Sunday, June 28, 2015

Test-Driven Reform

Common Core Dilemma, carefully researched by Mercedes K. Schneider, reveals the maneuvering and scheming behind Common Core. She writes, "Common Core cannot work. It is destined for the education reform trash heap, just like the punitive, test-driven No Child Left Behind (NCLB) from which it emerged." She says that the idea that Common Core is state-led was manufactured.

According to  Mercedes K. Schneider (Common Core Dilemma), "Common Core is the nucleus for test-driven reform."  She says that Phil Daro, William McCallum, and Jason Zimba were the leading writers of the math standards, and that all three were or became part of Student Achievement Partners (SAP). The core mission of SAP is to help teachers implement Common Core. In short, Common Core is SAP's bread and butter. Also, contrary to popular belief, classroom teachers were not in decision making roles in Common Core, says Schnieder. More later. Read Common Core Dilemma.

[Note. This is a random collection of compelling and/or often contrary cogitations regarding Common Core and education: December 2014 to July 2015.]

Richard Feynman used to say that you don't understand anything until practiced. Even then, your "understanding" could be minimal not only because "understanding" is an indeterminate idea that is difficult to measure, but also because "understanding" develops slowly over years of study. Incidentally, "understanding" does not produce mastery; practice does!  Furthermore, kids should not calculate single-digit number facts, they should memorize them starting in 1st grade, so they stick in long-term memory.  If we want to jumpstart kids in math then they need to be taught traditional arithmetic starting in 1st grade--not Common Core test-driven reform math. Furthermore, we must kick out accountability testing.

The test-based NCLB law failed. In many schools, the curriculum focuses too much on test-prep. (If it is not in Common Core, then skip it.) Sadly, the government keeps funding its own failures--from Head Start, to Title 1, to NCLB, etc. Common Core is a political narrative, not an education narrative. Common Core is intrinsically linked to test-based reform (NCLB), which, in my opinion, downgrades children to numbers [test scores]. A child is not a data point. The main assumption under NCLB was that test-based reforms would work with consequences built in (sanctions and punishments). The fundamental assumption was wrong.

Walk into most any elementary school classroom and you will find students with a wide range of ability and achievement--fast and slow learners together. For decades the elementary school classroom is like the old, inefficient one-room schoolhouse. The status-quo solution has been differentiated instruction within the same classroom (more inefficiency), but the best solution is common sense, that of putting faster learners together (tracking) with an accelerated, challenging math curriculum. ["Oh, we can't do that." It goes against the progressive ideology of fairness. But, what's fair about holding our best students back?] Common Core means that everyone gets the same, which is like the old inefficient one-room schoolhouse. The caveat is that kids are not the same. Putting high achievers in math and low achievers in the same math classroom has been a recipe for underachievement and a regression to mediocrity. The kids who learn math faster get bored and the kids who struggle stay behind. In my view, mainstreaming [inclusion] for math class has led to underperformance at all levels. In short, the system of heterogeneous classes for math is deeply flawed. Moreover, the Common Core standards, themselves, are flawed, especially when the curriculum distilled from them is taught as reform math. Starting in 1st grade, we should arrange kids into homogeneous math sections by achievement [knowledge] with different teachers. Don't worry about their self-esteem. Worry about their competency!

“[At a conference] The subgroup I was in was supposed to discuss the ethics of equality in education,” writes Richard Feynman (Surely You’re Joking, Mr. Feynman!). He writes, “In education you increase differences. If someone’s good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?” Feynman is right. A good education should increase differences and inequalities, but the contemporary education mind-set seems narrowly focused on closing gaps, not boosting individual achievement.

Our schools slow down the learning and achievement of many bright children, including minority students. Several elementary school teachers told me that they were glad I gave algebra lessons to their self-contained classes because they didn’t know what to do with the brighter kids. For example, in my 3rd, 4th, and 5th grade classes, I used an idea from mathematician Ian Stewart to show that division by zero results in a counterexample. Ian Stewart (In Pursuit of the Unknown) writes, “Division by zero is not an acceptable operation in arithmetic, because it has no unambiguous meaning. For example, 0 x 1 = 0 x 2 [true], since both are 0, but if we divided both sides of this equation by 0, we get 1 = 2, which is false.” In short, division by zero leads to false statements.

[Note. Our best students are shortchanged in self-contained elementary school classrooms. The advanced kids in math are mixed with low achieving math students, which is a recipe for mediocrity. Common Core means everyone gets the same.]

[Note. It is incorrect to blame classroom teachers for the state of affairs. Teachers did not create the gaps (achievement), policies (inclusion), mandates (NCLB), standards (Common Core), and testing culture (Achieve) we have in our public schools. Who did? Think reformists, politicians, departments of education, special interest groups, organizations, pundits, opportunists, "experts" (non-educators), governments (state and federal), and big money.]

In a recent post, Katharine Beals writes good math instruction is the “kind done in countries that outperform us in math.” She says that basics must be taught first and learned to mastery. Good math instruction, she says, begins with “basic arithmetic facts and procedures to automaticity.” Teachers should should focus on one efficient method to do each operation, and push ineffective, alternative strategies to the background where they belong.  “The best math classes focus on standard arithmetic and algebraic algorithms rather than drawing of groups of objects, digit splitting, skip counting, number bonds, repeated action, repeated subtractions, landmark numbers, and lattices.” Furthermore, students must gain background knowledge to do math and teachers should stick to content and practice it.

