trunk.”

“It’s a trunk. OK. So tell me how you know something is half again. What does half mean?”

“Half means they are even pieces and you cut one of the even pieces up and you leave the other one, then they’re even, and that’s half. The one that’s down and the one that’s up without the other one is half.”

“OK. I have a whole lot of questions. Are you ready? This is the first one. Tell me what you mean by even.”

“Even? They’re both the same amount of tall.”

“They’re both the same amount of tall.”

“Yeah.”

“Whoa.” (Long pause.) “You’re talking about trees. What about wide? Do they have to be the same amount of wide also, or just the same amount of tall?”

“Just the same amount of tall.”

“Just the same amount of tall. OK. So here’s my next question about halves. How many pieces do you have?”

“Only two!”

“Only two. How come?”

“Because you cut one off and leave one on.”

“And that’s halves?”

“Yeah.”

“What if you cut it into three pieces that were the same amount of tall?”

“That would be quarters.”

“How come?”

“Because they just are!”

“What’s a quarter?”

“A little bit of something.”

“A little bit of something. What if I cut it into, like, 20 pieces of the same amount of tall?”

“That would be like…inches.”

“Inches! OK. So, it goes halves, the next smallest is quarters, and then the next smallest is inches?”

“Yeah.”

“What if I cut it into, like 1,000 pieces?”

“It would be just specks.”

“Specks…Hmm. And you said it doesn’t matter, like, I’m looking at that tree over there, and it’s fatter at the bottom–the trunk is fatter at the bottom and skinnier at the top–but the top has lots of leaves and the bottom doesn’t. Does any of that matter, or it’s just the same amount of tall?”

“Where?”

“That tree with the orange leaves.”

sugar-maple-1-547x547
(It wasn’t this tree, but it was a sugar maple like this one.)

Long pause.

“That one has too many branches.”

“OK. So what kind of tree does it work for?”

“Trees…like that one!” (Pointing at a sapling.)

“OK, so what is it about that tree?”

“It’s smaller and it doesn’t have very many branches.”

“Hmm. What sorts of things can we cut in half?”

“Bread…flags…” She trailed off, and we were quiet for a while. I decided to revisit the thirds and quarters idea a little more.

“Can we think about cookies for a second? We made cookies yesterday. If you and Maya and I shared a cookie and we all had the same amount of cookie, how much cookie would we each have?”

“A quarter.”

“How come?”

“Because that’s a little bit of sugar and it’s all the same.”

“How many quarters does it take to make a whole cookie?”

“It matters how big the cookie is.”

“Tell me about that.”

“It would be a lot of pieces if it was a huge cookie, but just a little bit of pieces if it was small.”

“So, if I broke it into pieces so you got one piece, and Maya got one piece, and I got one piece of a small cookie, and then I took a big cookie, and I broke it so you got one piece, and Maya got one piece, and I got one piece of the big cookie, it would take different numbers of pieces, you’re saying?”

“Yes. Because they’re bigger.”

“So if it’s a bigger cookie that the three of us share evenly, we have more pieces?”

“Yeah.”

We kept going for a while longer, but you get the gist.

I love this conversation. I love it partly because it’s evidence that kids are mathematical thinkers who come to us full of ideas. Some of the ideas hold up to scrutiny and some don’t. Some are misconceptions we’ll need to address, and some are deep conceptual understandings and experiences we’ll want to build on.

What does Daphne know? We can divide wholes into parts. We can keep dividing into lots of pieces, and the different pieces have different names. We can use fractions to share. And, most importantly, we can look out the window and see mathematics, and wonder, and have conversations about our ideas.

What does Daphne not understand? Units, although I adore her sequence of halves, quarters, inches, and specks. She has a lot of fraction ideas intertwined. For example, even though she thinks we can have halves of big things, like trees, and halves of smaller things, like flags, she thinks a quarter is a size, and it’s small. There’s a lot of room to explore the distinction between the number of pieces and the size of the pieces. She has developing ideas about measurement and the idea of identifying a dimension: the “same amount of tall” goes in my math Hall of Fame. Volume didn’t occur to her here. That’s OK.

