Show Your Work

When I was in high school, there was nothing I hated more than math. And in math class, there was nothing I hated more than showing my work. It felt like a tedious chore with for no real purpose—a hoop somebody wanted me to jump through. Math was not open to poetic interpretation; the answer was either right or wrong. And no one ever offered me partial credit for getting an answer partially correct, or for demonstrating an interesting, if flawed, process. As far as I could tell, nobody even looked at my work. So why did I have to show it?

Today, if my high-school aged son is to be believed, the situation doesn’t seem much improved. And that’s a shame. The Common Core State Standards in mathematics--and the new math standards in many non-CCSS states—make clear that reasoning, arguing, and critiquing are vital skills within mathematics, and that teachers should be instilling in students a devotion to precision and perseverance. None of those mathematical habits of mind reveal themselves in the calculation sitting to the right of an equal sign and a question mark. They reveal themselves in the process—in the work a student does along the way to an answer. There are real and strong reasons to show your work. But if a teacher doesn’t understand those reasons and make use of them, the practice can easily remain an exercise in hoop-jumping.

As Elizabeth Green explains in her excellent book, Building a Better Teacher, the right answer is often the least interesting piece of data available in a classroom. It is the wrong answers that give teachers real information about how students are thinking and where they may be going astray. And while a seasoned veteran may be able to intuit a problem from nothing more than a final answer, most of us will have an easier time understanding what’s going on by making the student’s thought process visible. In written assignments, that’s precisely where have students show their work is important.  

What about in the heat of a class discussion? Students can show their work here, as well, by talking about their process. Asking “what did you do?” is far more valuable (and interesting) than asking “what did you get?”  It allows the class to explore multiple problem-solving strategies and assess what works and what doesn’t. It helps students become comfortable and fluent in “talking math.” And it can make error an interesting topic of conversation rather than something to be embarrassed about and try to hide from.

There’s a lot of talk about Growth Mindset in the education world. Well, a vital component of Growth Mindset is the willingness to fail, and try, and fail again—the understanding that failure is a natural part of the process of learning, a necessary step on the way to success. Athletes understand this perfectly well. A baseball player with a batting average of .300 is considered pretty good; a player who bats .400 is considered outstanding. Outstanding—with a 40% success rate! In basketball, perhaps the greatest player of my lifetime, Michael Jordan, said this:  “I’ve missed more than 9,000 shots in my career. I’ve lost almost 300 games. Twenty-six times, I’ve been trusted to take the game-winning shot, and missed. I’ve failed over and over and over again in my life. And that is why I succeed.” He doesn’t say that he succeeded in spite of his failures; he says he succeeded because of them. Because he was willing to put himself out there, and make a mistake, and learn from the mistake, and press on.

We need to find ways to model this more positive view of failure for our students. Instead of focusing solely on end-results, we need to show students the long road it took to reach that goal. How many drafts did it take for an author to reach the final manuscript? How many publishers rejected the book before it was accepted? How many experiments did the inventor go through before hitting on the right combination of ingredients? What challenges and setbacks did a historical figure endure before doing the great work for which they are remembered?

And what about us? Are we willing to model failure personally in front of our students? Can we admit to them when we’re wrong, or when we don’t know the answer, or when an activity we planned didn’t go as well as we had hoped? Can we show them that there’s no shame in falling down, as long as we get back up again, brush the dust off ourselves, and push on?

Cultivating Student Curiosity

I performed a magic trick at a recent workshop. I was working with a set of elementary-school teachers in Indianapolis: two workshops per day, over two days. With each of the four groups, I asked the teachers to show me what their students would draw if they were asked to picture a house, with a family and tree out front and the sun up in the sky. When they were finished, I said, “Now here’s my magic trick. I haven’t been anywhere near my computer while you’ve been drawing, but now I will reveal the picture that every one of you drew.” And I showed them this picture:

Sure enough, it was exactly what every one of them had drawn. And they were right—it’s exactly what all of their students would have drawn.

Why?

Un-boxing the Student Brain

Think about it for a moment. Why is every house a square with a triangle on top? Is that really an approximation of what their houses look like? And why are they all the same, when every house on the block looks a little different? Where are the two-story houses, the split-level houses, the apartment buildings? And why do the mothers all have long hair and the triangle that symbolizes dresses or skirts? Is that really what their mothers look like today? Who told them that a triangle symbolized Woman? Why does the tree look like a lollipop? Why is the sun a yellow circle with straight lines coming out of it—and, often, a smiley face?

The teachers offered up a wide range of interesting answers, including the following:

• It’s what they’ve seen in books and magazines over the years
• It’s what they’ve seen their peers do over the years
• When they see their peers doing it, they change their picture to match what looks “right”
• They get corrected by their teachers, who unwittingly get kids to conform to what looks “right”
• They get preemptive instruction from teachers, who suggest using simple shapes (squares, triangles, circles) to keep kids from becoming frustrated

When any of these things happen—and especially when they all happen in the classroom—the result will be conformity—and, in this case, conformity to something that doesn’t even resemble reality. A student who enjoys science and knows what the sun really looks like will be encouraged (on purpose or unwittingly) to stop trying to draw a reddish-orange ball of burning gas and will draw a yellow, smiley circle. A child whose mother has short hair and wears jeans will draw a mother who looks nothing like her actual mother. Little by little, they will learn to substitute an approved, common vision for their own, singular vision. And then, suddenly, sometime in middle school, we’ll start asking what happened to their creativity.

