Monthly Archives: April 2012

We Have Met the Enemy and He Is Us

Going back to the post I mentioned previously, the one in which Scott McLeod describes his experience in taking the ACT again at age 44, I was struck by one paragraph in the section titled ‘Concluding Thoughts’. I quote it here in its entirety:

  • You’re so brave! I was struck by the sheer number of comments that Jason and I received that expressed disbelief that we would do something like this. Typical statements included variations of ‘You’re so brave! I could never do that!’ and ‘You’re willing to report your score publicly? Really?’ and ‘There’s no way in hell I’d ever take that exam again!’ and so on. I’m still mulling over what it says about us, our schools, and our society when we’re willing and even eager to have our children submit to experiences that we’re not willing to engage in ourselves as adults.”

I for one, I am not mulling over it – to me the answer is clear – we, as a society, do not value math, we as a society think it’s understandable not to do well in math. Time and again, I heard parents at back-to-school night say something like “Well, of course, I wasn’t very good in math either…”, or “I can’t help him/her at home because math has never been my strong suit…” These parents “mean well feebly”. Their sentiments are not expressions of regret; they are matter of fact expressions with the undertone of “of course you understand…”.

Exactly what am I supposed to understand?  That it is normal not to know math? That not doing well in math is a genetic trait? That when a subject is ‘too hard’ we should make allowances for it?

When the unspoken undertone at home is that it is somehow acceptable to have trouble with math, then no wonder our students are not very hungry to excel in the subject and do not persist when encountering difficulties.

In Asia and in Europe, a student who excels in math is looked upon with respect and is somebody to be emulated, in the US s/he is looked upon as a curiosity, a freakish being. This is a fundamental cultural difference and it directly influences everything else related to teaching math: the caliber of people going into the profession, the expectations we have of our students and the caliber of the courses we teach.

All the efforts and ingenuity of the many enthusiastic teachers I see in the blogosphere will have only a marginal impact until we have kids wanting to excel in math as much as they now want to excel in sports. Of course, the larger issue is that of respect for academic excellence in general and that goes deeper into our national psyche.


Dictionary of Forbidden Words: Tracking

Act One. In my last post, I made the case for a math curriculum that demands higher order problem solving skills from the kids, starting in middle school. I was glad to come across a recent post by Scott McLeod that also addresses the what, rather than the how of teaching.

Scott’s main points are that “we’re herding many, many students through math classes that are largely irrelevant to their future life success” and that “high school math problems generally [are] either decontextualized pure math problems (students: who cares?) or pseudocontextual word problems (students: who cares?)…. There wasn’t much on the math test [ the ACT] that I think would be of interest to typical high school students outside of the artificial environments of classes and testing.” Commenting on the ACT (which he took again at age 44!), Scott wrote: “[The] exam does … focus heavily on procedural knowledge (and provides a variety of contexts in which students can show that knowledge). Occasionally it assesses – in a fairly limited bubble test way – some application, synthesis, analytical, or inferential skills. But for the most part, the exam does not get at higher-order thinking skills in any substantive, applied, hands-on, performance-based way.”

Scott is an Associate Professor at the University of Kentucky – his opinion carries weight as an education professional. There are other educators too who agree that there is a need to change the math curriculum to emphasize higher-order thinking skills – but how do we get change it?

Act Two. The current California Algebra II curriculum has 27 standards. In the textbook used at our school (Glencoe) – a fairly typical Algebra II textbook – these 27 standards are broken down into 97 lessons.  California typically offers 180 instructional school days, even though many districts – under budgetary pressure – are shortening the school year. With 180 days, that works out to about 1.86 days per lesson. Taking account assessments, exam feedback and other non-teaching activities, this may well bring down the average to about 1.8 days per lesson.  How can one introduce, for example ‘Exponential Growth and Decay’ – lesson 9.6, and then reach higher order thinking skills on this topic in 1.8 days?  As the Brits would say, “Not bloody likely”. Besides that, we don’t let the concepts “marinate” in the minds of the kids either – the next lesson in the sequence is ‘Midpoint and Distance Formula’. Connections anyone?

So what happens in real life? The curriculum gets cut down – we teach only the “key standards”.  We blow through some topics so that we can spend more time on others.  At the end of the year, all teachers agree that it is virtually impossible to teach all the topics in the book well, or even teach well all the topics tested on the state exams and we fall back on phrases such as “mile wide and inch deep curriculum”.