I don’t think we can equalize gaps in academic ability, knowledge, and skills. Kids come to school vary widely in academic ability, motivation, industriousness, background knowledge, etc. Schools cannot come close to compensating for all the differences, nor should they. It is not their job to be parents. One prevalent solution has been to equalize down (a socialist dogma), says Thomas Sowell, which is a pernicious idea, to make everyone the same, that is, bring down the kids at the top—not bring up the kids at the bottom.

Of course, there are limits to what kids with lower IQs can learn, but this seems to escape educators. No matter, I think most kids can learn standard arithmetic and introductory algebra, but gaps will always exist. And, as Richard Feynman said that when you spot someone with an ability, you try to nourish that ability and increase the child’s knowledge and skills. There will always be students who will not learn algebra or even arithmetic, no matter how hard they try. And, there will be always be students who don’t want to learn.

Will Fitzhugh writes, “Skills have taken the place of content [not only in History, but also in science and math]. Content, after all, can be such a pain. What if someone asks you something and you don't know what they are talking about? Now you can just say ‘I was educated in critical thinking skills, and we moved far beyond content in my day.’ Another advantage is that with the content largely removed, the hard work of choosing what the content of a curriculum should be no longer needs to be faced (addressed).” The reformists say that content is no longer important, yet content knowledge is the foundation for critical thinking and problem solving. In short, you can't solve chemistry equilibria problems unless you know content, can apply it, and are able to calculate equilibria in equations. 

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©2015 LT/ThinkAlgebra

Friday, June 12, 2015


According to Michael Wysession, Next Generation (NG) high school science boils down to about 1 semester of chemistry, 1 semester of physics, a school-year of life science, and a school-year of geoscience, which is a major departure from regular high school science. Michael Wysession (NG team leader) says that geoscience is legitimately on par with physical science. Surely, he must be joking! In my opinion, NG does not prepare students for college level science courses because important content in chemistry and physics has been weakened, reduced, or completely left out. Just as in the Common Core eight math "practices," higher level thinking skills dominate the science standards at the expense of content knowledge. The reformers think content is no longer that important. They are wrong! Content knowledge in long-term memory is the foundation for critical thinking and problem solving. In science, evidence is important, yet NG ignores the findings of cognitive science. (Source: Scientific American, "Kids Are Scientists," written by Michael Wysession, a professor of seismology, the leader for the NG Earth and Space Sciences team, and author of several Pearson Prentice Hall K-12 textbooks)

The writers of the Next Generation science standards left out key ideas from the past 100 years or so. What were these people thinking? Their knowledge is so fragile, as Feynman would say.  Reforms, such as Next Generation, Common Core, NCLB standardized testing, minimal teacher guidance approaches, laptops or tablets for all, group work (and so on), do little to advance actual achievement. Our kids are mediocre in math and science because of screwy ideas and cockamamie reforms that don't work. Jacob Bronowski wrote, "Science is the acceptance of what works and the rejection of what does not." In science, evidence is everything, but NG ignores the findings of cognitive science. To me, Next Generation, which is another test-centered reform like Common Core, means that science content is left behind and so are our students. 

Will Fitzhugh writes, “Skills have taken the place of content [not only in History, but also in science and math]. Content, after all, can be such a pain. What if someone asks you something and you don't know what they are talking about? Now you can just say ‘I was educated in critical thinking skills, and we moved far beyond content in my day.’ Another advantage is that with the content largely removed, the hard work of choosing what the content of a curriculum should be no longer needs to be faced (addressed).” This seems to be the case with Next Generation science. The reformists say that content is no longer important, yet content knowledge is the foundation for critical thinking and problem solving, that is, you can't solve chemical equilibrium problems unless you know chemical reaction and equilibrium content (knowledge), can apply it, and are able to do the calculations involved.

"Start with a small model that works." This is what we should do in education but we don't. Indeed, the Next Generation science standards and Common Core math standards are the latest large scale models that did not start with a small model that actually worked. Alan Manne (Stanford) writes, "To get a large model to work you must start with a small model that works, not a large model that doesn't work." I think the Next Generation science standards, which were built on the National Research Council science Framework, are flawed and unrealistic. There is scant evidence that the new standards will produce better science students or more informed citizens. Indeed, most of the key physics from the last 100 years or so [e.g., quantum theory] has been left out of Next Generation. Also, to me, the Next Generation favors issues more than content. In my opinion, the purpose of science and the scientific method is not to prove something right (or a point of view right) but to weed out bad ideas. "We need to get as much baloney out of our sandwiches as we can," says physicist Leonard Susskind. (TEDxCaltech: To honor Richard Feynman...) 

Nobel Prize in Physics
Quantum Electrodynamics QED
The late Richard Feynman often talked about scientific integrity. He says that scientists must report all the data, especially conflicting data or data that doesn't fit. Feynman writes, "I'm talking about a specific, extra type of integrity that is not lying, but bending over backwards to show how you're maybe wrong, that you ought to have when acting as a scientists." What has happened to scientific integrity in an issue-centered, fractional nation? Many scientists do not always act as scientists should act when they gather or interpret data. Some have political agendas or confirmation bias, and too many scientists fudge data. They don't live up to the Feynman concept of integrity.  [Note. There will always be disagreements or arguments among scientists, but these should stay within the scientific community and should not mushroom into huge political or partisan issues we often see today. Unfortunately, they have, thanks to a biased, uninformed, partisan media, know-nothing politicians, and predisposed scientists who set out to prove something right (confirmation bias). The politicalization of science is a very bad idea. Science is not an opinion. It weeds out bad ideas. 

