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Teaching and the Art of the Possible

A couple years ago, an administrative vice principal (AVP) walked into my classroom and asked, “I’m checking up on previously ELL students who were reclassified English proficient (RFEPed) to be sure they are getting enough support. What can you tell me about Santino?”

“Santino? He’s doing well. He’s gotten either a high F or a solid D on every test so far, which is a big step up for him.”

“Really. I guess we have different notions of what ‘doing well’ is.” She raised her eyebrow, refrained from sneering, made a check on her clipboard form, and walked back out before I could respond, which was probably a good thing.

Santino, a junior in my geometry class, was passing all of his classes at that time. He had finished his sophomore year with a sub-2.0 GPA, still higher than his freshman year. Ultimately, that AVP would determine that Santino’s upward progression was evidence of a valid reclassification. She didn’t walk back into my classroom and apologize for her nasty bit of snark, though.

Santino had a shock of black hair, big black glasses, and a rail-thin physique, which he clothed daily in black tees, skinny black jeans, and a light black hoodie. Think Hispanic Emo without the makeup. He almost never smiled in school. He didn’t amble but stalked silently through the halls, eyes always on the ground, occasionally with an equally silent and dark-garbed friend. His English was surprisingly fluent, considering his qualification for the Migrant Education Program and two parents who spoke no English, and his rarely used voice unexpectedly mellifluous.

As a sophomore, he sat silently through almost the entire first semester of my algebra class, doing nothing. He didn’t goof around. He just sat. He turned in empty tests. I’d called his home a couple times and tried to get him to work in class to no avail. Finally, I made contact. I vividly remember our first nearly wordless exchange.

In an early November test, Santino was just sitting there again, blank test in front of him. I stopped at his desk and picked up his pencil. Problem: Alycia made five times as many pumpkin pies as apple pies. If she made 24 pies, how many apple pies did she make?

I drew five circles, marked each one with a “P”, then handed him the pencil. He reread the problem and drew one circle, labeling it with an “A”. I waited. He thought. Drew another five “P” circles. Then another “A”. I smiled, and walked away. When I came back, the entire problem space was filled with circles, and the words “4 apple pies” was scrawled down in the corner.

“So let’s try this next one.” Problem: Julio was at the beach, and noticed that the ratio of seagulls to blackbirds was 3:2. If he counted 30 birds, how many blackbirds were there?

I drew 3 Ss and handed him the pencil. He drew 2 Bs, then three more Ss. “There you go.”

For the first time, Santino turned in a non-blank test. He got all three ratio problems correct, with pictures and “guess and check”. Everything else was still blank.

For this particular ratio unit, I had taught students both the algorithm and the visual method I’d just used with Santino. I’d taught it not once, but several times. I’d given the students a number of different techniques to help conceptualize a ratio, concepts that they’d already been taught extensively their previous year, either in Algebra or pre-Algebra. I had already dramatically simplified and slowed instruction for the class.

Yet Santino’s response to my intervention demonstrated that I still had students for whom the gap between the content I was teaching and the support they needed proved much larger than I’d imagined.

And so, over Christmas break, I winnowed algebra down to the fundamentals, designing an “algebra lite” curriculum for these students. Through trial and error, I settled on a method: start the day with something they could do instantly, without needing explanation. Introduce new material twenty minutes into class with simple practice problems—and I mean simple. Take my initial notion of simple, cut the difficulty in half, and then again. Then take just half the problems I’d planned on and I’d be in the ballpark. They’d work on those problems the rest of the class. The next day, they’d start with a basic—again, really REALLY basic—problem in the new material, and move forward on that. Although this method meant more work (remember, I was still teaching the usual course in the same class), it gave my weakest students a chance to progress. Most of them responded eagerly, grateful for work they knew how to do.

Santino’s math skills didn’t improve dramatically, but his engagement inched up several notches. He worked in class, and his tests (easier ones for this group) were no longer blank. He was noticeably stronger at word problems, and best of all at word ratios, that first type we’d worked on together.

As a sophomore, he took the required California High School Exit Exam (CAHSEE) in March of that year. I assigned my strongest freshmen to coach a sophomore for eight class days, using a tutorial I’d designed to help both freshman and sophomore understand how the test was constructed . For Santino, I chose Carl, a shy, sweet, white kid who wore his NERD teeshirt once a week. I told Carl that any score above 330 (passing was 350) would make Satino feel ready for a rematch, instead of hopeless. Carl understood, and as I wandered the room during that prep fortnight, I would often hear him reminding Santino to “estimate and eliminate”. Santino was always hunched over his practice questions, thinking hard, not sitting passively; he even ventured a question now and again to the far less threatening Carl.

