Wednesday, December 30, 2015

Adding It Up

Adapting Thinking: Adding It Up 

Why is a 2nd-3rd-grade question given to 8th graders? Expectations are low. 

Only 61% of 13-year olds selected the correct answer. In my view, it is a 2nd-grade question, not a middle school question, and clearly indicates the fundamental relationship between addition and subtraction that all 13-year olds should know, but, apparently, many don't. If arithmetic were taught well, then most 2nd graders would have selected the correct answer without calculating. The example from Adding It Up (2001) [1] shows how poorly arithmetic has been taught under NCTM reform math. However, in my opinion, Adding It Up seems to confirm many reform math practices while ignoring the science of learning. For decades, the use of calculators and so on, which are typical NCTM math reforms, have pushed aside standard arithmetic in K-8 schools, an error in judgment. 

Adding It Up: "Only 61% of 13-year-olds chose the right answer, which again is considerably lower than the percentage of students who can compute the result." What percentage might that be? 90%? 100%? Adding It Up erroneously assumes or suggests that students who practice standard algorithms for mastery have little understanding of number relationships. The reason that only 61% of the 8th graders selected the correct equation, rather than a higher percentage, is that the fundamentals of arithmetic (via NCTM reform math) have not been taught well. The relationship between addition and subtraction is basic arithmetic, but so is competence in calculating via the standard algorithms (paper-pencil). Also, G. Polya (How To Solve It) states that understanding in mathematics is in the doing of arithmetic, i.e., applying it. 

Note. The Adding It Up report (PreK-8) of 2001 from the National Research Council is hardly the final word, of course. The Adding It Up theory of proficiency in mathematics is a fabricated on five intertwined strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The far-reaching theory of proficiency is based more on judgment than on science and has never been tested. In fact, there are countries that clobber US students in math, yet the Adding It Up report asserts that "no country--not even those performing highest on international surveys of mathematics achievement do all students display mathematical proficiency as we have defined it in this report." Put simply, the five intertwined strands of proficiency are not practical and almost impossible for typical kids to achieve, even the best kids.   

Adding It Up provides cover for NCTM reform math programs, such as Investigations (TERC), a program that is still used in many schools and embodies the math reform movement that focuses more on understanding than on learning standard arithmetic. The Investigations curriculum uses "minimal guidance during instruction" methods, that is, child-centered discovery activities. After examining the 5th-grade materials, mathematician W. Stephen Wilson (Johns Hopkins University), wrote that Investigations was not standard arithmetic. He called it pre-arithmetic. Professor Wilson writes, "Arithmetic is the foundation. Arithmetic has to be a priority, and it has to be done right." Starting in 1st grade, Singapore math does it right most of the time [2]; however, Investigations and other reform math programs do not.

Adding It Up has had a profound influence in math education, and, often, not in a good way. Its central premise is that proficiency is too narrowly defined. The report states, "Mathematical proficiency, as we see it, has five (intertwined) strands." Really? More judgment, less research. Also, the report states, "Many educational questions, however, cannot be answered by research." Education depends on "judgments" that "often fall outside the domain of research," especially in curriculum and instruction. Really?

I disagree. Math is hierarchical: one idea builds another and everything fits together logically. A good math curriculum starts with standard arithmetic. We know the essential content and skills (the curriculum) needed to get kids off to a world-class start starting in 1st grade. Indeed, content and its associated skills are hierarchical, along with intellectual skills (Gagne: instructional design and prerequisites) [3]. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount [4].

Moreover, Gagne writes that "intellectual skills are arranged in a hierarchical order so that successful instruction begins with teaching lower-order skills and progresses upwards." Furthermore, in addition to arithmetic, the elementary school math curriculum should include parts of algebra, geometry, and measurement to prepare for a full course in algebra by middle school. Also, we know from cognitive science that direct instruction is strikingly more efficient than the favored minimal guidance methods of teaching, group work, nonstandard algorithms, manipulatives, and multiple representations, which are among the least effective. In short, the diverse group of Adding It Up writers ignores the cognitive science of learning.

Despite what you may have heard from reform math apostles, there is nothing intrinsically wrong with standard arithmetic. Indeed, it is the keystone for higher-level math. Therefore, very young students should practice standard algorithms for mastery, grasp the rules of arithmetic that govern the behavior of numbers, memorize math facts for auto recall in problem-solving, and apply math concepts to everyday problems.

Beginners need lots of factual and procedural knowledge to do the math, says Daniel Willingham, a cognitive scientist. Indeed, knowing and doing math well requires factual and procedural knowledge in long-term memory. The modern reform math methods of minimal guidance during instruction (e.g., discovery, inquiry, problem-based, etc.) don't work, say, Kirschner, Sweller, & Clark. They point out, "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture [working and long-term memories], expert-novice differences, and cognitive load." 

Children are not pint-sized mathematicians or experts; they are novices. Contrary to Adding It Up, beginners don't need to explain their reasoning, make drawings, or engage in group work to learn arithmetic well. Students need straightforward instruction via carefully thought out, coherent, hierarchically organized worked examples. The standard algorithms are the most efficient ways to do arithmetic, but for years, they have been under brutal attack. Reformists say the standard algorithms are too hard for some students to learn, threaten a student's growth in independent thinking, and are obsoleted by calculators. These are bogus arguments. The National Mathematics Advisory Panel (2008) explicitly stated that students must master standard arithmetic to prepare for algebra, not something that looks like arithmetic or something that has no long-term value. 

According to Adding It Up, "Nearly all second graders might be expected to make a useful drawing of the situation portrayed in an arithmetic word problem as a step toward solving it." In short, making a picture is the first step needed to solve a word problem. Nonsense! The idea of "making a drawing" as a necessary step for problem-solving has emerged as a best practice in reform math programs via NCTM and now Common Core state standards, etc. The idea is misguided.

