Classroom poster with students names and mastered math facts

Shame in classrooms

I ran across a blog post by Brené Brown earlier today. In the post she relates some of the kurfluffle around her comments to Oprah Winfrey about teachers and shame (original video now hard to find — its the last clip on this page). This all happened at the end of September, so I’m late to the party, but I had a few thoughts about what happened here . To my mind, the biggest place where Brown goes wrong is when she says that shame is a classroom management tool used in schools. By calling shame a tool, she implied that the use of shame was conscious, and since her whole thing is talking about how bad shame is, that’s a pretty damning way to call teachers out as being bad for kids. That turned her comments into a public shaming of teachers. Brown didn’t intend to shame teachers, but that’s the sneaky way shame works — it’s everywhere, and it is hard to avoid.

Shame is certainly present in every classroom — it is a nearly ubiquitous emotion, so it happens in classrooms like it happens everywhere. For the moment, let’s try to avoid shaming the emotion of shame — shame and the threat of shame are intertwined with nearly every connection we have with other people. That’s not good or bad, it just is. When our relationships are working, we are able to use subtle clues about shame in ourselves and others to figure out how to navigate the relationships without alienating ourselves or other people. But an excessive weight of shame, or shame that calcifies in certain areas can break our relationships, causing disconnection and isolation. In classrooms, both students and teachers can feel shame, and that shame is a signal to us that our relationships are either fractured, or in danger of being fractured. Teachers feel it when they are disrespected by parents, administrators, and other teachers. Students feel it when they understand that they aren’t worthy of connection with classmates, teachers, and specialist because they aren’t good enough. We have to watch for signs of shame in other people, and use it as a sign of relationship danger. Relationships can be repaired and shame can be healed.

Teachers care about kids, that’s why they become teachers. They work hard every day to help kids succeed. They also face a job that is so difficult, so painful, so demanding of every resource they have. They do it without enough pay, and with a whole world watching to see where they are going to screw up first. In other words, teachers do their jobs in an environment that is a shame pressure cooker. When people are in that kind of pressure cooker, they will push their shame onto other people, and it is easiest to push shame onto weaker people. I do it as a teacher to my students, and I do it as a parent to my kids. I don’t do it because I’m a bad person, or because I am sitting around cackling and thinking up ways to torture students and children. I do it because I am human, and fallible, and, because, as Brown said in her mea culpa post “learning is vulnerable and classrooms are tender places.” We do need to raise awareness of the presence of shame in classrooms because awareness is really one of the only ways to combat shame. But we have to raise that awareness gently and carefully because any time we start to really see how shame operates in our lives it is easy to become overwhelmed.

I do think there are some institutional practices that increase students vulnerability to shame in schools. Particularly, I am thinking about public accountability in classrooms. In elementary math classes, you will sometimes find charts of times tables and other math facts, where you can see how each student is doing on proving their mastery. Short timed tests like “mad minute” are used in most elementary classrooms, and those are also a form of public accountability since all the students know who finishes on time and who doesn’t because they can look around and see everyone, including those kids that inevitably shouted out as soon as they were finished, before the timer rang, stopping the kid who still had half a page of problems left. Public accountability charts are also used for behavior management and in other subjects. For some kids, this is highly motivating. Students want to do well and the competitive spin of public results help spur them to work. But for other students, being at the bottom (or even in the middle) week after week is demoralizing and shaming. I think that in order to support all students, we need to have students set goals and chart their progress, but that this should be private. This won’t provide the competition that helps some students focus, but it will avoid the discouragement and feelings of shame and stupidity that other students experience.