In case you’re wondering, I didn’t “address” any misconceptions, and I’m not worried about any of them. My goal here was not to have Daphne identify fractions with accuracy, or learn any definitions, or figure out how to measure something as complex as a tree, or answer my questions with an expected answer. My goal was to join her wondering, listen to her ideas, help her clarify her thinking here and there, and gain a deeper sense of how she’s seeing the world. I look forward to revisiting fractions and measurements with her many times, and to bearing witness as she makes sense of these ideas. It’s my privilege, as a parent and a teacher. See, math is fascinating. And Daphne is fascinating. Together? It’s the best.

Making Sense

Last week, between work with grade-level bands, I had a complete K-6 staff for about an hour. I wanted to tackle “story problems,” so I started by showing them How Old Is the Shepherd, by Robert Kaplinsky:

Seriously, take the 3:07 to watch it. You’ll see that 75% of 8th graders presented with the nonsensical problem, “There are 125 sheep and 5 dogs in the flock. How old is the shepherd?” answered with a number.

I had given my close friend and colleague, Debbie Nichols, a sneak peek at the video. She often helps me think things through, and is one of the teachers I am profiling heavily in my book. She decided to give the same problem to her 1st and 2nd graders to see what would happen. We both expected the younger kids to do better on the nonsense problem, mostly because they haven’t been trained to pluck-numbers-and-do-something-with-them the way the older kids have. In our experience, younger kids are great sense-makers.

Debbie changed the problem to read, “There are 25 sheep and 5 dogs in the flock. How old is the shepherd?” Partway through, she wondered about vocabulary, and added a second question, “There are 25 kids and 5 dogs in the classroom. How old is the painter?” Debbie interviewed her students individually, recorded it on video, and sent me her notes with the subject header WOW:

Response to sheep version Response to painter version Grade
Is this times or add? I have no idea. I do not get it. 33? 2
37? I thought, my mom is 37, so he might be 37. I was going to try to count to the highest number and then count on with the other number. I was gonna try to but then I changed my mind. Or I could just do that number and then count on. 1
7? I don’t really know, because I’m guessing. What’s a shepherd? 2
A shepherd? 5. I just, I don’t really know. I’m just really good at it and I like animals. 1
11. I’m trying to hold numbers in my head and I just counted. 20 years old? 1
I know 25 + 5 = 30. 30 2
25. You said 25 sheep. 25, because you said 25 kids. 1
30, because I added them up. 30 2
25. I was just thinking. I was counting. 30? 32 1
Shrugged shoulders. 30? 32? 7. I thought of the question. 2
8. I was counting. 8. I counted by 9s. 1
8. Is that correct? I thought it was a little boy so I chose 8. 20. I thought of another number. 1
5. I just got it out of my head. 20. It just came out of my head. 1
What are flocks? What’s a shepherd? 29? 59? 69? I got it out of my brain and my brain is made of pink worms. 1

After individual questioning, Debbie brought the kids to the rug and asked the whole class, “There are 25 sheep and 5 dogs in the classroom. How old is the teacher?” Students quickly fell in line behind some opinionated students, and came to consensus that the teacher was 30 because 25 + 5 = 30.

Wow indeed. What’s especially striking to me is how many students admitted they didn’t understand the problem, and still gave a numeric answer anyway. Some of the nonsense answers even sound kind of right–like they are imitating the way a math answer should sound. “The answer is 8. I counted by 9s.” (As much as I love the brain made of pink worms, the counting by 9s answer is the most fascinating one to me.)

The following day, Debbie posted, “There are 4 kids and 3 chickens in the room. How old is Mrs. McCabe?” (Mrs. McCabe is another teacher in the building.) Students wrote their answers on the chart paper. Take a peek:

photo (16)

43, 7, 7, 40, 64, 70, 2, 2, 10, 9, 30, 36, 100, 34, 7, 44.

Nobody wrote, “I can’t tell because this problem makes no sense.”

43 and 34 come from students taking the 4 children and 3 chickens and using them as digits in a new number.

7 came from students adding the 4 children and the 3 chickens.