Just over half of my algebra sophomores passed the CAHSEE, which is as much as I could ask for. Santino, who passed the English section with a 356, stunned me with a 348. The day the scores came out, we had the longest conversation of our two year acquaintance.

“I didn’t pass.”

“You got a 348! That is AMAZING! One question from passing! I am so proud of you!”

“I almost did it. I think I will do better next time, because I didn’t know geometry. I need to study again in November. I will pass it then.”

“You make sure to come back and see me in November and I’ll give you the practice material.”


Santino passed my “algebra lite” curriculum and I changed his first semester grade to reflect his new work. His junior year, he was assigned to my geometry class. Come November, even before I’d prompted him, he came up and said, “I will be taking the CAHSEE.”

“I know. Do you want a few days in class to practice?”

He worked independently and diligently. While he didn’t go so far as to ask me questions with, say, his voice, he had questions circled and would look up mutely when I stopped at his desk. I’d coach him on best methods for elimination; he’d nod and get back to work. He came to class the day of the test, a little less inscrutable than usual, just a bit anxious. When the announcement calling students to the testing room came over the speaker, I said, “Go get ‘em” and the class all cheered him and the others on.

He passed with a 356. His seatmates and I harassed him to look happy, until he finally turned the ends of his mouth up, reluctantly. But he did look pleased.

In geometry, Santino would still sit silently in front of a blank sheet if he was stuck, but he never turned in an empty test again. He’d peek up through his shock of hair as I walked by, and point to a problem when I stopped. Most importantly, he was passing the same course as the other students—passing with a D, but passing.

So you can see, perhaps, why I didn’t particularly appreciate the AVP’s snark. Santino was, indeed, “doing well”.

He did well enough in right triangle trigonometry, of all things, that I gave him either a D+ or C- for the second semester. Since he’d passed pre-algebra in summer school his freshman year, Santino had, at the end of junior year, the necessary three years of math to meet his graduation requirements. He was as quiet as ever. Despite our two years’ acquaintance, he never initiated a greeting when we passed by in the yard, although he would, if I waved, give me a “chin jut of recognition”, as Sheldon Cooper would call it.

I left the school that year, but often wondered if Santino would be able to “make the walk”. Would his credit gap force him to an alternative high school for senior year, or some online academy? He didn’t have the grades for the voc-ed program, so that wouldn’t be an option. I have beer or coffee with my ex-colleagues frequently, and would often nag them for any status about Santino. None of them could find his name in the system, but I kept hoping they’d just missed something.

One Tuesday in early August, I ran into another student from Santino’s school, and wished him luck in college. “Did you see my name in the paper? They had the whole graduating class!”

I went home and googled the Mercury News list of graduating seniors for the school. Santino’s name was on the list. He’d walked with his class.

A growing body of research suggests that non-cognitive factors—persistence, effort, motivation—are important in adult outcomes. Kirabo Jackson of Northwestern University analyzed the impact of teachers on both test scores and non-cognitive skills (as assessed by attendance, graduation rates, and so on). He found that “many teachers who are among the best at improving test scores may be among the worst at improving non-cognitive skills”. Moreover, “teacher effects on the non-cognitive factor explain significant variability in their effects on these longer-run outcomes that are not captured by their test score effects”.

In practice, valuing “non-cognitive skills” almost always means lowering academic standards. Many students struggle with advanced content but have no ability to choose easier courses, thanks to our well-meaning but misguided education policies. If teachers hold all our students to a strict reading of the course requirements, students who either don’t want to or can’t understand the rigorous material will fail. Obviously, reasonable standards would eliminate the need for that choice. Schools might provide a menu of classes of varying difficulty, allow students to choose course material they are capable of and interested in learning, rather than set a ferociously high bar and then make some teachers choose between failing most of their students or not covering the material with a rigor appropriate for the strongest kids. But in today’s world, fail or pass a student who can’t really do the work is our only choice.

On the other hand, Rishawn Biddle argues that teachers like me are the problem: “Behind all this gatekeeping is the view among many traditionalists that there are some kids who just aren’t capable of high-level work”, and that students of color are given a terrible foundation due to terrible teachers and weak curriculum.