Newton: Inverse Square Law
Feynman (paraphrase): Why is it that the force of gravity acting between any two objects is inversely proportional to the square of the distance between the two objects? It’s mathematical, and Newton taught us that we can make progress if we stop arguing about the “why” of it. Newton said he makes no hypothesis. He does't explain the gravity law. He correctly states that gravity follows the inverse square law. This is what the law is. It is a rule of nature. Newton implied that he didn't understand the why. Feynman opens his lectures at The University of Auckland (1979) with Newton's idea of gravity and Newton's idea that light is made of particles (photons). Newton's ideas led to Feynman's intense study of the interaction of photons (particles of light) and electrons (particles of matter) called Quantum Electrodynamics (QED). Feynman also states he doesn't understand the "why" of quantum mechanics (QED), just that QED is very strange or hard to believe (yet, very interesting), mathematical, and that it works! Feynman lectures, "Nobody understands why, but we've looked carefully...and that's the way it is. I'm not going to simplify it. I'm going to tell you what it really is like, and I hope you accept nature as she is--absurd." Comment. Kids do not need to understand "why" gravity follows the inverse square law to apply it, but they do need to know (i.e., be able to do) the calculations.  

The  Big Bang theory "doesn't really describe how the Universe was born," writes Ben Gilliland. "Instead, it describes the evolution of the Universe after it came into existence." Of course, under the new K-12 science framework, students will likely never learn this, or that the Planck era (in which the quark, electron, and the massless photon existed) was at the very beginning of the Big Bang (13.8 billion years ago) or that quarks instantly (0.0000001 seconds later) formed larger particles called protons and neutrons, which, in turn, would eventually form stable hydrogen and helium atoms 377,000 years later, and so on. The building of a universe is an electrifying and thrilling field of basic physics, which includes the great minds of Einstein (time is not absolute, motion is relative), Heisenberg (uncertainty principle), and Feynman (time-traveling electrons). Richard Feynman figured out how time-traveling virtual electrons "can make something from nothing" in spacetime. It is compelling because the quantum world is counterintuitive and weird! Yet it is likely that students will not learn much of anything about quantum theory under the new science standards because it has been left out. When the new K-12 science framework was formed, the progressive powers-that-be assumed that many of the key ideas (e.g., relativity, uncertainty principle, time-traveling electrons and quantum mechanics) would be beyond the comprehension of children. But, I don't agree. I think young children are more apt to accept counterintuitive thoughts and enjoy weird ideas that stretch and spark their imaginations. Think, Big Bird? Big Bird is counterintuitive. Real birds actually fly, don't grow that huge, and don't speak English. Einstein's ideas are counterintuitive, too. Objects shrink, time slows, and mass increases (as objects move very fast). (Note: The Australian Cassowary monster bird grows to 6 ft. tall and 130 lbs.)

The K-12 Science [conceptual] Framework from the National Research Council (July 2011) is a disappointment. It was used to develop the Next Generation Science Standards, which some states have already adopted. The framework 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 6-8 grade band. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) isn't found until grades 9-12. The sequence is off. 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), while the Framework writers gave Life Science content 20 pages. Chemistry and Physics are woefully underrepresented in the Framework, yet physics is the most basic, inclusive science. Everything in science boils down to physics, which is mathematical, such as Newton's law of universal gravitation or today's Standard Model of Atoms. 

The Framework is lean on core science content, especially chemistry and physics; downplays the major role of mathematics used in science; glosses over the great discoveries of scientists that moved science forward, and focuses more on practices and issues and less on actual science content. Also, the [progressive] powers-that-be have determined that hands-on science (discovery approach in group work) is the best way to learn science, but it isn't. Indeed, minimal guidance or discovery learning in small groups is an inferior way to learn science, or math, or any academic discipline. The new science Framework places process over content, which is an error in judgement. But bad ideas should not surprise us, especially since the Framework was supervised by National Research Council's Division of Behavioral and Social Sciences and Education with a committee made up of mostly educators, not actual scientists. In short, the new standards seem to stress issues more than core science. Indeed, learning science and engineering practices or arguing issues is not the same as learning science. Opinions about issues are not science.  But, for progressives, which control K-12 education, science is just another opinion and relative, write Berezow and Campbell (Science Left Behind). Well, there you have it.

Like Aristotle, the Framework writers relied too much on conventional ideas and common opinion--actually their [progressive-powers-that-be] opinion. Apparently, the trusted idea that children can learn science well from strong teacher-led instruction or by reading well-written science textbooks and books never occurred to the committee. Also, labs once a week are good for kids if used to reinforce scientific ideas they are studying. Dr. Mark A. McDaniel says that before engaging students in inquiry-based problem solving in science or mathematics [hands-on approach], they should have a sound knowledge base (background knowledge). Kids are novices; they should not reinvent science to learn science. Nor, should they argue issues they do not understand because they lack background knowledge (content) in long-term memory. Experiments and demonstrations should be used to reinforce key ideas kids are studying.

If we want students to understand the world and the universe in which they live, 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 (or 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 Committee's statement is packed with vague, unclear ideas that cannot be measured. Indeed, unclear generalizations are commonplace in education and, in this case, the opposite of what science is. What is sufficient knowledge or appreciation? Also, I don't believe that the average citizen is  going to read and study science and scientific studies after they graduate from school. To imply that they would is nonsense. The Framework’s expectations are fantasy. 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 6-8 grade band, and atomic structure (protons, electrons, and neutrons) is not introduced until high school. In short, content is diminished or restricted. (Surely, you’re joking!)

