Wednesday, April 15, 2015


Important Ideas from ThinkAlgebra 
1. Math Fundamentals Don't Change! 
2. Knowledge, both factual and procedural, in long-term memory fires up problem solving, innovation, and creativity.
3. Assault on Knowledge and Liberal Education.
4. Standards for Mathematical Practice (SMPs), the essence of Common Core, are unrealistic because they represent mathematical expertise. Kids are novices, not experts.

Comment. This post is in first draft form and was updated on 5-12-15. It is a collection of random thoughts, both new and old, in no particular order. Please excuse typos and other errors. Sometimes, I repeat and rephrase ideas. Repetition is a good thing! Quotes are from W. Stephen Wilson, Jason Zimba, David G. Bonagura Jr., Robert B. Davis, Ze've Wurman, Will Fitzhugh,  Lisa Hansel, Bruyckere-Kirschner-Hulshof, Steven Strogatz, Leslie Valiant, Barry Garelick, Ian Stewart, Zig Engelmann....  Reform math, including Common Core, which typically is implemented as reform math, has screwed up standard arithmetic by focusing on inefficient, alternative, non-standard procedures. It makes little sense to make standard arithmetic more complicated or difficult than it actually is, but this is what reform math does.  Plausibility, popular beliefs, and common opinion--without scientific evidence--often forge education policies. Comments can be made directly to

::::: Teach basic arithmetic to automaticity and focus on one efficient method to calculate each operation (the standard algorithms). To learn arithmetic well, students need to memorize single-digit number facts and practice standard algorithms, including long division. Essentially, this is what we used to do back in the 60s! Elementary school students also need to know place value, master fractions-decimals, use formulas and write/solve linear equations, grasp ratio-proportion, be able to work many types of word problems (problem solving), and learn parts of geometry, measurement, and algebra. The best way for children to learn these things is through explicit instruction by explaining carefully chosen worked examples and giving feedback to students. The teacher is not a facilitator but the academic leader in the classroom. 

Mathematics Professor W. Stephen Wilson, Johns Hopkins University, writes that without memorizing single-digit number facts for instant recall (factual knowledge) and gaining proficiency in standard algorithms (procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics.”  This is exactly the situation. The memorization of single-digit number facts (numbers) and the fluency in standard algorithms (whole number operations) "give students power over numbers" and are key "basic skills and knowledge that a solid elementary school mathematics foundation requires," writes Dr. Wilson. He continues, “The case for the importance of the standard algorithms for whole number operations cannot be overstated. They are amazingly powerful.... They give the operations structure.” Indeed, standard algorithms should be taught immediately. Inefficient, alternative algorithms should be pushed to the side as enrichment. 

Leslie Valiant (Probably Approximately Correct), Professor of Computer Science and Applied Mathematics at Harvard University, points out that even though there has been tension between learning and reasoning, the "primacy will be given to learning, but reasoning will still be essential." This fits the basic idea in cognitive science that learning knowledge in long-term memory is the solid rock on which higher-level thinking is made possible. Note. Jason Zimba, one of the two major writers of Common Core math standards, recently acknowledged, "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." Zimba's clarification connects well with the traditional teaching of arithmetic, often called old school. Old school math works well when taught well and is solidly supported by cognitive science, that knowledge, both factual and procedural, is the rock on which higher-order thinking is made possible. Just because something is old school does not mean it deserves to be tossed in the waste can, which is what hip reformers have done. Old school, in the teaching of arithmetic that prepares students for algebra, needs to be restored starting in 1st grade! The problem is that the "old school" teachers have all retired. There is hardly anyone left in teaching. 

Barry Garelick (Teaching in the 20th Century) is retired, has a math degree and real world experience applying math as an analyst. He embarked on a second career as a secondary school math teacher, but he has encountered difficulty finding a full-time teaching position where he lives in California. Administrators say he is too old to be hired as a full-time teacher or he is too "old school," or both, so he began as a substitute teacher for the 2013-2014 school year. He had two assignments: a long-term substitute math teacher at a middle school for 6 weeks and later at a high school for the entire 2nd semester.