Add It Up Box 5.15 Never teach these strategies. Teach the standard algorithm.

Adding It Up sharply criticizes the standard algorithm for division. I won't go into details, but the report is wrong.

Instead, the report offers two alternative versions. One is Box 5-15. The other is the "partial quotients" model shown in Box 5-16 below. 

No one uses the partial quotient method to do long division (Box 5-16) or the area model to do multiplication, etc., much less the methods shown in Box 6-15. Who would calculate this way? They are a waste of classroom time.

Add It Up Box 5-16 Teach the standard algorithm, not this.

Still, in modern reform math classrooms, a disproportionate amount of classroom time is spent on these and other similar calculation strategies leaving efficient standard algorithms, which are vital, on the back burner. In my opinion, nonstandard, complicated, multiple strategies to do simple arithmetic are usually a waste of valuable instructional time. Kids need to know the standard algorithms (Box 5-14)

The long division standard algorithm should start no later than 3rd grade with up to 4 digits divided by one digit, sometimes two digits.
This is the standard algorithm. Teach it first.

[4] Break a problem into smaller problems. 

This is a fundamental idea taught in mathematics, and it can carry over to everyday life. Also, the idea that new knowledge builds on old knowledge is central to learning math. Because math builds in long-term memory, the proper sequencing that creates coherence (a learning hierarchy) in a math curriculum is paramount. Here is a sequencing example from Science--A Process Approach (SAPA), which uses Gagne’s hierarchical approach. It is not hit and miss. The sequencing (learning hierarchy) must work in the classroom, which is the reason SAPA was tested extensively and rewritten several times before it was released to the public.

Part C is 2nd Grade - 1967
FYI: Integers were introduced in Part B, 1st Grade. 

[1] Adding It Up is a product of the National Research Council, specifically the Mathematics Learning Study Committee, Division of Behavioral and Social Sciences and Education (2001). According to the report, Adding It Up was written by a committee composed of "diverse backgrounds." Its theory of proficiency in mathematics is based on five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The theory is based more on judgment than on science and never tested. What, no mathematicians?

[2] Singapore 1st-grade students learn much more standard arithmetic than American 1st-grade students and so on up the grades. The curriculum is better than in most countries. Still, the Singaporean 1st-grade math curriculum isn't perfect. In my view, it lacks some essential content, especially algebra and integers, topics I typically teach to 1st-grade students. Moreover, the overemphasis on bar models (drawings) to solve arithmetic problems in Singaporean math can be distracting. A few kids might benefit from drawing bar models, but, for many kids, making a drawing slows up and disrupts cognitive activity. 

[3] Robert Gagne greatly influenced the hierarchy of Science A Process Approach (SAPA) by identifying the processes and prerequisites: observing, classifying, using numbers, measuring, predicting, inferring, formulating hypotheses, and interpreting data. But these processes are actually skills, which are essential to inquiry, analytic thinking, and problem-solving, explains Henry P. Cole (Process Education, 1972). The processes are actually skills of doing something, so they are measurable. The thinking is hidden in the doing.  

To Be Continued. 

©2016 LT/ThinkAlgebra


Friday, November 20, 2015

Mathematical Language

Mathematical Models (Equations)

Area Model [5th Grade] Add it up!
I hope teachers toss out 
this junk and focus on 
standard algorithms.

What has happened to simple arithmetic? The uncomplicated answer is that inferior methods from reform math have been in vogue for decades. For example, in reform math via State Standards and Common Core, students are often asked to make drawings, such as the area model (5.3 x 2.4), to calculate or justify their math, which I think is confusing, pointless, and useless for novices. Indeed, popular reform math methods make simple arithmetic unduly and ridiculously complicated. 

The modern "visual" approach, such as the area model (left), in my opinion, wastes valuable classroom time on pointless material that leads nowhere. Put simply, no one calculates this way! Why should kids labor over nonessentials? ( In contrast to American math instruction, kids in top-performing nations drill and memorize important math facts and practice for mastery efficient procedures for abstract operations starting in the 1st-grade. Indeed, 1st-graders in Singapore carry and borrow (regroup) in standard addition and subtraction calculations, do multiplication as repeated addition and write equations in three operations--addition, subtraction, and multiplication--from word problems. Our 1st-graders do not come close. )

The "visual" approach seems confusing and needlessly complex. It inhibits the learning of standard arithmetic by restricting the time spent on learning the standard algorithms. Standard algorithms are often minimized and portrayed as merely one of many ways to calculate--even discouraged in many classrooms. The efficient use of standard algorithms requires the memorization of single-digit math facts in long-term memory. Reformers claim that "sketching visuals" is needed to show understanding or as justification for answers, which is reform math hype. Regrettably, the importance of numerical relationships and their symbolic representations in mathematical language, which are important for understanding standard arithmetic, have been undervalued, trivialized, and delayed. Children are novices; they need to memorize single-digit math facts, practice standard algorithms for mastery, use mathematical language, and mathematize word problems. 

The symbols of math, detached from physical content, and the abstract operations of arithmetic are the very essence of algebra. Abstract rules, such as a + b = b + a, or (a + b) + c = a + (b + c), or a(b + c) = ab + ac, and others, govern all of arithmetic, including the standard algorithms. "The strength of arithmetic lies in its absolute generality. Its rules admit of no exceptions: they apply to all numbers," writes Tobias Dantzig (Number, 1930).  

Furthermore, an important part of understanding arithmetic-that-leads-to-algebra is the ability to do arithmetic quickly, efficiently, and effortless. Students should avoid doing calculations via complicated visuals or drawings (extra baggage), such as the area models, array models, charting models, bar models, or other drawings. In contrast, to do simple arithmetic, the student should use fast, efficient procedures that are the standard algorithms. The drawings often become the focus, and they distract from the straightforward mastery of basic arithmetic needed to advance to algebra and beyond. Doing math well is doing it as simply and as efficiently as possible, which is contrary to reform math methods. Reform math apostles argue that students understand math only when they can make a "drawing" or write an "explanation" as a justification for an answer. They are wrong!

