Wednesday, April 10, 2013

Do the authors of the Common Core really want us to replace step-by-step math with words, pictures, and applications?

Education Week, perhaps the most widely-read K12 education publication (circulation: 50,000+ paid copies per week; registered online users: approx. 725,000), has been consistently interpreting the Common Core as favoring Reform Math, discovery learning, and "real-life" applications--all staples of the dominant education paradigm known as Constructivism.

Here's the third of three articles in last week's education week that read Constructivist goals in the Common Core Math standards (and, in this particular case, also in the National Board Certification process): an article by Algebra 2 and AP Calculus teacher Alison Crowley. Describing her realization that "'teaching math' and 'covering textbook sections' were not synomymous," she writes [all boldfaces are mine]:

Before I started implementing, or had even heard about, the Common Core State Standards, I had already begun shifting my instructional practices to include more hands-on activities and group work, and less book work. Project-based learning began trending in my math teacher circles, and pursuing National Board certification forced me to rethink my instructional practices. Were my students actually learning the material for mastery or were they just good at following directions and memorizing steps?
One might ask the same of teachers. Here again we have the unquestioned contrast between "mere memorization"=book work vs. actual learning=hands-on group work + project-based learning.

Crowley asks us to consider the following situation:
If a teacher is explaining how to solve a system of equations using the substitution method, she might list on the board a set of steps for students to follow. Step 1: Solve one of the equations for one of the variables; Step 2: Substitute the value or equation found in Step 1 into the other equation. If you peeked inside her classroom on this particular day, you would likely see all of the students copying notes, and then probably completing a worksheet with problems similar to the example. From an observer's perspective, you might think the lesson was going very well.
But do the students really have a solid understanding of the mathematics they are using? And more importantly, do they understand why they're using it? Do they have a graphical understanding of what it means to solve a system of equations? Can they explain their methodology to another student? Can they apply it to real-world situations? Is their knowledge transferable so that they will be able to draw upon it when they are solving more difficult systems of equations in future math classes?
Again there's the assumption that putting things into words and applying them to the real world is the best test of understanding--even initial understanding. But why wouldn't initial guidance through solving systems of equations build a foundation for solving more difficult systems later on?

Crowley, naturally, guesses it doesn't--a mere guess being apparently enough of a foundation for an article in Education Week. Drawing on the untested assertions of Ann Shannon, a mathematics consultant with the Bill and Melinda Gates Foundation (itself a notorious promoter of Constructivist practices), Crowley writes:
My guess is that the answer to most of these questions is "no." What Ann Shannon would say is that in this particular situation, the students have been GPS-ed from problem to solution. Just like when I drive in a new city using my global positioning system, I can follow the directions and get to where I need to go. But I can't replicate the journey on my own. I don't have a real understanding of the layout of the city. If a road were blocked due to a parade, for example, I would be in trouble because I have no real understanding of the city's geography.
A cute metaphor, and I'm sure it's only a matter of time before "GPS-ing" will be buzzing all over the edworld. But cute and catching don't mean true. The GPS gives specific directions that aren't supposed to generalize, and they don't. "Drive 5.8 miles, then turn left on Cherry Street" is not analogous to "Step 1: Solve one of the equations for one of the variables; Step 2: Substitute the value or equation found in Step 1 into the other equation." If Cherry Street is blocked, the GPS's directions are useless; if the first equation has 5y instead of 2x, the teacher's directions still work. GPS-ing is not the same as step-by-step instruction.