Garelick explains that students should not be expected to solve far-fetched problems without prior knowledge and experience, only "variants of well-studied problems." He observes that good multiple-choice questions are valid ways to ascertain a student's math content achievement and reasoning. In addition, Garelick points out that the eight Common Core Standards for Mathematical Practice (SMPs) were written by non-math people and have been strongly criticized by several mathematicians and others. The SMPs represent mathematical expertise, but kids are novices. Also, during Common Core training, Garelick was told that "the content standards require students to work in groups, discuss, conjecture, critique each other arguments and that teachers are to be guides on the side." These ideas are unconvincing and do not come from content; they come from the SMPs, which are hardly effective in the teaching of mathematics to novices. In short, the SMPs are the essence of Common Core--not the content. However, in my opinion, the SMPs are unrealistic because they represent mathematical expertise. Kids are novices, so the SMPs are not a good fit. 
[Comment. The SMPs are nonsense mandates. Common Core reformers believe kids should become little mathematicians, that is, mathematical experts. Students should invent or discover math, work in groups, do lots of projects, critique mathematical arguments, and focus on novel or real-world problems that are far from the problem types they have studied. The mandates for mathematical expertise are unrealistic because kids are novices, not experts. Moreover, basic mathematics is not a matter of opinion or argument. It is based on true statements and consists of both factual and procedural knowledge. Ian Stewart points out, "Mathematics deduces new facts from old ones."

One SMP mandate is to make arithmetic [or any math] intentionally hard so that students struggle and learn to persevere. How stupid! The logic is flawed. Why make arithmetic harder and more complicated that it really is? Why insist that students learn inefficient, non-standard algorithms that they will never use for operations? Why give students far-fetched problems without the prior knowledge or experience needed to solve them? Is it any wonder that for multiple decades our kids have stumbled over simple arithmetic or algebra and lag far behind their peers in other nations?

Zig Engelmann rebuts the "dreaded" and "distasteful" SMPs especially for little kids: "Committees keep writing standards that are not based on empirical evidence of what children are able to learn about math and the specific technical details of instruction that cause the learning." Zig also points out, "The approaches that require less instructional time are superior to approaches that required more time." Thus, the popular inquiry, discovery, project, problem solving or other similar "minimal teacher guidance" approaches, along with the inefficient, alternative algorithms often taught as part of Common Core, misuse instructional time and often lead to "too little" learning. Efficiency in learning is vital, but not in the minds of reformers who stand by Common Core.  

Ian Stewart (Letters to a Young Mathematician) writes, "Mathematics deduces new facts from old ones.... New mathematical ideas build on older ones... Everything must fit together logically.... Mathematics happens to require a lot of basic knowledge and technique."End of Comment 

::::: Steven Strogatz, Professor of Applied Mathematics at Cornell University, writes, "Dad, can you show me how to do these multiplication problems?" Sure. "No Dad, that's not how we're supposed to do it. That's the old school method. Don't you know the lattice method?" No? Well, what about partial products?" [Comment. Like many parents, mathematicians, too, are deeply troubled with the content and methods being taught in school math. Inefficient, alternative methods for calculating (e.g., lattice, partial products, etc.) are commonplace in reform math, including Common Core, which doesn't stress standard algorithms. Reform math ideas, often championed by professors in schools of education, greatly influence the curriculum, but the ideas just don't work well and squander valuable instructional time. Why waste time with  non-standard algorithms that lead to remedial math? In contrast, the standard algorithms (old school) and the memorization of single-digit number facts (old school) get the job done when taught well. In elementary school, kids need to master the arithmetic that is required for algebra, but reform math slows progress and doesn't get them there. Under a reform math regime, kids never seem to master simple arithmetic. In 1st grade, students should begin to learn the five building blocks of arithmetic as applied to addition and subtraction.] See W. Stephen Wilson below. 

For at least 50 years, reformers, professors in schools of education, and others have belittled the traditional teaching of arithmetic, calling it obsolete and old school (throw it in the trash can), and branding it as poor teaching. I disagree with those who claim that traditional teaching of arithmetic is poor teaching or obsolete because the fundamentals of arithmetic and algebra, which need to be automated through memorization and  practice, don't change. 

