Target group

To whom do you aim the problems you are using to teach the lesson?
This question assumes that you teach by asking interesting questions and allowing students to figure out the math. If you do not teach in this mode, I am not sure this question applies. I am also not sure I have much to offer, because I think good teaching starts by recognizing that our job is to ask interesting questions and to help students figure out the math behind the questions. I think a teacher helps by watching, listening and letting the students do the work. Telling a student how it works does not work.
But I have talked about that before. This idea is new, I think. P.J. reminded me of this yesterday so I thought I would write about it before he did.

The problem should almost always be aimed at the top group of kids, say the top 25%. There is a myth that says teach to the middle. Do that, and over half of your students learn nothing. The ones above the middle already know it so you are wasting their time.

The problem needs to be accessible to everyone, but difficult enough, challenging enough, that no one can just solve it. That means it needs to be designed so the middle kids and below always have to stretch a lot, while the best kids are still challenged.

One thing to consider is that the top 25% one day will be a different group of students another day. The problems should not be aimed at a specific student but rather at a specific level of challenge. Problem solving has to happen for every student every day, or students will not learn how to solve problems.

"Oh, but students will give up, because it will be too hard for them," you say. I say, you are with them, you are walking around watching them work, and listening. You can push them in the right direction if they have a good thought, and redirect them if they don't. If everyone is about to give up, you will know it, and you can immediately fix it by intervention.

Differentiation does not mean make it really easy. It means teach students how to think. Good teaching means helping students learn what to do when they don't know what to do. That can only happen face to face in your classroom.
It is our job.

A True Story

Back in the year 2000, there was a lot of fuss about "Reform Calculus." The Advanced Placement committee had announced changes in the AP test, and there were several calculus books being offered that were quite different from the Thomas book that most high schools had been using for years and years. After much deliberation, Evanston decided to make the change, and we adopted the Ostebee Zorn calculus book for BC Calculus. There were two of us teaching the class that year: Ron Selke and myself. We approached the year with excitement and fear.

As we worked through chapter one with the students, we both learned a lot about how to make calculus meaningful and understandable to our students. We had decided to collaboratively write tests, and so we did. The first test covered the ideas in chapter 1 rather well we thought, and we were eager to see how students performed.

To say the first test was a disaster would be an understatement. There was not one student who even tried to work all of the problems. Many students left three or four blank. Ron and I looked at the test, and it measured what we thought was important, but because of the difficulty it measured little or nothing and created considerable discontent among our students. And these are the best students we had. We adjusted the grading scale on the test, admitting that we had totally failed to create a fair test, amd promised that we would do better for the next chapter.

In considering how to fix the problem, we had several ideas. One was to break the chapters into two tests. We rejected that because it would mean giving up too much instructional time for formal assessment; we would spend the year writing and grading tests. Another idea was to only test the easy stuff. We rejected that approach as not being in the best interests of our students. We were committed to making the class a rich mathematical experience that matched the wonderful way Ostebee and Zorn were allowing the course to unfold. Then one morning Ron came to me with one of the best ideas I have encountered. And I resisted at first. I offered reasons why it was a bad idea. After all of that, I agreed to try it. I have never looked back.

Ron's idea was to make a collaborative problem part of the test. The original plan said the day before the test, we would give each student a collaborative problem. This collaborative problem would consist of several parts and would encompass the main ideas of the chapter. The problem would allow us to address some of the subtle concepts or more complicated aspects of the material covered in the chapter. Each student would be required to work with at least two other people, and each person would turn in one copy of the team's perfect, well-organized, well-written solution when the student came in to take the in-class part. It would count as about one seventh of the test grade.

After a couple of tries, we modified the conditions a bit. In particular: we gave students the collaborative problem several days before the test, and always so it would be in their hands over a weekend. We posted the collaborative problem on our websites to allow absent students to access it. We helped shy students find collaborators. We encouraged collaboration with students in the other sections of BC Calculus.

The collaborative problem turned in to one of the best educational experiences in my career. Most of the work was correct, making them easy to grade. The in-class part was now manageable, but we were assessing all of the material. More importantly, students were learning mathematics while taking a test. What an amazing experience! Ron and I listened to their conversations as they worked the problems in the math lab, we heard them talking at the beginning of class, and we flat-out asked them about their experiences with the collaborative problem. All of what we heard was exciting. There was an outcry when we did not offer a collaborative problem for a test we gave on a half chapter. We had found a way to help them consolidate their ideas before they took the in-class part, so the in-class tests were also done better.

