Patient vs. Impatient Problem Solving

According to Dan Meyer, the problem with a steady diet of TV sitcoms is students learn to expect easy problems resolved in twenty-two minutes “with a laugh track.” We have now raised several generations of “impatient” problem solvers, and typical math textbooks pander to the syndrome instead of challenging it.

Mr. Meyer has a prescription for what ails our math teaching.



According to Mr. Meyer, there are two kinds of mathematics: computation, or “the step you forgot” and math reasoning. Within computation, there are a lot of tricks and gimmicks, like counting decimals places. The tricks work because of the underlying math reasoning. We teach the tricks, the non-math, and call it math. Good grades for non-math amount to “congratulating students for following the smooth path and stepping over the cracks.” No wonder our students display symptoms of impatient problem solving syndrome:

Lack of Initiative,
Lack of Perseverance,
Lack of Retention,
Aversion to Word Problems, and
Eagerness for Formulas.

The older your students the more likely you can be teach math reasoning well and still encounter not only the symptoms, but also resistance to the cure. Your students have been so conditioned by previous experience, that like chemical tolerance, they do not believe they can function mathematically any other way. It might be a good idea to show this video the first day of class to shock their systems into even entertaining the idea that math could be different.

His description of his presentation of the water tank problem is very like the way Japanese elementary teachers have been teaching math for decades (that I know about). They can easily spend a whole period on a single problem, but they actually save time, because they are not wasting it practicing forgettable procedure on twenty problems. They invest the time it requires to think about math, for as Mr. Meyer says, “Math is the vocabulary for your own intuition.”

Mr. Meyers suggests a five-part prescription:

Use Multimedia,
Encourage Student Intuition,
Ask the Shortest Possible Question,
Let Students Build the Problem, and
Be Less Helpful.

Teachers ignore many features of a problem as irrelevant without discussion as if we expect students to figure it out on their own. Many do, some do not. Asking what matters, says Mr. Meyer, id probably the most underrepresented question in math curriculum.

After, and only after, students have acquired the math reasoning should we give them shortcuts, tricks and mnemonics.
This video is an excellent example of a math teacher receiving accolades for teaching non-math.




And finally, just for fun.

Algebra in 2nd Grade?

Back in February, a teacher in Montana made EdWeek headlines because she was teaching algebra to second graders and had been doing so for five years. Why all the oohs and aahs?

Gregorio C. Sablan, CNMI Congressional Delegate, got it right. “Pre-algebra” starts early, or should.

At Broadwater Elementary School in Helena, Montana, algebra starts in second grade, where students learn the basics behind mathematical theory and reasoning to prepare them for high-level math classes in middle and high school.

Elementary math is supposed to prepare students for high-level math classes is middle and high school. Students should not need a dedicated pre-algebra class. When I was a kid, pre-algebra did not exist. Now it is part of every school's math course line-up. The author of a pre-algebra text wants students to build math reasoning skills.

But that doesn't always happen. Many teachers treat pre-algebra as a last chance for students to get those elementary math procedures down pat. Problem is, a student can be A+ in procedures and still not get algebra. In fact, students who are competent with procedure often believe they are good at math. It's not their fault. Our education system has been telling them for years that grades equal understanding. So if they get a good grade in math, naturally they conclude they are good at math.

Math has been misnamed. What passes for math in schools is often non-math. “Carry the one” is not a mathematical explanation. Students get good grades in non-math believing it's math. No wonder algebra is such a shock. Math reasoning skills actually matter in algebra.

Still a student with a good memory can get by, at least until they meet a new math monster, calculus. However, since middle and high school math also fail to teach math reasoning, students get to take pre-calculus, another relatively recent addition to course offerings. Without a major change of emphasis, pre-calculus prepares students no better for calculus than pre-algebra prepared them for algebra.

By now pre-calculus students have so internalized non-math that they complain to the instructor, “Just tell us how to get the answer. We don't want to know why.”

No Surprise There

“Algebra-for-All Policy Found to Raise Rates Of Failure in Chicago”

Math educators, with good reasons, have long recommended that students be required to study algebra. Many districts mandate algebra in the ninth grade. California, one-upping everyone else, currently requires eighth graders to take algebra. Japanese children begin studying algebra in the fifth grade. So How's it working out?

Findings from a study involving 160,000 Chicago high school students offer a cautionary tale of what can happen, in practice, when school systems require students to take algebra at a particular grade level.

160,000 is a lot of students, and normally the bigger the sample from the population, the more reliable the conclusions. Researchers studied eleven “waves” of students entering ninth grade from 1994 to 2005.

(Researchers) compared changes within schools from cohort to cohort during a period before the policy took effect with a period several years afterward. They also compared schools that underwent the changes with those that already had an “algebra for all” policy in place.

What did the researchers find?

The policy change may have yielded unintended effects, according to researchers from the Consortium on Chicago School Research, based at the University of Chicago. While algebra enrollment increased across the district, the percentages of students failing math in 9th grade also rose after the new policy took effect.

By the same token, the researchers say, the change did not seem to lead to any significant test-score gains for students in math or in sizeable increases in the percentages of students who went on to take higher-level math courses later on in high school.

