Wednesday, July 30, 2014


Single-Digit Math Facts Retrieval v.s. Calculating
According to Price, Mazzocco,& Ansari (The Journal of Neuroscience), students who memorize by rote single-digit math facts starting in 1st grade, rather than always calculating them, become good math students, and the benefits reach far into high school with much higher PSAT scores. Being able to retrieve 5 + 6 = 11 automatically from long-term memory is different from [and better than] calculating it using a strategy (e.g., 5 + 6 = 5 + 5 + 1 = 10 + 1 = 11), which is a different mental process that clutters working memory, leaving less space for problem solving. Even though mathematicians repeatedly have stated that kids must learn facts by rote, which requires a lot of memorization, practice, and constant use, many US educators have gone a different road, that of calculating by using strategies (e.g., NCTM reform math, now Common Core reform math). 

Why are Asisan students superior in math? 
A January 2013 article in The Journal of Neuroscience by Price, Mazzocco, & Ansari confirms the rote learners. The reason that Asian kids excel in math, while US kids struggle, is that Asian kids memorize by rote the single digit math facts early on (1st grade, even earlier) and become proficient in using the standard algorithms at the same time. Strategies (US approach) and calculators (Finland's approach) should not replace or delay the instant recall of single-digit math facts. Instant recall of single-digit math facts is necessary for proficiency in standard algorithms.

What about problem solving?
In TIMSS (4th & 8th grade students), students who scored at the Advanced Level are good problem solvers. Asian nations consistently have 40 to 50% of their 4th and 8th grade students scoring at the Advanced Level in TIMSS compared to only 7% of US students and 4% of Finland students. In fact, in TIMSS math, the US and Finnish students scored about the same [1].

The "rote-learners" from Asian nations are not only tops in international math tests (TIMSS, PISA-15 year olds), but also tops in "creative problem solving," a new test from PISA.  (The results were released earlier in 2014.)

Children who have auto recall will not be inclined to calculate basic facts; they will automatically retrieve them as needed. No matter, the strategy shown above (reasoning from doubles) won't work well without instant recall of 5 + 5. Kids often use strategies to remember 3 or 4 demon facts, which, at first, is okay. 

Too often, in US math programs the cart (Whys) is put ahead of the horse (Hows). US students are weak in procedures, such as standard algorithms, while Asian kids excel in them and in problem solving. 

Standard Algorithms are Key
Dr. H. Wu states, "The multiplication algorithm reduces the multiplication of all numbers to the multiplication of single-digit numbers, which is part of a general pattern. If you can add, subtract, multiply, and divide using single-digit numbers, then you can add, subtract, multiply, and divide all numbers no matter how big." This is the importance of standard algorithms: They are powerful and fast, and they are based on auto recall of math facts. Unfortunately, Common Core tends to ignore or delay standard algorithms, which is another way of saying that in Common Core reform math the memorization and practice of single-digit math facts early on are not that important. 

[1] If you analyze the TIMSS results (2011), the Finnish and US 4th and 8th grade students, on average, are almost the same with Finland having a very slight edge. However, 7% of US 8th grade students reach the TIMSS Advanced Benchmarks in math, compared to only 4% of Finland's 8th graders. [Problem solving is thought to be one of Finland's strengths, but this may not be the case.] In 4th grade, the percentages for the TIMSS Advanced Benchmarks are about the same: US 13%, Finland 12%. For perspective, 40% to 50% of 4th and 8th grade students in the top performing Asian nations reach the Advanced Benchmarks. See Finland

Recently, Pasi Sahlberg (How American Innovation Improved Finnish Education) wrote that Finland turned around its education system based on innovations in the US, which include progressive ideas (Deweyism: social interaction and group work), cooperative learning (group work), multiple intelligences, alternative assessments, and peer coaching. What turn around? Finnish 15-year-olds dropped from 1st to 12th in the matter of a decade (PISA). Their 4h and 8th graders are not any better than US kids (TIMSS). And, the innovations Dr. Sahlberg talks about were failures in US classrooms, which is a reason that our 4th and 8th graders are about the same in math achievement and problem solving.   

