Singapore Math – a revelation

It is funny how we build things up in our minds to be one thing, and then reality turns out to be quite different. A zen monk named Zuishin that I used to know once told us a story about his going to the dentist, and how the whole trip there he was agonizing about going. He told us how he hated sitting in the chair, rinsing his mouth and watching water dribble across his chin, feeling a needle as they put anesthetic in his mouth, or just feeling them cleaning his teeth. Then he realized that although he could not change his experience while at the dentist, he did not have to live through it twice, which is what he was doing by fretting.

The point? Our mind colors our perspectives with a million thoughts about what we think reality is, rather than seeing what it really is.

I had a vision of Singapore Math as a drill and kill type program which focused on procedural fluency at the expense of all the other things I feel are important in mathematics. Because this is a program some of our parents – and some very vocal critics of the current program we have in our elementary school – have pushed, I decided to purchase the book, The Singapore Model Method for Learning Mathematics, put out by the Singapore Ministry of Education.

Was I in for a surprise! I’m still reading the body of the book, but the initial chapter talks about the Singapore Mathematics Framework. Here is a key thingthey say,

Mathematical problem solving is central to mathematics learning… [it] is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudes, and Metacognition.

This describes exactly what I would like the focus of our math program to be. They go on to break down what is meant by each of the components, but really I’m still digesting it all.  Maybe there is something to this after all.

Perhaps the trip to the dentist won’t be as bad as I feared…

Required Reading for Math Teachers I « Research in Practice

Okay. This guy who writes the “Research in Practice” blog makes a lot of sense. Below is a quote that really resonates with me, but it is very much worth your while to read the rest of the post.

Take-home lesson: never underestimate your ability to fool yourself into believing your students understand something when really what they are doing is watching you. To force them to engage the material it is often necessary to restrict their access to you or systematically confound the signals they get from you.

I think this is a central issue for modern math teachers. We need to explicitly develop ways of question-posing and interacting with our classes and individual students that hide or disguise our intentions for how they are supposed to respond. This needs to be part of the core training of math teachers, much more than it already is.

Getting It Wrong: Surprising Tips on How to Learn: Scientific American

People remember things better, longer, if they are given very challenging tests on the material, tests at which they are bound to fail. In a series of experiments, they showed that if students make an unsuccessful attempt to retrieve information before receiving an answer, they remember the information better than in a control condition in which they simply study the information. Trying and failing to retrieve the answer is actually helpful to learning. It’s an idea that has obvious applications for education, but could be useful for anyone who is trying to learn new material of any kind.

via Getting It Wrong: Surprising Tips on How to Learn: Scientific American.

What an interesting article – and one that resonates with what I have seen and experienced in my classroom. I want my assessments to be opportunities for learning, not benchmarks to meet. This is not that I think there is no place for ‘mastery assessment’ type tests, but I feel that the bulk of the assessments students get in class should be of the type that they can learn from. I think this is what, over the years has led to my quests, problem solving tests, or any other assessment where I have deliberately chosen challenging problems that I knew would push the kids.

I often make a judgement ahead of time (before grading, and often even before giving, a test) regarding the difficulty level of the questions I am using and plan a curve based on that. After the students take the test, and I grade it, I apply the curve. Most of the time this results in a spread of grades that I think reflects where the students are. Some times the spread is low, and I re-evaluate the curve. Questions I ask are:

• Did I underestimate the difficulty of the question(s)?
• Did I not prepare the students adequately for the questions asked?
• What types of errors were made?
• Were most errors on material they were to expect to have ‘mastered’ or was it on the newer material and/or applications?

Afterward I decide if I need to adjust the curve. Sometimes I do, sometimes I don’t. It depends on the answers to the above (and other) questions. I never adjust the grades just to have a certain range of grades. If, after the curve, the grades are all A’s, I applaud the class and there prep and understanding. If they are all C’s (or lower) we have discussions about test prep, daily work, and other issues that may have contributed.

Is my method totally objective? I know it isn’t, but I am honest about that. I don’t think it is possible to have a totally objective test, although it is possible to consistently score a test. Even in what teachers think is a completely objective skill test, the choice of questions and the grading of each problem lets subjectivity creep in.

And I evaluate how I do each year and adjust.

Abstracts and 3 Problems

I realized that, for those of you (which is probably most of you!) not familiar with my classes, the references in my Geometry FAQ document to “abstracts” and “3 problems” were probably a little vague.