I suspect 70 comes from a student adding the 4 and 3, then deciding 7 was too young, so they made it 70.

40, 64, 30, 36, and 44 may come from students disregarding the information in the scenario, and just making a reasonable guess about Mrs. McCabe’s age (56). The 40 and the 30 might be similar to the 70, in that the numbers 4 and 3 were in the problem, and they were clearly too young, so 40 and 30 sounded reasonable. (In this group of answers, if students were thinking about reasonable ages for teachers, that’s something to build on. My next step would be to ask them for evidence in the question.)

2, 10, 9, and 100 remain mysteries to me.

I can see in the anchor chart that Debbie took out her purple marker and tried to help students recognize the lack of relationship between the number of chickens, the number of children, and Mrs. McCabe’s age. She had them try to come up with a question that would make sense from the chickens and children, like how many legs are there?

After reading Debbie’s notes, I wondered what my own children would do with this problem. I asked my 5-year old daughter, (who asked to be called D), “There were 4 children and 3 chickens in a room. How old is the teacher?”

“9.”

“Where’d the 9 come from?”

“No, wait, that’s not enough. 90.”

“Where’d the 9 come from and where’d the 90 come from?”

“Well, I added the 3 and the 4, and that made 9. But that’s not enough. Like, Maya is 7, and that’s 2 years less than 9, and she’s not old enough to be a teacher! So I made the 9 a 90.”

“Can you show me how you added 3 and 4?”

She counted on her fingers.

“Oops. It’s not 9. It’s 7. So she’s 70.”

I was quiet for a while, thinking about a question.

“We have 2 dogs and 1 fish, right?”

“Yeah.”

“So, we have 2 dogs and 1 fish in our house. How old is D?”

She laughed uproariously. “Mommy, that doesn’t make sense! The dogs don’t have anything to do with how old I am! 2 and 1 is 3, but I’m 5!”

“Really? OK. Let me try again, then. We have 2 dogs and 1 fish in our house. You are 5 years old. Let’s say we bought another fish. How old is D now?”

She was hysterical at this point. When she calmed down, I said, “OK, let’s go back to this question again. There were 4 children and 3 chickens in a room. How old is the teacher?”

“OH!”

I recognize that these nonsense word problems are contrived. I think they’re revealing, though. In particular, I think they show some problematic beliefs our students have about doing math:

  1. All math problems have to be answered with numbers.
  2. All math problems can be answered.
  3. It’s normal for math not to make sense.

Where do these beliefs come from?

In the last week, as I’ve been mulling all this over, I’ve been revisiting some of the books I have about the intersection of reading comprehension and math, like Comprehending Math by Arthur Hyde, Mathwise by Arthur and Pamela Hyde, and From Reading to Math by Maggie Siena. This quote of Siena’s about the foundations of reading jumped out at me:

“Children must…expect the things they read to mean something and expect to be satisfied by that meaning” (17).

Do we teach the same expectation in math? Or do we teach students to answer every problem with a number, guessing if they must, and it’s OK if it doesn’t make sense?

I’m still scratching my head over where this message comes from with my kids and Debbie’s students. D has grown up during the writing of my book, when I am hyper-aware of the math messages I am sending and hearing. Debbie’s students had a great year of kindergarten with a teacher who emphasizes making sense, and now they’re with Debbie, who teaches math for understanding. And yet, 100% of them answered the nonsense question with a number.

As a fan of CGI, I know children are naturally sense-makers. But I also know that reading mathematical problems is a special kind of reading, and students need instruction in it. Historically, teachers have used two different types of instruction for reading word problems:

  1. Teach students to “decode” math problems with “keywords,” like “in all means add.”
  2. Teach students to recognize unnecessary information, red herrings, and traps that “they put in the problems to trick you.”

On keywords, some of us have been having fun over on Twitter, creating a list of problems that show why it’s a doomed strategy. For example, Tommy buys 3 bags of avocados. There are 4 avocados in each bag. How many avocados did Tommy buy in all? Hmm. I thought in all meant add?