We began to notice that the collaborative groups became entities in themselves. The students started to get together just to study. Some of them met during a common free period every day in the math lab or the cafeteria and went over homework questions. Parents praised us for the learning they saw taking place in their homes as students gathered. Collaborative groups compared reaults with other collaborative groups. Students made friends and learned that learning is not an isolated activity.

I also taught a class in Multivariable Calculus and another in Linear Algebra for those students who had finished BC Calculus and had not graduated. There was a clamor for a collaborative problem in that class, so I happily agreed to their demands.

The second year, we realized that we did not have to rewrite the collaborative problem except to fix questiosn that didn't quite go where we thought they would. By now these questions are establsihed. The students understand their value as learning tools, and so there is little evidence that they are looking at old tests.

One moral to the story: when you try something new, it rarely works the way you
intended it. But the thing to do is not to throw it out--then you wind up
doing the same things you were dissatisfied with before that led you to make
the change. Rather, you need to try and identify what's not working and
fix that piece, By iterating several times, you come up with a new
strategy that does accomplish your goals--and even gets you places you hadn't
realized you wanted to be!


By the way, our students performed better than ever on the AP test and have ever since. It is nice when one's observations are valideted by an outside source.

Grading and Formulas

John, I agree with everything you said, except that I have found that grading eventually just wears me down.

An idea I took from George Milauskas takes the "only two points for a correct answer" one step further.  George would give his students "magic dots" (= the circles punched out from paper with a 3-hole punch) which they could redeem--in combination with a single point--for any formula required on a test.  That is, a student could ask "What's the midpoint formula?" (even though requests for that particular formula make me cringe) and get it, for a mere one point.  George's reasoning, which persuaded me instantly, was as follows:

1. If a student just writes down the correct formula--but does no other work--he or she will usually get one point of "partial" credit.  Most of the problem consists in using or thinking about the formula, not regurgitating it.  (That leads to a whole nother conversation, about problems that are just about formulas.)
2. Giving a student the formula allows the student to demonstrate what he or she can do with it.
3. Leaving the student stranded without the formula means that he or she can't demonstrate anything.

Thus, giving the student the correct formula for a one-point deduction allows me to assess what else he or she knows and can do with it in a fair and reasonable way.

Before you ask how they do on "real" tests outside my class, remember that AP, ACT, SAT all include lists of common formulas.  So my students have done just fine.

John, thanks for reminding us how important this work is.

== pjk

Grading Papers

Without exception, retired teachers I have spoken to agree that the best part of retirement is not having to grade papers, especially on Sunday night. And yet there was something very satisfying and rewarding about looking at student work in detail. Grading a student's paper was like spending a few minutes inside of that student's mind. It took a lot of time and it was hard, but carefully grading papers was an important part of the teaching and learning experience.

My last blog was about how grades influence the learning experience. This one is about how the details of arriving at one of those grades are equally important. Papers must be graded fairly. A graded paper should contribute to a child's education in a meaningful way, regardless of the grade attached to it. I would like to share some of my thoughts about the process of grading student work.

At the top of every paper I intend to grade is the sentence :

"You must show enough work so that I can reproduce your results."

I have found that this phrase solves a lot of practical problems about how much work a student needs to show. It also helps when the students solves a problem in an unexpected way. It allows a student to use technology intelligently as long as I am given enough information so that I can get the same result using the same technology. It enables me to effectivly evaluate the error a student has made and give the appropriate amount of credit for the work.

This leads to another aspect of grading a student's work that developed carefully over the thousands of problems I graded. I give points for correct mathematics. I do not take off points. When a student looks at the number of points earned for a particular problem, the student will see a +10, not a -2. The students got the 10 for doing several things correctly that would have lead to the correct answer. Unfortunately, the student made an arithmetic error when computing part of the answer and so did not earn the 2 points allotted for determining the correct answer. It should be noted that one consequence of this grading policy is that a bald answer without supporting work will get 2 of 12 possible points.

We are teaching mathematics. We are assessing the quality of mathematical reasoning a students is capable of. That means we need to see the process used to arrive at the solution, and it is the process we are evaluating. It is inappropriate and short-sighted to require the students to use the process we expect, but we cannot evaluate what we can't see. So, if a student uses a guess-and-check method, I need to see the guesses and the checks. If the student uses intuition and evaluation, I need to have the intuition explained, and I need to see the evaluation.

Consequently, it takes a long time to grade a set of tests. The effects, however, make it worth the effort. I have learned a lot of mathematics by following the work of a student who took an unexpected path. But more importantly, grading lots of papers teaches the grader what sort of misunderstandings students have, and that in turn enables the grader to try to find ways to eliminate thsoe errors next time around. At the very least, one learns to warn students about a mistake students often make on a particular kind of problem. At the most, the teacher can modify the problems used to teach the concept so the class will make the mistakes early on, exposing and hopefully eliminating the potential for those mistakes to occur.