Classroom poster with students names and mastered math facts

Here not only is your math prowess public, but it’s linked to ice cream

We can’t change the fact that in every classroom, the students know where they stand in academic rankings. All of the students know who is at the highest reading level and who is still struggling to read beginning books. They all know who finishes the math assignments before everyone else and who never finishes. We humans are constantly comparing ourselves to others, and figuring out where and how we rank. But when those rankings are publicly displayed, their importance is reinforced, even if the teacher is telling the students verbally that effort and progress are the most important things. We need to put the rankings away and to consistently remind students that they need to look to themselves to measure progress. They need to be better able to handle fractions at the end of the unit than they were at the beginning. They don’t need to be better than another student, they need to be better than they used to be. We also need to talk to students about how we handle that terrible feeling we have when we realize that we aren’t doing as well at something as we wish we were. We need to talk with our kids about how to manage the pain when we find out we sang the wrong note, or messed up all the problems, or our drawing wasn’t selected for the prize, or we realize that our friends are all reading at a higher grade level than we are. If we avoid emphasizing rankings and do some explicit teaching around how to handle the emotions that arise when we fail or don’t do as well as we would like, then we give students the tools to navigate pressure and criticism without falling into a pit of shame. Most teachers already do a lot of work to combat shame, but connecting the dots on the impact and mechanisms of shame can help teachers better see what they are doing and how to do it effectively.


Who Benefits from Educational Intervention?

Last Monday in Liberation Math, the class, we had a great conversation about Logan LaPlante’s “hackschooling” TED talk. It’s a great talk, and I highly recommend checking it out. LaPlante is a winner and is doing great things with his education. He makes it all look easy, but extraordinary resources go into making his hackschooling education possible. His family skis a lot, which tells us that they have an income that makes that possible. He has an extensive network of opportunities, which means that his family is aware of the opportunities, that they are available to people like LaPlante, and they have the time and funding to allow him to access all of those opportunities.

LaPlante is a kid who is likely to be successful no matter what system he finds himself in — he’s charming and well connected. We can’t use him as a test case if we want to figure out if hackschooling is useful for a broad range of students because he is not a typical student, and he’s certainly not one of the millions of students struggling to successfully complete their education. Of course, it could be argued that if LaPlante was forced into a soul-crushing educational system that he might have begun to have difficulties. A bad education might have hurt LaPlante and decreased his chances at success in education and in life.

The fact is that educational innovations and interventions work — nearly all of them work, which doesn’t mean that we should implement every educational fad. We spend a lot of money on educational “fixes” that don’t really change things very much, often at a high cost. And the impact of the intervention depends a great dean on the population you start with. Suppose we take 100 well-resourced students like LaPlante, 80% of whom were going to be successful without intervention. If we do an intervention with these students that shows a 25% rate of improving student outcomes, then we see the following:

100 students, 20% struggling, intervention helps 25% of students

You are a student in this group who is doing fine or better. Was it the intervention that made the difference for you? To calculate the probability that the intervention moved you from struggling to fine, given that you are a student doing fine or better, we have to take the number of students who are fine in the end but would have struggled (5 students), and divide by the total number of fine or better students (85 students). This gives a 6% chance that the intervention is what made the difference here. We have to provide this intervention to 20 students in order to move a single student from struggling to fine. If this is an expensive intervention, that may be impractical for the results that we get, but will be entirely worth it if you are that one students and you possess the resources for change

Now imagine a population of students that has far fewer resources and experiences greater struggles and a greater likelihood of failure. Let’s say this new population of students has only 20% who are going to be successful with no intervention. We’ll imagine the same intervention that helps 25% of students:

80% struggling, 25% intervention success rate

In this new situation, if you are a student doing fine or better, what is the likelihood that your performance is a result of the intervention? Here again, we take the number who were struggling and are now fine (20 students) and divide by the total number who are doing fine or better (40 students).  This gives a 50% chance that your positive outcome could be credited to the intervention, a big difference from the previous population of students! Here we need to provide the intervention to just 5 students to move a single student from struggling to fine, giving a much greater efficiency. Perhaps this hypothetical intervention now looks great, but if it is an expensive intervention, requiring a lot of human capital, we still may not be able to provide the intervention to a broad range of students who need it.

How effective does this intervention appear to be? If it is implemented with the first population, then 85 students are doing fine or better, so it appears to have an 85% success rate. But if the intervention is implemented with the second population, then just 40 students are doing fine or better by the end, so it appears to have a 40% success rate.

