# Shut the Hell Up, Dept.

In Education Week “Teacher”, a recent article reproduces the responses of four high school math teachers to the question: “What are some best practices for teaching high school mathematics?”

I got as far as the first teacher who starts off by telling us:

“Focus on the processes and connections between different processes rather than just finding the answer.”

Those who believe the big lie told at school board meetings and repeated ad nauseum by math reformers (in publications like Education Week and others of similar ilk) that “traditionally taught math failed thousands of students” likely nodded their heads at this with a big “Yup, that’s what’s wrong with how math is taught.” What goes unnoticed and unacknowledged is that a good textbook does everything claimed to be his, but in a more efficient way and one that focuses on individual success, not feel-good student reactions in class.

To be more specific, in elaborating on this idea about processes and connections, he starts off with: “Teach to big ideas rather than 180 disconnected lessons.”

I don’t know about you, but I haven’t heard of or witnessed any high school teacher who teaches 180 disconnected lessons. And though math books from previous eras like the 50’s and 60’s are held in disdain by today’s math reformers and others who claim the high ground in “best practices in math teaching”, the lessons contained therein were well sequenced, well connected and well scaffolded. Such textbooks taught to big ideas; in fact, my old algebra book was called “Algebra: Big Ideas and Basic Skills”.

But then he gets to the heart of what’s really bugging him:

The answer is important, but the processes that lead to answers are far more important to learn. The connections between processes and between different representations of that process are where the core mathematical ideas you want your children to learn reside. This means that while teaching processes, you should use these processes to teach students mathematical principles they can apply to solve problems, not how to solve problem types.

The connections he thinks are not happening have always been a part of traditional math. We learned big ideas from unit material. Given a governing equation, like distance equals rate times time, we solved straightforward problems with worked examples to get us started.  Then, in working the homework problems, there were many problem variations. We learned how to rearrange the equation and solve for the unknown, to draw diagrams, label unknowns, and start writing down known equations to find an equal number of equations as unknowns. There was never one type of problem for any new unit of material.

There was no special process or set of ideas that guaranteed transference other than practice. From the unit material homework variations, we saw patterns of solutions that could be applied to other types of problems. The variations took us well past the initial worked example and required us to stretch our prior knowledge and apply it to new situations within the context of the original problem type. What if you had five people working on a job at different rates? What if they started at different times? If they could only work for 5 hours, how much might be left over for you to do? (For more on the topic of non-routine problems within the context of problem types, see this article. )

The teacher next exhorts us to:

Use instructional routines to support all of your students in having access to the mathematics. Instructional routines shift the cognitive load for students as they focus less on what their role is and what they are supposed to next, since these tasks are delegated to long-term memory, and are therefore more able to focus on learning mathematics.

He provides a link to his blog to provide an example of instructional routines.  There, we see the typical “growing tile” type of problem presented as an instructional routine.  He states that

[t]he high level goals of Contemplate then Calculate are to support students in surfacing and naming mathematical structure, more broadly in pausing to think about the mathematics present in a task before attempting a solution strategy, and even more broadly in learning from other students’ different strategies for solving the same problem. A critical aspect of this and other instructional routines is that they embody a routine in which one makes teaching decisions, rather than scripting out all of the work a teacher is to do with her students.

I find that “growing tile” type problems promote inductive reasoning as if that is all that is needed in math and deductive reasoning takes a back seat if given any seat at all. That aside, he evidently believes that the effect of such problems is to get students to pause before leaping into solving something. In turn, this “thinking skill” is viewed as an inherent part of problem solving–i.e., a “habit of mind” like those of the eight Standards for Mathematical Practice contained in the Common Core Math Standards–that will allow students to solve problems independent of type and without the practice one needs to achieve any kind of flexible thinking.
One only needs to “learn how to learn”; all the rest can be Googled, I guess.
(Special thanks to SteveH for his thoughts on this topic which are incorporated into the above.)

# Articles I Unfortunately Finished Reading, Dept.

The opening to this article about students in Wake County, North Carolina, was almost enough to get me to stop reading, but apparently I felt the need to punish myself, so I pressed on.

Wake County students taking high-school-level math courses are now finding that getting the right answer isn’t always as important as the process they use to solve the problem.

Yes, as a math teacher, I often give partial credit on test questions where a student has made a numerical error but has set the problem up correctly.  But I do like right answers, and from what I’ve seen, the definition of “process” varies from teacher to teacher.  Drawing pictures amounting to a time consuming and inefficient process is sometimes considered to be just fine. In a high school level math course, in my opinion it is not just fine.