As for focusing on the traps “they” are putting in the problem, I am no fan of this strategy either. What sort of message are we sending kids with this teaching? That there are rooms full of nasty adults, rubbing their hands together, trying to set traps that catch nice little children taking math tests? Ahem. Though there may be some truth to that image, I refuse to cede mathematics to the standardized-test and curriculum writers who write crappy, trappy math problems. I want to snatch math back, and teach students to see the beauty and usefulness of math around them, and to enjoy the journey through a perplexing, puzzling problem. So red herrings and tricks be damned!

What do we do instead? How do we teach students to read math problems for understanding in a way that will yield empowered students who expect to make sense? I’m looking for resources on this question, so please pass them along in the comments. In the meantime, let me share three of my favorite approaches. They all have something in common, which is that they are all strategies to make it impossible for students to leap right to answering the question. All three approaches force students to slow down and make sense, first.

1. The Math Forum at Drexel University is a fantastic group of people who are all about teaching students to make sense of math. One of the strategies they have been promoting is Notice and Wonder, where teachers share a scenario without a question, and ask students what they notice and wonder. You can read about it in Max Ray‘s Powerful Problem Solving, by following Annie Fetter, or at a whole bunch of sites here, here, here, here, here, here, and here.

2. Brian Stockus wrote a great blog called “Numberless Word Problems,” in which he described a co-worker removing the numbers from a word problem. Again, this strategy eliminates the option of racing to an answer, and introduces students to the idea that we can do quite a bit of mathematical thinking about quantities without knowing what they are, which Kate Nowak framed as the the rich idea at the heart of algebra:

Screen Shot 2014-10-18 at 4.08.39 PM3. In Mathwise, Art and Pamela Hyde wrote, “Getting students to slow down and think about a problem is not always easy, especially if they are used to calculating answers quickly to one-step translation problems. We have found that students can be encouraged to think through their assumptions with an intriguing type of problem called “Fermi questions'” (66). Fermi Questions are mathematical questions where answers seem impossible, but we can get close by making some assumptions and then approximating:

  • How many piano tuners live in Chicago?
  • How many kids could fit in the gym with no furniture inside?
  • How many hairs are on your head?

From a teaching point of view, Fermi Questions can be fantastic for helping kids realize they are making assumptions and connections and using their prior knowledge in mathematics.

All three of these strategies–Notice and Wonder, Numberless Word Problems, and Fermi Questions–force students to slow down and make sense of the situation before worrying about the answer.

I’m hoping to learn more about high quality instructional strategies for math teachers that are rooted in what we know about teaching reading comprehension. Annie Fetter presented on this idea at NCTM in New Orleans, and I think it’s an idea with long, strong legs. The connection between making sense in literacy and math is something I talked about in the workshop last week, and it seemed to resonate with Shawna Coppola, a wonderful literacy specialist. I loved her notes:

image1

Making sense is the thread that ties everything together, in every content area. Right? If our students arrive having already internalized the message that making sense isn’t part of math, or that math doesn’t make sense, or that word problems are just a bunch of numbers hidden in words and traps, we have our work cut out for us. Time for some intentional, creative, inquiry-based teaching that empowers students to make sense.

The Math of Medical Decisions

I’ve spent the summer immersed in medical decisions around several loved ones, so this blog has been brewing for a while. At each appointment, I am struck by HOW MUCH MATH patients need to know in order to understand the choices framed by their doctors. There are always odds, statistics, costs, benefits, side effects, down sides, and probabilities to weigh. Nothing in medicine is ever 100%.

Scratch that. We actually went through an ordeal this summer where the number 100% came up twice, and the decision was still not simple. We had a bat flying around our house, out of our sight for a time, while our two kids were asleep in their room with the door open. Here is the background information we learned (from sites like CDC and WHO):

About 1 person dies in the U.S. each year from rabies. This number is much higher in countries where dogs are not vaccinated.

Of the deaths in the U.S., most are caused by bats.

Most bats do not have rabies: recent estimates are around 1%. (Bats are also endangered.)

If someone is asleep, there is no way to know if they’ve been bitten or scratched by a bat. The marks are microscopic.

So here’s where the 100% comes in:

If a person is bitten by a rabid bat and not treated before symptoms develop, he or she faces certain death. There is nothing that can be done. 100% fatality rate.