You don’t need a calculator to do math well, but you do need to memorize key facts and practice procedures a lot. "Algebra boils down to solving for x and working with formulas," writes mathematician Steven Strogatz (The Joy of x). You don’t need to be a genius or have a Ph.D. to do math. With a place-value system, ordinary children all over the world learn to do math. The catch is that you need to know some math in long-term memory, mostly arithmetic, algebra, and parts of measurement and geometry, to perform math and solve problems. In short, you need factual and procedural knowledge in long-term memory. And, the best way to get key math knowledge, such as single-digit number facts and standard algorithms, both of which are essential, into long-term memory is through memorization, practice, and repetition--not group work, discovery activities, discussions, or projects. In short, you have to memorize, study, and practice a lot of math calculations and word problem types. On the other hand, you do not need to know why "everything in math works the way it does," but it helps. "Even things one learns by rote represent the substrate, the raw material, of understanding," writes Barry Garelick (Teaching Math in the 21st Century)The more math you learn, the smarter you become. Also, it is fanciful to believe that students can do critical thinking to solve math problems without knowing the prerequisite math content.

The idea that you can do math without knowing math originates from trendy reform math programs such as Pearson's Investigations, which is still used in many schools. The reforms emphasize critical thinking at the expense of learning and mastering arithmetic and algebra [knowledge], which, in turn, are needed for problem solving. The "conventional wisdom" today is that kids don't need knowledge in long-term memory to be successful in the future, which is the reason that memorization and practice have been slowly disappearing from many classrooms over the years. The conventional wisdom, of course, is dead wrong!  The traditional teaching of arithmetic (old school) works and reform math ideas haven't worked well for multiple decades.  

::::: W. Stephen Wilson, Professor of Mathematics, Johns Hopkins University writes that the five building blocks of knowledge and skills in elementary school mathematics (K-5) are: 
(1) numbers
(2) place value
(3) whole number operations
(4) fractions and decimals, and 
(5) problem solving. 
Early elementary school basics or building blocks do not change and should be taught to all students. For example, the memorization of single-digit number facts (numbers) and the fluency in standard algorithms (whole number operations) "give students power over numbers" and are key "basic skills and knowledge that a solid elementary school mathematics foundation requires." Professor W. Stephen Wilson explains, "Mathematics is built level by level. Multi-digit addition and multiplication are built up from single digit operations using the place value system and the basic properties of numbers such as distributivity. The general operations reduce to the single-digit number facts." The whole-number standard operations (procedural knowledge) depend on instant recall of single-digit number facts (factual knowledge) from long-term memory. Teach basic arithmetic first (to automaticity), and focus on one efficient method to calculate each operation (the standard algorithms).

After harsh criticism, Jason Zimba, one of the two main writers of the Common Core math standards, acknowledges flexibility in interpreting the standards. He points out, "The standards also allow for approaches in which the standard algorithm is introduced in grade 1, and in which only a single algorithm is taught for each operation." David G. Bonagura Jr. writes in the Wall Street Journal, "Contrary to today's education theories, memorization is critical in the classroom and life." He also writes in the National Review Online, "All students must learn to perform the basic mathematical operations of addition, subtraction, multiplication, and division in order to function well in society. Knowing why these operations work as they do is a great benefit, but it is not essential. In mathematics, concepts are often grasped long after students have mastered content--not before." Indeed, understanding grows gradually and changes over time, says the late Robert B. Davis (The Madison Project). 
Memorization plays a major role in learning basics.

Comment. Many claim that Common Core is about critical thinking, but the truth is, students can't do much higher-level thinking without a solid bedrock of lower-level thinking. Basic mathematical knowledge, both factual and procedural, which are lower-level thinking, need to be in long-term memory to enable higher-level thinking. Bonagura reminds us that higher-level thinking skills such as critical thinking or analysis are "impossible without first acquiring rock-solid knowledge of the foundational elements upon which the pyramid of cognition rests. Memorization is the most effective means to build that foundation." Indeed, "Memorization deserves to be reinstated to its foundational role in learning," not only in mathematics, but also in other disciplines such as science, English grammar, spelling, literature, and history, etc.

Regrettably, memorization and practice needed to automate fundamentals of arithmetic in long-term memory have fallen out of favor in many modern classrooms. Indeed, Common Core does not stress standard algorithms and substitutes inefficient, alternative algorithms for whole number operations, which is a faulty approach. 

Without memorizing single-digit number facts for instant recall (factual knowledge) and gaining proficiency in standard algorithms (procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics," writes Professor W. Stephen Wilson. 