Population General Educational Intervention GraphBut of course I made up the numbers about the population and the effectiveness of the intervention out of my head! We can model the situation in general with a population in which F out of 100 students are fine and we implement an intervention which is I percent effective. In this general situation the success rate will be F+(100-F)*I/100, which we can graph F on the horizontal axis and I on the vertical axis, coloring each point in the plane with the success rate as below, where lighter colors means a higher success rate. You can see that both the base success rate in the population and the effectiveness of the intervention constrain the overall outcome. For very successful populations, nearly any intervention will appear to be successful, but not all interventions that “work” with a naturally successful population will work with a struggling population.  And for an intervention that is nearly 100% effective, you can achieve amazing things with nearly any population. The trouble, to my mind, is that there aren’t any interventions that are 90% effective or better. Even the best interventions are going to be unlikely to get much over the 25% mark. Of course you have to keep in mind that this entire scenario is simply a “toy model” — it’s not reasonable to measure effectiveness simply by reporting a percent or lump students into two categories of “struggling” or “fine.”

Educational success is not always easy to achieve. Interventions, even when successful, aren’t going to solve everyone’s problems. When we see amazing educational success, we need to ask ourselves who is successful and why they are successful. Our educational system holds out the idea of advancement for all, but the reality is often that the greatest advances are made by those who were already set up for success. I’m really trying to wrap my head around these things and connect them to other ideas, so I’d love to have your thoughts!

Liberation Math Week 7: Easy Now

I have a tendency to make things hard, and I wonder if you have the same issue.

Almost everyone in an academic environment gets behind at some point during the semester. I have watched students do this for years. Typically a student starts to get behind, feels bad about themselves, and the bad feeling makes it hard to work and the student falls further behind. After an episode like this, a student will often come to me apologetically, promising to do better, and telling me about how they are going to get caught up soon and will keep up after that. The trouble with that is that it rarely happens. I think it’s a bit like how dieting causes weight gain. For the most part, we all want to do well when we take a class, and when our actions run counter to that goal, then something is going on. Maybe we really don’t want to engage with the course, and so we avoid the work because we don’t want to do it. Maybe we have more on our plate that we can handle, so we start letting go of things, coping with feeling overwhelmed through avoidance. Maybe we feel bad when we do work for the course because we don’t think our work is good enough, so we avoid possible failure by avoiding the work. I do each and every one of these, and more. We all do, because we are only human.

The Fables of Aesop

We all know we’re supposed to be the ant, right? (Photo credit: dierken).  Currently listening to Easy Now by Edie Carey 

I think we actually use our tendency to get angry at and disappointed with ourselves to give us an excuse to do even more avoiding. Sure, you may be letting yourself down, but if you feel really awful about it, then that gives you a little distraction from the fact that you really do have a dream or a goal and your actions are hurting that dream. We can’t get out of the hole by feeling really awful, or by making a vow that now we are going to be perfect and we will never fall behind again (that’s making promises that our future self doesn’t want to keep any more than our present self does). There’s really only one way out: Take one step. Taking an action that points your feet in the right direction, an action that gets you moving, that’s the only way out. Any one real action right now can make things better.

I’m talking about this as if it’s a student problem, and it’s not. Everyone does this, and we all deal with it in different ways. I deal by staying really busy and pushing myself hard. Then when I get overwhelmed I have an excuse to throw up my hands and give up since there’s not enough time to really set things right. If it is true that the only way through is to take the next step, then maybe I should make the next step easy, rather than hard, which is my natural inclination. In fact, I’ve decided to let everything be easy. Ironically, that’s not easy to do. It runs counter to what I’ve been taught in my years of school and what I’ve taught my students since I graduated and became a “source of knowledge” (note to readers: use an ironic tone in your mind when reading that last phrase). We all know that the secret to success is hard work and that “practice makes perfect.” Particularly in math, I have always believed that lots of practicing is absolutely essential if you want to do math. The trouble is that I’m starting to think it’s a bit more complicated that that.

In K-18 education, we have students practice by giving them homework, and there are arguments both that homework is “good for kids” and homework is “bad for kids” with research about the impact of homework on achievement (grades, test scores) backing up the different sides. But is achievement what we really care about? In K-18 we care about grades and tests because grades and tests will serve as a signal to future schools and employers that they should pick us for their team. If we can get the right GPA, degree, or test score,  the promise is that we can have something that we want in to future (like a great job), so achievement is something we care about when we think about the future.