1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math) instead of tried and true standard algorithms to do arithmetic. 

2. See PiagetianMyth
Constructivism does not work in the classroom. 

3. See Memorization & Practice
Very young children can learn a lot more than we think or are prepared to teach. American children underperform in math! Parents should drill math facts at home before kids are school age. It is easy to show 4-year-olds addition using a number line. Early on, parents should provide the environment for achievement and teach their kids how to be successful in school. Early arithmetic at home pays off.     

4. See CommonCore 
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education. 
To Be Revised
7-30-14, 8-1-14
© 2004 LT/ThinkAlgebra

Thursday, July 10, 2014


[Note. I add information, rearrange parts, change a word or a sentence, so the post is not as coherent as I would like. I repeat myself. Maybe, repetition is a good thing. Please excuse typos and errors in grammar. This is a rough draft. It is longer than I want. If you have comments, email them to LT/ThinkAlgebra]  

Common Core 
Read Outcomes
Teach the real thing: arithmetic--not something that looks like arithmetic, but isn't.

The New Math (70s) & Common Core Reform Math (Today). 
There are a few similarities between the New Math of the 70s and the new Common Core Reform Math, which was adopted by most states before the final version became available, an unintelligent decision. The New Math fell out of favor because teachers didn’t know the content to teach it [so they resisted] and parents couldn’t make sense of it, says mathematician Dr. Evelyn Boyd Granville in a recent interview. Common Core follows the same path--teachers don't know enough content to teach it well, says mathematician H. Wu, and parents [and students] cannot make sense of it. See Quiz below. The standards may be better than before for many states, but they are not world-class, ignore STEM, are badly written, and insert reform math strategies (multiple models) that change simple arithmetic into arduous, non-essential clutter. Reform math has screwed up arithmetic for decades; consequently, most kids learn very little substantive arithmetic that prepares them for algebra by middle school. They cannot move forward or excel. Our best students are not challenged.

Richard Feynman
The late Richard Feynman (Nobel Prize in Physics), who served on the California State Curriculum Commission to rate textbooks, said that the New Math elementary textbooks were "lousy" because there weren’t enough word problems or enough applications. Moreover, “None of them [the textbooks] said anything about using arithmetic in science,” observed Feynman. Many definitions were wrong or vague. What good is learning another base or changing one base to another when the student doesn’t understand base 10? It’s ridiculous! The books were weak [on when to] add, subtract, multiply, or divide to solve word problems, he explains  [Students must learn patterns for different types of problems. Pattern recognition is the key to problem solving, says Ray Kurzweil, MIT, How to Create a Mind.] Moreover, children didn’t need most of the set theory stuff. Why do elementary school children learn or practice stuff that is useless? asked Feynman. Making a few changes in the New Math would not have helped much because teachers were not trained in math content in schools of education. Guess what? K-6 teachers are generalists and don’t know enough math content to teach Common Core well contends Dr. H. Wu in a recent article. Moreover,  parents are confused and baffled because they cannot understand the convoluted math or what is going on. Kids are frustrated and don't understand it. 

Not much has changed since the 70s. In Common Core, why ask children to learn stuff that is useless, such as "Find 5.3 x 2.4 using an area model. Show your work."? The CC math curriculum is often extreme and part of a reformist progressive agenda that equalizes downward in an attempt to close the achievement gap, suggests Thomas SowellThe classroom curriculum is based on the Common Core math standards, which were deficiently written and embedded with reform math pedagogy [strategies] that failed in the past. The problems starts in 1st grade. 

Carol Burris (Four Common Core 'flimflams') lists unjustifiable statements promoted by Common Core. Bill Gates, who financed Common Core, has paid teacher organizations, think tanks, and influential individuals to support and praise Common Core.
1. The Common Core standards are internationally benchmarked and grounded in research. [No, they are not.]
2. The standards are merely goal posts and do not tell teachers how to teach.[No, they tell teachers how to teach because strategies (pedagogy) are embedded in the standards].
3. The Common Core will close the achievement gap. [There is no evidence that this can happen.] 
4. The problems with the Common Core standards can be fixed at the state and local level. ["Nothing could be further from the truth." The standards are copyrighted and cannot be changed.] 

