K-12 Science Framework

This is in draft form and redundant. I wrote it in the summer (2011). Parts of this draft are found in Science in Elementary School and on my science page. 

The new K-12 Science [conceptual] framework from the National Research Council (July 2011) is a disappointment. The framework committee merged science with engineering, skimped over math needed to do science, stressed science practices over content, and required little content knowledge in elementary and middle school, which is the same problem we have had for decades. For instance, the word “atom” is not used until the 6-8 grade band. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) is reserved for grades 9-12. The sequence is off-target. But, in my view, the biggest blunder, in addition to a lack of math and mixing science and engineering, is combining chemistry and physics (total 24 pages). Life science content takes up 20 pages. Chemistry and physics are woefully underrepresented.

If we want students to understand the world, then we should teach them substantially more physics and mathematics early on. Furthermore, we should establish math and science standards that, at the least, match the benchmarks from the nations that excel in these academic disciplines. The new K-12 Science framework does not do this. The Science Framework is the latest version (vision) of science education, but it is off-target because it requires very little knowledge of math needed to do science and very little science content knowledge. The committee’s frame of mind in composing the framework is troublesome.

The Committee on a Conceptual Framework for New Science Education Standards was charged with developing a framework that articulates a broad set of expectations for students in science. The overarching goal of our framework for K-12 science education is to ensure that by the end of 12th grade, all students have some appreciation of the beauty and wonder of science; possess sufficient knowledge of science and engineering to engage in public discussions on related issues; are careful consumers of scientific and technological information related to their everyday lives; are able to continue to learn about science outside school; and have the skills to enter careers of their choice, including (but not limited to) careers in science, engineering, and technology. (p. 14)

It sounds great! The caveat is that the statement is packed with unclear ideas that cannot be measured. Indeed, unclear generalizations are commonplace in education stuff and, in this case, the opposite of what science is. What is sufficient knowledge or appreciation? Sure, I bet the average citizen is going to read and study science and scientific studies after they graduate from school. To imply this is nonsense. The Framework’s expectations are speculation. How can you have expectations that are nonspecific and not measurable? I guess the mostly “non-scientific” committee thought they could. For example, the word “atom” is not used until the 6-8 grade band, and the atomic structure (protons, electrons, and neutrons) is not introduced until high school. (Surely, you’re joking!) 

Table by LT, ThinkAlgebra. It is based on Chemistry/Physics.

We infer that quarks exist even though we have never seen an individual quark. And, we infer that electrons exist even though we have never seen an electron. These inferences are based on solid measurements. 

Larry Cuban writes that the Framework is a “science for living.”  A blogger’s reply rephrases Cuban by saying that the Framework represents “issues-orientated, inquiry-based science.” Ze’ve Wurman concludes, “The document simply teaches students science appreciation, rather than science.”  

I call it the document to nowhere. The science framework lacks sufficient chemistry and physics content and depth and states unclear goals; e.g., "all students have some appreciation of the beauty and wonder of science."

A few years ago I wrote that mathematics must be brought back into elementary and middle school science. Today’s science programs or textbooks seem to skimp on the math needed to do the science. In the Sputnik era, the United States produced superior, coherent science programs. For example, Science--A Process Approach (1967) stressed the process in the context of the content, along with the mathematics used to do science. For instance, in Part B (First Grade) four of the six science processes were math or math-related [Using Numbers (arithmetic), Measuring, Communicating (graphing), and Using Space/Time Relationships]. In short, the math needed to do science was a major part of the SAPA science program, starting in the 1st grade. Moreover, the math taught in the program was very specific and ahead of grade level. 

College professor James S. Walker (Physics) writes, “The goal of physics is to gain a deeper understanding of the world in which we live.”  Indeed, the goal of science is to gain a deeper understanding of the world. Richard Feynman says that students should study physics because it plays a basic role in all phenomena. But, the Framework does not specifically state this view as its main premise. In fact, the Framework stresses “practices” of scientists; however, learning the processes of scientists does not imply that the student is learning content. Critical thinking requires considerable content knowledge. Learning what a scientist does is not the same as learning content. Kids are novices, not experts. They need to learn content, lots of it. In math class, I do not expect students to learn what mathematician do. I expect them to learn how mathematics works--how one idea links to or builds on another idea. I want students to learn essential content and skills so they can work math problems from different disciplines, including physics.   