Biddle’s assertions aren’t borne out by any reality I’ve ever seen. In Santino’s case, he had a solid grasp of math facts but struggled tremendously with abstractions. He attended the same K-8 schools that many of my strongest students, both Hispanic and white, attended. But while I’m no traditionalist, it’s certainly true that I thought Santino was incapable or uninterested, at that point in his life, of deep understanding of algebra and geometry. So I modified both his work and my expectations to give him passing grades. I would do it again. In fact, I have done it again.

While the education philanthropists like Whitney Tilson hold that “kids will live up – or live down – to whatever expectations are set for them”, reality plays out very differently. Many kids simply don’t try. Many kids try but simply can’t do the work. And quite a few kids fall somewhere in between. Fail the kids like Santino, and they either drop out, or settle for a GED, or go to some credit-recovery room, separate from their peers. Pass the kids like Santino, and they get to feel normal, even in an environment designed to make them feel inadequate. The Whitney Tilsons believe that failing a kid simply makes him work harder at an achievable task. But what if they’re wrong, as the majority of teachers who work with low ability, low incentive kids would argue? The data shows that, given the same level of academic achievement, kids are better off graduating than dropping out, or even getting a GED.

So, the question: Do you teach the course or teach the kids? Many math teachers hold that higher standards are essential, that the only way to ensure that our classes accurately reflect their descriptions is to fail those students who don’t perform with the expected rigor. I understand that argument but ultimately, I agree with a colleague who said once, “Look. If half your class is failing, blame the person you see in the mirror.” I simply can’t fail half my kids in classes they didn’t choose to take.

As for Santino, I know this: He almost certainly would not have passed algebra and geometry with a different teacher. This alone gave him a better shot at graduating normally. Without the need to repeat math classes, he had more slots on his schedule to repeat earlier failed classes and make up even more credits. The more he could see graduation becoming a possibility, the more he was willing to work to achieve it.

God speed, Santino. Go get ‘em.

This piece was originally written in 2013.

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23 thoughts on “Teaching and the Art of the Possible

  1. I agree with the point you are making. Everyone wants children to believe the sky is the limit for them, and that’s great. I would never tell a child “eh, you’re mediocre at best”… but I am a firm believer that not everyone has what it takes to be an academic success (not when everyone is graded on the same scale and in the same way). I appreciate your approach.


  2. I don’t see why you can’t both teach the course and the kids. You just need to have courses tailored to different abilities. One course for people who might take longer to get algebra. Another for those who can get it when they’re 9 and yet another for those who need to be 13-14 before they get it.


    • Partly because budget (each curriculum costs money to develop, etc.), and IMHO partly it’s the idea that if you allow kids to have an “easy path” option, they will take it, even if they have the ability to succeed along the more difficult path.


      • 1. This is penny-wise pound foolishness. Developing separate curricula for slower, average and faster students pays dividends later on.

        2. Kids won’t want to take the “easy” path if they knew that it would close off most college courses for them.


      • This is not strictly true. My daughter has consistently chosen the most difficult courses open her, to the point where she’s taking 4 AP courses next year.

        My son doesn’t especially like English, but he chose to 6 level class instead of 5 level that he could skate through, even though it will have no impact on his ability to get into the college courses he’s interested in, which are biology/science.

        In both cases the primary equation was engaging vs boring which was a more important consideration than how easy the class would be.


  3. When I was in fourth grade a new kid came to our school. Yateen was from India, spoke not a lick of English, and though he made friends, struggled academically the whole time. By the end of the year, he spoke a bit (immersion for the win) every bit of the educational aspects of grade school was a challenge. And no wonder, trying to work with a new, barely understood language while attempting to navigate a school set up for faculty brats is a recipe for disaster. So the teacher recommended he be held back a year, so as not to miss any of the building blocks that come at that point.

    He was still there on the playground, just making new friends, speaking better and hopefully learning, as opposed to being moved on up the ladder. This isn’t to say what you did was wrong, or that I disapprove in any way. It is simply to note that there are as many ways to teach as there are kids. And that without a teacher who cared Yateen would have been lost in the system, as small as it was in my hometown (SLO).


  4. A lot of this experience reminds me of something that happened earlier on in my teaching here. I am at a small regional college, one in what is sometimes euphemised as an “underserved” area. We’re rural, we get a lot of students from small, not-great high schools. We’re not open-admissions, but we’re close to it.

