My district switched to so-called integrated math a few years ago, and at my school we've added a new course each year. We started with Integrated Math 1 two years ago, added Integrated Math 2 and 2+ last year (2+ is a fast track to calculus), and this year we added Integrated Math 3 and 3+ (students in 3+ will take calculus next year, by passing pre-calculus).
Would I rather have stuck with Algebra 1, Geometry, and Algebra 2? Heck yes! And so would the vast majority of my district's math teachers. That wasn't the input the district suits wanted, though....
Anyway, my department co-chair and I are teaching the two 3+ classes. We've already planned out which lessons are to be covered each day for the entire school year so that we can cover the material needed to prepare students for calculus. Not only is there no time for reteaching, there's not enough time for teaching in the first place.
I've been somewhat stressed lately, feeling like I'm not a very good teacher in that course. These are exceptionally capable students, and I'm just shoveling information at them as fast as they can take it. If they can take it, what's the matter, right? Aren't I usually the person who says we should let students accelerate as fast as they can handle? Yet here I am, in a super-accelerated class, and I feel like I'm not really teaching. Since I pride myself on my teaching, this class has me down.
Something's not right, and I couldn't quite put my finger on it--until yesterday.
As I said, I'm shoveling those students the information as fast as I can. The problem is that I'm not teaching. I'm showing students how to solve problems, teaching them what they need to know how to do, but I'm not teaching them why what I'm teaching them works. I'm not giving them the background information that explains an algorithm or amplifies a concept. Here's the task, learn it, move on.
All that deep understanding, all the Common Core stuff? That's what I'm not doing. Believe me when I tell you that there isn't time to do so. I've stated that I have these kids drinking from a fire hose, and that analogy isn't so extreme. There isn't time in a 60 minute class to teach more.
That's what's been bothering me.
My kids' school is telling the parents that the kids can get through Math 1, 2, and 3 and won't need to take precalc. I don't believe them. It is very informative to read your blog.
ReplyDeleteSomewhat off point...
ReplyDeleteOne of the longest (and, of course, therefor, most pointless) arguments I've had on the web in the last couple years was of a supposedly Core-aligned test question and how it showed the teaching of deeper meaning in the CC.
The question was about a simple linear graph which related one measurement to another. No causal relationship between x and y were intended, nor would there be based on the test question.
The question asked about the *meaning* of the slope. How very Common Core: we don't just teach rote learning, we want kids to understand the meaning behind the math!
However, one of the multiple-choice answers was something like: the slope shows how y varies as x varies. But another answer was: that it shows how x varies as y varies.
I argued for days that both answers were correct, since the slope is a simple ratio and, if there is no causal relationship, you can flip the axes and the slope still shows the same relationship. In fact, even if there *is* a causal relationship you can flip the two: it is only a *convention* (aka, devoid of deeper mathematical concepts) that we put the independent variable along the x-axis and the dependent along the y.
The only way the first answer was the *only* correct one was to rely on rote learning: y varies with x. I pointed out that despite Common Core's claims to get kids to dive deeper and understand more, this was just relying on memorization of graphing rules. In fact, kids with better understanding would be tripped up by the question, since they would know that both answers are correct. They would be forced to answer the question based not on the actual math and meaning, but on what they thought the test writers wanted.
Many questions I've seen are testing that latter skill: how well can you interpret the test writers' intentions.
Fascinating post. I'd like a little more clarification. Most of the complaints that I've read about the Common Core math sequence is that it is too slow, but you seem to be suggesting that it pushes students too fast. Also, calculus by the senior year has been a common high school goal for years. Why does getting the kids up to speed for calculus now seem to require rushing them?
ReplyDeleteI'm not a high school math teacher, so I don't understand all the underlying issues. Your post is very interesting, and I would like to understand it better.
Let's differentiate between Common Core and Integrated Math.
ReplyDeleteCommon Core doesn't require integrated math; our district chose to drop the traditional sequence of algebra/geometry/algebra and adopt integrated math instead.
Common Core adherents suggest that students not be allowed to accelerate too much--I guess one way to end the achievement gap is to bring the top students down. Anyway, we don't want to let students accelerate, so as freshmen they're not supposed to be put into a class beyond Integrated Math 2 or 2+. We can accelerate them a bit more in high school, which is why we have 2+ and 3+ courses. 2+ and 3+ shave some non-calculus-related topics off and cover just enough IM2, IM3, and pre-calculus topics to prepare students for calculus.
We *create* the problem by not allowing students to accelerate much before high school, and them slam them in high school by forcing them into classes that go 50% faster than they should--with 2+ and 3+ we're covering almost 3 years of math in 2 years.
That is the problem I'm up against.
I teach at a liberal arts university, and I teach a one-semester precalculus course that starts with functions and covers algebra of functions, solving polynomials, exponential and logarithmic equations, simultaneous equations, trig functions, their graphs, solving triangles, vectors, polar coordinates and conic sections - all the typical precalc concepts. We meet four times a week, for 50 minutes at a time, for 15 weeks. That's 60 class periods; every fourth one is a Q&A session, and three periods are exams. So really, it's about 40ish classes to deliver instruction - about 33 hours total (less than _one_ standard work week). This is an entire school year's worth of material.
ReplyDeleteI really think all of college is of the "drink from the firehose" situation. The real learning, and deeper understanding, comes from conversations in office hours, and making sure the students work on the material in a meaningful way _outside_ of the class periods. It's not any easier when you do have more time as there is always another tangent, and another deeper discussion that you can have. I'm reading older algebra books (right now, it's Bruce Meserve's Fundamental Concepts of Algebra, from 1953) and learning new tricks and techniques, and I am very sad that I can't spend a week or two showing these to my students...
I understand what you're saying, but it's a little different when you're teaching 15- and 16-year-olds who are carrying 5 or 6 classes at a time.
ReplyDeleteSherman Stein once gave me the copyright 1984 version of his Elementary Aglebra: A Guided Inquiry, which also has copyright dates of 1969 and 1970. It's Algebra 1 with some Algebra 2 thrown in, notably leaving out conics.
My first encounter with the new "New Math" made it clear to me that like "Whole Language" the larger educational infrastructure is invested in making sure only a few really understand math. https://sumofallthingsaccording2me.blogspot.com/2017/09/new-new-math.html
ReplyDelete