Richard Feynman writes, “Physics is the most fundamental and all-inclusive of the sciences, and has had a profound effect on all scientific development.” To Feynman, the scientific method is “observation, reason, and experiment.” This is what we should teach kids. Feynman refers to rules of the game, which scientists guess and check by experiment. Feynman stresses, “The sole test of the validity of any idea is an experiment.” Untestable ideas do not make sense in science. Ian Stewart, a mathematician, writes, “Mathematics has played a central role in the physical sciences for hundreds of years.” The framework does not emphasize the intrinsic link between science and mathematics.  

What about biology? There are plenty of numerical patterns in biology (e.g., Fibonacci numbers, golden number, etc.), but Ian Stewart (The Mathematics of Life), points out that “Mathematics is being used not just to help biologists manage their data [e.g., enormous DNA genome databases] or improve their instruments, but on a deeper level: to provide significant insights into the science itself, to help explain how life work. Over the past ten years, there has been a massive growth in biomathematics--mathematical biology.”

Physics is the fundamental science, but it is mistreated in the Framework. First, it is lumped together with chemistry. If life science is treated as a separate topic, then chemistry and physics should be separated and expanded. Life science content takes up about 20 pages, while physics and chemistry combined take up 24 pages. The choices made by the Framework’s committee show a bias--life science content is more important than chemistry or physics. The committee writers say that chemistry and physics have too much in common to be treated separately, but I can make the case that chemistry and life science have much in common, too.   

The framework committee tries to justify lumping science with engineering rather than with mathematics. This is the new vision: the committee skimps over math. 

The laws of thermodynamics are nowhere to be found. The word “atom” is not used until 6-8. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) is reserved for grades 9-12. The sequence is off-target. 

The committee justifies its decisions on content by saying that the document is broad-based and for all students. It uses very general statements. It is merely a structure to composed standards. In my view, the framework falls woefully short because it leaves out important ideas in both chemistry and physics--from thermodynamics to relativity. 

The new science framework from the National Research Council (Framework for K-12 Science Education) has never been tested. Ironically, experimental testability is a fundamental principle in science. Yet, educators are asked to accept the framework “on authority,” something Galileo Galilei argued against. The framework is a guide for states and schools to write new science standards. It is not Common Core, which is [or will be] working on its own set of science standards, presumably using this framework. The new science framework, oddly enough, was written by the Division of Behavioral and Social Sciences and Education and its committee, which is made up of mostly of educators, not real scientists. How good is the framework? Don’t ask. (I think I hear the late Richard Feynman grumbling, “If it disagrees with experiment, then it is wrong.”) Accepting something on authority takes us back to the days before Galileo. 

The writers insist that the framework is for all students, broad-based, and not a grade-by-grade or course description. Its function is to develop a new set of science standards. And, it lumps science with engineering right for the start.

The science framework lacks sufficient chemistry and physics content and depth and states unclear goals; e.g., "all students have some appreciation of the beauty and wonder of science."

The framework writers “anticipate” that all students will be able to “to engage in public discussions on science-related issues, to be critical consumers of scientific information related to their everyday lives, and to continue to learn about science throughout their lives.” Lastly, the writers write, “We hope that a science education based on the Framework will motivate and inspire a greater number of people [to go into the science fields]” and its allied subjects, such as psychology, computer science, and economics.” Let me point out that these well-intended expectations are assumptions and assumptions are just that. There is absolutely no evidence that the Framework's new vision will produce better science standards or better science students. But, this is the committee’s “hope,” which means that there is no supportive evidence. Moreover, the Framework committee says that students should continue to take honors and AP courses in the sciences. But, this may not be possible because students have not learned enough content. 

The college-educated citizen, much less the average citizen, does not have the expertise needed to understand many of the scientific issues or studies that arise. This will not change. Often, I have trouble comprehending some of the articles in Scientific American or parts of M-theory (The Grand Design, Hawking, Mlodinow). 

What is troublesome is that the design teams (content experts in the sciences) were excluded from the committee’s final decisions. The document states (p. 17), “No members of the design teams participated in the discussions during which the committee reached consensus on the content of the final draft.” This alone makes the framework suspect. 

