Improvment

In my first several years of teaching, when things did not go as expected,  I thought about how to improve results. This [what] often meant adding another rule or expectation [to what], because my students were not doing what I perceived they needed to do in order to learn. After a few years, I had created a bureaucracy that was unmanageable for me as well as for my students. The worst part was that these rules and expectations had not helped improve instruction.

I came to realize that my job was not to tell students what to do. My job was also not to show them how to do a problem. My job was to create interesting situations where they could think about mathematics and learn from the work and discussions that followed. My job was to assist students in anyway I could to develop their own understandings of important mathematics.  I have documented these thoughts and processes many times in earlier posts. Here is a new thought for me.

This same process applies to teacher improvement. Those in charge of teachers at the local, state and national level are ill served by piling on more and more rules and mandates about how teachers should teach. If they really want to improve instruction, they must follow the same model that I followed in my classroom. The supervisors need to see that their job is to do whatever they can to facilitate teacher learning (therefor student learning), as opposed to requiring teachers to do certain things in a certain way. Teachers, with proper resources, will find ways to reach students. As things stand now, those in charge are working very hard to make a teacher's job as hard as possible.

Administrators need to observe teachers teaching. Administrators need to listen to what teachers have to say about the difficulties they have, and administrators need to work hard to help the teachers solve their problems. I have had to good fortune of working for a few such supervisors, and it makes more of a difference than I would have ever imagined. And the very best supervisors worked hard to help the top-level administrators understand that learning to teach well is a very difficult process, that it takes time, and that it requires support and nurturing. Learning to teach well does not require demands and punishment on the part of the Administration.

It is part of the job of experienced classroom teachers to facilitate this process with newcomers and to help administrators understand what they need to do to be effective. [why this last sentence? I don't quite follow you here. can you say more about what you mean here?]
[as it is now, this last sentence weakens your argument. you have focused on the way in which teachers help students in a way that is like what administrators should do for teachers and don't; you haven't talked about peer-to-peer learning at all, so to say "seasoned teachers need to help too" may be true but blurs the focus of your point. ]

The importance of Content knowledge

I was asked a while ago to write a blog for the National Council on Teacher Quality. I agreed and here it is. This is also posted on their website nctq.org/commentary/blog. This is re posted here with their permission.

During my forty-two years of teaching high school mathematics in Evanston, Illinois, I concluded that an essential ingredient for providing quality learning is that the teacher be well versed in the subject that the student is learning as well as the content that comes before and after the subject being learned. This may sound obvious, but it often happens that teachers have mastered what is in the textbook they are using without having knowledge far beyond. I believe this greatly inhibits their ability to help students make connections and often such teachers make poor choices about instruction because they fail to see the entire picture.

I taught a two semester Algebra 1 class, Empirical Geometry, Mathematics--A Human Endeavor, as well as Traditional Euclidean Geometry, Trig, Calculus, Multivariable Calculus and Linear Algebra. I found deep knowledge to be useful at all levels, all the time.

A teacher who has mastered the material well beyond the course being taught will understand why certain topics are presented the way they are and will anticipate what's next. A less prepared teacher will emphasize tricks and shortcuts that will get the students through Friday's test, but leave them ill-prepared for future courses.  For example, a student who learns to multiply binomials using FOIL (First, Outer, Inner, Last) instead of the distributive property of multiplication over addition may do well on the problems involving multiplication of two binomials, but will be hopelessly confused when multiplying more than two, or when one of the factors is a trinomial.

Part of good teaching involves understanding the importance of what is being taught and how it can be applied. Sometimes application of the content does not come until the student studies physics, or calculus, but a teacher who is not well versed in those subjects will not understand their importance. For instance, a teacher who is not familiar with Linear Algebra will not understand the importance of row-reduction of matrices and probably will not present it as the tool of choice for solving systems. In fact, many of those teachers will never ask their students to solve two equations with three unknowns because they do not see the big picture, limiting their students.

I have also observed that students can have remarkable insights into the subject at hand, but those insights may not be well formed. A teacher with deep content knowledge will be able to see the gem the student has noticed and clarify it for the rest of the class. A less prepared teacher will not. I found that by giving students a problem and walking around observing their work, I could find the teachable moment for the concept I was trying to teach, and I could make intelligent use of student work in bringing that moment to life in the class. This would have been very difficult if I was not confident in recognizing good and bad mathematical work.

To ensure that our students receive a rich math education rather than a string of rules, I think we should move forward by insisting that certified math teachers know a lot more mathematics than what they will be expected to teach and that they know it well. 