But what about our current selves? Do we really have to wait to a diploma or degree to have what we want in life? Sometimes the answer is “yes” — for instance, if a master’s degree will get you a promotion and raise, then you really want that marker of achievement, and that may be enough of a goal to sustain you along the way. But for many of us the game of grades and tests is stressful, scary, unpleasant, and hard, even if it is necessary. Many of us need something to care about something besides achievement in order to make it through all of those difficult tasks, to make things a bit easier. We also arguably need to care about something other than achievement in order to make our school experiences truly transformational. So, if its not achievement that turns our cranks, then what is the point of homework, practicing, and all that hard work? If we have an authentic purpose, aside from achievement maybe we don’t have to slog through the drudgery. For instance, if you want to launch a rocket, you might need to test out configurations and do hundreds of calculations, but its not practice and it doesn’t have to be hard. True, it may take time, your path to that rocket launch may not be direct, and you may sometimes be very frustrated, but you don’t have to drive yourself forward, convinced that if you don’t keep your nose to the grindstone, you’ll never get there. Some of your best ideas will come when you distract yourself and take time for play, and you can have faith in your dream and keep taking that next step.

As a teacher, when I worry that my students aren’t “getting it,” my inclination is to do to my students just what I do to myself: push harder. I do the same thing to my kids; when things aren’t going well I make more demands, thinking that pressure is really what they need (that, and lectures too). I do the same thing to myself — when I feel that I am “behind” or that I want to be doing something more or different, I remind myself that I’m lazy and that I really need to push myself hard if I want to have my dreams. After all, I did watch three TV shows last night rather than working on this blog post.

What if I stopped doing this? What if when I feel really bad about what I’m not doing, I think back to my big goal, remind myself that the work really is easy and pleasurable, and just get myself to engage for 15 minutes and then take my TV break? What if instead of lecturing and threatening my kids, I remind them of how great they really are at the things I’m wanting them to do? And in the classroom, what if I point out to my students what they are doing well and find a way to increase my connection to them, believing for them that it is going to be easy to re-engage with the class and get over the obstacles in their path? No, none of these would be perfect solutions, but expecting things to be perfect never really gets me anywhere.

My questions for all of you: Why do teachers have students do work outside of class? That is, what is the purpose of practicing the math concept, reading the article, writing the paper, or whatever else we are asking students to do in K-18 classrooms? Is the work we assign the most effective way to reach our goals? Is the solution to difficulty to work harder? Do you believe that you should work harder? What do you do when you “get behind?”

Anger and Shame in my Teaching, a sort of anti-liberation-math

English: A metaphorical visualization of the w...

Photo credit: Wikipedia

Last week I wrote a post on another blog about being disappointed in students, which is something I struggle with and want to eradicate. Yesterday I realized that I had misnamed the problem. It’s not disappointment I struggle with, it’s anger. Anger isn’t a professional emotion as a teacher, and it makes me feel distant and disconnected with my students. My mission in life is connecting with students, so why would I repeatedly entertain an emotion that hurts my chances at living my dreams? And I do mean that I entertain the emotion of anger. I tolerate it, nurture it, and feed it. And I don’t just do it as a teacher. I do it with my kids, getting angry when I could be connecting. I do it with my spouse and my family, getting angry when they need me.

I have been studying shame for some time now, particularly shame around mathematics, which typically means shame around math in a school setting. Shame and anger frequently occur together, often in a cycle, so I should have suspected that my problem was really anger. Here is how it works, and how it connects to shame for me.

I ask students to do certain work and I have certain expectations of them and the work they do. I can do this because of the power that I have in the classroom. This is the power that I have over their grades, and the power that I have because they have been raised in a school culture of compulsion and obedience. When they do not do the work, or they do not meet my expectations, I feel that as a threat to my social self. I feel disrespected and I feel a version of shame because, in my eyes, the has student decided that my expectation was not important to them, and thus that I was not worthwhile. The work I ask for and expectations that I have are the clearest evidence of my relationship with the students, so when that is threatened, my relationship is threatened.