In my opinion, instructional time is better spent on standard arithmetic, writing equations and finding solutions to solve problems (aka mathematizing). It is important for students to express numerical relationships in abstract, mathematical language, such as y = 3x - 1. 

The understanding of mathematics is rooted in the meaning of symbolic mathematical language, not in drawing visuals, etc. It is rooted in the abstract, not in the concrete. 

The 4th-grade student figured out the rule
and wrote an equation. 

Basic algebra is accessible to very young children when it is fused to the fundamentals of standard arithmetic. My "early algebra" program (Teach Kids Algebra) is an attempt to do this in actual classrooms. I use x-y tables as a stepping stone for figuring out function rules that lead to writing equations in two variables. Students can find function rules to complete x-y tables, write linear equations, work backward with inverse concepts (undo) to find x when given y, and graph tables on a coordinate plane. The key part is writing an equation. I have used function rules and building tables as methods for writing equations as early as 1st-grade in my Teach Kids Algebra program. By 4th-grade, the equations are more difficult (See table: left). The inverse concept is an important and useful mathematical concept, not only for table building but also for solving equations. Unfortunately, inverses are seldom taught in early elementary school. It seems that educators think that inverse ideas to solve equations are too advanced. I disagree. Even 1st-grade students can understand simple inverse ideas: 6 + 4 - 4 = 6 or n - 5 + 5 = n. It is simple to demonstrate and easy to learn by reasoning. 

"Mathematical expressions and sentences [equations] can be applied to real life situations to describe numerical relationships. The same mathematical expression or sentence [aka equation] may represent the numerical facts in more than one situation." (Dolciani & Wooton, Modern Algebra, 1970 ) The numerical relationships are mathematical models.

[Special Note. Reform math people think the best way to make sense of math is through creating visuals, such as charts, graphs, and diagrams, not symbolic representations. I think this is superficial because students should focus on the symbolic representation, not drawings. The reformers claim a better way to learn algebra is through tables and graphs made on a graphing calculator. If that were true, then our students would be the best algebra students in the world. Unfortunately, the graphing calculator is required for most algebra classes and the SAT. The overuse of graphing calculators to solve problems in algebra textbooks is often at the expense of symbolic representations and manipulation. 

What is important, I think, is that young students use mathematical language to write expressions and equations and use algebra concepts to solve equations. Too often, our math programs do not stress abstract, symbolic representation enough. Moreover, we should not insist that students make drawings or write an explanation to justify answers. Students need to mathematize word problems via the language of math (aka equations). Then, they need to apply algebra concepts (inverses) to solve the equations. In short, students need to write and solve equations. For example, my 1st-grade students figured out function rules, wrote equations, built tables, and plotted the graphs in Q-1.] 

Mathematical models are written in mathematical language: symbols for the known numbers, the unknown (variables), operations, calculations, solutions, etc. Furthermore, mathematical models imply a level of understanding, knowledge, and skills, which are substantially better than making visuals or writing an explanation as an afterthought. Students must be able to convert (mathematize) a word problem into mathematical language correctly. In short, the student must be able to abstract what is known (the numbers), the unknown, the operations needed, and then synthesize them into a coherent whole, that is, an equation that corresponds to or models directly the word problem.

Writing equations in the 1st grade to describe
word problem situations is a key skill. 
Converting a word problem into mathematical language (aka an equation) should start in early elementary school. To solve equations, 1st-grade students can use guess and check, the rule for substituting and memorized math facts to find the "unknown." By the 3rd- or 4th-grade, well-trained students should change over to ab efficient algebraic technique, which is "unpacking" via inverse operations (UNDO). The technique is very important in algebra. 

Writing and solving equations that lead directly to a correct solution requires adequate (prerequisite) mathematical knowledge and thinking. The steps are enough to imply understanding. Indeed, writing an equation and solving it in a sequence of logical steps to find the unknown is the root of understanding in mathematics.

To mathematize a word problem, students need to separate the known numbers, the operation(s) needed, and the unknown from the words and then put the symbols together in an equation that corresponds to the problem and, when solved or calculated, leads to a correct solution.

Teaching children to write and think in mathematical language is not an easy task. Traditionally, mathematizing has been a troublesome area in our math programs. Still, kids must learn to write and think systematically in mathematical language, beginning in 1st-grade, which is when Singapore children start to write equations in one variable.

Note. In the equation n + 12 = 45, to undo add 12, subtract 12 from both sides of the equation to isolate n and maintain the balance (equality). Algebra focuses on equations. It is vital that young children turn sentences or word problems into math language (aka an equation) and practice for mastery the algebra techniques for solving equations, which means to isolate the variable. To isolate the variable is to get the variable n by itself on one side of the equation, such as n = 33. To do it (method), we undo operations. "For the two sides of the equation to stay equal, whatever we do to one side has to be done to the other."


Sample 3rd-Grade Problem
Sally has some pencils. Ben gives her 79 more pencils. Now Sally has 167 pencils. How many pencils did Sally have at the beginning? (Think this way: The known numbers are 79 and 167the operation is +, and the unknown is n. With these abstract symbols (aka math language) think up an equation that closely models the word problem and leads to a correct solution when solved. See below.) 

Mathematical Models
Applying Mathematical Language & Thinking

n + 79 = 167
n + 79 = 167 (calculating, solving)
     -79    -79
n +  0   = 167 - 79
n = 88

88 + 79 = 167 (checking)
167 = 167 (true)

The model in mathematical language becomes 88 + 79 = 167.
The steps shown above imply adequate understanding.
Drawings or explanations are not needed or helpful.
Nonmathematical Language: Sally started with 88 pencils.