— John Benson

Directions

The Shanghai Metro website is terrific: you can look up stops, fares from point to point, in nearly-flawless English.  But while doing some browsing I came across the instructions you see at right.  In case you're having trouble reading the image, they are:

Take the Metro

1. get into the station
2. buy the tickets
3. move to the platform through turnstile with ticket
4. wait for the train
5. get on the train
6. get off the train
7. move out of the platform through turnstile with ticket
8. get out of the station

What struck me as funny about these instructions was that I couldn't figure out who they might be for.  I mean, if you can't figure out that after going through the turnstile, you have to wait for the train, what use is that instruction going to be?  I could imagine one of our students with autism using these kinds of instructions ... but also with lots of practice and review.  What I can't imagine is someone who really needs these instructions being able to go on the web, download them, and then use them to actually successfully navigate the subway.  Anyone who can do all that can probably figure out the subway.

To be fair, another page on the Shanghai Metro website gives really helpful, step-by-step directions with warnings and pictures.  It's still a little funny to me to think of someone needing to be told things like:

but I can imagine saying them, so I guess it's worthwhile.  (And the instructions for using the ticket machines are actually excellent.)

But I was left wondering how often we as teachers make this exact same mistake, namely, give directions that would only be useful for people who don't really need them.  For example, when we "teach" kids to write research papers ("teach" being a term I use loosely in this context), we often say things like:

Step 1: Identify a topic.  Pick something that interests you that you can write about.

Step 2: Research the topic.  Keep track of your sources so that you can footnote them in your text.

I'm being a little facetious here, but not very.  Instruction about selecting a topic might include some platitudes about not being too broad or too narrow, but how often do teachers actually sit with each kid and talk about the topic for 3-5 minutes to help the kid learn what is too broad or too narrow, and how to widen, narrow, or pivot the scope?  We teach the mechanics of how to research ("This is how you use the online database" "This is the card catalog") but do we actually model the process of finding a source and using it to find others, or to supply background knowledge, or questions for further inquiry?   Do we model the process of constructing a paragraph in which information from two different sources is combined in a synthetic way, so that students can actually see the difference between copy-paste and genuine research?

The same is true about other kinds of products.  I've never yet seen an elementary school teacher workshop students' written fiction.  Neither of my children has actually designed an experiment in science class.  These challenging processes require actual instruction--not just assessment--as much as any other.  We need to be sure that the directions we give are useful to the students receiving them.

Questions are Expensive!

A passage from this great posting on standards-based grading caught my attention:

Am I tempted to include one question on my test to send the message “HEY! We spent a day on this in class and we had a homework assignment on it, so you better do it because I said it would be on the test!” Yah, I’d probably do this. But the real message it sends is “I use my tests to reinforce that you should be doing my homework for arbitrary reasons and to punish you when you don’t”

What the author realized can be summed up in the title of this post:  questions are expensive!

In my work on a test development committee, one of the first things I learned was that a standardized test  ("item") that appears on the ACT and SAT represents a substantial investment of cash and time.  It costs upwards of $1,000 a question (in fact, by some estimates, more than $10,000) to develop, vet, and pre-test a single item of the hundred-plus items that appear on a typical SAT.  In the context of test creation and administration, this fact makes test editors somewhat conservative: deciding to change (and re-vet) or throw out a test item in the late stages is actually a major financial commitment.  But what I'm suggesting here is that, as teachers, we should all be somewhat conservative about what we put on tests, because every test (or quiz, or homework, or project) item is expensive.

"Expensive how?" you ask.  The costs abound.  It takes time to write, proofread, and format the test.  It takes time for your students to do the test, time that could be spent in doing other questions or assignments or just (imagine!) having fun.  It takes time to check and grade the test, and then you have to figure out what to do with the scores and information about student performance.  Most important, every item you include on a test represents a decision not to include something else:  you can't give a class of fifth-graders a six-hour math exam.  Unless you teach the most boring class ever, the chances are that in the course of a single unit, you've had your students work on many different skills in literally dozens of tasks and contexts.  You can't rehash all of that on a test, so every item that gets on the test has pushed four or five or six more off.

So you need to be a little conservative.  By that I don't mean that you can only assign items that you've already reviewed in class, or that you should never change a test--quite the opposite.  You do need to choose your items carefully, thinking about them more as an incredibly expensive data sample -- or a trip to a very expensive gym or tourist destination -- rather than as simply a collection of objects that more-or-less mimics some of the things you've done in class.  You need to ask:

• What will I learn from doing this about what my students know?  What skills and concepts does this item assess?  How are my students likely to respond to the item, and what will I learn about my students from those responses?
  
• What will my students learn from doing this?  Is this an opportunity for them to grow and stretch in some interesting ways, or just a check that they can spit out what we've put in?  Will students come away from the experience with a better sense of what they themselves know and can do?  And will they come away with a better sense of what it is I'm trying to teach them?