A while ago now I began to get fed up with assigning traditional math homework. Not because I thought it was inherently bad or that I cared too much about their self-esteem, or really anything like that. I got fed up because they would do the bare minimum to get it done. Period. If I assigned ten well-chosen problems designed to bring them from a simple, repetitive example to one which explored deeper concepts the reality was that they did the least they could do to get me 10 answers. Of course I expected to see work, so they had to scribble something down before putting said answer. Seeing more than one student copying the homework from someone else made me begin re-evaluating how assigned homework.

In the beginning I found creative ways to beat them at their own game. Copy homework from someone else? I called it cheating and didn’t accept it. Finish it ten minutes before class by quickly scribbling out some work? I began giving homework quizzes and assessed the quality of the shown work (which I allowed them to copy directly from the homework). And the list goes on.

So I decided to rethink why I had students do homework. I know the typical reason is for students to practice what they learned. My problem with that has always been that practice without pretty immediate feedback is not consistently fruitful and often a waste of time at best and excruciatingly frustrating and painful at worst. What I really wanted students to do was reflect on what they were learning, develop the habits of mind needed to study what was important, and be able to communicate about the deeper ideas.

“Abstracts” were one of the earliest fruits of this thought process. I actually got the idea from the book Using Writing to Teach Mathematics, edited by Andrew Sterrett. In my class it has evolved into a short 2-3 paragraph essay due nearly every week of the year. I collect it on our block day (Wednesday or Thursday) and its focus is the work from the previous calendar week (Mon-Fri). I ask the students to briefly right about what they felt were the most important points/problems/concepts about the week before. I tell them not to give me a day-to-day accounting of the week – that is what my planbook is for – but to describe what they learned, what made sense, or what they are still questioning. I also encourage them to share what is going on that week outside of class if they feel it may impact their work such as sports, play practice, lessons, etc. Because they have 4-5 days from the end of the week it is about to when it is due, and I insist it remain brief, I don’t count it in much to the time homework takes them. All together it should really be no more than 15 minutes.

I have learned a lot about my students, both as a group and individually, from reading their abstracts. I comment on what they write, make corrections if they are wrong about something, or answer questions. It is one thing I anticipate doing in some form for years.

The other assignment, referred to as the “3 problems,” is a literal description of what I ask them to turn in. Since I am using a problem based approach this year where the daily homework is to work on 7-10 problems which individual students then present the next day they do not always get specific feedback from me, particularly if I feel the discussion around the question was sufficient. At the end of each week I ask them to pick three problems from among the recent problems (I allow them to go back as far as the previous test) and turn them in for me to respond to. The only guidelines I give them are that they need to (1) write out the question; (2) show their work – which they can just copy from their notebook; and (3) write 1-2 sentences about what they did, how they approached the problem, or any difficulties they are having. I then read them, make comments on their work, and turn them back.

The trouble I have been having with the 3 problems is that I encourage the students to include problems they are still struggling with so that I can really give them feedback on their difficulties. Some of the students, however, seem determined to show me their three best problems every time. It makes it pretty easy to grade – I have little to comment on – but I know it is not as helpful as it could be. I’m going to continue thinking about this.

I assign both of these in my Geometry Honors classes and my Calculus BC class. Remarkably I find that the level of reflection of my BC students is not sufficiently more advanced than my Geometry students, although this may be partially because they rarely do this in the other math classes leading up to mine.

Murphy, sickness, and calculus

So today starts an interesting day. I was already going in late because I have a doctor’s appointment. I was only missing a free period and then part (or possibly all) of a dept. chair meeting. No big deal. I would still make it in time for my Calculus BC class.

However, Murphy has a way of showing up when you least expect it and are not ready for house guests. My daughter has not been feeling well the last couple days, and she has been home from school. Fortunately she is old enough now – and I live on the campus I teach at – that leaving her at home is not a problem. This morning, though, my wife and I took her temperature after feeling her skin literally radiating heat to find her with a fever of 104!

So plans change. I’ll still get to go to my doctors appt in a little bit because I can call her doctor immediately after, but this will clearly change the rest of the day. I had three different meetings scheduled for today. Fortunately I am only leading one of them, and that nominally.

However, I will need to miss my Calculus BC class almost certainly. The good news is that I can leave work for them to do, and they are responsible enough to not just blow it off.

So here is a problem from their next set:

Verify the identity . Use this result to help you solve the logisitic differential equation , which you should recognize as separable.

(a) Show that the solution can be written in the form .

(b) Given the initial condition f(0)=0.36, express f(t) as an explicit function of t. Evaluate the limiting value of f(t) as t approaches infinity.

At this point they have already been introduced to logisitic differential equations through a couple previous problems (one of which also introduced them to the concepts behind Euler’s method). In addition they have also seen a similar partial fractions problem, although in a similar vein – verifying, not finding the fractions yet.

We’ll see how they do!