According to research by the Right Question Institute, as children become verbal, the number of questions they ask in school each day shoots through the roof, but then begins a slow decline starting at around age three. By age 18, they are hardly asking any questions at all. Now, one could argue (as their research shows) that because this decline happens at about the same rate as their reading and writing skills develop, they have a decreasing need to ask questions in school, because there are finding their own answers.  But anyone who has ever taught middle or high school would snort with amused disbelief at that argument. In fact, far too many of our students become increasingly un-curious as they make their way through school. They learn not to ask questions, because they learn that their questions are not considered important. The adults mandate what will be studied, and the adults decide what questions matter. The job of students is to answer questions, not ask them. The job of students is to be compliant and responsive and well-behaved. There is no room for curiosity in the lesson plan.

In recent years, however, room has been made for curiosity in our learning standards. The Common Core State Standards, as well as other new state standards, ask students to form and write personal opinions (through grade 5) and evidence-based arguments (starting in grade 6) about a wide variety of texts. Informational writing is still important, but far less important than developing the skill of argument. You can’t make an argument if you don’t take a position—and you can’t make an interesting argument if you aren’t curious enough about what you’re studying to develop a point of view. The standards of mathematical practice likewise talk about the importance of argument. The first standard asks students (at all ages) to “reason abstractly and quantitatively,” and the second standard asks students to “construct viable arguments and critique the reasoning of others.”

If that’s what we expect students to do, how can we make sure that we’re not inadvertently steering them away from those goals as they grow up with us?

Changing What We Ask For

An easy first step is to change the way we phrase our learning expectations. Every teacher learns, as part of her training, that lesson objectives must be clear, concrete, and measurable. And most of us learned to write our objectives using SWBAT language: “Students will be able to…” followed by that clear, concrete, measurable goal. For example:

Students will be able to support a topic sentence with evidence.

There’s nothing wrong with that objective…except that it tells students what they will do, rather than inviting or challenging them to do something. And that’s not an insignificant difference. If we want students to be curious, not just compliant, than we need to give them something to be curious about.  Imagine if we phrased that learning objective as a question instead of a command:

How can you convince readers that your argument is valid?

Think about how differently those two sentences register and resonate in your head, when you hear them. The statement is impersonal and commanding, where the question is personal and inviting. The question connects the academic content to the student’s own world. The question challenges them and poses them a problem to be solved. The statement simply tells them to do stuff.

Now imagine what you could do if that lesson objective was part of a larger unit—maybe even an interdisciplinary unit—that looked at all the different ways we have of figuring things out, as humans. What if we used a question to frame the entire unit—something like this?

How can we know what’s true?

You can still keep the SWBAT language in your lesson planner, to remind you what the concrete goals are. But why not invite the students to be curious and interested in what you’re teaching? After all, what you’re teaching is interesting and important!  If it wasn’t, we wouldn’t keep teaching it, generation after generation. Let’s try to remind ourselves—and show our students—why the stuff of school actually matters. And who knows? Maybe they’ll ask a question that no one has ever asked before—or find a solution to a problem that no one has been able to solve before. 

NAEP Results: Less “Bang for Our Buck” (But Plenty of Whimpers)

“Between the idea and the reality . . . falls the shadow.”      —T.S. Eliot

A new report from our friends at the National Assessment of Educational Progress (NAEP), also known as the “Nation’s Report Card,” provides data on student performance in reading and mathematics across multiple grade levels across the country. This latest report shows us how well American twelfth graders performed in 2015 as compared with the last test administration in 2013.

Those were two years of contentious Common Core adoption in many states, or resistance to Common Core and reliance on existing standards in other states—two years of hard work at educational reform and improvement, wherever you lived, whether you were changing course or staying the course; two years of teaching, reaching, explaining, begging, and maybe even bribing students to achieve.

So, what’s the result of our efforts over the past two years? Well, according to NAEP:

In comparison to 2013, the national average mathematics score in 2015 for twelfth-grade students was lower and the average reading score was not significantly different.
In comparison to the first year of the current trendline, 2005, the average mathematics score in 2015 did not significantly differ. In comparison to the initial reading assessment year, 1992, the 2015 average reading score was lower.

Not what we wanted to hear, is it? Either nothing changed, or things got a little worse – hence the quote at the top of the page. Between our aspirations and our actuality, there always seems to be a little shadow—a little gap—a little, “Sorry, not quite.”

Why? What went wrong? Or, more accurately, what’s going wrong, day after day, week after week? Why is that shadow falling between what we’re trying to do and what we’re getting?