Wake began rolling out this school year new classroom materials from the Mathematics Vision Project, a nonprofit education group that provides resources designed to align with the Common Core learning standards. Wake school leaders and teachers say the new materials have led to major changes in how math classes are run to shift from lecturing to having students work in groups to learn concepts through problem solving.

“We have to teach failure as a part of learning,” said Brian Kingsley, Wake’s assistant superintendent for academics. “If we’re going to be college and career ready, the answer isn’t always the most important thing.

Well, to be college and career ready, let me just say that the right answer is a lot more important than what Mr. Kingsley and others of his ilk seem to think.  Learning how to work with algebraic expressions may seem like senseless manipulations of meaningless symbols to those subscribed to the group-think this article extols. But to the rest of us, it’s rather important to learn how to factor, and work with algebraic fractions.  Students will undoubtedly make mistakes, so if “failure” is a goal of these people, the traditional method they hold in such disdain provides opportunities to make mistakes.  Making mistakes is not the province of the “new way of teaching math”, I can assure you.

Math I students got workbooks this school year, but a lot of what they experienced was unexpected. Instead of students copying what their teachers write on the board, they’re working in groups to solve problems such as how long it would take to drain water from a pool. Teachers are guiding the students instead of lecturing them on subjects like quadratic functions.

“Instead of me saying, ‘Here is a linear equation: It’s y=mx+b,’ it’s much more, ‘Let’s get to this equation,’ ” Herndon said. “It’s very much a different approach. Rather than me giving them all the answers, they’re having to work towards them themselves.”

The opinion of those who believe traditional math has never worked is that teachers give the students all the answers, and students do not do any work for themselves.  There is general belief among the prevailing educational forces that initial worked examples with plenty of practice is nothing more than “rote memorization” of procedures and that the process produces “math zombies” who cannot think for themselves.

I have observed the teaching methods described in this article.  A problem about how long to drain water from a pool is rather straightforward, but I’ve seen problems like this take the better part of a class period when with proper direct instruction and worked example, it should take about 10 to 15 minutes.  I’ve seen a class in which students worked on slope, graphing and the connection to linear equations for five to six weeks. There is no reason for taking that long and I wonder what evidence there is that such students have “deeper understanding” (a term that remains undefined, but generally means a “rote understanding” of explanations that students learn to give the teacher in order to satisfy them) than students taught in the traditional manner.

Do the people in the article have any evidence that their methods are producing better results than the methods they hold not to work?  I mean other than their opinions and seeing what they want to see.

# Silver Lining, Dept.

In this recent article, we learn about a math ed professor from Southern Illinois University who is going to Tapei, Taiwan to bring them American methods of teaching math.

“I’m looking forward to sharing my perspective on mathematics knowledge and education with an international audience,” Lin said. “I will be telling them about American teaching techniques and learning more about their methods. I want to know more about their recent teacher training and compare it to ours.”

Lin noted that in recent years, American mathematics education has shifted toward a focus on teaching for understanding, assuring that students comprehend what they are doing rather than just learning to apply formulas and procedures. The teaching approach used in Taiwan is a more traditional format that relies heavily on written computation.

To readers who are unaware of the battles over math education in this country, this sounds just dandy.  There is the usual misconception packaged as an implicit assumption, that in the US we have never taught for understanding, and that students are proceeding by rote (i.e., “math zombies” as some clever bandwagon math teachers have dubbed it).

To readers who are more aware of what has been happening, the obsession over “understanding” has been going on for more than a century.  But during the last 3 decades, such obsession has grown legs, due in large part to the National Council of Teachers of Mathematics’ (NCTM’s) standards written in 1989 and rewritten in 2000. Students in lower grades must now demonstrate understanding of the mathematical concepts in simple arithmetic operations and are taught the standard algorithm later.

I’m seeing the results of such “deep learning”, “deep understanding”, “deep dives into math” and other 21st Century nonsensical edu-jargon in the 7th grade math class that I teach. Students are reluctant to do double digit multiplication, or to divide. Many of those who do double digit multiplication use an inefficient “partial products” method, and are relatively unaware of the standard algorithm method for doing so.  Math facts for many are not mastered, and I get many requests to use a calculator.

Through a series of lectures, workshops, seminars and other activities, Lin will work with teachers-in-training, teachers and administrators regarding theories and practices used successfully in mathematics education instruction in the United States. He will also assist in developing appropriate tools for testing mathematical skills and knowledge and help analyze the results. In addition, Lin will assist colleagues in preparing students for Taiwan’s new compulsory national tests for graduating teachers and will make recommendations for course revisions or other changes to assist students.

Last time I looked, Asian countries were doing just fine on international math tests despite the highly disregarded and despised “traditional methods”.  I guess the silver lining is that if American methods are adopted hook, line, and sinker in Asia, we’ll come out smelling like a rose!