W. Stephen Wilson, who teaches mathematics at Johns Hopkins University, according to his assessment of 5th Grade Investigations/TERC (constructivist reform math), asserts that kids never get to arithmeticWhat they learn looks like arithmetic, but it isn’t the real thing. It's constructivist reform math. 

Insert -----
5th Grade Common Core Reform Math Quiz [1].
1. Find 15.7 + 9.72 by decomposing the number 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. 
Kaplan: A Parent's Guide to the Common Core, Grade 5

The reform math quiz is trash-can junk and useless. How are the strategies essential for learning basic arithmetic? In the quiz, there is no mention of using standard algorithms to do arithmetic. Our kids stink at basic arithmetic because they don't actually learn basic arithmetic, asserts W. Stephen Wilson, who teaches mathematics at Johns Hopkins University, according to his assessment of Investigations/TERC (constructivist reform math). Instead, students are taught strategies [multiple models], which are confusing, convoluted, arduous, inefficient alternatives to standard algorithms, to do simple arithmetic. Common Core, as constructivist reform math, in my opinion, is a misrepresentation of and a poor substitute for the real thing; i.e., basic arithmetic. 
End Insert ----- 

An old soul--Richard Feynman--is reaching out to us: Surely you're joking! What have the reformers done to simple arithmetic? Diane Ravitch says that the standardized test reforms of NCLB, which are an intrinsic part of the Common Core Package and imposed on all public schools by the powers that be, are meant to discredit, even punish, schools and teachers. Indeed, the road to "equality of results" is littered with bits and pieces of incoherent content and bad ideas, such as constructivist reform math, in an effort to make sure no child gets ahead. Thomas Sowell (Dismantling America) calls it equalizing downward, a pernicious idea taught in education schools. Ian Stewart (Letters to a Young Mathematician) says that practice does not cause talent. No amount of study and practice will turn a typical student into a mathematically talented student, but most students can achieve an acceptable competency at math, especially in arithmetic and algebra. Equally, mathematician Ian Stewart wrote that no amount of training in guitar would have turned him into a Jimi Hendrix, but that didn't stop him from playing lead guitar in a rock band. The mathematician writes that he has mathematical talent, not musical talent. 

Common Core, which is taught as constructivist reform math, is a mismatch to the needs of most students. Kids are often taught inferior or weak methods that are not practical or useful. The purpose for learning math is to do mathematics efficiently to solve problems. Why ask kids to practice stuff that is useless or trivial? Why not focus on essential arithmetic rather than inefficient strategies? Doing math starts the process of understanding math, which grows gradually over the years with practice and experience.

"Arithmetic fluency [competency] plays a key role in the acquisition of higher math skills"(Price, Mazzoccco, & Ansari). Auto fact retrieval is essential for learning arithmetic and for learning higher math. "The importance of early arithmetic skills for math competence" is critical and requires memorization, practice, and fluency in standard algorithms starting in 1st grade. [Parents are fleeing public schools because Common Core wants children to reinvent arithmetic using minimal teacher guidance and inquiry/discovery group work, which are the tenets of constructivism. I simply call it reform math.]

Note. Frankly, I don't think the content, itself, is too hard for most kids to learn if taught straightforward, practiced for mastery, and if the embedded strategies [multiple models] are eliminated. Kids in other nations learn more content, starting in 1st grade. However, I believe the way the content is taught [as constructivist reform math] in America makes it unduly and excessively difficult for no good reason. Why should a 1st grader learn 3 or 4 different ways to add? Kids are novices. Furthermore, the standard algorithms for addition and subtraction are "tried and true" and have always worked well in lower elementary school, yet they are postponed until 4th grade. Moreover, cognitive or academic ability varies widely, so the curriculum should vary in proportion to the students who walk through the school door, but Common Core doesn't allow for this. Everyone gets the same no matter what! 