Act Three. Since we do cut down the curriculum already, does an Algebra II course truly need to cover all these topics?  What math should we teach? Specifically, do we make math relevant or do we say that math is intrinsically important, that it creates habits of logical thinking and that relevancy is not a concept that has merit here?

The truth, as I see it, is that this is not an either-or proposition. A student who has in mind a career as a dental technician probably does not need Cramer’s rule, multiplying and dividing rational expressions, or infinite geometric series.  A student planning to go into engineering probably will need to get a good handle on all of these (except possibly Cramer’s rule, to be replaced by computer algorithms).

The point is that we should not have one curriculum for all students.

For reasons of ability, desire or any other factors our kids do not arrive at Algebra II with the same degree of math knowledge or math reasoning skills.  In my experience, and that of my fellow teachers, this course serves to further differentiate students by performance.

The situation is analogous to that of a jeweler that receives raw materials from one source: chains, stones, rings, etc. Not all these raw materials are of equal quality and workmanship. The jewelry maker has two choices: either invest in a lot of pre-processing to raise the quality of the below-par raw materials to some acceptable level or differentiate the merchandise that s/he sells. If s/he chooses to differentiate, the jeweler can select the best raw materials, process them into beautiful jewelry and sell them to high-end stores. The below-par raw materials can be made into cheaper jewelry that can be shipped to stores that do not cater to the high-end clientele.

Differentiating your products and aiming these products to specific target markets is the technique that most businesses will use under these conditions – pre-processing is too costly and often does not result in high quality products anyway. In education, the equivalent of product differentiation is called tracking.

And…tracks can cross.

Are you a Math Teacher or a Teacher of Math?

Why is it that I would not dream of giving my students a problem like:

“Evaluate:   ” ?

The problem appears in a section called “Logarithmic Evaluation without Tables” in Smith and Fagan’s “Mathematics Review Exercises” (Ginn and Co., 1968). This is an American book, written as a supplementary text for high school students. If I were to assign this problem, (a) most students would decide just by looking at it that it is too hard and they would give up and (b) even if they did try it, they would not be able to do it, most likely because of the fractions and the relationship between (4/9) and (27/8). So the problem is too hard, right?  Then how come it appeared in the middle of other similar problems in an American book written for American students of 44 years ago?

To my mind, the answer is clear – we have dumbed down the curriculum and our expectations. In particular, middle school math (even when the kids complete Algebra 1 in 8th grade) does not prepare the kids for any kind of higher-level thinking.  From what I have observed, middle school math teaching, even when it claims that it is rigorous, graduates kids that have only a sketchy idea of math and spoon feeds the kids at every step of the way. One of the reasons for the poor performance of these kids in high school is “the system”– often middle school kids can not be held back if they fail a subject – it is deemed too damaging to their self-esteem.  Another possible reason is that middle school teachers are driven by the state testing to cover all the topics in the book and there is no time to teach any concept in depth.

The result is a process where the middle school teachers I observed give the same kind of simple problems 50 times in a row, with the aim of having the kids master the procedures by ad nauseam repetition. The (unintended) consequences are kids who are bored, kids who do not see connections between the various math topics, kids who are not forced by the material to focus on a problem more than for about 4 minutes and kids who are not asked to use the procedures in a creative, higher level thinking way. These kids are lost to math – indeed, I would argue that they are lost to critical thinking – and the situation only gets worse as these kids “progress” through high school and college. Statewide, in California, roughly 30% of the students entering junior colleges have to take remedial math – essentially Algebra II. Locally, where I teach, that figure is over 40%.  What use is instilling a “college going culture”, if the kids end up failing in college?

When I look through teachers’ blogs, such as the ones I follow, I am struck by the amount of effort and ingenuity that goes into procedures for teaching and grading and how little is said about what we ought to teach.  Of course, one can not divorce what we teach from how we teach it. However, I would argue that what we teach should take precedence.

George Cobb, a highly esteemed statistics teacher and author, said: “Judge a book by its exercises and you cannot go far wrong”. I would paraphrase this as “good math teaching should be judged by the problems we set out for our kids”. Thinking, “hard” problems should start in middle school – we should abandon the idea that these are still children that need constant spoon-feeding. When are they supposed to grow up? If we start to demand and stick to our demands for critical thinking, we will get it despite all the grumbling that teenagers are prone to.

The resources are there: the Exeter problems, in Geometry Weeks and Adkins, in Algebra Dressler and Rich, the Dolciani books, Weeks and Adkins Algebra 2 and Smith and Fagan. Singapore books, such as New Additional Mathematics and the series on New Syllabus Mathematics, provide exercises far superior to those in American textbooks.