I respond to that feeling of shame by getting angry. Shame is an extremely uncomfortable emotion, so people will usually seek to cover it with another emotion, and anger is a popular choice. When I am angry, I can focus on the student as the problem, rather than the hurt, disconnection, or shame that I feel as the leader of the class. Furthermore, I can blame my feelings entirely on the students. The compulsory nature of education means that they should be doing what I say, and if they don’t, it is the students that are in the wrong, not me, the helpful teacher. And even worse than that, once I am angry, I don’t just want to let the problem slide, I want the students to know that I am right and they are wrong, to punish them and to have them accept that punishment as their rightful due.

Honestly, its a gross thing to admit to, and is the opposite of everything I want to be. I suspect that we all have this kind of shadow side, working against our dreams in the background even while we strive toward them in the foreground. I’m a good and caring teacher with a mission to connect with students — why would I keep this anger and meanness around? It’s my defense against being too vulnerable. I don’t want to look at what it means when students don’t do assignments, stop participating in class, or give up. Yes, sometimes it means that the students have things going on that don’t have anything to do with me. However, it can also mean that I am failing to build my relationships with the students and failing to find genuine ways to help them build power with and through math. It can mean that I don’t have real relationships with the students at all, only a dance with the students in which they carefully display certain signals in order to get the maximum grade with the minimum of effort. I’m responsible for that dance at least as much as they are, if not a little more. Why should they not be doing this when I have to admit to often falling back on my default job as a teacher, which is to compel students to do a certain series of tasks, to assess them on those tasks, and to assign them a letter grade that they take with them as a credential or a black mark. It can mean that I’m actually hurting them through my participation in a particularly oppressive subject within an oppressive educational system.

I sure want all of your thoughts on this one, whether you are a student, a teacher, or anyone who gets in the way of their own dreams and connections!

Disappointment and Hiding in the Classroom

I’ve been noticing lately my disappointment in students. I don’t want to feel disappointed in students. Honestly, I don’t want to feel disappointed in anyone. Who does? But you might argue that we have certain expectations for how the people around us will act, and that people don’t always meet those expectations. When they don’t, I am justified in feeling disappointed, at least provided that my expectations were reasonable. The trouble is that  disappointment is counterproductive, and for me it is part of an overall tendency I have to disconnect with people.

Let me look at this a little closer. I have certain expectations for my students. I set those out for the students by giving them specific assignments (“turn this worksheet in on Monday” or “write a blog post about your problem-solving process”), and I lay them out on the course syllabus by telling students to come to class, check their email regularly, participate, and so forth. There are also a collection of expectations that go unspoken by me. I expect that students will be thinking about what they need to do to prepare for upcoming exams, even if I don’t give them explicit assignments. I expect that students will ask for help and support when they don’t understand something after class. I expect that students will monitor what they do and don’t understand. I expect that students will give me their best work, and won’t piece together something at the last minute. I often say things which imply these expectations, but I’m not always explicit about them. Also notice that not all of these expectations are realistic.

If a student doesn’t meet these expectations, I get cranky. In between classes, if I am expecting work and participation from students that I don’t see, I start to worry, and to run my “disappointment tape.” Typically it involves me getting frustrated and making up a lot of things that I imagine to be happening with the students. I imagine them as uninterested in the course, not dedicated, not hard-working, wanting to get away with not doing work, not caring about thinking deeply, not caring about interacting with me or other students. Yes, there’s some really ugly stuff hiding in there. The thing is that I don’t know that any of that is really happening. Mostly, I think what is happening with me is that I want this connection with students, and most of what I have to connect with is their work. When the work isn’t there, I feel rejected. I imagine the students pulling away from me, and I rush to pull away from them first, by getting “disappointed” in them. Most of the time, I can get back my connection with the students simply by being around them — it is the time in between classes that provides a space for these feelings to grow.

Students don’t always do what we teachers what them to do. In fact, people in general don’t always do what other people what them to do. So we get anxious about our relationships and our standing with other people. In school, this means teachers get frustrated with and disappointed in students. What do students do? Students learn to hide from the disappointment of teachers. They hide and they lie so they can save themselves from the consequences of expectations unmet. Students hide so that they’re grades aren’t in jeopardy and they hide so that they can maintain positive relationships with the powerful people that are important to them. Students get into a habit of hiding, so that it seems as natural as breathing. I remember it well from the last time I was a student — doing work I wasn’t proud of and hoping it would slip by without notice, making up excuses for doing work late or stretching excuses that were technically true but not really accurate, trying to look good in order to get away with things. As a teacher, I know that students are doing these things, but I ignore it, acting as if students are going to meet all of my expectations, and then getting disappointed when they don’t. Because I am required to assign grades to students, I maintain and perpetuate the fiction that grades mean something objective, when the reality is that they’re just a somewhat arbitrary record of how well a student met my somewhat arbitrary standards about a somewhat arbitrary collection of activities and topics.