One 140 page booklet for both
the 3rd & 4th grades. 1877
1. This is Arithmetic! 
If 12 peaches are worth 84 apples, and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches?  

G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." This requires knowledge, experience, and skill through lots of practice. Comment. We don't teach arithmetic well to kids in grade school or to teachers in schools of education. If ES, MS, or HS teachers cannot figure out the problem above within 1-2 minutes, then they don't understand basic arithmetic or how to apply it. No calculators. And remember, the problem above was written for typical 3rd and 4th graders in 1877 as a mental arithmetic problem. It is clear that arithmetic content has been dumbed down over the decades. At one time, our kids were very good at essential arithmetic, but those days are long-long-long gone. [Note. The "peaches problem" is from Ray's New Intellectual Arithmetic: 3rd & 4th Grades, 1877.] 

Note. You won't find arithmetic problems like this in "reform math" or Common Core elementary school textbooks or the "thinking" that is needed to solve them. In fact, today's elementary school textbooks are low grade compared to Ray's textbooks. Incidentally, fractions and long division are key building blocks in a young child's numerical development. Also, for decades US reform math programs have marginalized their importance, but research shows that this has been an epic mistake. 

2nd Grade Algebra (TKA)
Kids must memorize times tables, which should start in 2nd grade, in order to do long division, fractions, and algebra. Algebra comes out of arithmetic. Furthermore, our best students are not supported ($) or challenged with an accelerated curriculum. Furthermore, Common Core reform math has no STEM. Starting in grade 1, the CC reform math curriculum is not at the Asian level. More depressing is that, under Common Core, kids are often taught inferior methods that are not practical or useful (i.e., reform math). Note. Photo: Algebra comes out of arithmetic, and many algebra ideas can be taught in the lower grades, including 1st and 2nd grade. 

The "many ways" (i.e., multiple models stressed in reform math operations), frustrates and confuses kids because kids are novices with limited working memories. Moreover, you don't draw a picture or write an explanation to do basic arithmetic, yet this is what kids are asked to do under Common Core reform math, which is a distorted form of arithmetic. Very young children should use mathematical symbols, not draw visuals, to do arithmetic, starting in the 1st grade. Putting dots in circles, etc. is not practical or useful. Students should learn math facts for auto recall and practice standard algorithms for fluency, but Common Core delays them for no good reason. [How about 13 x 57? Who would draw 13 circles and put 57 dots in each, then count all the dots, or add 57 to itself 13 times (repeated addition). It's silly. A 4th grader wrote 27 in a column 20 times (20 x 27) and started to add them, but I caught her in time. All she knew was repeated addition. A highly intelligent 5th grade student, new to the school, had a simple multiplication and started to draw a lattice. Kids are often taught inferior or weak methods that are not practical or useful. This is the mad, mad world of math education under Common Core reform math. And, I want to state clearly that students should not use mental space (working memory) to calculate single digit number facts; they should practice them until they stick in long-term memory to free up mental space for actual problem solving. Students who have instant recall become the better math students later on when math becomes more complex. See example [calculating single-digit facts] at the end of this page.] Read SingleDigits.

Let me reassure you that there is nothing wrong with traditional arithmetic when it is taught properly and explained well, but progressive reformers, such as Common Core reform math crusaders, trivialize it.  The standard algorithms [traditional arithmetic] should be taught first and be the primary methods for calculating.  

Note. If the reform [progressive] methods of the past 25 years actually worked well, then our kids would be at the top in international math testing. The new Common Core is implemented as a version of reform math. 

In contrast, to learn arithmetic well, students should memorize single-digit math facts. The standard algorithms should be taught first and be the primary methods for calculating. Students need practice-practice-practice. They don't need to "draw a picture" or "write an explanation," or "calculate single-digit number facts" as needed (Kids need to memorize them).