Note Well. First-grade and second-grade students should start with true/false concepts, guess and check, rule for substituting, and memorized math facts to solve simple equations, such as x + x - 3 = 7, but experienced 2nd-grade and most 3rd-grade students should leave guess and check behind and advance to a fundamental algebra idea that equations can be solved by "undo" operations (inverses) to isolate the variable. Thus, for the equation n + 79 = 167, to undo add 79, subtract 79. Insist that students show the steps and standard calculations needed to communicate and express their understanding. Addition and subtraction are opposites or inverses of each other and undo each other. Thus, n + 79 - 79 is n + 0 or n. Also, the operation of subtracting 79 must be applied to both sides of the equation to keep the equation balanced (equal). Multiplication and division are inverses, too, and undo each other. When working with equations, students should always Think Like A Balance. 

Note. The model 88 + 79 = 167 applies to many different mathematical situations, including the pencil problem above. It is the power of abstraction in mathematics. A simple math fact, such as 2+3=5, is a mathematical model but, because it is commonplace, we do not think of it as a model. Despite its simplicity, the 2+3=5 addition fact is a very powerful model and applies to many different concrete situations found in the real world, which is the point the late Morris Kline makes in his book Mathematics for the Nonmathematician, 1967.

[Aside. Morris Kline, a mathematician, writes, "When a child learns that 5 + 5 = 10 [or 36 ÷ 9 = 4, etc.], he acquires in one swoop a fact which applies to hundreds of situations. Part of the secret of the power of mathematics is that it deals with abstractionsWhole numbers and fractions and the various operations with whole numbers and fractions are abstractions."]

Put simply, arithmetic is abstract and should be taught through conceptual symbols.
Its understanding is rooted in abstract, symbolic language. The problem with arithmetic today is that we have gotten away from its symbolic structure and substituted pseudo-mathematical models and thought such as the area model or the array model.

Unfortunately, we have underestimated the key importance of standard algorithms (fast, efficient procedures), which are part of the wonderful, abstract tapestry of mathematics, and the memorization of single-digit math facts for auto recall in problem-solving, which is a basic tenet of cognitive science (the relationship between working memory and long-term memory). Practice and memorization are part of learning the structure and method of arithmetic and algebra.

Put simply, for decades we have been teaching math poorly. The reason is that progressive reform math has been taught while tried-and-true standard arithmetic, memorization, and practice have been deemphasized. We need to restore traditional arithmetic, both structure and methods, that prepares students for algebra in middle school.


[Extra. Demonstrate that the expressions 56 • 1/8 and 7 are equivalent by stating the reason that justifies each step. Showing the steps implies adequate understanding. The steps, themselves, are enough; however, students should also know the reasons that govern each step, so they do correct mathematics. 

An equation consists of two expressions set equal to each other. The rules of arithmetic and algebra consist of assumptions, principles, definitions, axioms, properties, and conventions. Some equations contain variables such as x + 3x = 200. The coefficients of x and 3x are 1 and 3 respectively. We do not write 1x just x. Also, 3x is a product and means 3 • x or x + x + x. Students must know the rules and use them. Expressions and equations show numerical relationships.] 


Dr. Katharine Beals (Out in Left Field), summarizes reform math: "Along with group work, group discovery, multiple solutions, and, of course, explaining answers to easy problems, there's doing math visually." I think Beals is right. The problem with all this is that kids do not learn much content. Early on, elementary students are not required to work with abstract, symbolic representations, which are so vital to understanding arithmetic and algebra. 

Relying too much on visual representations often downplays the importance of mathematical language that is the writing of equations from word problems (mathematizing), the use of algebra concepts to solve the equations, and the learning of conceptual symbols to do arithmetic. The problem starts in the lower elementary school where students use counting strategies (via visuals, pictures, manipulatives, etc.) to do simple arithmetic. I am disturbed that students are often required to make a drawing for a world problem or write an explanation for easy arithmetic. The meaning of mathematics is rooted in mathematical language, not in making visuals, etc.  It is rooted in the abstract, not in the concrete. 

To Be Revised

Last update: 11-24-15, 11-28-15, 11-29-15, 11-30-15

Model: Jayne
Some ideas of mathematizing from Numbers by Alfred S. Posamentier & Bernd Thaller
The idea of showing that 5 •1/8 and 7 are equivalent expressions is from Modern Algebra by Dolciani & Wooton 
Area Model from Kaplan: A Parent's Guide to the Common Core, Grade 5

Tuesday, September 22, 2015

Random Thoughts #4

Random Thoughts #4 (in no particular order)

Our kids are locked into a 
Common Core test culture! 

The Decline of Academics 
The Rise of Anti-Intellectualism 

In the US, sports trump academics, and jocks outclass geeks by a light year. Frequently, the decisions people make are about 30% rational and 70% emotional, says Jim Clifton, CEO Gallup. In fact, our feeling-based society often devalues individual academic achievement and academics in general. In many classrooms, excelling is not cool. "Reading is for losers. Math is for geeks," explains Greg Gutfeld (Not Cool). Scientific evidence is viewed as just another opinion, which, perhaps, partially explains why reformists seem to ignore the Science of Learning, 2015.

Also, there's a lot of talk about good schools, improving education, especially by using the latest technology, and so forth, but the narrative is mostly about good intentions and untested fads (aka innovations), such as Common Core. Feel-good education policies, fads, and notions originate from good intentions, I'm sure, but most innovations (notions, fads, etc.) fail because they lack solid evidence of effectiveness. Indeed, evidence or facts don't seem to matter much to progressive reformists who have an unrealistic passion for the new (aka innovation) and an irrational hostility for the old. Many widespread, favored classroom practices of today are not supported by evidence and are among the least effective, yet reform math people don't seen to care.