The post-exposure rabies vaccine is 100% effective when administered correctly.

So what’s your choice? If it were you, would you vaccinate the children or not? What are the odds that the one bat in the house was one of the 1% that’s rabid, and actually made it into the kids’ room, and bit them? Is that the number we should calculate?

Does it make a difference that the vaccine is a 4-part series of painful shots given over a few weeks?

Does it make a difference in your calculation that the first visit to the Emergency Department for the first round of shots billed insurance for….wait for it…more than $11,000?

When is it worth it? When is it feasible? What’s the right decision?

Last week, I was in the mammography waiting room, chatting with a woman who, like me, is 41 with a mother who had or has breast cancer. We’re both on the same high-risk protocol, although her insurance isn’t as good as mine, so she only gets MRIs every other year instead of annually, as recommended. While we waited in our blue robes, she told me about her mother’s journey through breast cancer. She said, “My mom waived chemo because of vanity. They told her her 10-year survival rate would go from 81% to 92% with chemo. In that appointment, my mom said, ‘Those are both over 50, right? Those are good, right? Then I’m not doing it.'”

Each medical decision needs to be made by a specific person, according to their values, in their context. I am in no way judging her decision. What concerned me was that I wasn’t sure her mother understood the math involved. What does she think “over 50%” means? Does she understand what an 11% difference in her survival rate means?

Later that day, I was told I have breast cancer. I know from my experience with my mother that there will be mathematically based choices to make at every stage of this journey. For example, my tumors will likely be Oncotype tested, which will yield a score my medical oncologist will use to decide whether I need chemotherapy or not. The Oncotype test supposedly gives a score from 0 – 100, but look how it’s used (from breastcancer.org):Screen Shot 2014-10-02 at 10.34.00 PM

If a patient hears lower is better, and she has a score of 32 out of 100, that sounds pretty low, doesn’t it? Except, no.

So, here’s where I’m going with all of this. Are we equipping our students to reason through these decisions? To make sense of risks vs. benefits, odds, data, or scales from 1 – 10, 1 – 100, 1 – 5000, all used in different ways? When we wring our hands about “real world” applications of math, we talk about making change and saving for retirement. But, at some point, every one of our students will be a patient.

As I face the medical road ahead of me, I am well aware of the privileges I have going in to treatment:

  • I have the open doors that come from my socioeconomic status: access to great care and the means to pay for it.
  • I have a job I can keep through this treatment, and don’t have to choose between care and financial ruin.
  • I have a home where I feel safe and am supported by my family.
  • We have a larger community and support network that will help us.
  • I am not likely to be discriminated against during my treatment, or face language barriers.

But I have a second set of assets too:

  • I have the literacy skills to make sense of a new glossary of medical terms, read technical articles, and communicate my needs with my support network.
  • I have the science background to understand protocols based on clinical trials and data collection. I have enough biology to build a general understanding of what’s happening in my body, enough chemistry to have a sense of what these medications do, and enough physics to understand some of the testing I’ve been through. (Speaking of, thank you to the physicists and engineers behind MRI, which may have saved my life.)
  • I have the mathematical background to understand the results I’m given and reason quantitatively through the decisions I’ll need to make.

That second set of assets comes from my education. Which begs the question, what should we do, as teachers, to equip our students with these skills? Assets that can never be taken away from them? Let me be perfectly clear: I am not suggesting we replace half of the tedious, textbook cell phone cost and rate word problems with scary cancer cost and rate word problems. For the love of Pete, that’s NOT what I’m saying. What I’m saying—asking really—is what are we doing to develop the kind of numeracy and quantitative reasoning our students can use in novel, complex, vital, contextualized decisions?

Could we start by remembering that’s one of our goals?

I’ll be getting a big round of pathology results back next Thursday. I have a lot to be scared of, but at least I am confident I will understand the results. I know my family and I will be able to talk through the options with my doctors, listen carefully to their counsel, and make the best decisions we can for me, at my age, with my history, in my context.

I want our students to have the same comfort and confidence when it’s their turn to make medical decisions.