Ze’ve Wurman writes, “I noticed something odd. The Framework does not expect students to use any kind of analytical mathematics while studying science.” In short, kids do not use mathematics to solve science problems. No algebra, no trig, no calculus. Wurman notes, “There is nothing about actually being able to model a system by equations, or solve it using mathematical techniques.”  Wurman searched for words like algebra in the 280 pages of “lofty prose.” Nothing, well almost. 

Wurman did find a reference to one equation, which starts as a word equation (distance traveled = velocity multiplied by time elapsed) and is then symbolized as s = vt. I am not sure students understand what velocity means in science, because students are seldom taught vectors and do not learn how to “resolve” the components of a vector to solve physics problems. 

The Framework does not require students to use mathematics to model systems. Wurman points out, “Only statistics and computer applications (e.g., simulations, spreadsheets) seem to have a place in this strange document.” Wurman concludes, "The document simply teaches students science appreciation, rather than science.” 

End
Draft 1
Needs revision.
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Science in Elementary School

Kids are left behind in science.

What has happened to science in elementary school? 
With an emphasis on math and reading, science has been pushed to the side in many elementary school classrooms [1]. There is no time and little equipment. Often, K-8 teachers do not know enough science to teach it well. Furthermore, the math needed to do science is seldom taught or introduced. Science is highly mathematical, yet K-8 science textbooks and teachers seem to limit the math. None of this is new. Science has been neglected for decades. 

The thinking required in science, however, is different from the thinking done in school math. Mathematical statements are shown to be true or false by following a set of rules called number properties (axioms) of operations, equality, etc. [2]. One true statement forms the basis for another true statement and so on. This is the way math knowledge builds: one idea builds on another. In short, the rules (properties of numbers) in math do not change. The definitions in our number system do not change either. A fraction (rational number) will always be represented by the quotient of two integers (a/b, b ≠ 0). Equivalents like 3/4 = .75 = 75% will not change; 7 is always 6 + 1, etc. Equivalency and substitution are important ideas in mathematics. 

In science, however, there is no "true or false" like in school math. Instead, there are observations and inferences. There are no absolutes in science. The rules in science can change based on "partitioning by scale." In short, Newton's laws of motion still work, but not at the atomic (very tiny) scale.

There are facts, such as the number of protons in the hydrogen atom, etc. Kids must know facts (background knowledge). In addition, they also must know how to measure (make observations), how to draw valid inferences (conclusions) based on observations (data), how to minimize confirmation bias and errors in experiments, how to do the required math, how to communicate results with charts and graphs, and how to distinguish between correlation and cause-effect. Above all, students must learn to distinguish between observation and inference. But, this is not the organizing principle of science textbooks. Too often, TV programs, documentaries, news programs, textbooks, and other materials blend the two. Often, students interpret an inference as fact. 

Students should do science projects that have clearly defined independent and dependent variables and control.  Moreover, students should be taught what the late Richard Feynman calls intellectual honesty in science, something that is often lacking. In real science, we bend over backward to prove our conjectures wrong through experiments. We also present data that does not support our conjectures. We do not fudge data. And, we do not extrapolate beyond known data, i.e., make an inference based on an inference. An inference based on another inference is a misguided conclusion. Such extrapolation of data is more common than most people think. A common example would be a computer model of a complex system [stock market, weather, etc.] that attempts to forecast the future [based on the past] and makes unproven claims. This is not science; it is speculation


Feynman states, in a lecture, that physicists guess theory, then they test it. "If [we guess a theory that] disagrees with experiment, then it is wrong." Science is not based on, authority, opinion, consensus, or political agenda. It is based on an experiment. Furthermore, experiments must be repeatable and peer-reviewed. Lisa Randall, a particle physicist, writes, "People too often confuse evolving scientific knowledge with no knowledge at all and mistake a situation in which we are discovering new physical laws with a total absence of reliable rules." 


Dr. Randall clarifies, "Science evolves as old ideas get incorporated into more fundamental theories. The old ideas still apply. The wisdom and methods we acquired in the past survive. Today's methodology began in the seventeenth century." Kids should study Newton's laws of motion because they still apply. Randall says that we can still measure pressure, temperature, and volume because they are real quantities. In short, fundamental scientific knowledge is important and should be stressed in school. 


