Tuesday, July 14, 2015

Going "deep" with Common Core algebra

It's time for another exegesis of an article in Edweek, this one on what the Common Core is doing for algebra students:

Under the Common Core State Standards, Algebra 1 is a much tougher course than what was taught previously in most states, teachers and standards experts say, in part because many of the concepts that historically were covered in that high school class have been bumped down into middle school math.
Actual Common Core-inspired algebra problems tell a different story.
Some say those changes could complicate efforts around the country to put 8th graders in Algebra 1—a still-debated trend that's grown over the past two decades.
And thus the Common Core, besides all the other fashionable K12 practices it endorses, has become yet another excuse not to accelerate students, but rather to impose a one-size-fits-all on everyone based on their calendar age.
And while that kind of move can disappoint some parents, educators point out it doesn't mean 8th graders aren't learning algebra.
Well, that depends on what you mean by "algebra."
"There's big confusion between the Algebra 1 course with a capital A and algebra, the mathematical subject," said William G. McCallum, a mathematics education professor at the University of Arizona, in Tucson, and one of the lead writers of the common standards. "If you follow common core, there's now tons of algebra content in the 8th grade."
Well, that depends on what you mean by "algebra."
"Traditionally in Algebra 1, a lot of time was spent looking at linear functions," said Diane J. Briars, the president of the Reston, Va.-based National Council of Teachers of Mathematics. "But a lot of that work now has been moved into 8th grade common core."
Right. Kids in 8th grade (and younger) are filling in input and outputs on function charts, guessing and checking and plotting points on grids, and solving for x in simple equations involving minimal mental (or algebraic) manipulation. Then there's ratio and proportion.
"The common core… built much higher expectations for conceptual understanding regarding ratio and proportional relationships [in 8th grade] to prepare students to understand the ideas of slope and rate of change."
It sounds pretty impressive:
Simultaneous linear equations and functions and their graphs—concepts also typically taught in Algebra 1—are now also taught in 8th grade under the common core.
The catch is that those simultaneous equations never have more than two variables in them, even in today's Algebra 2, and that they require little in the way of symbolic manipulation.
The idea is that by the time students get to Algebra 1, they will have developed deep understanding of some basic algebraic concepts, and can dive into more complicated coursework.
The "complicated coursework," though, is about applying these concepts to "real-world" situations, not to situations of mathematical complexity:
Students are focused on applying the algebra they're learning, rather than seeing it as a series of procedures and algorithms.
For instance, rather than doing a set of problems from the textbook, Mr. Ryan said, students in Algebra 1 might collect data on the weight of students' backpacks, plot them on a graph, and model them with an algebraic function.
What's challenging about math, however, often isn't the concepts themselves. How hard, after all, is the concept of a function, or a slope, or an algebraic "model" of a simple, real-world situation like the weight of students' backpacks?

What's challenging about math, rather, is the mathematical complexity and abstraction that emerges from applying functions, slope, and algebraic models, not to real-world situations, but to mathematically complex situations, with layer upon layer of abstractions, abstract patterns, and symbolic manipulations. What students need practice mastering is this emergent property of complex, abstract math, and not the building-block concepts themselves--or their real-world applications. Yes, students need to engage with the concepts in action, but most especially in an abstract, mathematical way that few are seeing today.

Here are some examples of what they are missing, and will miss--throughout their years of Common Core-inspired high school math:

--simultaneous equations involving more than two variables.
--simultaneous quadratic equations.
--equations involving abstract quadratic patterns.
--real-world situations that take some real mathematical thinking to model mathematically.

4 comments:

Barry Garelick said...

Kids in 8th grade (and younger) are filling in input and outputs on function charts, guessing and checking and plotting points on grids, and solving for x in simple equations involving minimal mental (or algebraic) manipulation.

See also here.

R. Craigen said...

On this business of 'real world problem' somehow indicating that things are going deeper, I like to point out what the PISA 2012 report says about this:
http://www.oecd.org/pisa/keyfindings/pisa-2012-results-volume-I.pdf

Apparently (p. 150) emphasis on exposing students to formal mathematics is worth about 50 points on the PISA scale, well over a full year-equivalent of teaching, whereas emphasizing word problems is worth about 4 points, which for many countries is around the margin of error (95% level) of their actual score -- and so borders on statistical insignificance.

Table 1.3.1a shows the frequency with which students were exposed to word problems. Notice that the top-performing countries in PISA all cluster at the BOTTOM of this table -- even though PISA testing is purportedly based on the Freudenthal Institute's Real Mathematical Education philosophy, which says that students should be taught-and tested-in "authentic" real-world situtations. You'd think it would be the other way around; whod'a thunk that those well-grounded in the basics would be better at all this "deep" stuff than those whose education majored in the deep, neglecting the, uh, actual abstract subject matter of math itself?

If there's any doubt from that exercise, compare table 1.3.1b which shows the frequency with which students were taught formal mathematics. Suddenly those same countries are clustered at the top! Whod'a thunk?

FedUpMom said...

I don't get it. How would you make an algebraic function to model the weight of kids' backpacks? The functions I'm used to have related x and y values. If y is the weight of the backpack, what's x? The kid's weight? The kid's height? The kid's grade? The kid's grade-point average? What?

SteveH said...

"The idea is that by the time students get to Algebra 1, they will have developed deep understanding of some basic algebraic concepts, and can dive into more complicated coursework."

Most schools have (for ages) provided different tracks in math starting in 7th grade. Those deemed prepared were put into a proper pre-algebra course and others had one or more lower levels of expectations. Even when I was in school, the lower track provided instruction that allowed students to "dive into more complicated coursework" in high school.

What I find in the comments above are "deep understanding" apologists for low expectations. Whatever happened to the big talk of STEM? Why is the Common Core solution to expect kids to take summer courses or to double up in math in high school to bridge the gap? (It won't happen.) Why does PARCC specifically say that it doesn't do STEM? All of this lets K-6 and full inclusion off of the hook.

Do they really believe their blather about "deep understanding?" Don't they ever look at the specific unit-by-unit content curriculum continuity from pre-algebra onwards? This is not just an issue of "discovery" pedagogy. It's an issue of educational competence. Our town used CMP for ages in middle school and did nothing about the curriculum gap for students who wanted to be prepared for geometry in 9th grade.

Do they just look at the students who make it to calculus and assume that the process has no fundamental flaws? Do they ever ask the parents of those students what they had to do at home or with tutors?

Life goes on. Students are still tracked in math, parents help at home or with tutors, and K-8 pedagogues continue to fool themselves into thinking that they know what parents and tutors do. I got to calculus in high school with absolutely no help from my parents. I doubt that any of my classmates had help. That is very unlikely to happen now no matter how much they talk about "deep understanding."

Since I was in school (K-8), the big negative changes have been full inclusion and social promotion. Add to that the ed school indoctrination against mere facts and superficial knowledge. We now have K-6 schools that set lower and fuzzier expectations (covered up with blather about "deep thinking") and a much larger non-linear transition in 7th and 8th grades. They are creating a bigger academic gap and letting K-6 off the hook.