Not much upside. More students failed, test scores were flat, and the percentage of students motivated to take advanced math course did not rise, but, gee, “algebra enrollment increased.” The district says more students will fail when required to take harder courses without supports in place. Yet the district made attempts include supports over the last seven or eight years.

Steps include developing curricular materials introducing students to algebra concepts in grades K-8, requiring struggling 9th graders to take double periods of algebra, and providing more professional development in math to middle and high school teachers..

One of the researchers thinks that test scores did not improve because teachers may have “watered down” the content because “math classes included children with a wider range of ability levels following the change.”

But Japanese elementary schools are not tracked. All children study exactly the same material with such predictability that some observers have quipped that every child in Japan is on the same page of the textbook on any given day. I have successfully taught Algebra 1 to high school special education students, or to give due credit, special education students have successfully learned Algebra 1 under my guidance.

The problem is with issuing mandates without a coherent, integrated societal commitment to the foundations of education, mathematics in particular. I have seen Montessori preschool students exploring algebra with manipulatives. I have often said that lots of profound math can be learned without any resort to pencil and paper. Children do not necessarily need numerals to understand number.

There is one other thing. Japanese children from kindergarten age regularly take abacus lessons the way American children take piano or ballet. Generating a sum with the abacus is different than generating a sum using the written algorithm. The activity of thinking about number and computation in more than one way leads to greater mathematical flexibility. Japanese students can therefore more readily absorb and manifest algebraic thinking. That's my hypothesis anyway and maybe the Gates Foundation or somebody else will provide me a grant to test it.

(P.S. You may enjoy reading the comments at the end of the link.)

Other links from the cited story:
“Calif. Judge Blocks Requirement For 8th Graders to Take Algebra,” January 7, 2009.
“Experts Question Calif.’s Algebra Edict,” July 30, 2008.
“Chat: California's Algebra 1 Mandate for 8th Graders,” July 30, 2008.

The Place of Place Value

Perhaps one of the most important foundational concepts in mathematics is place value. As the Massachusetts Department of Education rightly observes, “The subtly powerful invention known as place value enables all (my emphasis) of modern mathematics, science, and engineering. A thorough understanding removes the mystery from computational algorithms, decimals, estimation, scientific notation, and—later—polynomials” (Massachusetts Department of Education (2007). In fact, it is when students first meet polynomials in algebra, that the lack of a proper grounding in place value becomes painfully apparent. Most likely a significant number of the difficulties that students experience with math may be traced to place value.

I reviewed the state standards of various states with regard to place value. I looked for an explicit reference to “regrouping,” the current term for what we used to call “borrowing” and “carrying.” Personally, I prefer to call it “filling the cup” and “dumping the cup.” In the same vein, I like to call the “ones” place the “loose ones.” My survey of state standards resulted in a mixed bag. Some states require students to do little more than name the place value of a particular digit. Other states expect students to use various means to model place value. Alaska asks students to not only perform the operations of addition and subtraction, but to explain those operations.

State standards have their utility, but apparently whatever the specific state standard, students are able to follow the regrouping recipe without having any real understanding of why the recipe works. In fact, adults of all ages add and subtract by mindlessly following the recipe. Most adults, and of course, all children could do with a solid grounding in place value.

I have a number of activities I use to make place value explicit. Tomorrow I will tell you about an activity I like to call “The Chocolate Factory.”

Are You Good at Non-Math?

One of the most persistent issues in math education has been the reliance on non-mathematical explanations of mathematical principles. For example, we tell students that when multiplying positive and negative numbers “two negatives make a positive.” Such an explanation clarifies nothing about how the numbers behave or why an ostensibly English grammar rule should apply to math.

What is worse, we tell students who successfully master such non-math explanations that they understand math, or that they are good at math, when really what they are good at is non-math. Young children have no way to distinguish non-math from math. They believe, because we have told them, that they are learning math, when in fact they are learning non-math. If it does not catch up to them earlier, it often catches up to them in algebra class where historically “A” students may find themselves inexplicably failing to understand the subject material.

Children rely on adult teachers to initiate them into the joys and delights of math, but often teachers make math a difficult subject, usually because they themselves understand non-math rather than math. After all, if numbers are running around, it must be math, right? Even sadder are the number of elementary teachers who lack an interest in acquiring what math education researcher Liping Ma called “the profound understand of fundamental mathematics” even while believing that they “know” math.

Many colleges of education and community colleges have sought to address the serious weaknesses in the mathematical understanding of elementary teachers by either requiring, or at least offering, coursework in mathematics for elementary teachers. I am quite sure a survey of professors teaching such required courses would report remarkable levels of student resentment at being forced to take a class in something they think they already know, to “jump hoops” as they say . Some of these students may wake up and get motivated to learn the math concepts. Some seethe inwardly as they pass the class. However, most students will pass the class and eventually be certified to teach regardless of their attitude toward or understanding of the vital core subject of mathematics.

Only later, once they are in the classroom, will they be likely to regret the squandered opportunity to finally get math. They may grow to appreciate the professor who tried to give them the gift of mathematical understanding, a gift they resisted at the time.