But "new" doesn't necessarily mean "better." In education, we have been spending billions and billions on innovations (reforms, fads, etc.)--especially on the latest technology--and hoping for the best. Over the decades, increasing technology use in the classroom has not turned into better student achievement. In fact, many popular and trendy reforms or fads in education are counterproductive. In contrast, some of the old stuff (i.e., old school), such as standard arithmetic and explicit instruction with worked examples are very effective when taught well. It is not enough to know some math, which is a good start; it is also important to know how to apply what you know in math. 

[Aside. This post consists of random, often contrarian thoughts in no particular order. It is in rough draft form, so please excuse typos and errors. I repeat myself, a lot. Latest additions or updates: 10-11-15]

"The best way to know if an idea is right is to see if it predicts the future," writes Steve Pinker (Harvard). The fallacy of many education policies and innovations (fads) is that they start as good intentions (feeling-based), not via the science of learning (cognitive science). We often rely on the ability of so-called experts to predict the future, but the presumed experts often make policies and claims that fall flat because they are supported by ideology or beliefs, not valid evidence.

Furthermore, most teachers were not taught the cognitive science of learning, which is an "evidence-based core of what educators should know about learning," such as the critical role of practice to push knowledge into long-term memory, or the "understanding of new ideas via examples," or the fact that children are novices and don't think like experienced adults, etc. (Quotes/Ideas: The Science of Learning, 2015)

Moreover, I am not surprised that more technology use in schools has been linked to lower test scores, according to the OECD, but this will not slow the tech stampede into our schools at an enormous cost, with little value to actual student achievement. Tech is not the silver bullet. (I include calculators, smartboards, software, etc. as technology.) Also, I cringe every time a policy or notion is advanced and publicized as "for the kids," which--when put under scrutiny--is more "for the adults or special interests." [Aside. The OECD implements PISA, an international test for 15-year olds. According to the OECD, one weakness is, "U.S. students have particular problems with mathematical literacy tasks where the students have to use the mathematics they [should] have learned in a well-founded manner." In short, too many students don't know basic arithmetic skills, such as "using the number pi in calculations." Our students are incredibly shortchanged, not only in basic math knowledge, both factual and procedural but also in being able to apply or utilize that knowledge.]

Our kids are locked in a Common Core test culture (Click). 
Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Equally wrong is expecting students to do critical thinking without sufficient background knowledge in long-term memory. Sadly, schooling has been entrenched in accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. This approach may be okay for business, but K-12 schooling is not a business, and it is not okay. 

"I am a novice, not a pint-size mathematician."
We overload the working memory of beginners with extras. 

Children learning arithmetic or algebra should not be using calculators or over-burdened with questionable and annoying extras, such as indeterminate "deep" understanding, confusing and inefficient multiple models/strategies (as stressed in reform math), unrealistic Common Core Mathematical Practices, unjustifiable group work/collaboration, far-fetched, misguided real-world problems, time-wasting discovery/inquiry activities, or paragraph writing. All these extras are from reform math.  

[Aside. The "extras" are based on adult "thinking," not in the cognitive science of learning.Students are novices. They are not experts; they are not peer math teachers; they are not writers of math; they are not little mathematicians; they are not miniature adults. "Novices and experts cannot think in all the same ways (The Science of Learning, 2015)." 

We teach reform math via Common Core instead of traditional arithmetic. Reform math people oppose standard algorithms and substitute many different, inefficient, non-standard alternatives as the primary methods of calculation. Consequently, there is little time left to spend on standard algorithms. In short, many students do not automate basic K-6 arithmetic, which is necessary for a valid algebra course.  

Children are beginners, not pint-size mathematicians.
"Don't expect novices to learn by doing what experts do," writes cognitive scientist Daniel Willingham (Why Don't Students...). The reasons are simple. Kids lack both background knowledge and experience to do anything even remotely close to what mathematicians and scientists do. Furthermore, children do not think like experienced adults. They are not little mathematicians, junior scientists, or little adults. A good example of flawed thinking--that kids should emulate what experts do--is the Standards for Mathematical Practices, which are lodged in progressive constructivism and are the silent backbone of Common Core reform math. Willingham explains, "There are significant differences between how experts and novices think." Consequently, instructing students to be creative, pint-sized mathematicians, that is, to emulate what mathematicians do, seems rather pointless. It is just another empty-headed idea that does not agree with cognitive science.  

Multiplication Facts should not be calculated as needed; they should be memorized.
In contrast to Common Core, students should first learn and practice the essentials of standard arithmetic for automaticity and solve routine problems first, not wordy, complicated word problems with extra information and certainly not far-fetched real-world problems, which are often championed by Common Core, even though students lack sufficient background knowledge and experience. For math facts, "Memory is more reliable than calculations (The Science of Learning, 2015).The multiplication math facts, for example, are implanted in the standard algorithms for multiplication and long division. Math facts should be memorized and repeatedly used over a period to stick in long term memory, not calculated as needed, which wastes time, increases errors, clutters working memory, and inhibits fluency in using standard algorithms. New knowledge builds on old knowledge. The more math content you know, the more content you can learn and the faster you can learn it, says Daniel T. Willingham.

Held Captive
In education, we are held captive to bad ideas, counterproductive reforms, and untested innovations [e.g., accountability, metrics, benchmarks, performance indicators, inclusion, NCLB, sameness, Race to the Top, Common Core, standardized testing, the 4Cs (critical thinking, collaboration, communication, and creativity), mathematical practices, etc.], and many of us, as educators, have convinced ourselves that the current reform approach (via Common Core, standardized testing, NCLB, etc.) is probably okay for kids; however, we cannot logically justify the reasons that kids get a steady diet of test prep (hence, not much education) and that Common Core reform math is the same [one size] for all students, without regard to abilities or achievement, which, in my opinion, is equalizing downward.