In my opinion--not counting the stumbling blocks of testing, the implementation costs, teacher training, and government mandates--there are major problems within Common Core reform math, itself. For decades, the drawbacks in the mad, mad world of math education have been a weak curriculum, inefficient instruction, and teachers who are weak in both math and science, which has been known since the early 1960s or before. For decades, arithmetic has not been taught for mastery to prepare most students for algebra-one in middle school. We can do better, so why haven't we? I am fully convinced that the way arithmetic is taught in our schools (as reform math) by teachers who are weak in math keeps kids from achieving like their peers in other nations and regions. In short, teachers often are told to teach flawed reforms starting in grade 1, and they are not as well educated in math and science as teachers in top-performing nations and regions, which is the fault of progressive "schools of education" and the states that certify teachers.

1. The instructional methods often used to teach reform math are inefficient (e.g., minimal teacher guidance methods, typically inquiry or discovery activities in group work, etc.)
Note. Piaget's theory (constructivism) states that children should reinvent arithmetic through inquiry and discovery activities in small groups (i.e., reform math). Common Core is often interpreted as reform math. Piaget's theory has been discredited and does not work in the classroom, yet it is preached in schools of education. Look into almost any classroom. Kids are arranged in small groups. Reform math downplays traditional math, starting with arithmetic in 1st grade. Rather than explaining arithmetic clearly with explicit examples (traditional math), students are asked to discovery it on their own in inquiry/discovery activities in small groups (reform math). It has not worked for decades. 
2. The math content of Common Core is below the Asian level and does not have STEM level math. In short, it is not world class starting in 1st grade. 
Kids are frustrated with Common Core & 
Testing. What has happened to the the
 idea of "reducing the complex to the simple," 
which is the context for learning 
standard algorithms? writes Professor H. Wu. 
The standard algorithms 
[traditional arithmetic] 
should be  taught first and be the 
primary methods for calculating.
3. Children are asked to use multiple ways (strategies) that make a simple calculation complex, tedious, confusing, and time-consuming. Often, kids are asked to draw a picture, make an array, construct a chart, etc. to do arithmetic, but this often makes it more complex and less accessible. Unfortunately, the implementation or interpretation of Common Core is pedagogically biased toward reform math, which failed. It is a bad idea repackaged and recycled. Read more.

2. It is "pretend" arithmetic! [Multiple Models: Common Core Reform Math] It makes little kids cry and hate school.
A quiz [1] from Grade 5 on decimal operations shows what I mean. (I included a solution for problem #3.) Note. Teachers think they are teaching content, but they are actually teaching multiple strategies that often confuse kids. In my view, teachers should first teach the standard algorithms [starting in 1st grade], which require the automation of math facts, not strategies, which are tangential.
1. Find 15.7 + 9.72 by decomposing the number 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.

Let me see: decomposing by place value model, place value chart model, area model, number line model, and bar model. There are other strategies, such as partial quotient model, etc. The multiple ways (often called strategies, models, representations, diagrams, or even algorithms) to do simple arithmetic are enough to drive kids nuts and to tears. I am already nuts just thinking about it! Who makes up this stuff? It does not lead to a better understanding of anything. Instead, it confuses and frustrates kids, parents, too. Nothing is said about the standard algorithms. Using the standard algorithm is not an option in any of the calculations above.  Instead of standard algorithms, kids are taught inefficient, alternative models (strategies), representations, or procedures. It's called reform math. In my view, this is a waste of classroom instruction time and a prime reason our kids underperform. The arithmetic curriculum is too slow because of the "many ways" that are taught to do simple calculations, starting in the first grade. The focus is not on standard algorithms, which are delayed for no good reason. Instead, kids are taught models (strategies), etc. 

Kaplan: A Parents Guide to CC  5th Grade

Okay, here is a Common Core reform math answer to #3. Make an area model and add everything up. Wait. Take a closer look. How can 1/100 of the area be almost the same area as 1/10? I wonder how long it would take for a student to draw the area model and fill it in? (I am guessing at least 10-15 minutes to do everything.) But, let's ignore all this. Let me see, there are 10 ones, 26 tenths, and 12 hundredths. How do I add these if I don't know the standard algorithm for decimal addition? Maybe I should use the "decomposing the number by place value" model, such as in quiz problem #1. This is pointless and idiotic. Yet, this is the math our kids are often taught. It is the Common Core Way, I am told. And, in my opinion, it belongs in the trash can. I don't believe progressive methods work, but teachers have to do what they are told. 