(Note. Instead of teaching the standard algorithms straightforward, which requires students to memorize single-digit number facts for instant recall, Common Core often introduces inefficient or alternative algorithms, including the partial product and partial quotient algorithms. Consequently, students never get to or master the standard algorithms. The long-division standard algorithm, for example, is a lot more than just doing division. Students must have instant recall of single-digit multiplication facts, be able to subtract correctly, and be fluent in using the standard algorithm for multiplication. In addition, fractions are quotients, and long division changes them to decimal form. Why is 1/4 = .25? For example, 3/7 means 3 ÷ 7, which, through long division, shows an interesting repeating pattern: .428571428571.... "The long-division standard algorithm easily shows that rational numbers give rise to repeating decimals. It, by its very nature, also teaches estimation and begins to prepare students to understand convergence, a basic step toward calculus," writes Dr. W. Stephen Wilson.)

Reform Arithmetic vs Traditional Arithmetic
(Investigations, Grade 5, a popular reform math program)
Solution 2 is the standard algorithm, which should be the first and primary procedure for doing multiplication. It requires instant recall of single-digit multiplication facts and place value knowledge. In contrast, Solution 1 is an example of an inefficient, cumbersome procedure from reform math. Similar inefficient, alternative algorithms are promoted by Common Core. Common Core does not stress standard algorithms.

Note. Incidentally, the 5th grade multiplication problem from a popular reform math program is a typical 3rd grade practice problem in the traditional teaching of arithmetic. One of the best ways to practice the retrieval of single-digit number facts from long term memory is by stressing and practicing the standard algorithm, which is based on auto recall of single-digit number facts. 

In many of today's classrooms, literacy and numeracy are not that important. The trend against knowledge has been an epic error. Ze've Wurman says that teachers coming out of schools of education have an "intellectual emptiness" and are taught that "working well in groups is more important than knowing arithmetic, or that students being excited about learning trumps knowing history or science." But, engagement is engagement and not the same as learning, which makes changes in long-term memory. Learning requires memorization, practice, and repetition. Working in groups has nothing to do with learning math. The focus in school should be on literacy and numeracy, but, too often, it is not. (Quote Source: Ze've Wurman's "Forward" to Barry Garelick's new book Teaching Math in the 21st Century.) 

Professor Wilson says that we have lost the pro-arithmetic war. “How can I teach serious college level mathematics to students who are ill-prepared?" Ill-prepared means that the students don't know enough arithmetic or algebra to do college level mathematics. This is part of a larger trend in education, an assault on knowledge and liberal education, which extends from K-12 into the colleges and universities. The university should be more than a vocational school, says Fareed Zakaria (In Defense of a Liberal Education).

::::: Assault on a Liberal Education: Math, Science, and the Humanities
Many of today's "digital" students and reformers do not grasp or savor the intrinsic merit of math, science, and the humanities such as literature, history, art and music. Students no longer read books to advance their knowledge. The reformers say the era of books is gone. Will Fitzhugh (The Concord Review, 2005) writes, “Some educators have decided that we are in a “post-literate” era, where books, writing, and all those things we associate with the literate era of the past, are no longer so important." He continues, "Many educators also seem to have acquired the idea that students need almost no knowledge, because they can always “look it up,” presumably online." Regrettably, reformers have waged war on knowledge, even though domain knowledge in long-term memory is the rock on which higher-level thinking is made possible. There has been a shift from a knowledge-based education to a skills-based, practical education, along with a unintelligent bias of "out with the old" and "in with the new." 

"Out with the old," even if it worked, is twisted thinking and progressive ideology. If an idea, approach, or strategy is new, or uses the latest technology, or is technologically hip, then it must be innovative and good for digital kids. Yet, in education, almost all new ideas, reforms, or innovations have not grown to be productive programs. They are trendy fads that come and go. Similar to many progressive reform ideas that failed in the past, evidence isn't needed, just good intentions. Reformers assert that education needs to enter the 21st century--not redo the 20th century or the 19th century--no matter what. Old stuff, they say, has little value in today's world. Reformers chuck out old technology (blackboards, white boards) and stuff they don't like, such as long division or cursive writing. They cut back on the humanities and belittle arithmetic and algebra, etc. Liberal arts colleges are now less popular. The reformers, I think, are wrong! How dull this world would be without liberal education (science, math, and the humanities such as literature, languages, history, art, music, etc.). 