Table by LT, ThinkAlgebra. It is based on Chemistry/Physics NRC Framework 2011

[Comment. At the 5th grade level, according to Next Generation (NG): "Boundary: At this grade level, mass and weight are not distinguished, and no attempt is made to define the unseen particles or explain the atomic-scale mechanism of evaporation and condensation. (5-PS1-3)" Also, "The amount (weight) of matter is conserved when it changes form, even in transitions in which it seems to vanish. (5-PS1-2)" From the beginning, the standards should have distinguish weight, which is a force, and mass, which is an amount of matter. I had no trouble teaching this idea to 1st graders. 

It is nonsense not to introduce the periodic table or the atomic model in grades 3-8, even if it is an old model, with protons and neutrons in a nucleus and electrons revolving in orbits. Incidentally, in high school NG, the old model for the atom is presented: "Each atom has a charged substructure consisting of a nucleus, which is made of protons and neutrons, surrounded by electrons. (HS-PS1-1)" But, this is the  model I would use for 3rd graders. It is as if Bohr, Einstein, Heisenberg, Feynman, Murray Gell-Mann, and many others never existed. Kids won't be prepared to take physics or chemistry in college.  I recommend that teachers fill in the missing science and math content using old textbooks. I don't see geoscience being a required one-year course or chemistry reduced to a semester course.  

To the standards credit, the Big Bang is mentioned along with CMB or Cosmic Microwave Background. But, the standards also mislead, "Other than the hydrogen and helium formed at the time of the Big Bang, nuclear fusion within stars produces all atomic nuclei lighter than and including iron, and the process releases electromagnetic energy." (HS-ESS1- 2), (HS-ESS1-3) The problem is that hydrogen and helium, which eventuality became stable products of the Big Bang, did not exist at the time of the Big Bang as stated in the core standard--only quarks, photons, and electrons existed at the time of the Big Bang, then for only a fraction (of a fraction...) of a second.

As I skimmed some of the Next Generation core and often used the pdf Find Function, I see no mention of the strong nuclear force, the weak nuclear force, or a 5th fundamental force called the Higgs Field. The Higgs field gives particles mass and its force carrier (or boson) is the Higgs boson (found in 2012 using the Large Hardon Collider or LHC). I see no mention that physicists "recreated the conditions at the time of the Big Bang" with mini-Big Bangs. And, so on. Except for Newton, the names of scientists have been left out of the document.]  

MIT's "Introduction to Chemistry" Lecture
(Binding Energy of an electron using the Rydberg
Constant for Hydrogen)
Next Generation will not prepare students for college level chemistry or physics.

There are too many fundamentals left out in the high school 40-page physical science [Next Generation] Evidence Statements, written by Jennifer Childress, Ph.D. from Achieve. In my view, Dr. Childress is not qualified to write standards or Evidence Statements in chemistry and physics because her Ph.D. is in biomedical science. Achieve proclaims, “All students should graduate from high school ready for college, careers, and citizenship,” which, to me, is an extraordinary claim. Carl Sagan once said, "Extraordinary claims require extraordinary evidence." So, where is the evidence or proof that Next Generation [or Common Core] will accomplish its mission? There isn't any!

Ze'Ve Wurman says the Next Generation science standards, just like Common Core math standards, are undemanding intellectually (i.e., dumbed-down). He explains (June 3, 2015), "I'd call traditional algebra as supporting future STEM learning while the functional algebra [Common Core Algebra I] supports math appreciation you can talk about it, but you can't do anything with it." Wurman points out, "Common Core Functional Algebra I does not support the Next Generation Science Standards (NGSS). This is true because NG standards are empty of any expectations of using mathematics to solve science problems beyond plotting graphs on computers [or graphing calculators] and sagely pronouncing what those graphs seems to say. Dumbed-down math seems appropriate for dumbed-down science." Wurman states that Common Core functional algebra stresses the functional representations but not the solving of the equations. He writes, "Students are not expected to develop the analytical skills of handling most such functions beyond feeding them to programs that will graph them." For example, exponential equations appear in Common Core Algebra 1. Students graph equations on a graphing calculator, but they can't solve the equations using traditional algebra. In contrast, traditional algebra prepares students for actual science (chemistry, physics) and Algebra 2 and trig, etc.  

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 should 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 or SAPA (1967) stressed process in the context of 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 (metric), Communicating (graphing), and Using Space/Time Relationships (geometry)]. In short, the math needed to do the science was a major part of the SAPA science program. Moreover, the math taught in the program was very specific and ahead of grade level.

[Note. In the Next Generation 2nd grade standards, "Assessment of quantitative measurements is limited to length,” but my 1st graders in the early 80s measured mass in g, length in cm, and liquid volume in mL. They even found the volume of irregular objects like a small rock by water displacement in a graduated cylinder because they learned the relationship that1 mL water balanced 1 centimeter cube. Thus, if the rock displaced 4 mL of water, then its volume was 4 cubic centimeters.]