What if I stopped doing this? It’s hard to imagine. Could I stop having expectations of students? What would happen to me and to the students if I did? What if I kept having my expectations, but was more honest about the fact that I know students won’t always meet them? What’s so bad about the students not meeting them anyways? Could I keep the expectations, but let go of the disappointment, simply connecting with students about what happened and deciding what to do next? Could I let my students be honest with me about the unrealistic nature of my expectations and with what really happens for them in a class? Could I let students formulate their own expectations, help them to make those expectations realistic, and then help them to live up to those expectations? Could I create a classroom environment in which I helped my students evaluate themselves? Wouldn’t this cause the very foundation of objective and rational subjects like math and science crumble because students would start writing expressive poetry about how math makes them feel and giving themselves an A++ on every assignment?


Woman with hands in air

Showing Work in Mathematics

Last week, Karen Young stopped by this blog and made a comment that led to a great discussion that has taught me a lot, so I decided to pull it out and capture it in this post. First, she said in the first comment on this post:

As a kid I could do math in my head, doing basic mathematics without using a calculator or pencil (when I do laps I still do fractions in my head). Every test, there were marks off for not showing my work. If I answered a question on the board I had to show my work. Well sometimes I couldn’t do that because my brain “knew” the answer.

I replied to that point (and yes, I edited a typo in my original comment):

I really think your point about showing work is interesting — I know that the curricula used in my area are all about showing work, often showing multiple methods. I can get that we want students to be able to communicate their thinking, but can we do that without making it drudgery?

To which Karen made a response that really blew my mind:

Angela, when did we decide, as educators, that we had to show work in math? My grandfather was an accountant and brilliant with figures but he did them in his head. That is how I did math when I was young. Having to show it actually made me have to rethink my answer, leading me to doubt my accuracy, which lead to me “showing my work” and making more mistakes. Math was intuitive, almost instinctive prior to that point. Now it is something I fumble over, except when I am swimming laps and almost in a trance. Why is intuition in learning a bad thing?

She further expands this in a later comment:

We always remember how to tie our shoe because of motor memory, so if we’ve created a motor memory (or song memory) through the physical teaching of math, doesn’t that link remain? Especially if we exercise it everyday? At some point, if we approach a subject in the wrong way, I think we can break the ” old link” in the brain, by creating the new “show your work” path. And do the two conflict? In my case, yes. I am intuitive by nature and am used to my brain making what appears to be sudden connections but I am in fact just allowing it free rein to make associations that lead to “aha’ moments.

So, why do we ask student to show their work in math? Here are some reasons I came up with (if you have more reasons  or more interpretation of these reasons, please lay them out in the comments):