My friend and mentor Diane Herrmann speaks sarcastically about the "sponge theory" of teaching:  you start the term with a dry sponge and spend the term pouring water into it.  At the end of the semester, you squeeze out the sponge into a measuring cup (graduated cylinder, whatever): the student's grade is the percentage of the poured-in water that you can successfully squeeze out.  I think that theory drives a lot of the garbage-y tests kids wind up taking--tests with 50 or 75 or 100 items to be done in 45 minutes, tests that ask similar questions again and again.  By contrast, if you think about teaching as developing a kind of mental fitness--with certain types of habits, strengths, and skills--then you realize that a test is not just a way to find out what a kid can do, it's a chance for you to provide the kid another growth experience.  And that's the real value added.

How Not to Return a Quiz

This is how not to return a quiz.

1. Walk around the room handing each student his or her quiz while everyone else is coming in or sitting down doing nothing.  Even better, wait until after the class has done your opener (or "bell ringer") and fully settled down.
2. Don't post the answers, either in class or online.  That way, students will have to follow your in-class explanations to learn how to fix their mistakes.  
3. Go over every problem that anyone got wrong.
4. When you go over a problem, make sure that you're the one giving the explanation.  If possible, give the same explanation you gave the first time.  Don't let students who got the problem right give an explanation at length, and if a student does start explaining a problem, make sure you talk over him or restate his explanation.  Don't give students who got the problem wrong the opportunity to explain their misconceptions to the rest of the class.
5. After going over a problem, don't give students an opportunity to do a similar problem.  You already have an assessment of what they know and don't need another one.

There are lots of ways to return quizzes.  Some things I've learned:

1. Get mailboxes for your room.  Use them.  Put papers in them before or after school and let kids pick up their work on their way into class.
2. Post or hand out solutions; don't go over them in class. Suppose eight students got question #3 wrong.    Of those eight, two have errors they can see immediately from your corrections, and don't need further explanation.  Of the other six, whatever it was that they didn't get the first time, they're unlikely to get by having the same explanation a second time.  Either watching someone else do the problem doesn't work for them (does it work for anyone?), or they've got some underlying misconception that made the first time through not so effective.  Whatever the cause, it's unrealistic to expect more than half of them to actually correct their error when you go over it.  So you spend four or five minutes doing something that only benefits three students.  What's the point?
3. If there's a problem that a majority of students couldn't do, briefly illustrate the main point or issue, then give a followup problem.  Or if students got wrong answers, post some popular wrong answers and have students explain what's wrong with them.
4. Anything that's important enough to talk about in class is important enough to re-assess, sooner rather than later.

Adolescence, and other mental "disease or defect"

My non-teaching friends who knew me "back when" are probably surprised that I love teaching adolescents as much as I do:  as a high school student, I often felt disconnected--even alienated--from my peers, and certainly my best friends were typically a few years older, not kids my age.  And yet here I am, loving being with kids: on field trips, I tend to sit in the back of the bus, with the students.

Part of that is probably my mild but ongoing sense of alienation from my "peers":  I'd often rather spend time with high school kids than with people my own age, partly because I don't necessarily like people my own age.  But the major shift is perspective: I'm able to enjoy the company of adolescents much more now than I did when I was an adolescent precisely because I'm not one of them.  The things that kids do that I found weird or incomprehensible or eneverating are now more curiosities than anything else.  I can understand and appreciate where they're coming from, even if it's not where I was coming from at that age.  (Every so often I'll say to kid X about kid Y, "I totally get that kid Y is driving you crazy, but ... ")  The grade-panics, the living from dramatic cliffhanger to dramatic cliffhanger, the sometimes near-total oblivion about the "big picture"--I now see these things as part of the pathology of adolescence, not personal defects.  They're not bad people, and I've come to realize that the many of the things they do and say--even to me--aren't really about me.  Those behaviors are about where the kid is at this moment.  There are some truly antisocial behaviors, but often I can do something about those.

In the big picture, adolescence doesn't bother me, because it's just adolescence.

I find myself thinking about this appreciation-with-distance as I think about teaching and working with students who have learning disabilities or mental illness.  A kid who can't get it together to get work done and in on time, who can't be relied upon to seek out extra help (even after multiple suggestions), or with whom I find myself having the same conversation for the fifth or sixth time--that kid can get under my skin.  As teachers, it's much easier for us to accommodate obvious physical disabilities.  I can teach a blind kid geometry: I provide raised-print materials and manipulatives, allow him to talk through problems that would be too hard to write out, give him a partner that can help with the manipulative stuff or describing diagrams.  (Actually, my two blind geometry students have been among my strongest, perhaps because they expend so much effort on retaining and adjusting a mental representation of the physical space around them.  But I digress....)  I know what's causing his problem, and I know what I can do to work around it.  But the etiology of these just-not-enough-effort-applied-in-the-right-way behaviors -- I don't know it, I can't see it, and I don't know what to do with it.  And that's frustrating.

