Monthly Archives: May 2012

Looking Back, Looking Forward…

School ends June 8 and it is good and appropriate to look back at this year. My experiences are obviously colored by my training (I started as an engineer and I have been teaching for only ten years), my education (Europe and US) and my experience as a teacher (same high school, in a relatively small district with two middle schools and one high school).

First and foremost, I need to reconcile myself to the fact that the students I get are (by enlarge) poorly prepared, uninterested in math and sorely lacking in study skills. The latter deficiency extends to most academic subjects, but its consequences are perhaps more acutely seen in math. As I have blogged before, I do not think that we as a nation are very much attuned to academic success and especially to mathematics.

Second, I need to reconcile myself to the fact that my math colleagues are not very much interested in innovation – I would call them lemming-like.  They accept the state tests as a measure of student knowledge and would not dream of challenging that notion.  Their exams and their pacing -guides are literally by the book, and they do not bring extraneous or innovative material in the classroom.  Their “projects” are infantile. They accept the guidance of administration that lessons should follow an identical plan (given to the school by a consulting firm), and that they should not deviate from that format. The scariest part is when they come to me and ask me math questions that a teacher should know! Yikes!

Third, I need to reconcile myself to the fact that I work in a district where parents have a lot of power. If they do not like a teacher, they will successfully demand that their child be transferred to another class that is “easier” (i.e. where it is easier to get a good grade) or the teacher be changed (it did happen to a colleague this year). The counselors, and the school administrators cotton to the parents because they know that if they do not, those parents will go to the board. The yin and yang of a small district!

Therefore, the things that I want to do and that I think are worth doing, I need to do on my own, with little or no discussion with my peers or administrators.  I need to steer my own course and if they don’t like it they can slap my hand later.

What are the things that I want to improve on? First, I need to stress a lot more the conceptual components of math than I have done so far. I need to move away from the standard, current algorithmic approach to math teaching (“Here are the steps to learn if you want to solve this type of problem”) to a more big-idea type of approach (“What are the key elements in linear functions? How do they differ from, say, exponential or quadratic functions?”).  One of the great challenges will be to balance the need for algorithmic-ease with the need for conceptual understanding.

This ties to my second goal, which is to check for deep-understanding, i.e. understanding the principles. My thinking currently is that peer instruction (PI) and Just in Time Teaching (JITT) might be the best ways to accomplish these goals. I have watched Eric Mazur’s You Tube videos and they were eye openers. Adapting these techniques to high school teaching seems a worthy if challenging goal.

Of course, all of this depends on the courses admin will give me – if I get Algebra I, it will be more  about classroom management than about teaching math. PI and JITT lend itself more to Math Analysis or AP Statistics, but I may not get those courses.

To be continued…






There is no x and y in the real world

Five months ago, when we were studying systems of linear equations in Algebra II, I gave my kids the following problem on a test:

x + y = 20

5x + 3y = 70

I remember distinctly that most students – over 80% – got the correct answer. I was pleased – to my mind they understood what I had been teaching and were able to apply it successfully to a problem they had not seen before.

Fast forward to last week, when in preparation for the finals, I had just concluded a week of review of linear functions, equations and systems.  Thursday was the test on the review material. As part of the “free response” part, I put the following problem on the exam.

“In a room there are four-legged chairs and three legged stools. There is a total of 40 seats and 145 legs. How many chairs and how many stools are there in the room? (SHOW WORK FOR FULL CREDIT).”

Again, about 80% of the students got the correct answer, but what a difference! Most got the answer by trial and error – I could clearly see on their papers how they scribbled  different combinations of stools and chairs filling up the answer space. What is going on here?

When I asked my kids what was so difficult about that problem, what I got from their various answers is that many (a majority?) are so unused to word problems that they practically shut down when they see one.  My “big” exams always do have two or three free response problems, but usually they are more involved than this one. The student do more poorly on the free response part than they do on the multiple choice, but I had blamed that difference in performance on the caliber of the problems.

Not so – or not completely so. It is the intrinsic fact of having to deal with a word problem that causes the poor performance. Perhaps the kids are not to blame. The whole California Algebra II State Test is multiple choice. Teachers who want to prepare the kids for the test – and who is not under pressure to do so? – tend to also give simple multiple choice exams that mimic the State test.  Plus, let’s face it – multiple choice tests are so much easier to grade! As far as I know, I am the only Algebra II teacher in our department who gives free response problems.

Unfortunately, there is no x and y in the real world. In the real world we deal with variables that have names, represent quantities and have units associated with them.  Heck – in the “real real” world,  half the battle is just formulating the problem.  No wonder the kids ask “When am I ever going to use this?” Honestly, the answer is “never”. Given all the abstraction and recipe math we are throwing at them, no wonder so many of our students are bored. We need to resolve the tension between teaching algorithmic methods and teaching how to set up problems.

“I beseech you, in the bowels of Christ…

In 1650, the Scots joined with Charles II and prepared, once more, to invade England.  The General Assembly of the Church of Scotland (the Scottish Kirr) sent Cromwell a justification for its acts and in return, Cromwell sent a letter that included the memorable phrase “I beseech you, in the bowels of Christ, think it possible you may be mistaken.”