Regrettably, many of the elementary school science textbooks I have seen are incomplete and often perpetuate misconceptions. They are almost math-less, which misleads students. The real world is explained (modeled) through equations. The textbooks do not teach what science really is. One fundamental idea is that scientists try to prove ideas wrong, not right, by carefully crafted experiments; i.e., science does not prove anything right. Scientists seek out counterexamples and correct itself by getting rid of false ideas. According to Karl Popper, every theory must be falsifiable.

New Science Framework

The new science Framework from the National Research Council, oddly enough, was written by the Division of Behavioral and Social Sciences and Education and its committee, which is made up of mostly of educators, not real scientists. In fact, the "science content" experts (i.e., the design team) were not allowed in meetings in which the final decisions (consensus) regarding content were made (p. 17). Surely, You're Joking. No! Also, read the K-12 Science Framework.

I was disappointed after reading parts of the new K-12 science framework from the National Research Council (July 19, 2011).  The Framework committee merges science with engineering and technology, skimps over math needed to do science, stresses scientific processes (called "practices" in the document) over content, requires little content knowledge in elementary and middle school, and combines chemistry and physics, leaving important content out. 

In my view, the new Science Framework from the National Research Council is flat. It does not challenge children, and it does not paint a true picture of what science is all about. The fundamental idea, that science does not prove anything right, is missing. The Framework also lacks a historical perspective. Missing are the great scientists and how they changed the focus of science, e.g., Galileo, Dalton, Maxwell, Bohr, Heisenberg, Plank, Dirac, Einstein, Feynman, Higgs, etc. Moreover, the Framework lumps technology and engineering together; however, they are applications or products of science, not science. The framework skimps on chemistry and physics and the math needed to do the science. Is this the best we can do? It is disappointing! 

Note. I wrote an analysis of the Framework last summer (July 2011). It is very long. Here is a snippet: The Framework writers insist that a hypothesis (or theory) is not a guess. It is. This is what scientists do--they guess or make conjectures, then they test to see if the guess can be shown false. If an idea (guess) is not testable, then it is not science. We need to teach the testability principle by experiment to kids learning science. David Deutsch (The Beginning of Infinity, 2011) writes that conjecture (making a guess) is the real source of all our theories. Theories must be testable. He writes, “Knowledge must be first conjectured and then tested.” In science, we do not rely on authority or opinion. We have “a tradition of criticism,” says Deutsch. The bottom line, according to physicist Richard Feynman is, “If it does not agree with experiment, then it is wrong.” In other words, real science self-corrects itself over time. Ideology does not. Science does not prove ideas right; it eliminates wrong ideas. 

Many old ideas (e.g., Newton's laws of motion) are correct but incomplete. The laws work well at one scale, but not at another scale (e.g., atomic). We should not toss out Newton because his "laws" are incomplete. Lisa Randall, a particle physicist, says that many of the old ideas apply and have practical applications at the right scale ("appropriate conditions"). This [the scales] is what we should teach kids. Scales are an organizing principle in science. 

Elementary and middle school kids should learn Newton's laws of motion and the historical contributions of scientists like Galileo. The radical methods pioneered by Galileo in the 17th century are still used today: proof by experimentation (not authority, opinion, or consensus), thought experiments, and the use of technology to extend our senses to make better observations. For Galileo, the technology was the telescope. Technology plays an important role in science, but it is not science. by LT, ThinkAlgebra, July 2011 

If we want students to understand the world, then we should teach them substantially more physics and mathematics early on. Furthermore, we should establish math and science standards that, at the least, match the benchmarks from nations that excel in these academic disciplines. In my view, the new K-12 Science Framework does not do this. The Science Framework is the latest version of science education written by a committee made up of mostly nonscientists. It is off-target because it requires very little knowledge of math needed to do science and very little science content knowledge. The committee's makeup and its frame of mind in composing the framework are troublesome. It is not the best we can do.  It is not even close. And, as ZE"ve Wurman, a critic of Common Core math standards, explains, the conceptual science Framework is "science appreciation" all over again.