I don't think I'm alone.  While the teachers at my school do a great job with students with autism, or with disabilities that come with clear-cut accommodations (written directions, access to technology or reference materials during tests, etc.), I find our hardest conversations revolve around kids who "just aren't doing it"--even when we know that those kids have processing disorders, or are struggling with major depression.

One reason is that these kids are capable of consuming an almost infinite amount of one-on-one resources, which are scarce at the best of times.  That scarcity comes from our basic teaching model:  somewhere between 20 and 35 kids together in a room, working on roughly the same thing at the same time.  (The fundamental inadequacy of this model is why I find resources like Khan Academy worth investigating and thinking about, despite their overemphasis on rote or procedural knowledge.)  So I have a choice about whether to spend fifteen minutes or half an hour with a single kid, often with no obvious results beyond the task at hand, or to spend that time doing something that will help the other 19-34 kids in the room, or the other 119-150 kids in my courses.  And so it's hard to put that time in with that one kid and not feel like I'm taking something away from everyone else.

But a more fundamental problem is that it's hard to see the student's disability--whether a learning/cognitive disability, or part of a mental illness--as a symptom rather than as a character flaw.  That same breakthrough that I've had about my adolescents' adolescence is harder to attain.  Part of that is background knowledge:  unless you've had or spent considerable time with someone undergoing clinical depression, it's hard to see how debilitating that condition can be.  But part is that we work so hard on communicating the news about these proactive student behaviors to our classes that when someone seems like they're "not catching on", we can't really understand why.  "What's the mystery?  Just get it done!" we say, although I'd never say to a kid struggling with quadratics "What's the mystery?  Just use the quadratic formula!"

Kids with these problems can be almost infinitely frustrating--I want to say "annoying", even though I know that's not fair.  And that frustration makes the whole problem harder, because it makes it harder for me to achieve that distance where I say "This is not about me, or the assignment, or even the kid.  It's about this disease."  It makes it harder for me to devote that time to that student without feeling like I'm (or he is) ripping my other students off.  And when I do that spend that time, two things tend to happen.  First, the student's behavior doesn't change much right away, and then it's hard not to take the continuing "apathy" personally:  I spent all this time on you, and you won't even meet me halfway.  Second, it's hard to know when or where to stop:  I find myself having the same conversations over and over again, without any evidence of progress.

I don't have an answer to this problem as a learning problem, but I do know this:  the kid isn't just the symptom, or the collection of symptoms.  Every kid wants to feel like a whole person.  So the one thing I can do is communicate that fact to the kid, that regardless of how they're behaving now, I still think they're a whole, valuable person; that I know they're not just this one set of symptoms; that I still love them.  I'm not sure how much saying that helps in the short term, or even in the medium term.  But over the long term, I think it's the only thing that can.

Catch-all Catch-up

I've been delinquent for a couple of weeks, but here are a few things that have crossed my path that are worth thinking about.

• The New York Times reported last Friday that EdX, the MOOC consortium run by Harvard and MIT, has created and plans to release open-source software that professors can use to grade college essays.  John Markoff's thoughtful article points out that, in addition to the obvious cost savings over "regular" TAs, computer-grading would make it possible for professors to assign more writing and for students to resubmit papers multiple times, possibly increasing learning.  He quotes the usual cries of "Pattern recognition is different from grading!" (not according to the Church-Turing Hypothesis, but I digress), but then closes with the following astute observation:  

"Mark D. Shermis, a professor at the University of Akron in Ohio, supervised the Hewlett Foundation’s contest on automated essay scoring and wrote a paper about the experiment. In his view, the technology — though imperfect — has a place in educational settings.
"With increasingly large classes, it is impossible for most teachers to give students meaningful feedback on writing assignments, he said. Plus, he noted, critics of the technology have tended to come from the nation’s best universities, where the level of pedagogy is much better than at most schools.
" 'Often they come from very prestigious institutions where, in fact, they do a much better job of providing feedback than a machine ever could,' Dr. Shermis said. 'There seems to be a lack of appreciation of what is actually going on in the real world.' "

• On first take, Sir Ken Robinson's famous TED Talk about creativity in education can sound like a standard "Infuse more arts into the curriculum."  But reading his book,  Out of Our Minds: Learning to Be Creative gives a different story.  "At the same time, other disciplines, including science and mathematics can be just as creative as music and dance. Creativity is possible whenever we’re using our intelligence."  I'm really enjoying the rest of the book -- it's smooth, a quick read, lots of fun, and very thought provoking.

• On a related note, John's former colleague Zach Herrmann writes in his blog that it's not just what we teach but whether we teach it in a way that fosters creative, actual thought:  "How often do we as teachers deprive our students of the excitement of learning by the way we ask our students to learn within our classrooms? I believe we can positively impact students’ perceptions of learning by being mindful of the problems we give and the way we ask them to participate, while rethinking our role as a teacher in their learning process."

That's all for this week!