In contrast, the Framework glosses over mathematics and does't mention the metric system. "Create a computational model or simulation of a phenomenon, designed device, process, or system." (HSESS3-3) "Use a computational representation of phenomena or design solutions to describe and/or support claims and/or explanations." (HS-ESS3-6) These statements do not mean that the student actually does mathematics (writes and solves equations) to find solutions to problems, such as using trig to find the distance to an object in space (parallax]. Solving equations is not mentioned in the Framework or in the Next Generation standards. In fact, I could not find a single equation, not even distance = rate x time or E = mc^2. 

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 all science is to gain an in-depth  grasp of our 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 should not imply that students are learning science content. Critical thinking in any discipline requires considerable content knowledge. Learning what a scientist does is not the same as learning science content. Likewise, learning what a mathematician does is not the same as learning math 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 math content and how mathematics works--how one idea links to or builds on another idea. I want students to learn essential content, skills and uses 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 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. Leonard Mlodinow (The Upright Thinkers) writes, "Today, at least among scientists, there is virtually universal agreement about the validity of the mathematical approach to understanding the physical world. Yet it took a very long time for that view to prevail."

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 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. The word “atom” is not used until grades 6-8. The idea that atoms are composed of electrons, protons, and neutrons (atomic structure) is reserved for grades 9-12. This makes no sense.

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, such as relativity and quantum theory. The Framework asks students to pretend to be scientists (or engineers) by stressing science and engineering practices. In short, the Next Generation standards are not about learning and applying core content.

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. 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 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 puts us back to the days before Galileo.

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.” (You must be joking!) 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.” (Psychology and economics are not true sciences.) 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 is code for "no supportive evidence."

Moreover, the Framework committee says that students should continue to take honors and AP courses in the sciences. Why? Apparently, the standards based on the Framework have serious flaws. In fact, it is likely that the Next Generation standards will not prepare students for traditional honors or AP courses. Actually, many AP courses are below the college-level. For STEM students, I recommend that high schools establish traditional chemistry and algebra-based physics courses and use college-level textbooks to make up for the flaws and deficiencies in the new science standards. There should also establish a strong pre-college science and math sequence in middle school. Kids must know content and be able to apply it.

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 more complex 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 Framework 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. 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 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.

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.”

This post first appeared on April 7, 2012. I have changed a few sentences and added additional parts. Some parts have been edited on 6-26-15 and additions made on 7-18-15.


Draft 1
Last Update: 6-17-15, 6-24-15, 6-26-15, 7-18-15

© 2012, 2015 LT/

Monday, June 8, 2015


Dana Goldstein (The Teacher Wars) opens her book with an observation from 2011: "Public school teaching had become the most controversial profession in America." She also asserts, "The federal Department of Education has no power over state legislatures or education departments." Yet government overreach--federal, state, and local--is real (e.g., NCLB, Common Core, standardized testing, funding that favors certain reforms, unwise policies by state and local school boards, etc.). In many schools, almost every aspect of instruction is governed by Common Core embedded in  progressive ideology (the powers-that-be), which is the reason that Common Core has become the most contentious education reform issue of our time.  

[Comment. Larry Cuban points out that current reformers are on thin ice because the research doesn't support them. He writes, "The pumped up language accompanying “personalized learning” resonates like the slap of high-fives between earlier Progressive educators and current reformers. Rhetoric aside, however, issues of research and accountability continue to bedevil those clanging the cymbals for “student-centered” instruction and learning. The research supporting “personalized” or “blended learning” is, at best thin. Then again, few innovators, past or present, seldom invoked research support for their initiatives."]

Please excuse typos, errors, and embedded Notes and Comments. Draft 1.

Reform Math Fails. Over the years, multiple reforms have been imposed on schools. Indeed, educators have been "reformed to death," says Diane Ravitch. None of the math reforms have worked well and for good reason. Believing an idea will work because it sounds good and seems reasonable is not the same as having solid evidence. It appears that facts and scientific evidence don't matter much. Indeed, a lot of "studies" in education are junk science and often promote disinformation. In education, studies are rarely duplicated to verify results. Indeed, education is filled with ideas, common notions, and popular practices that have not been tested scientifically. Thus, we have had bad idea after bad idea introduced into our classrooms. The consequences of bad ideas are often calamitous. In one urban school district, 87% of the students who had enrolled in community college in 2014 needed remedial courses in mathematics. [Another school district is 88%; Another district is 83%; etc.--all in the Tucson area.] These kids are products of the K-12 reform math movement, which took hold in the 90s with NCTM standards. The new reformers say that Common Core will fix rampant remediation, yet Common Core is typically interpreted through the lens of reform math. Repeating or repackaging the mistakes from the past does not move math achievement forward. Also, test-prep, technology, curriculum narrowing, threats to withhold funding, minimal guidance methods, group work, and so on, will not erase the achievement woes. 

Jordan Ellenberg (How Not To Be Wrong: The Power of Mathematical Thinking, 2014) writes, "Some reformist go so far as to say that the classical algorithms (like add two multi-digit numbers by stacking one atop the other and carrying the one when necessary) should be taken out of the classroom, lest they interfere with the students' process of discovering the properties of mathematical objects on their own. That seems like a terrible idea to me: these algorithm are useful tools that people worked hard to make, and there's no reason we should have to start completely from scratch." Professor Ellenberg also points out, "It is pretty hard to understand mathematics without doing some mathematics."