    Anyway. One cohort of incoming students was averaging about 30th percentile on the math portion of the ACT. WELL below average. And I can attest to many students being underprepared in math; I have learned I cannot assume people know how to calculate an average, or that they know some of the math terminology…

    We have a portion of the junior class re-take sections of the ACT, as a way of tracking progress.

    The cohort that came in at 30th percentile scored 45th percentile after two-and-a-half years of college. In my department, we were ecstatic: look at that improvement!

    But the administration came to us – “Why…why can’t you get these students to achieve more?” they say, not looking at the level of improvement they had experienced. One person even actually said, “Why can’t you get them up to where they are above average?”

    (I left it to the other person who teaches stats in the department to explain to that admin what “average” means and why you can’t expect everyone to be “above average.”)

    But yeah. It’s frustrating when YOU see a lot of progress, but someone who isn’t working with a person every day comes to you and ignores the progress and just asks why the person hasn’t set some arbitrary waypoint that they think should happen.


    • Just as an aside, math is my Achilles. I was in AP or honors classes in high school for English, History, and French. I was in the “tech” math classes. I could not do algebra to save my life- it was the only F I ever received on a report card. Even in tech math, I had a hard time achieving even a B in the dumbed-down version of algebra (I fared better in geometry, got an A.)
      I’m a smart person, I think, but I have a major math block. To the point that I switched majors in college to avoid it. I started out a poly sci major. But poly sci required taking statistics, which required Algebra I. I switched to English, and the only math class I took in college was “Mathematical Concepts for Non-Science Majors”.


      • I have TA’ed a number of people just like you. One had such serious math block she froze like a deer in headlights as soon as I put an equation on the board. I still recall her in office hours gasping at the solution to what (to me) was the simplest possible problem and saying “X is 2? But it was 4 yesterday!” I did, after replacing x with the concept of a closed box whose contents we were figuring out, get her past that. In fact she made it through the only college math class she needed with a C-. I later learned she was an Honors student in her major, so really it had little to do with intelligence and everything to do with a mental block about math.

        On the other side of things, one of the guys I knew in high school was a true math genius. He was acing advanced math classes at a local college when he was junior in high school. But there was a problem with the way his eyes tracked (not dyslexia, but something related) that made it nearly impossible for him do the reading and writing required for English and History. Again, nothing to do with intelligence, but tied to the way his brain was wired.


        • Yeah, I found sometimes explaining things differently or trying to find an application helped some of my math-phobic students.

          Also, I find it ironic, but about 80% of the students who come to me before Biostats starts up and declare a “fear” of math, when I tell them, “Just keep up with the homework assignments, if you are confused by anything come to my office hours, if you get stuff wrong on the homework and still don’t think you understand it after I’ve gone over the homework, come in and talk to me and I’ll tutor you” wind up doing FINE – many of them earning Bs or even As.

          It’s the people who walk in thinking they know everything and who blow off the homeworks, by and large, who earn poor grades. And I make a big deal the first few days of class about how the homework is how I monitor not just how they’re understanding but how I’m teaching: if 75% of the class earns a 10/10, that means I’m probably doing OK and it’s just that a few people are confused and might benefit from coming in for extra help. But if 75% of the class is earning less than 5/10, I probably need to go back and re-teach the topic in a different way.


          • Yeah, I found sometimes explaining things differently or trying to find an application helped some of my math-phobic students.

            One thing that really helped me in algebra was my teacher saying something to the effect of, “there is no subtraction and there is no division. Instead, you’re just adding negative numbers and multiplying by the reciprocals.” For some reason–I don’t know why–that really helped me. I’ve never been a math maven, but thinking of things that way seemed to work, at least for me.


        • I’m not sure about other subjects, but it’s pretty clear to me that we can not pretend that a singular, or small set, of mathematics pedagogy is sufficient. I wonder, at times, if anyone has tried to catalog all the myriad approaches to teaching math that have worked, especially with kids who are not ‘math people’?


          • This is made harder because most math classes — at least by the time you get to algebra — are pretty much forced to have at least two different purposes, that are somewhat at odds with each other. On the one side in algebra, is setting up and solving word problems: a way of thinking about the world, of finding a number or numbers that satisfy certain conditions. On the other side is sets and functions as first-class objects that is a precursor for calculus. On the third side is sheer skill at symbolic manipulation that will be necessary to deal with doing calculus by hand. Same thing in calculus: setting up and solving practical problems, still more of functions as first-class objects, plus a foundation in mathematical proofs for those students going on.