The road to mediocrity, decline and failure is paved with good intentions; feeling-based policies, mandates, reforms, notions, and trendy, evidence-lacking fads (often called innovations). I think, education, especially the latest vision of math curriculum and instruction, has been on this road before. I think teachers and parents can disrupt the most recent vision (reform math), which isn't new because it started in 1989 with the NCTM reform math era. Repackaging old failures as innovations seems commonplace in education. 

The Re-definition
Under Common Core, math education has undergone a "re-definition" that focuses on real-life or real-world problems; hence, it required group work and calculators early on and diminished knowledge of standard arithmetic. Indeed, Common Core's expectation is a calculator dependent and dominated math curriculum. But H. Wu, mathematician (UC-Berkeley), refutes this narrow perspective. We should not "think of mathematics exclusively as a tool for solving real-world problems." Mathematics is a complex system, an "edifice," says Wu. It is a symbolic language in which the "symbols and equations of mathematics express not just ideas but the relations between ideas," writes Leonard Mlodinow (The Upright Thinkers). Note. "re-definition" is Wu's term regarding the New Math, but it is also applicable to the reform math Era of the NCTM then and Common Core now. 

Wrong Message: "I wasn't good at math either."
Sports do not operate in isolation, says Amanda Ripley (The Smartest Kids in the World): "Combined with less rigorous material, higher rates of child poverty and lower levels of teacher selectivity and training, the glorification of sports chipped away at the academic drive among US kids." Many kids believe math is merely one of several competing options and not high up on the list. Math is more abstract and, therefore, difficult than other subjects; consequently, many students avoid math, limiting their future. Many kids believe that they will get better at reading by practicing, but not in math. "You are either good at math, or you are not," which is a counterproductive belief not supported by cognitive science. Indeed, most kids, I think, can learn standard arithmetic and algebra well, but they have to work at it (drive) and be persistent (conscientiousness). And, according to Carol Dweck (Growth Mindset), we as educators and parents need to establish a proper mindset and stop telling kids: 
(1) Not everybody is good at math. Just do your best. 
(2) That's OK; maybe math is not one of your strengths. 
(3) Don't worry, you'll get it if you keep trying.
(4) Great Effort! You tried your best.
(5) I wasn't good at math either. 
Another conundrum is that modern reform math via Common Core is typically taught, not standard arithmetic. We need to prioritize math content and streamline the curriculum so that only essentials are taught and learned to automation. Not everything is important, but standard arithmetic is.  [Aside. As computer use increases, we need skilled workers who are reliable and competent, that is, we need "workers who are smarter, better trained, and more conscientious," writes Tyler Cowen (Average Is Over). "The premium is on conscientiousness." But, this is not the narrative we are fed.]

Arithmetic That Is Arithmetic: 1912
8th Grade Exam 1912 (The Arithmetic Part), Bullitt County Schools, KY
The arithmetic taught in 1912 is harder, in my opinion than the arithmetic taught in the late 20th century under NCTM reform math standards or, today, under Common Core reform math standards.

In 1912, kids did arithmetic using standard algorithms and paper pencil. No calculators, of course. In short, they were taught to calculate quickly, recognize key problem types, and apply straightforward arithmetic to solve questions.

[Aside. Here is a mental arithmetic question for 3rd/4th graders from Ray's Intellectual Arithmetic (1877), a 140 page textbook for 3rd/4th grade combined: If 12 peaches are worth 84 apples and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches? Indeed, a 140 page textbook for two grade levels is a novel idea compared to today's 4th grade 500-page enVisionMath.

Understanding Is a Matter of Degree.
A child's understanding of something is not the same as an adult's. A 1st grader's understanding of place value is not the same as a 5th grader's, etc.  
Understanding should be inferred via a student's ability to solve arithmetic questions, that is, by doing arithmetic, and not on selecting a nonstandard, "understanding-type" algorithm, or making a drawing, or writing an explanation, which are arguable points of reform math, such as in Common Core. And, to do arithmetic well presupposes that the student knows arithmetic well through study, memorization, and practice (drill for skill). Standard arithmetic knowledge in long-term memory, both factual, conceptual, and efficient procedural, is imperative, yet, kids, today, are not always required to master standard arithmetic. For example, standard algorithms are often delayed, marginalized, or not practiced enough. Typically, what is taught under the yoke of Common Core is the latest revision of NCTM reform math--a progressive ideology of sameness or equalizing downward and an ed theory of constructivism--via inefficient minimal guidance methods, such as discovery, or project, or problem-solving learning in group work; complicated, cumbersome multiple models or strategies to do simple arithmetic; so-called real-world questions that require calculators, etc. Below is an example of a 5th-grade parents guide to Common Core.

The Common Core Brand of K-12 Reform Math
#3. Multiply 5.3 by 2.4 using the area model (5th Grade Quiz). Show our work. 
[Aside. Unfortunately, educators are told to teach multiple models [many ways] to do simple arithmetic, not the standard algorithms that are efficient, easy to learn, and always work. Standard algorithms should be taught first, not put on the back burner. The area model shown below is total nonsense. When would a student use an area model to calculate products? It is pointless, useless, and ridiculous.] 
Screenshot above from my Math Notes in 2014: 

The New SAT Continues the Common Core Reform Math Brand.
The new SAT (2016) is a product of Common Core. It has nearly twice as many calculator questions as non-calculator questions. The heavy use of graphing calculators reflects the overall Common Core scheme: Let's concentrate school math on solving real-world problems so that kids use calculators. Calculator use in elementary school, as early as kindergarten, dates back to the failed NCTM reform math standards of 1989. The new SAT of 2016, which locksteps to K-12 Common Core, also overemphasizes data analysis, probability, and statistics. Students must rely on TI-84 graphing calculators for these topics, especially the statistics functions. The inclusion of these topics (and others) is another tactic used by reformists to defend calculator use among young students, even if their arithmetic and algebra knowledge and skills are weak.