Higher Math is Needed. There are a lot of smart kids, but the Common Core reforms do not prepare enough of them for college--not with watered-down math and science courses. There are exceptions of course. We do have pre-college kids who hold their own with the best high school math students from other nations in international contests. That said, many smart kids are not prepared for college level STEM courses, especially in mathematics and science. Indeed, Common Core ignores STEM, even though "high math" is needed markedly more today and in the future than in previous generations. The reforms are illogical and flawed. "The ability to create algorithms that imitate, better, and eventually replace humans is a paramount skill of the next one hundred years," explains Christopher Steiner (Automate This: How Algorithms Came to Rule Our World). The bots (algorithms) are coming for your job. 

Kids Are Ill-Prepared. Steiner says the two economic-growth-drivers over the next 50 years will be health care and tech, so how well are we preparing kids in chemistry, physics, and mathematics? We aren’t! A "dearth of engineers" is the reason the Silicon Valley imports talent. “The problem,” writes, Steiner, “is that not enough US kids get that foundation of upper-level math before arriving at college.” It is unfortunate that many smart kids can’t hack the STEM college courses in chemistry, physics, math, and computer science—the real stuff--because they lack sufficient preparation in high school. [CommentThe problem of inadequate preparation does not suddenly pop up in high school. It begins early in elementary school with the way arithmetic has been taught (as reform math). Today, Common Core is often interpreted in terms of reform math by the "progressive" powers that be. Indeed, memorizing single-digit number facts, learning standard algorithms, and practicing essential math to mastery have been intentionally lessened, even ignored, in reform math approaches.]

[Note. The reform math people have exploited the eight Standards for Mathematical Practice in Common Core, which are vague and open to various interpretations, as a means to devalue the traditional teaching of mathematics and promote their own constructivist pedagogy and progressive agenda. In short, the people in charge impose their favored progressive reforms.]

Teach Standard Arithmetic Straightforward.  Many teachers get hung up on understanding, critical thinking, and group work, which are some of the elements of the reform math movement. Consequently, they miss the target. Kids learn very little math in small-group-centered reform math "discovery" activities. Instead of reform math, teachers should teach standard arithmetic content straightforward and require that students practice it for mastery beginning in 1st grade. Indeed, the standard algorithm for addition can be taught to 1st graders in the 1st month of school when children begin to study place value, that 25 is 10+10 + 5, or 2t + 5, or 2tens+5ones, and memorize the addition facts, such as add 2 number facts, which students easily figure out on the number line. Soon, students can add 23 + 42 in columns using memorized single-digit addition facts [i.e., the standard addition algorithm]. Later, well-taught 1st graders can add 40 + 129 + 24 in columns. They also learn subtraction and much more.

[NoteReform math has had many different names over the years. One feature of reform math is minimal teacher guidance, that is, the teacher's role is passive and diminished to being a facilitator so that students work in groups. Examples of reform math include constructivist, discovery, problem-based, experiential, inquiry-based, etc. Kirschner-Sweller-Clark say that minimal guidance is inferior and has failed, yet reform math elements are still championed in schools of education and extend to popular interpretations of Common Core. According to cognitive science, problem solving in math is not possible without prerequisite factual and procedural knowledge in long-term memory. For example, you can't solve a trig problem unless you know trig, and knowing trig requires practice-practice-practice. (Likewise, you can't translate Latin without knowing some Latin.) In mathematics, the emphasis should be on students acquiring factual and procedural knowledge in order to do and/or apply mathematics to solve problems. You don't get good at math without substantial practice.]

Standard Arithmetic--not reform math--should be the basis of instruction. This includes axioms (basic rules of arithmetic), the automation of single-digit number facts, the standard algorithms, routine word problems and their variants, mathematical reasoning, and so on. I don’t want novices trying to discover or reinvent arithmetic in group work or learn inefficient, non-standard algorithms at the expense of standard algorithms. In reform math, standard algorithms are discouraged and get scant coverage. 

John von Neumann is alleged to have said, "In mathematics, you don't understand things. You just get used to them." Indeed, children need to learn to deal with and get used to abstractions because "whole numbers, fractions, and the various operations with whole number and fractions are abstractions," writes mathematician Morris Kline (Mathematics for the Nonmathematician, 1967).  Dr. Hung-Hsi Wu writes that kids should use symbols to convey mathematical ideas and operations.

Professor Kline writes about a man who walks into a store to buy three pairs of shoes at $10 each. "The difficulty," says Kline," is that you can't multiply shoes and dollars," so you have to abstract from the particulars and "multiply the number 3 by the number 10 to get the number 30."  Kline explains, "One must distinguish between the purely mathematical operation of multiplying 3 by 10 and the physical objects with which these numbers may be associated." Starting in 1st grade, children need to learn to pull out the numbers and operation needed to find the solution to a word problem. Translating a word problem into abstract symbols that conveys numbers and operations in the form of an equation, which shows a solution, requires plenty of practice. Singapore kids write equations in 1st grade. 

At the basic level,  the fact 4 + 7 = 11 is an abstraction that can be applied to hundreds of situations, says KlineAbstraction is a powerful idea in mathematics. The fraction 3/4 is an abstraction. It can be represented as a point on the number line just like a whole number. But operating on fractions seems "arbitrary and mysterious." Kline explains, "Operations with fractions are formulated [made up] to fit experience."

We make up algorithms to fit our experience in the real world, but they must be efficient for practical use. (All the standard algorithms for whole number operations are based on single-digit number facts.)  If we take 10 oranges and split each orange in half, then we would we have 20 halves. That is, 10 ÷ 1/2 = 20 must give the same answer as 10 x 2 = 20; therefore, 10 ÷ 1/2 = 10 x 2. From this abstract idea and others, such as the idea of reciprocals (1/2 and 2 are reciprocals because their product is ONE), we should be able to make up an efficient algorithm for division of fractions that always works in the real world, and we have: To divide by a fraction, multiply by its reciprocal, which is the standard algorithm. It fits our experience. The proof, however, is left to mathematicians, not to novice kids. 

