Monthly Archives: August 2011


Yesterday’s Op-Ed piece in the New York Times, “How to Fix Our Math Education” is by two card-carrying mathematicians, Sol Garfunkel and David Mumford. To my surprise they argue that “different sets of math skills are useful for different careers, and our math education should be changed to reflect this fact.” (Italics mine) They propose that, as an example, we should be “replacing the sequence of algebra, geometry and calculus with a sequence of finance, data and basic engineering. In the finance course, students would learn the exponential function, use formulas in spreadsheets and study the budgets of people, companies and governments. In the data course, students would gather their own data sets and learn how, in fields as diverse as sports and medicine, larger samples give better estimates of averages. In the basic engineering course, students would learn the workings of engines, sound waves, TV signals and computers. Science and math were originally discovered together, and they are best learned together now.” At some point in the article they ask the rhetorical question “how often do most adults encounter a situation in which they need to solve a quadratic equation?”

My immediate reaction was “hogwash”, but upon further thought I can see some nuances in the argument. For example, the authors agree that “[o]f course professional mathematicians, physicists and engineers need to know all this “ (they mean algebra, geometry, complex variables, etc), “but most citizens would be better served by studying how mortgages are priced, how computers are programmed and how the statistical results of a medical trial are to be understood.”

Is this then, an argument for different math tracks? If it is, it presumes a career choice that must be made way before most kids know anything realistic about various careers. I know that many other countries (Europe, Asia) do in fact force the students to make this choice. In these countries, high stakes exams and course grades determine whether a student will be on the university path or on a non-academic one. Do we want to go that way? I can see arguments on both sides. My guess is that parents (i.e. voters) will not care as long as their kids get accepted into college.

As far as encountering a situation in which the average person needs to solve a quadratic equation, well…when does the average person encounter a situation in which s/he needs to do squats? Push-ups? We teach math the way we do because it exercises the logic part of the brain, just like squats and push-ups exercise the body.  But is this argument valid?

I have yet to see a controlled experiment where the “logic IQ” of persons who “had” math is compared to that of those who did not have it. Personally, I believe that math does change pathways in your brain, towards some sort of more logical thinking (especially if you take a statistics course), but whether these pathways stay there without continuous exercise, I do not know.

Color me puzzled.


Cri de coeur

Only one more week before the kids come in. Four classes of Algebra II (three of them are for freshmen who passed Algebra I in middle school) and one class of AP Stats. So far the rosters are manageable, between 25 and 30 kids per class. What do I want to do with them? What do I want them to get out of their respective courses by June?

I want them to delight in thinking. I want them to delight in thinking mathematically. I want them to stretch their brains. I want them to grow  mathematically. With all due respect to all the great middle school math teachers out there, I think that to some degree we infantilize these kids. A poster with glitter, with an illustration of the Pythagorean Theorem and the formula next to it is NOT the same as understanding how to apply the Pythagorean Theorem.

Math is about solving problems – that is all. Too often what we do in math is boring and repetitious. The textbooks rarely have challenging problems. We are giving the kids recipes, but we are not teaching them to cook. We are teaching the kids to make a French onion soup, we have them do it 30 times until they get bored with it, but we never ask them “What do you do if you run out of onions?”

The situation is less dire in AP Stats – I continue to have a tremendous respect for those who put the course together, who wrote the main textbooks and who are coordinating and grading the AP Stats exam. But in Algebra, arguably the foundation stone of mathematics, we (as a country) are falling behind.

Take a look at some of the textbooks American schools used in the 60s.  Take a look at Smith and Fagan’s “Mathematics Review Exercises”, at Weeks and Adkins’ “A Course in Geometry”. Those had truly challenging problems. What happened? Have we become dumber? Take a look at Singapore’s “New Syllabus Mathematics.” Those are good problems!  Take a look at today’s Exeter problems. How come we don’t have more teachers using them?

Of course there are explanations. As a nation, our classes are more heterogeneous, and in general we expect everybody to graduate from high school with Algebra II and Geometry level mathematics (at least), whereas 40 years ago we did not expect everybody to do so. We have provided opportunity for more, but we have at the same time lowered the quality and rigor of our courses. And of course there are the standards – and we are evaluated on how our kids perform on those standards. We are truly in Lake Woebegone – where “all children are above average.”

Indeed there are many causes that one can list to explain the decline in math education – from parents, to administrators, to politicians, to video games and so on. But I think we – the teachers – bear a heavy burden also.

We have failed to hold the line.

We are coddling the kids, we try to make it interesting, painless,visual, relevant and in the process we have sacrificed rigor. We need to ask more of them…and more of us.

On Assessment (IV) – SBG Gradebook

After all these posts on assessment, I needed to choose an SBG gradebook.  I quickly realized that the software I was looking at …. well, it just didn’t do it for me.  Both ActiveGrade and LearnBoost are excellent pieces of software. However, I wanted more flexibility than either one provided. I ended up kludging my own Google Docs.