Also, read  Most Kids Don't Understand Science by ThinkAlgebra

Endnotes
[1] A report supporting my observations was released at the end of October (Strengthening Science Education in California). The report states the obvious: little science is taught in elementary school. But, neglecting science in grade school is not new. In my experience, not much science has been taught in elementary school for decades. Middle school science has gone downhill, too. There is not enough stress on basic science content, reading science, and learning the math needed to do science. Furthermore, many teachers are ill-prepared to teach science. 10-28-11


[2] There are not that many properties. A few of the basic properties [axioms] of numbers that should be learned in first grade in the first month or two of school are: add zero [identity] property, add one property, commutative property of addition (2 + 3 = 3 + 2), equality [or equivalency] property (2 + 3 = 1 + 4), "add in any order" property (3 + 4 + 7 is 10 + 4 or 14), etc. The idea that teaching arithmetic to 1st graders should use a framework based on number properties, rather than counting, is absent in American programs. Morris Kline writes, "Axioms are suggested by experience and observation. Kline also writes, "Operations on numbers [must] give a result that fits our experience." He states that "axioms are useful when our experience fails us or leaves us in doubt." Indeed, axioms (number properties) come in handy as kids learn arithmetic. For example, 3 + 5 = 10 - 2 is a true statement because of the transitive property of equality. In "little kids" talk, both 3 + 5 and 10 - 2 name the same point on the number line and, therefore, are equal to each other (equivalent). 


Mathematician Morris Kline (Mathematics for the Nonmathematician) states that operations (let's say, fractions) are designed to "fit experience." Arithmetic facts and operations are learned mostly by rote, but students should also be aware of the axioms or properties (e.g., commutative property of addition and multiplication) that govern operations to determine whether or not the mathematics is correct. For example, I can explain why 1/2 of 1/3 is 1/6 on the number line, but this type of understanding does not come into play when students are multiplying fractions (e.g., 2/3 x 3/4). In short, when applying the multiplication of fractions algorithm, students do not think in terms of marking off 3/4 of one whole on a number line, then dividing each fourth into thirds, which gives 12ths (but from 0 to 3/4, there are nine equal parts or ninths. Converting 2/3 to 9ths = 6/9). Counting over 6 tick marks, you end up at 6/12 or 1/2, etc. Sounds confusing? It is to many kids. 


Furthermore, making a number line model for fractions with larger numerators or denominators becomes a total mess. The multiplication of fractions algorithm can be inferred from a number line demonstration. Students should be taught to depend on efficient methods (algorithms, step-by-step procedures, operations on numbers, or recipes) that produce correct answers fast. 


The multiplication of fractions algorithm can be "formulated" by the number line idea and other clues by the 3rd or 4th grade. For example, ½ of a number (e.g., 1/2 of 10) produces a smaller number, not a larger number (½ of 10 is 5; it means ½ x 10 = 5). We know this by experience and develop a multiplication of fractions algorithm so that the answer is always correct. (See Example 2 below)


In division, 5 oranges divided into halves is 10 (halves). In arithmetic, this is 5 ÷ ½ = 10. To divide by ½ gives the same result as multiplying by 2/1 (the reciprocal of the divisor. This is invert and multiply). The algorithm for the division of fractions is formulated to fit experience. In short, 5 ÷ 1/2 = 5 x 2/1. Thus, to divide by any number, multiply the number by the reciprocal of the divisor and then apply the multiplication of fractions algorithm. In short, students change division to multiplication. 

In Example (1), adding the fractions should produce a larger fraction. Thus, the idea of adding the numerators and adding the denominators does not work because it does not fit our experience. Adding fractions can be represented by adding lengths on the number line. The answer is greater than one, not less than 1. The algorithm does not work. 

In Example (2), multiplying the numerators and multiplying the denominators works 

(fits our experience). It is the algorithm that kids are taught to use when multiplying fractions. A fractional part of any number (fractional part must be less than 1) produces a smaller number. (But, 3/2 x 7/5 will produce a larger number because 3/2 is 1 + 1/2 and 7/5 is 1 + 2/5. This is consistent with our experience when both factors are greater than 1.)

Algorithms (operations on numbers) must fit experience, be efficient, and produce the correct answer. 
10-24-11, 10-28-11, 11-1-11, 12-223-11


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Photo Credit: Hannah by LT
©2011 LT/ThinkAlgebra

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 1^st 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]. 

Research

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 Wurman, in 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 1^st grade, it is repeated in 2^nd, 3^rd, and 4^th grade (it spirals). Indeed, in the new Common Core math standards (left), students are not expected to be fluent in addition and subtraction until 4^th 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. 

Endnotes

[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