G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics.This requires sufficient knowledge in long-term memory, experience, and skill development through practice. It implies that if  a student cannot do the math, then the student doesn't really understand it or know it. This is related to Richard Feynman's insight: "You don't know anything until you have practiced."

I think it is difficult to discuss understanding with any consensus because the term is vague and hard to measure and analyze, and, in my opinion, overrated in US math programs. I think teachers should talk in terms of doing or applying mathematics, which is observable and measurable. In order to do or apply arithmetic, kids must know some arithmetic--key factual and efficient procedural knowledge. The idea is to master the important stuff, which includes number facts and standard algorithms. 

David Ruelle (The Mathematician’s Brain) explains mathematical intuition, “When we study a mathematical topic, we develop an intuition for it. We put in our [long-term] memory a large number of facts that we can access readily and even unconsciously. Since part of our mathematical thinking is unconscious and part nonverbal, it is convenient to say that we proceed intuitively. This means that processes of mathematical thought are difficult to analyze.” Dr. Ruelle also observes, "Mathematics is a matter of knowledge, not of opinion.” (I think understanding falls under opinion.)

Ruelle writes that “mathematicians put a lot of facts in their long-term memory through long days of study.” In this sense, I want kids to be more like mathematicians (or musicians), not little mathematicians, that is, I want kids practicing the essentials so they stick in long-term memory and become automatic. Basic knowledge of arithmetic (ideas, skills, and uses) enables mathematical thought. I don’t want novices trying to discover or reinvent arithmetic in group work or waste time with non-standard algorithms at the expense of standard algorithms. As Hersh & John-Steiner would say, learning mathematics well takes drill and practice, lots of it. There is no workaround. If students can apply the math they have learned, then this cognitive outcome likely implies that they have some understanding of it, but I cannot measure it.

Perhaps, the closest we can come to a student’s actual mathematical thought process, which is often unconscious and nonverbal (Ruelle), is when the student actually performs math that leads to a solution or correct answer, whether it be solving an equation or executing the standard multiplication algorithm, etc. In my opinion, written explanations or drawings are nonessential. Showing steps is sufficient to demonstrate a child's knowledge.

While it is difficult to figure out a student's actual understanding, we can, I think, confirm some level of performance or competency by looking at the steps the student writes on paper to get to a solution. Also, speed is an important variable in competence. As teachers, we should depend more on observable [measurable] behavior and not so much on understanding, which is uncertain,  equivocal, and difficult to measure. Here is an example of observable behavior. A 3rd grade student has 4 of 5 long division problems correct within 5 minutes.

In short, teachers should focus more on specific, measurable cognitive outcomes, as described by behavioral objectives, rather than on inexact, hard to measure qualities such as understanding, appreciation, soft cognitive skills, and so on.

First Draft. Please excuse typos and other errors. To Be Revised. June 13, 2015

© 2015 LT/ThinkAlgebra

Wednesday, June 3, 2015


The topics are in no particular order.
Explicit Instruction: Children need strong, teacher-guided instruction.
Minimal-guided instruction has been an epic flop.
Progressive school reforms haven't worked.
Standard Algorithms: We make them up to fit the real world.
"Understanding" is a slippery slope because it is hard to measure.

Look at what reformers have done to simple arithmetic! It is bizarre! Common Core math has been interpreted and/or implemented as reform math by those in power over education in an effort to weaken "tried and true" traditional teaching of standard math. Below is a 5th grade Common Core math quiz from Kaplan. It represents a typical misinterpretation of Common Core as reform math.

Kaplan's answer to #3: Area Model

1. Find 15.7 + 9.72 by decomposing the numbers by place value. Show your work.
2. Find 9.53 - 4.6 using a place value chart. Show your work.
3. Find 5.3 x 2.4 using an area model. Show your work.
4. Find 4.8 / 0.8 using a number line model. Show your work.
5. Find 3.6  / 12 using a bar model. Show your work.

Who multiplies whole numbers, fractions, or decimals using an area model? Frankly, I have trouble believing that teachers would actually teach this junk and diminish the importance of standard algorithms, yet this seems to be the case in many classrooms. Standard algorithms get scant coverage, if they are taught at all. They are seldom  practiced for mastery. Without memorizing single-digit number facts for instant recall (key factual knowledge) and gaining proficiency in standard algorithms (efficient procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics," warns Professor W. Stephen Wilson.

Dr. W. Stephen Wilson, a mathematician at Johns Hopkins University, critiqued a popular reform math program (Pearson's Investigations, 5th Grade) and said it was actually "pre-arithmetic." Students never get to arithmetic, implying that students do not focus on memorizing single-digit number facts or on practicing the standard algorithms for mastery. In short, instead of  focusing on standard algorithms, students are often taught many inferior or weak methods or strategies that are not practical or useful. By pre-arithmetic, Professor Wilson means that kids learn something that looks like arithmetic, but it isn't the arithmetic that students need to know to advance to algebra by 8th grade. Is it any wonder that most students struggle with basic arithmetic and math in general? 

