Friday, July 15, 2011

Problem Solving

Reform math programs launch into "problem-solving" backward by de-emphasizing the grade-by-grade mastery of basic arithmetic knowledge, both facts, and procedures, which are the very essentials needed for problem-solving. Polya's problem-solving strategies work only if the student has sufficient prerequisite knowledge needed to solve a specific problem. Thus, problem-solving is always deeply rooted in background knowledge in long-term memory. The primacy of background knowledge cannot be over-emphasized. Elementary school teachers should focus on making sure students master the fundamentals of arithmetic, both factual and procedural knowledge, starting in grade 1. A curriculum that is focused on problem-solving strategies and light on content knowledge does not cut it.  ThinkAlgebra [Draft I]
Let’s start here . . .
“Mathematics has the dubious honor of being the least popular subject in the curriculum . . . Future teachers pass through the elementary schools learning to detest mathematics . . . They return to the elementary school to teach a new generation to detest it.” You would think that this was written in 2011, but it wasn't. G. Polya was so concerned about math education that he wrote it in the 2nd edition preface (1956) of his famous book, How to Solve It. Actually, Polya quoted it from a study reported in Time magazine. Not much has changed in the past half-century. I think some teachers are doing a great job teaching math. We just do not have enough of them.    

This cycle has been entrenched in education for at least five decades because schools of education are not selective or academically demanding. I do not blame teachers; I blame those in charge of selecting, training, and certifying teachers. If we want better teachers in elementary and middle school math, starting in 1st grade, then we need to educate and train them better and weed out teachers who dislike mathematics, who are mathphobic, or who demote the importance of mathematical [content] knowledge. Poyla believes a teacher's knowledge of mathematics and attitude toward mathematics rub off on students. He writes, "Yet it should not be forgotten that a teacher of mathematics should know some mathematics and that a teacher wishing to impart the right attitude of mind toward [math] problems to his students should have acquired that attitude himself." The book was first published in the U.S. in 1945.

Polya poses a problem.
The length of the perimeter of a right triangle is 60 inches and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides?

A student cannot solve this problem without substantial knowledge of high school mathematics (algebra and geometry). But, isn't prerequisite knowledge necessary for any math problem, at any level, even for routine problems? Knowledge first! In mathematics, students should start with basic arithmetic and routine problems to build a storehouse of knowledge and experience in long-term memory before moving to more complex problems that take more insight. The idea that students can do problem-solving without fundamentals in place is illogical and backwards, yet this is what many teachers think. Elementary students should focus on mastering basic arithmetic and routine word problems, grade by grade. This requires solid practice.

I pose a chemistry problem.
Calculate the grams of hydrogen required to produce 82.000 grams of ammonia from nitrogen and hydrogen gasses.

Would you attempt to solve this routine chemistry problem without knowing the fundamentals of high school chemistry? Of course not! Solving problems requires domain-specific knowledge. Moreover, learning to solve routine problems, whether they be in chemistry or elementary school arithmetic, presupposes both knowledge and practice.   

I pose a Latin problem.
Ego vos hortor ut amicitiam ombibus rebus humanis anteponatis. Sentio equidem, excepta sapientia, nihil melius homini a deis immortablibus datum esse. 

Would you attempt to translate Latin without knowing the fundamentals of Latin? Of course not. Translating Latin requires domain-specific knowledge. 

Knowing builds the foundation for higher-level thinking.
You cannot apply something you do not know well.

Knowing builds the foundation for higher-level thinking . . .
The range of cognitive skills (right), starting with a strong base of Knowing, is similar to Bloom's taxonomy. Applying requires Knowing, and Reasoning implies both Knowing and Applying. Teachers should start at the bottom and focus on Knowing (both factual and procedural knowledge in arithmetic and algebra). This builds the foundation for higher thinking, such as Applying and problem-solving. In TIMSS, Applying is solving routine problems. (This is problem-solving.) Furthermore, knowing something takes substantial practice. You cannot apply something you do not know well [in long-term memory]. 
We tend to believe what we think, but our assumptions are often wrong. 
Sweller, Clark, and Kirschner [2] write that the results of research in problem-solving in mathematics are “both counterintuitive and contrary to many widely held views. For example, many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge.” The assumption is wrong and unproven. It is a gross misinterpretation of Polya by many math educators, special interest groups (e.g., P21), and others. 
Practice Fundamentals
Knowledge Is Key
According to Polya, when a student attempts to solve an unfamiliar problem, the student should think of a related problem and then, by analogy, try to solve the original problem. He writes, “We may consider ourselves lucky when, trying to solve a problem, we succeed in discovering a simpler analogous problem.” But, thinking of a simpler analogous mathematical problem, solving it, and applying it (“extrapolating” it to solve the original problem) require specific content knowledge. Moreover, a simpler analogous problem can have both similarities and dissimilarities. As one reads Polya’s book, it becomes clear that students need extensive mathematical knowledge and “determination” to do problem-solving. In short, problem-solving in mathematics requires sufficient mathematical knowledge in long-term memory, practice, and “determination.” There are no shortcuts. 