But, when you have a great new buzzword, who cares? Especially you find justification for it in the Common Core math standards:
So, how can we keep from GPS-ing our students, so that they understand the mathematics behind a series of steps? How can teachers help them grasp the why, instead of just the how?
The good news is that the common-core standards provide an open playing field that encourages teachers to move away from the step-by-step model.
Consider the following standard for solving systems of equations:
A.R.EI.11 Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
It should be noted that the Common Core standards also include A.REI.5 Solve systems of equations using the elimination method (sometimes called linear combinations) and A.REI.5 Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation).Furthermore, these two standards are the first two under the "Systems of Equations" section; there are five more intervening standards before the standard Crowley cites. But, even though the her very own source suggests that it should come later on in the curriculum, Crowley prefers the graphical to the symbolic, and the real-world application to the step by step procedure:
Remember the earlier example about the teacher showing the students how to solve a system of equations using a set of steps? The first sentence of the new standard, "Explain why the x-coordinates where the graphs intersect are the solutions," really pushes the teacher to introduce and explain a new concept in a way that goes beyond one-dimensional instruction. What is an x-intercept and what does it look like on a graph? How is that related to the algebraic equation? Perhaps rather than starting a lesson with the steps for solving the equations, the teacher might first have students consider graphs of related equations, or better yet, a real-world example of a system of equations and what the values of the x-intercepts mean in that situation. This standard also challenges the teacher to present multiple types of equations from the beginning of the lesson so that the students can apply the concept of an x-intercept to many types of functions.
Multiple types of equations all at once as an introduction to systems of equations? This sounds like a pedagogical nightmare for all concerned.
For many teachers, myself included, this is a fairly significant change in instructional practice. Although I have taught lessons on solving systems of equations using real-world applications and emphasizing graphical connections, I have not yet truly focused my instruction on the "why" behind the mathematics or given students opportunities to create their own understanding.
But the "why" isn't inherently graphical; nor does it necessarily emerge from real-world applications any more perspicuously than from algebraic applications. As for students creating their own understanding, this is one of those things that looks better in theory than in real-world application.
So how do math teachers make that shift away from GPS-ing students a reality? It won't be easy, and we can't do it alone. We need opportunities to collaborate, plan, and reflect with colleagues, both in our buildings and nationwide. We need quality resources and relevant, engaging professional development.
The more I read, the more it strikes me that the big winners under the Common Core aren't the children, but the professional developers (and the developers of all those "quality resources").
We need time to learn from teachers who are already successfully implementing the common core, like Kansas educator Marsha Ratzel, who recently shared insights in her piece "The Talking Cure: Mathematical Discourse," on what happened when she gave her students time and space to have conversations about math. We need administrators and parents to support us and play an active role in helping us transform our classrooms into places where students are truly engaged in what they are learning.
True engagement with x = having converations about x. Now we just need to solve for x.
In my daily classroom instruction, I am still sometimes guilty of GPS-ing students. But I am hopeful that as I learn how to fully implement the common standards, I will become less and less dependent on steps and crossing standards off a poster. After all, my students really deserve to navigate themselves.
Do the authors of the Common Core also agree? It would seem so. As I noted above, Education Week may be the most widely read publication on K12 education. If its articles are misinterpreting the Common Core, surely the CC authors would have noticed and spoken up by now.


Auntie Ann said...

Instead of "precision teaching", this: multiple types of equations all at once as an introduction to systems of equations? This sounds like a pedagogical nightmare for all concerned, is more like imprecision teaching.

Teach by confusion, chaos, and distraction.

Barry Garelick said...

I've had Bill McCallum, lead writer of the CC math standards, twice state publicly that the Common Core standards do NOT prescribe or encourage any particular type of pedagogy. Once in a comment on my Atlantic article, and once in response to a comment I made on an interview with Jason Zimba in Rick Hess' blog that Ed Week publishes.

The CC party line is that the standards are pedagogically neutral. That people choose to interpret them to require a constructivist implementation is, in the view of the CC authors, the teachers' choice not theirs. The teachers, schools, school districts, and PD vendors, if pressed to explain their choice, will say "The CC standards require a deep conceptual understanding as well as procedural fluency and the traditional method of math teaching doesn't do a good job of that." The CC authors are mum on that, and will defer in the usual manner: "It's the teacher's choice, not ours".

But in effect, the choice HAS been made subtly. The standards reflect a pedagogical bias, by emphasizing "understanding" and "explaining". The standards feed into the momentum of reform math ideology over the last 20+ years that holds that the traditional methods simply are not effective at getting to "deep understanding". The reform math thoughtworld of what is "deep understanding", ironically, comes down to a rote approach to concepts via pictures and certain buzzwords designed to win points on the open-ended and "authentic" assessments that are being developed for this brave new deeply conceptual world.

Deirdre Mundy said...

I have a homeschooled fourth grader (she finished her third grade work last month, so poof! She graduated!) Her math has ALWAYS been mostly book work (Saxon). What I've seen is that once she masters a skill (multiplication, or finding the missing addend, or simple fractions and reducing) in the book, she THEN starts using it in the real world. Then, in turn, the book work becomes more automatic and easier.

But the real world applications that make sense for her and WORK are the ones she sees by herself.

Kids have the real world all day. Lets do some math in math class!

Lynn Guelzow said...

I've seen the Constructivist CC in action as my daughter's 8th grade algebra teacher has implemented the standards this year. We are in the midst of substitution now. From my child's view of the world - most of the kids are completely lost.

There was almost no instruction at all - but the kids were given a "real world" problem to solve involving 5 variables and told to figure it out on their own that night. Because we'd been doing algebra on our own using Dolciani - we were able to easily solve the 5 variable problem

The other kids weren't so lucky and the entire next class period was taken up with going over the problem. So now, we've lost 2 entire class periods to solve one complicated (but mathematically easy) problem with lots and lots of confusion.

Using Dolciani - that one problem would have taken up 15 minutes at most with an entire class.

So eventually, they will "learn" substitution, but the method used by the teacher is extremely time consuming (so much discussion and flailing around in groups takes a lot of time), and the teacher will move on to the other methods of solving a system without ever having given them a problem involving a fraction, a negative number, or anything with numbers bigger than 10.

In case you are wondering, all of the other kids solved the 5 variable problem using guess and check - the numbers were that easy. So there's your conceptual understanding, I suppose.

Barry Garelick said...

Here's yet another piece
that reads constructivist goals in the Common Core Math Standards.