Another consequence of grading this way is that the grader learns a lot about the particular habits of mind of the student. This information in turn is very helpful when the student or the parent wishes to know what can be done to improve. The teacher is aware of a lack of organization, or not checking work,or poor computation skills, or careless marking of a diagram or any of the other habits that interfere with success and can communicate those habits to the students and parent.

Comments to the student congratulating a clever move will have more impact than criticism about a bad move. A teacher can take the opportunity to point out specifically to a student what might have resolved the error. The feedback is up-close and personal and has impact.

But perhaps the most important aspect of grading papers this way is that it conveys a sense of value to students. It implicitly tells them that the important part of mathematical work is the process. They will get points for failed attempts if those attempts are appropriate and reasonable. They will get more points for a clever observation than for remembering a few steps from a previous problem. The students will learn that mathematics is about logical reasoning and making connections more than mathematics is about remembering rules and following them carefully. And that is a big deal.

Grades

Grades are the elephant in the room when it comes to learning and teaching. They are an ever-present part of the relationship between teacher and student in most educational situations. Grades can interfere with learning if students believe that no matter how hard they work, their grade will not improve, or if students believe that they can get an A without doing much work. Grades can destroy that partnership if students think the teacher is unfair. Grades consume a large percentage of a teachers time.

Perhaps the worst part is that frequently grades become the goal and overshadow learning. Teachers sometimes use grades to coerce students to comply. Teachers do things like not giving credit for math work done in pen, even if the work is exemplary. Students use grades as an excuse for not doing work. Students will not investigate a problem further bacause they know it will not be tested.

And yet, grades persist. I think they can play an important role in education. Students and parents need feedback about their accomplishments, they need advice about how to improve, and they benefit from legitimate praise and criticism.

Allow me to share two very personal experiences that are vivid memories of my grade school days. For two years in a row, I was told by the music teacher that I could not sing. (She was right.) I was then instructed to mouth the words as the other children sang. As I grew up, I realized that I love music, but I can't sing, so I don't try. I have been told by several people that anyone can learn to sing. I don't believe them, because I was told at a very young age that if I sang, it would interfere with the singing process in class. I can't do singing. The second experience happened in eighth grade when my social studies teacher was trying to explain inflation. I rasied my hand and asked a question about the consequences of inflation. He looked at me and said, "You must be really good at math." Fifty years later, I remember that moment in class; I have devoted my life's work to mathematics.

Both teachers were assessing my work. Neither assessment had anything to do with grades. My grades in school were never very good because I frequently didn't comply with the teacher's wishes about how to do the work (I really liked doing math with a pen, for example). But I did learn and so consider myself to have had a good education in spite of all those C grades.

When I started teaching, I had the good fortune to be in situations where I had the freedom to decide how grades were going to be given. I spent a lot of time thinking about it, tried many plans, and eventually hit on one or two that worked. I think my grading schemes helped me become a better teacher. I would like to share some of my thoughts about grades in the next few blogs.

I have always thought that in order to earn a good grade, a student should demonstrate knowledge of the subject and the ability to apply that knowledge in a variety of situations. That means that each assessment should include some routine exercises to see if the student has learned the basic material, problems right out of the book with different numbers. The students should also be expected to do problems similar to some of the really hard problems that we did in class. And the student who wishes to earn an A ought to demonstrate the ability to apply the information from the unit, as well as from the entire course studied so far, to a new situation. So, my tests are usually about 50% routine problems, 25% difficult problems that are similar to problems they have worked on , and 25% original problems. I give them one period to work on the problems unless they are legally entitled to more time. I carefully look at their work and give credit for correct mathematics relevent to the problem.

Solving a difficult problem takes time, and there is often a certain amount of luck involved. A promising approach may lead to a dead end through no fault of the problem solver, while an equally promising approach may work just right. If we intend to assess out students' success as problem-solvers, we must ask them to solve problems on tests, not just do exercises. That in turn influences how we associate a grade with work done.

I would like to know who decided that 95% was the benchmark for excellent work and what sort of work were they thinking of. Nothing I can think of that is reasonably difficult can be done correctly 95% of the time. The best baseball players that ever lived were successful at getting on base if they could get on base 40% of the time. Most players don't even come close, because hitting a baseball is very hard to do. A 30% success-rate is outstanding.

One of the national standards of excellence in the U.S., the Advanced Placement test, gives only five grades: 5,4,3,2, or 1. In order to get a 5, a student needs to get approximately 72% of the test correct. That level of excellence will often earn college credit for the course in question.