In the process, I started thinking what ought to be the main characteristics of a good SBG gradebook. Here’s what I came up with.

(1)   The gradebook should make it very easy for the teacher to enter grades.

(2)   The gradebook should have flexibility in the grading scheme – it should allow the teacher to designate his/her own formula for computing grades.

(3)   Since this would be and SBG gradebook, it ought to be able to compute grades by Learning Objectives. Once a new Objective starts, the whole scheme for grade calculation should be applied to the new Objective, while keeping the details and average for the previous objective.

(4)   There should be a class view and a student view for the grades. The software must address the issue of grade confidentiality – a student should be able to access only his/her grades.

(5)   I am a fan of color coding grade ranges. I think many students would be proud to say “I moved from yellow to green” and it might just encourage some of them to try harder. In addition, by looking at the class view, the teacher can assess which LOs gave most trouble and re-teach them.

Points (2) and (4) gave me the most trouble. If one reads teacher blogs, it quickly becomes clear that there are many grading philosophies – from straight good old-fashioned averaging to “comments only”. Therefore, I can only speak for myself when I explain the grading scheme I came up with.

My goals were perhaps mutually exclusive: I wanted to encourage continuous, sustained effort, but I also wanted to incentivize the kids who did remediation work and re-assessment. I am planning to do cumulative assessment, so that each LO might be assessed 4 or 5 times, at different times during the semester.  To signal the importance of sustained effort, the last grade on an LO counts as 60% of the total grade for that LO.  The other 40% is an average of the previous grades for that LO. Therefore:

Current grade for a given LO = 0.4*(average of previous grades in that LO) + .6* (last grade)

Why .4 and .6? I don’t know – it could be .3 and .7, or 50-50, or plain averaging. I just wanted the flexibility to enter a formula and not to be locked in into a scheme embedded in a piece of software.

Here is the result of my effort.

Weaknesses? One that I see right now is that if a student did not take the quiz where the LO was tested the last time, I need to change my formula manually. For example, LO 1.1 appeared on a quiz last time on Nov 9, but “Jesse Darling” did not take it that day. Therefore the last grade he got in LO 1.1 was on the quiz taken on September 26 – this becomes the “last” grade and I have to change the formula manually to reflect this. I just don’t know how to code for the last value in a row (or column). Any help would be appreciated.

The other problem is with the student view. For the time being I decide to give each student a class ID which is not alphabetical or their own school ID. They will receive this ID from me, in their private school mail account. Then after each quiz, I delete the names and sort the rows by ID number. Then the students can see the whole class, but they would not know another student’s ID and grades except their own.

Like I said – a kludge. However, that’s the best I can do before school starts in ten days.

On Assessment (III) – Grading

Grading is where the rubber meets the road. We can talk about how and when to assess, but at the end of the day we need to give the kids some feedback. What form should that feedback take?

I do not believe that we can talk in universals when we talk about feedback/grading.  We deal with different populations of kids and different administrative policies. There may also be subject specific influences on evaluation policies. Therefore, I am going to limit myself to my experience in math and show why I arrived at the feedback system I use now.

First, the population of kids that I deal with is probably neither the best nor the worst. What is however a constant, is the fact that all kids (from failing freshmen to college bound seniors) will try to get away with as little work as possible. Anecdotally, this is as much a teenager characteristic as it is due to the lack of demanding and rigorous teaching in the middle schools. (Again, I can only talk about the district that I teach in). I have had many seniors in my AP Statistics class telling me that this was the first time they had to study seriously.  These kids have gotten away with not doing too much homework and not doing very demanding work until the AP courses (and even then some of them are pretty lackadaisical about doing sustained, high level work).

A deeper discussion as to why we have students like this is for another time. For now,  given this particular population of students, my feeling is that there is no way I am going to evaluate these kids by giving them “written feedback” or making homework, exams and so on voluntary. Neither am I going to evaluate their work through “collaborative projects”, a poster or a Power Point presentation. At this point in their lives these kids do not have a sense of responsibility for their education – sadly, they have not been trained for it. Besides, evaluation in math is very simple – can you solve math problems on your own?

I know the argument that in the real world we do not work as individuals, but as members of teams and therefore evaluation should consist (at least partly) of group projects and so on. To a great extent this is a specious argument. We may well work as members of teams in the real world, but our evaluation – in that same real world – is by what we as individuals bring to the table. Our peers and our bosses evaluate us by what we contribute as individuals to the team. It may well be that a cohesive group is more than the sum of its parts, but there have to be parts to begin with.

Therefore, my evaluation of students is based on individual numerical grades in exams, quizzes and on homework assignments. Period. I do recognize however, that as important as it is for a student to be able to solve a problem (individually), life necessitates that we communicate our results. Therefore, especially in AP Stats, a well written paragraph – succinct, clear, logical and correct – is another metric for evaluation and grades.