Traditional arithmetic works well when taught well. Students become better at mental math because they have memorized basic number facts. Furthermore, the standard algorithm always works. We keep forgetting that little kids are novices. They don't think like adults. Children need to memorize and practice to put mathematical knowledge, both factual and procedural, into long-term memory for instant use in problem solving. Students cannot do mathematics without knowing some mathematics. Also, understanding is a slow process. It does not produce competency, practice does. Do not expect instant understanding or hold kids back because their understanding is partial or incomplete. Furthermore, Jason Zimba, one of the two major writers of CC math standards, clarifies, "The standards also allow for approaches in which the standard algorithm is instructed in grade 1, and in which only a single algorithm is taught for each operation."

Note. Isaac Newton invented a fast way to calculate answers to physics problems, called calculus. It always worked (i.e., it was consistent with experimental data), but he didn't understand why the calculus, itself, worked; it just did. The "why" would take another 200 years. Indeed, maybe "understanding" is overrated and a slippery slope, something to stay away from. Newton was a true polymath, but, under Common Core reform math, he would not have been able to write a paragraph that explains why his procedures work; however, Newton might say through inductive reasoning, "It works in all the cases I have tired, therefore it must be correct."

Algorithms--we make them up to fit our experience in the real world. If we take 10 oranges and split each orange in half, then how many halves would we have? {20} That is, 10 ÷ 1/2 = 20 gives the same answer as 10 x 2 = 20. From this fact and others, together with the idea of reciprocals (1/2 and 2 are reciprocals because their product is ONE), we could make up an efficient algorithm for division of fractions that always works in the real world, and we have: To divide by a fraction, multiply by its reciprocal, which is the standard algorithm. It fits our experience. The proof, however, is left to mathematicians, not to novice kids. Kids should not be expected to reinvent arithmetic or be required to write explanations, but they do need to apply rules procedures and perform steps correctly. Furthermore, writing a "why" paragraph does not imply "deep understanding," whatever that is. Indeed, understanding is vague and very difficult to measure or pin down. In math, understanding varies widely, is always imperfect, and grows slowly over time with practice. It is a slippery slope best left alone.

Explicit teaching, which uses a carefully-planned sequence of worked examples in math works well for almost all students. Students learn concepts through examples, lots of practice, and repetition, says Zig Engelmann. However, since the 60s, teacher-led instruction has been called "old school" or the opposite of “good” teaching. Explicit, teacher-led instruction (using examples, practice, and repetition) “contradicts much of what educators are taught to believe about good teaching,” writes J. E. Stone (Clear Teaching).

Stone says that explicit teaching [the teacher is the academic leader that leads instruction by explaining examples on the board, etc.] has not been popular in K-8 schools, not because it didn't work but because it goes against progressive reform ideology taught in schools of education. The Progressive Era revolution of the 60s affected education by attacking teacher-led exercises, scripted lessons, skill grouping, choral responding, repetition, etc., says Stone. “Thus, education professors and theorists denigrate teacher-led practice as ‘drill and kill,’ its high expectations as ‘developmentally inappropriate,’ and its emphasis on building a solid foundation of skills as ‘rote learning’,” writes Stone. In short, kids have not been taught a solid foundation of arithmetic for multiple decades.

Today we have teachers as facilitators, not academic leaders; mainstreaming (inclusion); a weak, incoherent, narrowed curriculum; low expectations for students; popular reform methods of instruction (i.e., minimal guided, not teacher-led) that are inferior; reforms such as Common Core, intrinsically linked to standardized testing; etc. Education is no longer a "work hard and achieve" narrative; it is a political, test-centered, money-driven narrative.  

Note. I have quoted this study (Kirchner-Sweller-Clark: Why Minimal Guidance During Instruction Does Not Work...) since it first appeared in Educational Psychologist in 2006. The instructional methods in classrooms across the US--mostly group work activities with minimal teacher guidance or no teacher guidance--have failed our students for decades. The minimal guidance instructional methods, which come in different names or favors over the years (e.g., discovery, constructivist, problem-based, inquiry-based, etc.) and have been championed by schools of education, extend to Common Core. They are part of the progressive movement in education, starting with Dewey. Kids do a lot of group work, use manipulatives, etc. Their desks are in groups of 3 or 4, so kids face each other. Instead of being the academic leader in the classroom, the teacher's role has diminished to a "facilitator" of learning. In short, the teacher no longer teaches.

Kirchner-Sweller-Clark (Why Minimal Guidance During Instruction Does Not Work...) write [long quote], "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional design models that support guidance during instruction are briefly described."

Kirchner-Sweller-Clark write, "Cognitive load. Sweller and others (Mayer, 2001; Paas, Renkl, & Sweller, 2003, 2004; Sweller, 1999, 2004; Winn, 2003) noted that despite the alleged advantages of unguided environments to help students to derive meaning from learning materials, cognitive load theory suggests that the free exploration of a highly complex environment may generate a heavy working memory load that is detrimental to learning. This suggestion is particularly important in the case of novice learners, who lack proper schemas to integrate the new information with their prior knowledge."

Understanding is overrated and a slippery slope. If students can apply the math they have learned, then this cognitive outcome likely implies that they have some understanding of it, but I don’t know how much because I cannot measure it. I also think that understanding varies widely as does academic ability and that the two are correlated. Often, I hear reformers claim that novices need "deep understanding." What is that? How is it measured?

Reprinted [with additions and changes] from Strong Teacher Guidance, February 25, 2015, Math Notes by ThinkAlgebra

© 2015 LT, ThinkAlgebra,org