A score of 100 or more out of 150 is considered outstanding on the National Mathematics Exam offered by the Mathematical Association of America every year. That score qualifies a student to move to the next level of competition and often means that the student was in the top 1% of students taking the exam.

P.J reminded me about Dr. Paul Sally's rubric: "If you're getting 50%, you're doing well." Dr. Sally taught Honors Analysis at the University of Chicago. No one ever accused Dr. Sally of having low standards.

That brings me to another thought about grades. My first two years I computed grades two ways. I kept track of total points earned by students, and I also assigned a letter grade to each assessment and then used the letter grades to determine a final grade. It became apparent theat the letter grade method was far superior in two respects. First, students always seemed to have a feeling for where they were. Second and more important, the letter system was fairer, the letter system meant the grade was less influenced by a really bad test, and the letter system allowed me to assign points to problems without regard to making the total come out to a pre-specified number.

Another principle I followed without exception: Every problem was worth the same number of points. I didn't want students to have to worry about how much time to spend on this one or that one bacause one problem was worth more points than another. I did want my students to look the problems over and work the ones they were most confident about first. Since I established the cutoff points, this was a very good and fair policy.

The point is that the elephant is there, and it matters. Grades influence our effectiveness as teachers, and we must spend considerable time and effort working out systems that enhance our teaching, emphasize the things we think are important, inform parents and students about the quality of their work, and are even-handed and fair.

Next time I will share with you some specific things that worked for me. Until then, please reflect on the grading policy that you are using and how it alters your ability to teach mathematics. No matter what you think, it does make a difference.

More later.

"I think I discovered a theorem, Mr. Karafiol!"

This past week, one of my students stopped me on my first pass around the class with the exciting words "I think I discovered a theorem!  In a right triangle, the angles the median makes with the hypotenuse are twice the other angles!"  We had discussed the median to the hypotenuse in the previous class, but I'd never thought about the angles before.  So I started doing algebra in my head, confirming the relationship he was describing:

Before I could get very far, Vik showed me the elegant diagram below:

With the inscribed angle theorem, the proof is immediate.

Eureka!

Low Tech, High Impact

Although my department has a reputation for integrating all kinds of whiz-bang, high-tech devices into our classes, we've become increasingly excited by one decidedly low-tech tool:  large whiteboards.

The model is one I learned from our AP Physics teachers several years ago.  Contrary to my usual one-problem-at-a-time practice, I give students a handout with several problems, and instruct them to work on one that they find tough--so that they can build their mental muscles.  Then, each group gets a large whiteboard, and writes up a good solution.  Groups present their solutions and field questions from the rest of the class.

When I first heard about this technique, I thought it was a poor second-best to having every student work on every problem.  But I tried it, and now I've seen some substantial benefits

1. Students get to see a variety of problems worked out carefully, and ask questions in a nonthreatening setting.  And while they're not grappling with each individual problem, the more questions they ask and answer, the more they think through the problems presented--even ones they didn't work on.
2. Students work on communication skills.  In other instructional models, it's rare to have an entire student response of this length subjected to a full-class critique.
3. Students focus on one problem for enough time that they can think deeply about it, and transfer the ideas to other contexts.

The class above is my AP Calculus AB class, working on optimization problems.  After a brief introduction, groups chose between four:  minimizing the distance from a point on a line to a given point not on the line, minimizing the surface area of a can with fixed volume, maximizing the area of a rectangular pen with dividers, and the traditional "box problem" shown above.  In this case, I had two major goals:

1. Students needed to understand and be able to recreate the major steps in solving any calculus optimization problem: identify the constraints and the objective function, rewrite the objective function using a single variable, and apply calculus to find the maximum or minimum value.
2. Students needed exposure to standard problems, both in case they should encounter those very same problems later, and to provide foundational experiences from which they could build out to other, similar but perhaps more-complicated situations.

Whiteboarding was perfect: it was enough for kids to see and discuss the different setups--some of which they had encountered in precalculus--without having to struggle through creating each one themselves.  (Assessment today: they were pretty comfortable doing another distance minimization with a harder graph.  So it worked!)  And the in-depth thought about the process really paid off: again, in today's followup class, students seemed able to articulate the major steps and follow them with little confusion about sequence or "what comes next?" kinds of questions.  Finally, their writing and communication were much stronger than I would have anticipated:  students presenting today wrote clearly and identified major steps and results along the way, and other students used vocabulary correctly without hesitation.

So there you have it: for less than $15 per board all-in, a mechanism to facilitate thoughtful collaboration, presentation, and discussion. 

Postscript: if you're looking to acquire some, we get ours from whiteboardsusa.com--they ship quickly and have reasonable prices.