  • We believe that explaining the mathematics is an integral part of understanding the mathematics. That is, just as Karen says, we mistrust intuition. After this conversation with Karen, I think that this mistrust may be wrong-headed. I’ve known plenty of students that could see answers that they couldn’t fully explain, and I think it is patronizing of me to not believe in their understanding simply because they can’t walk me through a solution step-by-step in a way that I expect their mathematical learning should have trained them to do.I see both my own 6-year-old and a fourth grader I tutor working with the TERC Investigations curriculum which asks them to draw out solutions to problems (for the first grader) and to show two different methods for a problem (for the fourth grader). These both seem like they might artificially interfere with a student’s process. You should of course draw out solutions to a problem, but only if that’s the way you solve the problem — if you count on your fingers or see the answer in your head, the drawing step is artificial, meaningless, and can possibly get in the way of your own method. And you should explore different methods for doing, say, three-digit addition so that you can understand the process and settle on a method for yourself that is actually meaningful, but why would you need to show two different methods for the same problem simultaneously? (And see Karen’s comment below this — if you were comparing answers with a peer you would get exposure to multiple methods for the same problem in a more authentic way.)
  • Some problems are complex enough that they require record-keeping. This might be careful recording of useful data, recording results and methods so that the problem can be tackled again after a break without losing momentum, or writing an explanation to be shared with others. Asking students to show work even on less complex problems may help to train them to do this kind of recording so they have it available as they problems they are going to tackle get more intense. I think this is actually a good reason to show work, and the question for me becomes whether we can do this kind of training in an authentic way that still honors intuition (which is really just deep, non-verbal understanding), but helps students gain facility with communicating.
  • We don’t trust that the students are really doing the work we set out for them, or we don’t trust that they are using the methods we want them to use. This could be because we think the students are cheating or because we think they are using techniques or technology that we don’t want them to use on the problem. I have been noticing lately when I make choices as a teacher because I don’t trust students. It is more often than I would have thought, and I want to find a way to stop and increase my trust of students.
  • As instructors, we want a window on students’ thinking in order to help them. If we only have an answer, and that answer is wrong, then we don’t know anything about where the student went wrong — it could be as simple as an error in what numbers were used, but it could be a misunderstanding about the mathematical concepts. This is, I think, another good reason for having students show work, but it is really only needed if students are struggling.

I laid out these reasons for Karen, more or less, as part of the comment thread then asked:

How do we support student intuition and still help them to develop record-keeping and communication skills that will serve them well as problems get more complex? And if we are having them “show work” to help develop those kind of skills can we make sure that it is authentic, not made up after the real work is done to satisfy requirements?

Karen answered:

When we look at teaching math we are, in a sense, trying to develop two skills within math once the basic math foundations are in place. The ability to edit numerically (see your mistakes) and to be able to reason mathematically (knowledge, logic and intuition). This is predicated on the foundation being strong (but we both know sometimes it isn’t.) In English, I have had many students who have a wonderful writing style, a true “voice”, but their grammatical skills are terrible. I have always counseled them to keep writing and find themselves a good editor. Not every student can see their mistakes, which is why we have groups share papers to help proof at all grade levels. So why can you not have math conversations and math proofing shared between students? If sharing is how we build knowledge and understanding this would help support student math learning and math intuition.

I think this is a great suggestion, and a profound one in mathematics. If students are in conversation with each other, then communicating mathematics and showing a record of your work become authentic tasks that allow you exchange ideas with other people.

Thanks, Karen, for the great conversation! If you have any thoughts about your own experiences with showing work or asking students to show work, or if you have thoughts about why students should (or shouldn’t) be asked to show work, chime in!

Reading: Grading Student Writing by Peter Elbow

On the recommendation of Jesse Stommel, I’m reading this paper about grading student writing by Peter Elbow, and I’m trying to figure out what it might say about my own grading practices. First, let me say that that the problems of grading writing may be qualitatively different that the problems of grading mathematics. Mathematics has this wonderful and horrible right/wrong duality in it, and it is often set up as an objective arbiter. Emotions and opinions don’t come into mathematics grading, because how can they? It is always true that 2+2=4, and it is never true that 2+2=0 (except of course if you are working mod 4, but that’s just me being an obnoxious mathematician). I suppose that is the true truth if you are either machine grading, or are grading with no “partial credit.” But that is almost never true for me, because I don’t think that the most important thing about a solution on an exam or in homework is the final answer. The process is far more important, and gives me more clues about what the student is thinking and what they have learned. Add to that the fact that I typically give projects and other more subjective assignments for at least part of a student’s grade, and the situation gets quite muddled. And of course I accumulate a large list of quantitative measures during a semester and combine them together in an arbitrary way that I determine at the beginning of the semester, with each grade making up a certain percent of the final grade. All of that is the say: mea culpa, I may need a better theoretical framework here.

Right away the paper grabs me, then with the discussion of the difficulty and unreliability of grading, and even more with the wall it puts up between teacher and student. As Elbow says on the first page, “Students resent the grades we give or haggle over them and, in general, see us as people they have to deceive and hide from rather than people they want to take into their confidence.” I’m in, but what do I do?