MIND THE GAP

Part of the performance gap comes from an implementation gap—the shadow that falls between our plans and the way we put those plans into effect. New standards or textbooks or pieces of whiz-bang software may be brilliant and revolutionary in theory, but if they’re rolled out to schools ineffectively, or haphazardly, or without real buy-in and understanding from teachers. Eventually they wind up on the trash heap and reinforce our feeling that nothing ever works.

Were all of those abandoned programs and initiatives really terrible? Probably not. In fact, most of them were probably fine—maybe even better than fine…in theory. We just didn’t use them properly, or hold onto them long enough to see a result. Anything new requires a little patience, a little persistence. You would never buy a packet of apple seeds on Monday, plant them on Tuesday, and expect a glorious, fruit-laden tree by Friday. But that’s pretty much what we do in our schools, year after year. If the new thing doesn’t work in its first year of implementation, we give up on it and go back to whatever it was we were doing before. Our “flavor of the month” approach to reforms and resources may be one of our problems.

Another problem is that we aim for real thought from our students, but too often settle for mere response. If we’re not aware of that gap, it can cast a lethal shadow over all of our “college and career readiness” initiatives. Here are a few recent examples I came across in my travels:

THE VOCABULARY QUIZ

I visited with a high school teacher several weeks ago—a bright and capable young man who teaches Japanese in an excellent private school. He has been successfully teaching Japanese for a number of years already; his kids get good grades and their parents have been happy. He usually teaches in a fairly traditional style—a lot of lecture, a lot of worksheets, a lot of rote memorization. Pretty standard stuff.

But after a PD session at his school, he decided, as an experiment, to change how he assessed vocabulary. Instead of giving his students a traditional quiz on the words they had been given to learn (here’s the word; choose the correct definition from among four choices), he asked his students to use each word in a couple of sentences. The result was disastrous. They could identify the meaning of the words, but they couldn’t use them. They couldn’t do anything with what they knew.

WHAT’S IN A NAME?

In an elementary classroom at another school, I saw a teacher leading an activity in which students generated lists of nouns, adjectives, and verbs, which the teacher placed up on a large, color-coded chart. The students were asked to create sentences by choosing a couple of the nouns, one of the adjectives, and one of the verbs. When I arrived in the classroom, the students were almost done; most of them were drawing pictures to illustrate their sentences. All the kids looked happy and successful.

But when I leaned over and asked one of the girls which word was the verb, she had no idea…even though it was colored blue on her paper, just as it was colored blue on the chart, under the title, “Verbs.” When I asked her what a verb did a sentence, she didn’t know. When I told her what a verb did in a sentence and then asked her which word in her sentence was doing that, she didn’t know. And she wasn’t the only one who was having this problem. For many of the students, the sentences looked fine. Their sentences were, in fact, correct. They were able to respond successfully to the instructions and complete the activity to the teacher’s satisfaction. But they couldn’t talk about what they were doing, and they seemed not to understand what it is they had done.

SUNK IN CARBON

When I came home from my most recent travels, I saw my sixth grader hard at work on a science assignment. I asked him what he was doing, and he showed me a worksheet about the carbon cycle. The question he had just completed read, “How does deforestation affect the carbon cycle?” His answer was, “Trees are carbon sinks.” He was very happy with his answer, because it was factually correct. He could even show me, in his textbook, where that factual detail lived. But his answer, while true, didn’t respond to the question he had been asked. It took several minutes of (gentle) browbeating and asking “so what?” to get him to connect his fact to the idea of deforestation. He had a lot of facts ready at hand—in his brain and in his notes—but he wasn’t sure what to do with them.
It’s an easy miss on a homework assignment, but it’s exactly the kind of thing we need our teachers to tease out with students. Are our teachers taking the time to help students connect thought to thought, and idea to action, in a way that helps them make the things they’re learning useful?

WHAT DOES THE DATA MEAN?

Every test is a transfer task—you have to take what you learned in the classroom and apply it somewhere else. But no test prep can prepare you for every question or question type you may encounter in the world. You have to be able to come at any challenge with a deep understanding of the relevant content and an ability to be flexible in the way you use it. You have to be ready to improvise at a moment’s notice. You have to be able to think about what you know.
This is why athletes need more than drills. They need scrimmages—practice games—to get the experience of making decisions and using their skills in the crazy, unpredictable, changeable context of a game.

The question for us is: are we deficient in our skills drills, or in our scrimmages?

I don’t believe that a stagnation or slight downturn in NAEP scores means that our teaching is deteriorating, or that a particular class of students isn’t as bright as the class that came before it. We’ve been at the lower levels of Bloom’s Taxonomy—teaching, questioning, and testing at the levels of basic knowledge—and we’re still pretty good at that level. But we’re less effective at getting kids to think about what we’re teaching them so that they can use what they learn confidently and in a variety of new ways.

The more our assessments move away from basic question-and-response—the more they try to present students with authentic thinking and reasoning tasks, the more we are liable to see a shadow fall between what we’ve taught them and what they can do with what they know.