  5. The system is still far from perfect, but we do have different levels in the local school system here: 4, 5, 6, and HP. Most students are in 5, which is for the average/expected proficiency student. 4 is for those who struggle or need remedial work (there are also tutoring and special education classes for those who have difficulty at 4 level). 6 is accelerated for those ahead of 5, and HP (high potential) is for what we used to call ‘Gifted Students’.

    These are not blanket levels, where a student is always in just one of them, but applied subject-by-subject. So, for instance, there are kids in 6 for math and 4 for English, and vice versa.

    Of course, this is a huge school district (the high school holds only 10-12th grades and is still ~5000 students) so having to provide many multiple classes in any subject anyway allows them to more easily break core subjects into levels.


  6. I’ve mentioned before that I went to a high school with an ESL track. I guess it was considered the best way to handle this sort of thing at the time. The semi-literate immigrants got shuttled off to an upstairs wing at the end of the school that the Regents and Honors kids never had reason to go to. It’s somewhat weird, though. We had a number of students going to the school who spoke English as a Second Language (we had a bunch of kids from Jamaica who grew up speaking the Patois, for example). The only people in ESL were Spanish-speakers from Mexico or from Central America. The Jamaicans were in the regular classes with the rest of everybody.

    When I was in Elementary School, we had a kid show up who wasn’t good at English and the teachers pulled all of us aside and told us that we had to include him in our playground games and that he wouldn’t get better at English if we didn’t include him. So we included him and he did his part and watched Voltron after school so that he could partake in our rabbinical discussions surrounding the show.

    I don’t know what the official policy ought to be (depends on what we want, I guess) but the ESL kids in the first example didn’t ever integrate with the rest of the school. The kid in the second example integrated very well.


    • It is also easier (and more of a necessity) to integrate when you’re the only kid not speaking the language well, it is altogether different when it is a group that can speak with eachother in Spanish and can fall back on each other for social life.

      Immersion works best for learning a language and culture, but is very hard to pull off with larger groups.


  7. A growing body of research suggests that non-cognitive factors—persistence, effort, motivation—are important in adult outcomes.

    Sounds right, also sounds like we should be measuring this too.

    While the education philanthropists like Whitney Tilson hold that “kids will live up – or live down – to whatever expectations are set for them”, reality plays out very differently.

    Setting high expectations has worked really well in my family. Over the years, the big problem I’ve (repeatedly) had with the system is their expectations have not been high enough. Just letting the system do their thing would have resulted in “success” that wasn’t anywhere close to my kids potential.

    In practice, valuing “non-cognitive skills” almost always means lowering academic standards.

    Why is this?

    Schools might provide a menu of classes of varying difficulty, allow students to choose course material they are capable of and interested in learning, rather than set a ferociously high bar and then make some teachers choose between failing most of their students or not covering the material with a rigor appropriate for the strongest kids.

    This sounds promising… but I’d thought research showed the rest of the class is better off with the strongest kids there?


  8. “I don’t see why you can’t both teach the course and the kids. You just need to have courses tailored to different abilities. One course for people who might take longer to get algebra. Another for those who can get it when they’re 9 and yet another for those who need to be 13-14 before they get it.”

    You really don’t see why we aren’t allowed to do this? Look up the tracking wars of the 90s. Has nothing to do with budget.

    Fillyjonk: as to your first comment, preach. It’s very frustrating. Also annoying that by high school, the tests are designed far above the bulk of kids’ ability, so you can’t use them to capture progress.

    It is not, however, my experience that kids who are afraid of math end up doing well, unless the teacher rewards effort over ability.

    Em, there’s always someone who doesn’t grok their own personal restriction of range.

    Jay, Santino had surprisingly good English skills. ESL is almost never an issue with math. Even kids who can’t understand the language at all can advance if they are capable of understanding the math.

    Oscar: “I wonder, at times, if anyone has tried to catalog all the myriad approaches to teaching math that have worked, especially with kids who are not ‘math people’?”

    We’re not really allowed to pretend one method doesn’t work for everyone, and the “myriad approaches” pretty much boil down to progressive vs. traditional, after you scratch the surface.


    • At least some battles which were lost in the 90s may be winnable now. SSM is an example. It also seems that many of the education battles which were hot-topic in the 90s and early 00s are just off the radar now, not because the overton window has changed, but because the news cycle is much shorter. Given the different political environment, it might be winnable.


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