It is a salutary thing to remember – we may be wrong, despite our strong beliefs.

For the last couple of years I have been down on my math department colleagues for their teaching. I thought they were not challenging the kids sufficiently, that they went too much by the book, that the “projects” in their classes were infantile and that in general they are what I call teachers of math rather than math teachers.

But… what if I am wrong? What if their teaching is appropriate for the level of kids we have? After all, most of these kids will never likely have to solve for x in the rest of their lives.  Why give them challenging problems about things like conic sections, geometric series or systems of three equations? Why not just teach the basics, just enough to get them through the state testing with enough of a performance to make the school look good? Who cares if they don’t remember it the next day? Who cares if they take remedial math in college?  – they are no longer “ours”.

There is something to be said for teaching at a level appropriate to our “raw materials”.

But then… another memorable quote comes to my mind, this time much, much newer. It is best known in an interpretation by Peggy Lee: “Is that all there is?”

Yin and yang.

H.O.T.? – I Think Not! (Watering Down the Common Core Standards – II)

One of the threads that runs through the CCSS is the need for more rigor in math education. The Algebra II textbook reviewed here (Glencoe/McGraw Hill) responds to this need by introducing “H.O.T.” problems in each section. H.O.T. stands for higher order thinking skills and, like the rest of this book, these problems are a huge disappointment.

Consider the following example. In the section on logarithms and logarithmic functions, one such “H.O.T.” problem is an error analysis. Given that

which is the correct next step:

Frankly, I think this is an insult to the students (and to math) to call this high-order thinking. What kind of standards do we establish if we label a definition as high order thinking?

Unfortunately, this mislabeling of definitions as high-order thinking is widespread. I looked at random at Error Analysis/Critique  H.O.T. problems in the sections on function composition, graphing exponential functions and multiplying and dividing rational expressions. In all of these sections, what should be definitions are labeled as higher-order thinking problems.

There is no way one can call this book “informed by other top performing countries” as the CCSS wishes.  Just to stay with logs, here is first an example from a foreign high school collection of problems and second, a less difficult one, from an older US high school textbook:

Solve the system for x and y:   

Show that

Why is it that we don’t think our students can do these kinds of problems? Why don’t the authors put in truly higher-order thinking problems in their book? Where is the CCSS rigor?

All in all, this book is a travesty of the Common Core standards. One can only hope that other publishers will do a much better job.

Watering Down the Common Core Standards (I)

There is hope – at least in some quarters – that the Common Core State Standards (CCSS) will bring an improvement in math education.  According to the CCSS website, the new standards “[i]nclude rigorous content and application of knowledge through high-order skills” and “[a]re informed by other top performing countries, so that all students are prepared to succeed in our global economy and society”. It sounds exciting and indeed, when one reads the standards, one does see an attempt to connect concepts and to ask students to think rather than memorize a set of unrelated facts.

At our last staff meeting, I was understandably excited to see for my first time a book that claims to be aligned to the common core state standards (Algebra 2 – Glencoe/McGraw Hill).  I reviewed the book for the last three days and …what a huge disappointment!

A serious concern is that, in this book, some of the CCSS standards are missing either in body or in spirit. Here is an example. F-LE1 wants students to “[d]istinguish between situations that can be modeled with linear functions and with exponential functions”. The key in this standard is to distinguish between quantities that grow by equal differences over equal intervals and those that change at a constant rate per unit interval.  In my opinion this is an important concept – it goes to the fundamental idea that functions model real phenomena in nature and that these phenomena represent various rates of change – linear, exponential (both growth and  decay), zero and so on.

Not only is F-LE1 not in the book’s index of standards, but the concept of constant rate growth is stressed in the section on series, not in that on exponential functions. There is no connection established in the book between exponential functions and geometric series and the two are separated by 3 chapters (1 – 2 months worth of school days) so the idea of exponential growth is likely forgotten by the time students get to geometric series.

Here is another example. Standard A-CED.4 wants students to “[r]earrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.” Again this is an important concept, especially for those who later go into engineering, math or the sciences. In my experience students have difficulty with this – they are not trained in middle school to manipulate abstract variables.

Standard A-CED4 appears twice in the book’s index of standards.  The first time it appears in the section introducing circles, but the rearranging of formulas here is not in the same spirit as Ohm’s law example – it has more to do with completing the square. The second time the standard appears is in the discussion of geometric sequences and series and again I fail to see how this standard is applied in the spirit it was written. Therefore, for all practical purposes, the standard is missing.

It looks like the standards were fitted to the book, rather than the book written around the standards. Perhaps this is not surprising, given that 7 authors of the previous book’s edition (non-CCSS) are also the authors of the new one. From a business point of view it’s good to be early in the market in response to a market need – unfortunately this product leaves a lot to be desired.

In my next post I will address the rigor of the book and how it compares to other texts (domestic and international). For now, I am concerned that standards which aim to be rigorous and indeed world-class will be watered down by textbook publishers and possibly by those that write the state exams.