The NEW SAT question type (#16) is, in my opinion, a 7th-grade pre-algebra level question, not a high school level. Note. A few of my Title I fifth graders in my Teach Kids Algebra program could figure this out, too. A well-prepared 7th-grade pre-algebra student should find the answer simply by examining the graph. No calculator is needed, just knowledge. The answer has to be either C or D because both have a y-intercept at -4. A quick "rise to run" check (1 to 3) means the slope is 1/3, not 3. In short, no calculator is needed, so why is this a SAT calculator-allowed question?  I can only guess, but, apparently, under Common Core reform math, the expectation is that most high school students will not gain sufficient knowledge of algebra fundamentals to figure this out without using a graphing calculator. It This is yet another example of dumbing down the math. The NEW SAT has nearly twice as many "calculator allowed" questions as "no calculator allowed" questions. It is cause for alarm! (Question Source: Kaplan 2016 SAT)
[Late Note. Perhaps, the question is considered a nonroutine problem because you need to know stuff, such as the y-intercept, slope, linear equation form, etc. and know how to figure these out from a visual as you apply the concepts to develop an equation in y = mx + b form. Gee that is knowing and using what you know. And it starts with knowledge.]

In my opinion, this calculator problem (#16) clearly illustrates the sharp difference between knowing basic math in long-term memory (and applying it) and Common Core's expectation of a calculator-dependent-dominated math curriculum. Unlike their peers in top-performing nations, American students are lost without calculators. They don't know math. They can't do simple calculations, such as 2.54 x 1000 or -7 + (13/17) + 7 without reaching for the calculator. (Answers: 2540; the fraction 13/17) Over the years, calculators have dumbed down math content via US reform math programs. [Note. For -7 + (13/17) + 7, the calculator spits out .7647058824. What does that mean? The student didn't know that -7 and 7 are opposites (inverses) and equal zero when added via the commutative and associative rules-7 + 7 = 0 or 7 + -7 = 0.]

[Aside. Common Core, like NCTM reform math, believes that real-world problem solving should be the focal point for K-12 school mathematics. What a terrible idea! Of course, this reform ideology "justifies" the frequent use of calculators, especially graphing calculators. What is lost? Fundamental math ideas and skills (factual, conceptual, and procedural knowledge) that are not linked directly to the real world problems are marginalized. With a calculator, it is now possible to calculate problems without knowing much math in long-term memory, so why memorize, or practice, or drill for skill? It is nonsense, of course. It is not the way novices master mathematics. Reform math, in my opinion, is an anti-knowledge approach

A similar anti-knowledge approach is found at the university level. For example, in Harvard Calculus, the student can pass calculus by using a graphing calculator and without knowing much algebra, says H. Wu, a mathematician at UC-Berkeley, who is anti-reform. It is a terrible idea! Also, students can pass a College Algebra course by using a graphing calculator. (FYI: Many College Algebra courses are about the same as a good high school Algebra 2 course.) Why should we be concerned about the reform math in K-12, Harvard Calculus, AP Calculus, etc.? Our kids, including many of our best kids, are weak in math fundamentals because we made them that way under the facade of reform math, first with NCTM, and now with Common Core. Arithmetic is no longer arithmetic, algebra is no longer algebra, and calculus is no longer calculus. Like most high school math courses, AP Calculus relies heavily on graphing calculators, which is one reason that a growing number of universities and colleges don't accept AP Calculus for college credit, not even a 5. Another reason is that AP skips important content. Also, there are no proofs required in AP calculus. In short, AP is simply not up to the university level, not only in calculus but also in several other subjects. End]

Indeed, calculators have dumbed down content in mathematics. In short, the graphing calculator has replaced knowledge and skills. Our kids are weak in math, and we made them that way. H. Wu adds that under reform math, there is "a serious lack of essential technical facility—the ability  to undertake numerical and algebraic calculation with fluency and accuracy." Common Core follows the same reform trend and avoids or delays standard algorithms.

Pattern Recognition
Pattern recognition comes from the experience of doing and studying many types of math questions. "Oh, this is a percent of change problem." Understanding the problem (pattern recognition) is much more important than understanding "why" an algorithm works, and I think, this is what reform math people and Common Core fail to grasp, but G. Polya (How to Solve it) did. He lectured, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." Polya points out: "[Give students] plenty of opportunity for imitation and practice.... You learn to do problems by doing them." Pattern recognition is the key to solving math questions. Understanding a standard algorithm is ancillary because understanding develops slowly. Understanding does not produce mastery; practice does!

Students need to learn patterns for different types of problems. Learning patterns requires substantial imitation, practice, and experience. Pattern recognition is the key to problem-solving, observes Ray Kurzweil, MIT, How to Create a Mind. Moreover, kids are novices and do not need to learn multiple model stuff or do discovery lessons in groups. As Richard Feynman once exclaimed: Why do elementary school children learn or practice stuff that has little value? Furthermore, K-6 teachers are generalists and don’t know enough math content to teach standard arithmetic well (or Common Core reform math) contends Dr. H. Wu in a recent article. Additionally, parents are confused and baffled because they cannot understand the convoluted reform math or what is going on. Likewise, kids are confused, frustrated, and don't get it.

Speed, Efficiency, & the Correct Answer
Furthermore, I have always thought that the purpose of learning standard arithmetic was to solve questions as efficiently and quickly as possible, but not according to reform math (via Common Core), which stresses learning many different ways to solve the same problem, most of which are time-consuming, confusing, and more complicated than standard arithmetic. When Common Core is interpreted as reform math, the standard algorithm is merely one of many ways to solve math questions, and it is often pushed to the back burner to focus on what I call the “understanding” ways to do math, which include drawings, writing explanations, nonstandard "understanding" procedures, invented algorithms, intermediate algorithms, etc. I think most of this "understanding stuff" is math education for teachers rather than standard arithmetic for kids. We should not be training kids to be little math teachers or young mathematicians.