The second reason why I give numerical and letter grades is that such evaluations are part of the common currency of college admissions. Sure, college admission officers look at essays and recommendations, but first and foremost in their decision process are grades and SAT/ACT scores – basically numerical evaluations. For better or worse, just like with the dollar, numerical/letter scores are a rapid, more or less accurate way of establishing worth, of weighing performance. A numeric/letter grade, just like a dollar sign, is a signpost that catches our attention.

I want to underline that this discussion is limited to how to evaluate not what. As I said in a previous post I am very much in favor of assessing conceptual knowledge, not just algorithmic one. I want the kids to stretch; I want to evaluate them on problems they have not seen before. This discussion is how to evaluate that stretching.

On Assessment (II) – WITIWYG

Some years ago, Hugh Burkhardt said that “What You Test Is What You Get” – WITIWYG. So what should we be testing for in high school mathematics? There are obviously many answers, but given the ethos of our times – when everybody wants to go to college – one valid answer would be “test students so they are prepared for college”. Unfortunately too small a proportion of our high school graduates go into careers that emphasize math and science, so mathematical preparation for college must have a broad meaning.

For me, that the essential objective that math education should achieve is to create students who can think flexibly, based on logical reasoning and who are able to communicate clearly their thinking processes. By flexibility, I mean the ability to solve problems they have not seen before, problems that necessitate connections. To put it another way, our graduates not only must be VERY familiar with basic mathematical concepts and operations, but they must also be able to get away from “plug-in” problems.

I will be teaching two courses in the fall: AP Statistics and Algebra II. From previous experience, AP Statistics largely achieves the flexibility, logical reasoning and communication triad. Here I am going to concentrate more on Algebra II, especially since at our school Algebra II is a freshman course (Geometry follows it) and therefore it is a formative course in mathematics for our students.

Politics intrudes here, since for the next few years, until Common Core becomes translated into assessments, what we are still going to hear is “state standards, state standards, state standards”. I am not a great fan of these standards – they are limited to the lower levels of Bloom’s taxonomy, they have low bars for “Proficient” or “Advanced” and they are completely multiple choice.

In my opinion, our assessments should be based on what we, as professionals, know that students need to achieve. For example, from my experience in engineering school (undergrad and grad) as well in engineering practice, an Algebra course should not touch completing the square in the quadratics unit. It is more important to have students understand the meaning of roots of a polynomial function, than memorize –b/2a. Graphing and computers find the axis of symmetry and the vertex without any problem. To understand what happens to a function at its roots, why we call them “zeros” and how this is applied in practical problems – this is way more important than completing the square.

Perhaps even more important than content, is the type of problems we should use in our assessments. If we are going to educate critical thinkers, multiple choice problems can have only a limited role.  Here are some types of problems that I think should be part our assessments in Algebra and that I plan to use in the quizzes for SBG.

(a)   Function generation from patterns.


(1) How many circles (black and white) are in Box 9?

(2) Write a formula to find the number of circles in Box n. Explain your results.

(3) The total number of circles in a box is 265. What is the box number? Show work.

The mathematics here are not very difficult. It is rather the form of the question and the fact that we connect the mathematical idea of a function with a pattern that is more important. A more challenging variation is to just to have number (3) above as the problem.

(b)   Conceptual understanding

Example: In each of the four equations below, the solution depends on the constant a. If a > 0, what is the effect of increasing a on the solution of each equation – does the solution increases, decreases or stay the same?

(1)   x – a  =  0

(2)   ax = 1

(3)   ax = a

(4)   x/a = 1

Again, solving these equations is pretty simple, but how often do we check for student understanding of what we mean by a solution? How often do we check that our students understand  how and why solutions can vary?

(c)    “Realistic” problems

Some educators assume that if we put in homeworks or in exams “real world” problems, the kids will understand why math is important and they will be eager to solve such problems. Unfortunately, research (Willingham, Boaler) shows this is not necessarily the case.  This research shows that “problem context [can distract students] from [the problem’s] deeper mathematical structure”. (Shannon). Therefore we should NOT trivialize the math, just to make up a “real world” problem.

Example: For health reasons people should limit their efforts, for instance during sports, in order not to exceed a certain heartbeat frequency. For years the relationship between a person’s maximum heart rate and the person’s age was described by the formula:

Recommended maximum heart rate = 220 age

Recent research showed that this formula should be modified slightly as follows:

Recommended maximum heart rate = 208 (0.7 * age)

A newspaper article stated “A result of using the new formula instead of the old one is that the recommended maximum heart rate  (heartbeats per minute) for young people decreases slightly and for old people it increases slightly”.

(1)   From which age onwards does the recommended maximum heart rate increase as the result of the introduction of the new formula?

(2)   The new formula helps determine when physical training is most effective: this is assumed to be when the heartbeat is 80% of the recommended maximum heart rate. Construct a formula for calculating the heart rate for most effective physical training, expressed in terms of age.

Realistic? Yes!  Trivial? NO!

We can use examples such as the last one in groups, for small projects – whatever. The important thing is to expose students to such problems.