Sweller, Clark, and Kirschner point out, “There is no body of research based on randomized, controlled experiment indicating that such teaching [generalized problem-solving strategies] leads to better problem-solving.”  They observe, “Recent reform curricula both ignore the absence of supporting data and completely misunderstand the role of problem-solving in cognition.” 
Note. Reform math champions and elevates the idea of “general problem-solving skills,” group work, spiraling of content, and calculator use. At the same time, these “problem-based” programs minimize or demote essential content knowledge. This is no accident; it is by design. In my view, this upside-down relationship is a crucial flaw in reform math programs. Students must practice content to learn it and to apply it. 

Ze'ev Wurmanin a recent blog (7-16-11), writes, "The overwhelming majority of children can reasonably easily learn what we teach in our K-12 schools, given competent teachers and effective teaching methods." But this is not what happens in K-12. Most of our students remain mediocre at best (TIMSS) and only about 30% are proficient in math (NAEP). Furthermore, tens of thousands of incoming students flood remedial math courses at community colleges. Wurman says, "[The] cause must be in how we teach our students in school and outside it." We do not come close to teaching content that nearly all Singapore students learn in grades 1-9. By 9th grade, virtually all Singapore students (99.9%) have covered all of Algebra 1 and Geometry, according to Wurman. (Note. Ze'ev Wurman was one of the writers of the California math standards adopted in 1997. While most states continue to use the NCTM reform math framework, California dropped it in 1997 because its test scores plummeted. The 1997 California math standards were among the best in the United States. Furthermore, they were benchmarked to top-performing nations. Kids in top-performing nations do algebra in middle school. The California 1997 standards put Algebra 1 in 8th grade and Geometry in 9th grade. However, recently, I am sad to say, California replaced its excellent standards with mediocre Common Core math standards. Wurman has been an outspoken critic of Common Core math standards.)
Without explicit guidance
The belief in reform math is that, if students can learn problem-solving strategies and “discover” solutions to problems “without explicit guidance,” knowledge, or instruction, then learning math content (e.g., basic arithmetic) is not that urgent or important. This methodology is not “the most effective or efficient way to learn mathematics” and partly explains why most students are not proficient in mathematics (NAEP). What happens in reform curricula is that students do not learn enough content to support cognitive problem-solving. There are many math programs that emphasize a problem-based approach. This sells textbooks but does not produce “excelling” math students. In fact, reform math programs have produced a flood of remedial math students. For example, in 2009, nearly 90% of incoming students at Pima Community College (Tucson) were required to take remedial mathematics. 
Inverse [Upside Down] Relationship
An emphasis on problem-based curricula often displaces mastery of key math content. This inverse or upside-down relationship is found in most reform math programs. Not only is this inverse idea a basic philosophy in NCTM math standards, but it also carries over to Common Core by delaying fluency. [3] Math educators call it spiraling

In reform math, grade-level mastery of arithmetic is not the goal. If a student does not learn addition in 1st grade, it is repeated in 2nd, 3rd, and 4th grade (it spirals). Indeed, in the new Common Core math standards (left), students are not expected to be fluent in addition and subtraction until 4th grade. This is a nonsense approach. 
To be effective, a curriculum that is strong is problem-solving must also be strong in computational skills and fundamentals. Knowing is the foundation for higher-level thinking. So-called “generalized problem-solving skills” are not a substitute for mastery of fundamental content.  
In arithmetic and algebra, problem-solving is deeply rooted in the background or content knowledge (both factual and procedural) and not in general [problem-solving] strategies as embraced and advocated by many math educators, textbook writers, ed school professors, and special interest groups.
Problem solving in math is domain-specific, but math educators act as if it is not. 
Skill in problem-solving in mathematics requires substantial domain-specific schema [background knowledge], not “domain-general.” Students cannot apply the mathematics they do not know well. Students should be taught to solve routine problems to build an arsenal of background knowledge for more complex problems.

Worked Examples
Sweller, Clark, and Kirschner write, “But domain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies.” In other words, students can learn problem-solving skills by studying worked examples that exemplify them. This requires diligent practice. In mathematics, problem-solving skills are deeply rooted in content [knowledge]. Moreover, these skills take time to develop.  Sweller, Clark, and Kirschner write, “There are no separate, general problem-solving strategies that can be learned.” 

21st-century skills (P21), the latest foolish fad
The idea in 21-century skills is that these skills can be learned outside of domain-specific content knowledge. I think not. 
[1] G. Polya's How To Solve It was first published in the United States in 1945. Polya was concerned about math education, the training of teachers, and how teachers influence the attitudes of students.  
[2] Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics by Sweller, Clark, and Kirschner, in Doceamus, November 2010. 
[3] The Common Core Math Standards (2010) continue the NCTM spiraling approach. Delaying fluency makes no sense. 

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Updates: 7/15/11, 7/16/11, 7-17-11, 8-4-11