We teachers, especially the middle school teachers, need to be math teachers first and not just teachers of math – the priority should be on the subject and only then truly good teaching will follow.

Video Games for Math Learning

In a recent series of articles, Professor Keith Devlin of Stanford University outlines some of the issues related to creating video games to enhance math learning. Part 6 of this series – the most recent one – gathered 15 comments at the time of this writing.  The comments, while related to video game design, also speak to some more fundamental issues in teaching math. I quote some parts of the discussion, as a way of illustrating what I see are two important pedagogical issues.

(a)   “Are you going to approach mathematics as a collection of procedures or as a way of thinking? These are not completely separate classifications; indeed, the latter is in many ways a broader conception than the former. But they do tend to cash out in very different forms of pedagogy.”(Devlin, part 5)

(b)   “Even if K-8 math is taught extremely well, how can a video games of this kind teach higher level math (precalculus and calculus) – which is where even those who’ve done well in K-8 math struggle?” (Jeremy Millington, Comments, part 6)

(c)    “…we don’t need games (or other new learning methods) merely to teach fractions or to bootstrap kids into algebra. We could really use them at higher levels, too. At those higher levels, the gap between conceptual understanding and symbolic understanding seems bigger to me.” (Anonymouse, Comments, part 6)

(d)   “In terms of gameplay not in conjunction with a teacher-led class, this strikes me as harder, since you need to provide the player/student with good feedback on what they have done.” (Devlin, Reply to Jacob Klein, part 6)

Procedures vs. a way of thinking [(a) above] is, in my opinion, a fundamental pedagogical issue. The ways textbooks are written, the way most teachers teach, math is a collection of procedures. Most teachers announce the topic, give some definitions, work out some examples in class, assign homework and then go to the next topic. Even if exams are cumulative, in the minds of the students, the topics are not related – yesterday we did exponents, tomorrow we do rational expressions. There is no attempt (or no time) to present a bigger picture.

As an example, 90% of the California standards for Algebra II deal with functions (the exceptions are series and probability). However, most teachers, in my experience, do not emphasize the commonality that exists between lines, quadratics, exponentials, higher order polynomials, etc. In my experience, most teachers do not show that all of these are functions that represent different models of reality. Neither do most teachers emphasize the common theme of finding the roots of these functions, why we try to find the roots or what do these roots mean.

Because we teach sequentially, often with an eye toward the released questions of the state test, we do not often assign synthesis problems. For example, if in the same problem we would ask students to find the roots of (1) a line, (2) a quadratic, (3) a cubic, that would strengthen the concept that in all cases the x-intercepts are the roots and that the function goes to zero at that point.

In my personal experience, this lack of a broader emphasis on the patters of mathematics is often due to students’ lack of symbolic understanding and difficulty in manipulating these symbols [(b) and (c) in the comments quoted above].  When the students do not understand the procedures very well, it is pretty hard to emphasize the concepts; it is natural for teachers to try and “fix” the basics first, before talking about the broad principles of algebra.  I will write about what I think might be some potential solutions to this conundrum in a future post.





Red Meat, Chocolate, AP Statistics and Teaching

In a recent article in Discover magazine, Gary Taubes presents a lucid analysis of some recent nutrition news articles – articles that conclude that “meat is bad, chocolate is good”. Taubes’ argument is that the results of these research reports violate a principle that we teach in AP Stats: “correlation does not imply causation”. There is a world of difference between OBSERVING that people who eat meat on a regular basis have a higher death rate than those who do not and CONCLUDING that the higher incidence of death rate is DUE to eating red meat. Taubes argues – as we all do in AP Stats – that the gold standard in drawing conclusions is randomized trials and he mentions that when these trials were indeed performed, eating red meat improved the mortality rate (Atkins diet). [Unfortunately, for me, Taubes also shows that eating chocolate is not the healthy thing that these reports show].

The article is long, but it would be a very useful reading when we teach experimental design in AP Statistics and I intend to use it if I teach Stats next year. However, Taubes’ argument can be extended to what happens currently in high school teaching in general. From personal experience at my school and from what I read, there is increasing pressure that all teachers of a given course, say Algebra II, teach the same thing, the same day, IN THE SAME WAY. The pressure to do so comes from the disease of standardized testing. That all teachers – again, say in Algebra II – should cover the same curriculum at reasonably the same pace is to be expected when we are all judged by a test given to all, at the same time.

What is more disturbing to me is the fact that we are moving toward standardizing THE WAY we teach. In my district, administration has hired a consulting company that developed a way of structuring lessons based on “brain research”. The company is training the teachers in how to use this lesson structure and administrators observe us in how we apply it.