In contrast, scientists did not toss out Newton's Laws of Motion & Universal Gravitation because they are old. They work at the right scale and are key parts of classical physics. Likewise, educators should defend, maintain, and build on what has worked in the past, not discard it. Indeed, long-established teaching of traditional arithmetic has worked well when taught well and practiced well (i.e., for mastery). In traditional teaching of arithmetic, for example, the automation of factual and efficient procedural knowledge needed for problem solving is a good fit with the findings in cognitive science. Yet, the reformers call the traditional teaching of arithmetic obsolete and old school and brand it as poor teaching. 

::::: Fundamentals in math don't change. 
The math content in school math, such as, but not limited to arithmetic, algebra, and geometry, is old. Nonetheless, the traditional teaching of math is certainly not obsolete or outdated, as many reformers claim, because it forms a necessary foundation for higher math. The newer discoveries in math are for mathematicians (Ph.D. level) and, therefore, are far, far above the K-12 school or regular collegiate math curriculum levels, so there is little chance that the latest math content will reshape or influence the math fundamentals taught in K-12 school math. The idea that math knowledge is changing so rapidly that it is impossible to keep up with is nonsense at the K-12 school math level because the basics do not change. The fundamentals may be old, but they won't change because math content is hierarchical, cumulative and logical, that is, one idea builds on another, etc. The generation of the natural numbers start with two assumptions that 0 + 1 = 1 and 1 + 1 = 2 or n +1 = n'.

While technology can advance rapidly, the fundamentals of arithmetic and algebra stay the same. For example, multiplication is still multiplication, but simplified and much improved (i.e., standard algorithm) over the ancient Babylonian method. Unlike the ancient Babylonian method of multiplication using a table of squares (about 2000 BC), which is complicated, today's standard method of multiplication uses memorized single-digit multiplication facts, which makes it fast, efficient, and easy to learn. Students should learn and practice for mastery the most efficient procedural methods for operations, which, typically, are called the standard algorithms. (Unfortunately, Common Core reform math adds unnecessary complications that slow the acquisition of key content in long-term memory. Students learn inefficient, alternative algorithms. Common Core reform math does not simplify or improve mathematics. My message to reformers is simple. Don't make simple arithmetic harder than it is.) In short, no matter what the reformers claim or believe, the fundamentals of arithmetic and algebra are essential and need to be taught efficiently and mastered by most students to prepare for the future.   

FYI: One formula for Babylonian multiplication is 
ab = [(a + b)^2 - (a - b)^2]/4. 


There have been many attempts to upend knowledge in schools and replace it with [out of context] skills. A basic tenet of cognitive science is that broad knowledge is needed for problems solving and critical thinking to meet the needs of the future. The knowledge that kids learn in school has not changed very much over time. Even though kids are bombarded with information, broad knowledge can only be obtained through the thoughtful study of the liberal arts.  Thus, the progressive, anti-knowledge movement, which is prevalent in schools today, is irrational. Knowledge in long-term memory fires up creativity, innovation, and problem solving. This is basic cognitive science, that higher level thinking (problem solving or critical thinking) is the product of lower level thinking (knowledge).

Lisa Hansel (The Core Knowledge Blog) wrote, "I hear this all the time that information is growing at a shocking rate, and that today's knowledge will be out of date before students graduate. Obviously, students don't need knowledge, they need to learn how to find knowledge." The solution is Google, reformers say. Well, actually, no! Kids need to study the liberal arts in a well-rounded curriculum to gain organized knowledge in long-term memory. Googling information is not the same as acquiring knowledge in long-term memory, which takes practice.

In short, very little of the information born from the latest, cutting-edge research affects elementary and secondary education or even college, say Bruyckere, Kirschner, & Hulshof (Urban Myths about Learning and Education). "The fact is that much or most of what has passed for knowledge in previous generations is still valid and useful."  In my opinion, it is irresponsible to disparage knowledge or to say that kids don't need knowledge. Knowledge is the rock on which higher level thinking is made possible.

Please excuse typos and other errors in this post. Rough Draft
Model Credit: Remi

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