Unfortunately, children seldom practice for mastery the best strategies, which typically are old-school arithmetic, such as the standard algorithms. The importance of standard algorithms has been marginalized by reformers, even though the intrinsic merits and fundamental importance of automating standard algorithms for novices have been substantiated by many mathematicians, including W. Stephen Wilson, H. H. Wu, and so on. While math questions can often be solved in different ways, teachers should emphasize speed and efficiency to get the right answer. In short, novice students should learn the most efficient ways to solve arithmetic questions from the get go. Don't clutter the minds of beginners with a bunch of non-standard algorithms (many ways) or ask them to make drawings or write explanations. Instead, teach students pattern recognition of problem types and calculating using standard algorithms first and later some tricks (shortcuts).  

Unfortunately, students are often asked to make drawings, or write explanations, or use inefficient algorithms or models that purport to show their understanding, such as the area model for multiplication or the partial quotient method for a division, etc. But, I think, this should not be the reason to study arithmetic. Usually, straightforward standard arithmetic or algebra is the best strategy, but it has been marginalized in Common Core and early reform math programs. Indeed, the “many ways” for “understanding” do not stand up to the scrutiny of cognitive science or even common sense. Why make arithmetic harder than it is?

I agree to disagree. 
Wagner & Dintersmith (Most Likely to Succeed) focus on preparing kids for the innovation era as if the 20th century were not an innovation era. It is what I think: We are not producing enough home-grown talent. Our exploding tech companies are short on STEM talent; hence, for decades, major high-tech companies have imported foreign talent because many could not find enough home-grown talent. Furthermore, many high-tech companies locate branches where the talent is: Asian nations. And, while our STEM graduate schools are simply the best in the world--at least for now--they still attract a host of international students. Some stay in the US to work while a growing number return to their native roots.

I agree with Wagner & Dintersmith that the business model and standardized test approach are counterproductive and should be discontinued. But, sadly, Dintersmith repackages progressive child-centered-ideology that failed in the past—child driven discussion and child-centered assessment, etc. Lectures (explaining with worked examples) are out and replaced by a project-based approach in which teachers are facilitators, not academic leaders who know the content. Dintersmith's approach is anti-knowledge. He thinks content [knowledge] people are old school.

I agree with Dintersmith that children should master core academic content, but I do not agree with his approach, which is child-centered-project-based learning. The progressive ideology of the four Cs (critical thinking, collaboration, communication, and creativity) has displaced memorization, repetition, and practice. In short, gaining knowledge in long-term memory is not that important. Frankly, kids don’t study arithmetic because it will make them more creative or collaborative, and so on. Ordinary thinking makes them creative. Kids study arithmetic to learn it (automate it) and to use it to solve problems.

Sameness Ideology
Progressives [aka liberals] postulate that fairness and equality should dominate education, not individual achievement or excellence. Thus, in my opinion, sameness or uniformity has become the mantra of the education business, which is test-driven. Indeed, Common Core is the centerpiece of the test-based reforms, which drives both curriculum and assessment and makes lofty promises of college and career readiness without evidence. Kids are told to follow their passion in college, but many kids end up without a job,  crushing student debt and live with their parents.

Math Talent is not being developed.
"It's Your Brain That Count!" 
I am not sure we know what to do to change the culture in low-income Title 1 schools. The liberal answer has been to put more money into these schools (e.g., Title 1 funds). But, we have neglected the most important reform, which is equalizing funding. But, even if this would happen, I am skeptical that there would be leapfrog-type improvements. I am sure of one thing, however.  I have found that teaching algebra lessons in the early grades (1-5) in a low-income Title 1 school works well when I explicitly teach complex content through worked examples with practice sheets I make up. And, even though these kids had come from low-income families, I discovered a lot of math talent distributed among them. It’s there, but it just isn’t being developed through test-based accountability reforms such as Common Core’s EnableNY. I agree with Amanda Ripley, who asserts, "Poor kids could learn more than they were learning." But, I disagree with the narrative from   Diane Ravitch and others that the main problem in our schools is poverty. In Finland, Heikki Vuorinen says the opposite. He is quoted by Amanda Ripley: "Wealth doesn't mean a thing. It's your brain that counts." Vuorinen's message conflicts with the popular poverty narrative found in the US. The Finns didn't wait until poverty was cured to change their educational system.

Smart low-income kids cannot get to real Algebra 1 in middle school because Common Core reform math pushes Algebra 1 to high school. Also, CC is not set up for STEM. When I taught my algebra enrichment program (Teach Kids Algebra--TKA), I found many bright minority students, but, under the grip of Common Core in which everyone gets the same, they won't get to real Algebra 1 in middle school. No students will.

What does fluency mean in Common Core or to progressive reformists? It doesn’t mean practicing the standard algorithms for automation. To the Common Core people, fluency means to do something in many different ways, which allegedly implies a deeper understanding.  In Common Core, as I understand it, the standard algorithm is merely one of the many ways, often not the preferred way. That said, standard algorithms are not practiced for mastery. The fact that standard multiplication algorithm is pushed from 3rd to 5th grade validates the motives of the writers that the standard algorithms are just not all that important, which is a shift away from standard arithmetic. All those cognitive models, or strategies or many ways in Common Core's EngageNY scripted curriculum apparently are much more important because they allegedly demonstrate fluency and deeper understanding, even though evidence supporting such claims is lacking when placed under scrutiny. In contrast, standard arithmetic, when explained well and taught for automation (fluency), has always worked well for most kids.

Coming Soon

©2015 LT/

Credits: Caitlin