Now, brain research is a new catchword in reforming teaching, and it is becoming more and more en vogue, despite the fact that our knowledge of how the brain works is extremely poor for any kind of practical applications.  Administrators are generally not qualified to evaluate this so-called brain research, but usually they try to find some new hook to hang their hat on, and boy, does “brain research based teaching” sound good!

It so happens that the consultants hired by my district are very professional, experienced and their lesson plan is sound and reasonable. So what is wrong with using their method, outside of a visceral reaction that we are all not the same and that students benefit when we put our different personalities in our teaching?

I think the rub is that there is no evidence that our kids will do better if taught by this method than by any other. There are no randomized trials where the results of this method were compared with the results of another method. By whatever metric we apply – state tests, class grades – we cannot argue that one way of presenting the material is better than another in the absence of randomized trials.

As a recent post (Apr. 4) indicates, engagement in class does not necessarily result in doing the homework or in understanding the material better. My argument is that, up to a point, the effect of teachers is GENERALLY not as high as we would want it to be. I refer specifically to high school where students’ study habits and basic knowledge (in math) are already formed. Students who do not want to learn, who are not eager to learn, will not do better or worse when the material is presented to them in one structure or another. From my experience, truly understanding the basic math facts in middle school is much more important to high achievement in high school than any method or structure we use to formulate our lesson plan.

I would argue that “good teaching” (IN HIGH SCHOOL) gives more bang for the buck in the higher-level courses (Math Analysis and AP) than in the lower ones.  In my opinion, developing critical thinking depends much more on the teacher and the method of teaching than developing algorithmic facility. However, the distinction between the two is the subject for another post.

Why I have not posted recently

It has been almost six months since I have written and I feel that I ought to give an explanation.  My wife was diagnosed with cancer and we spent these last months working through radiation and chemotherapy and there is more treatment to come. We still don’t know how effective the treatment has been, but we take each day at a time.

Of course, it is my wife who did and will do the heavy lifting – it is she who got the radiation and the chemo, – but it did affect me as well. Not only did I worry (and still do) about her health, but also this crisis brought a stark reminder of what is important in life and what can take – at best – a second place. Worrying about SBG is one of those second place things. So is school and educational policy, even more so when educational policy and politics in the US is enough to make a healthy person sick.

However, since this is a blog about teaching, I feel I should summarize some of the main activities and insights I gathered as a teacher in these last months. The first insight is how difficult it is to make a difference as a teacher. I do not mean that a teacher can not make a difference in the individual lives of his/her students – I am glad and proud that I have done so for some of my students; there is nothing more heart warming than a note from a former student saying that you really impacted their life. What I do mean is how difficult it is to make a systemic difference.

Take for example SBG which I tried in my Algebra courses. It lasted from September until late November. Parents started to complain that the reporting was too hard to understand. They were used to the school’s grading/communication system (School LoopÔ) and the Excel sheets I created with SBG were too complicated for most parents to understand. I tried to hold the line, but the parents went to the principal and… you can imagine the rest. In our district, parents have way too much power – most often to the detriment of education. Sometimes – from talking to some of the parents – I get the impression that they consider high school courses as “stamps of completion” so their child can get into college. What the kids learn or do not learn does not seem to matter so much as the course grade and how that grade affects the GPA the colleges will see on the kids’ applications.

Since September, I have been teaching four classes of Algebra II – three of freshmen and one of sophomores and juniors.  I found my freshmen so poorly prepared – even though most of them came from Algebra I Honors in middle school – that my predominant grades at the end of the first semester were D’s and F’s.  Again, parents were unhappy and they went in droves to counselors to have their kids classes changed to “easier teachers”. It got me curious as to why these kids (a) knew very little algebra (Algebra I) and (b) why they were so antagonistic to thinking problems.

I took it upon myself to try to build performance based predictive models and do a statistical analysis of past and current performance. Major conclusion: grade inflation in middle school leads to improper placement and sub par performance in high school. Secondary, but still important conclusions: (a) the state exam has very poor class-grade and mathematical ability predictive effects and (b) middle school education is geared towards a repetitive, algorithmic way of teaching and does not prepare students for any kind of critical thinking.

Coming out of middle school our students seem to see math as a series of various types of problems, each with a well defined set of rules (and somewhat arbitrary ones at that) for solving each type of problem. Start combining problems, or even worse put them in front of word problems and our students melt. If the problems I see at my school are true for the general, average high school in the US – we are in deep trouble.