Elbow recommends using minimal grades, like pass/fail or strong/satisfactory/weak. He recommends this for low stakes writing, and I could see it working perfectly for low-stakes assignments. In fact, I rarely grade homework. Most is graded on completion only, or if I actually want to provide feedback I use a 0-2 or a 0-3 scale. But really, maybe the words work better (only what do I write in my gradebook?). Elbow says that we can judiciously increase the number of levels in higher-stakes situations if we want, still without resorting to the eight levels of the traditional letter grades with pluses and minuses. Honestly for a test, this would be harder for me than what I already do. I tend to grade student work on each problem using a rough rubric that tells me how many points to give what kind of work — I might subtract points for certain kinds of errors, or give a certain number of points if the student made a correct start to a problem. So when grading is done, I have a bunch of number to add up, and presto, I have a grade! And arguably that grade gives me an idea of how well they were able to demonstrate their knowledge on that particular test. Moving to a more fuzzy system would be more work for me, but I can still see some advantages. I would likely still grade in much the same way, but I’d have a less fiddliness over the small numbers of points, with all questions being strong/satisfactory/weak. Then I have to think of a way to get the exam assessment overall.

Elbow can help here again, and maybe help with my poor Excel gradebook. He suggests to look at all of the grades in aggregate. Say you have a lot of low-stakes grades. Doing “satisfactory” on all of those might be a B, and then looking at the smaller numbers of higher-stakes pieces could pull that B up or down. Being a math person, that screams out to me to make up a formula, and you again get into the whole problem with grades. Wouldn’t a narrative evaluation simply be better and more nuanced, allowing me to say to a student “You did great with all of the lower-stakes pieces, but once the stakes were raised, you struggled to show your competence and understanding.” Then the student and I could both think about why that was. Perhaps the higher-stakes assignments required putting more concepts together, or maybe the pressure negatively impacted the student’s ability to think and communicate clearly.

Elbow advocates for portfolios, which I think are a good idea, but I have only occasionally used. He also discusses the use of contracts for grading, which I think I last encountered in high school. I could see contracts being a way of being up-front in my manipulation of students, as Elbow suggests. In doing so, I could clearly spell out my expectations for behaviors associated a passing grade. My only question there is what happens if the student has all of the behaviors associated with passing, but still doesn’t learn the material? What if they still can’t do any math? Honestly, I don’t think that really happens, at least not if I choose the right behaviors. But I worry about whether I have a clear leg to stand on if criticized for this kind of grading practice. Is it “rigorous enough?” Don’t I want students to come out of the class with some products, rather than just a process and effort? I think what I am struggling with is the student that just does the motions as they go through my class, appearing to really engage without really engaging. I suppose that such students pass through my classes all of the time, and there is no fool-proof method for bending them to my will and forcing them to engage in the ways that I desire. And when I put it that way, perhaps there shouldn’t be. Maybe the real problem is in trying to manipulate students into doing what I want them to do at all.

SJSA Grade Six -  The Year I Rebelled

Photo credit: Michael 1952

Elbow also suggests being explicit about criteria. I tend to have rubrics when I grade project work that spell out what I am looking for, and Elbow’s minimal grading would make this easier and less rigid. I could also give criteria on exam problems, or I could split up into multiple criteria. In a calculus class exam problem I might be looking for the method of solution, setting up the solution in a reasonable way, and executing that method including getting algebra correct. I could be clear about each of these criteria and evaluate each problem on each criteria.

Part of what makes grading hard is being the person that holds the power of judgment, and that’s just part of being a teacher. The power is mine to hold and negotiate, since I have to write down a letter grade at the end of the semester. I want to use student assessments in a way that is helpful to the students, and to determine letter grades in a way that doesn’t create excessive distance between me and the student, or between me and the task of judgement. Honestly, right now I use my grading system as a very long arm that allows me to avoid the uncomfortable position of judge. I don’t really determine the grades — a lot of numbers determine the grades, and I have very little to do with it. I can hide behind those numbers. I can even advocate for and advise students about how to beat those numbers, ignoring the fact that I’m the one writing down that letter grade. Once again, it all comes down to the relationships in my classroom and how I navigate them and engage with the students, and I can see that I have some work to do here.