I hope teachers would not require useless multiple ways, often more complicated, to do simple calculations. When would a student use an area model to calculate the product of two decimals? It's pointless and ridiculous. Students should be taught standard algorithms for operations [procedural knowledge] from the start, and teachers need to know how to explain how and why the standard algorithms work. To become good at using operations [arithmetic] to solve problems, students must automate math facts [factual knowledge] and practice, then practice some more.  

In contrast, It took me almost 13 seconds to multiply 5.3 by 2.4 using the standard algorithm, which includes writing down the problem and starting and stopping and timer. I think my time is slow. I am 71, and kids should be faster than I am. {Answer: 12.72} 

3. This 3rd Grade Common Core problem [Sandra....] is Pre-First-Grade Arithmetic.
-----> Here is a typical 3rd grade Common Core Reform Math Problem [3]. In the Zig Engelmann 1966 film, this problem would be a pre-first-grade problem. Zig's kids would not draw a picture to find the answer; they would count memorized multiples to find 4 x 6: 6-12-18-24. Singapore first graders would calculate 4 x 6 as repeated addition: 6 + 6 + 6 + 6. They are excellent at adding numbers. By 2nd grade Singapore students memorize at least half the times tables.]

:::: Sandra has made 4 gift baskets for her aunts. If each gift basket has 6 fruits, then how many fruits are in all the gift baskets? Draw a picture and write an equation to show your work. 

Common Core 3rd Grade Solution

Why draw a picture? This is nonsense and a waste of time for basic arithmetic. Apparently, students "draw a picture" to figure out that 4 groups of 6 are 24. I hope not, but this is what is implied. The math fact should have been memorized and the pattern in the word problem for multiplying should have been recognized immediately.  [Make a drawing, say Common Core reformers, is a very important skill in arithmetic, but mathematicians, such as Dr. H. Wu (UC-Berkeley), disagree.] Very young children should use mathematical symbols, not draw visuals, to do math. Putting dots in circles is not essential or practical. How about 12 x 127? 12 circles with 127 dots in each circle, or do repeated addition, a 1st grade idea, or the distributive. You don't draw a picture, or repeated addition, or the distributive to do basic arithmetic. You use the standard algorithm for multiplication. Of course, you can't do that unless you have memorized the times tables to automation. 

The Common Core 3rd grade question is a pre-first-grade calculation for Zig Engelmann's kids and a 1st grade level Singapore, except Singapore 1st grade students do not draw a circles with dots. [Neither do Zig's kids.] Singapore 1st graders write an equation: 4 x 6 = 24 and figure out 4 x 6 by repeated addition: 4 x 6 = 6 + 6 + 6 + 6 or 24. First graders could also figure out 4 x 6 by counting multiples of 6, 4 times: 6-12-18-24, which is reminiscent of Zig Engelmann's work with disadvantaged NYC pre-first-graders as shown in his 1966 film. Zig's kids memorized the multiples and counted by multiples to solve problems. What a great idea! 

First graders in Singapore start multiplication as repeated addition and write numerical equations that are solutions to word problems. In 2nd grade they leave repeated addition behind and memorize half the multiplication facts to write equations, etc. Writing an equation that shows a solution is "showing your work," but, apparently, that's not good enough in Common Core. Yet, the equation is the model; it does show your work. In algebra, students need to write and solve the equations without graphing calculators. They need to be able to plot all functions (paper-pencil), which squarely conflicts with the Common Core pedagogy of using graphing calculators and technology. 
My Title 1 urban first graders learned function rules, built function tables,
and plotted linear equations in my algebra enrichment classes. [1st Grade]

See TKA. Doing math starts the process of understanding math, which grows gradually over the years with practice and experience. A 1st grade student's understanding of linear functions is far different from an algebra-one student's. ["We begin with the hypothesis that any subject can be taught effectively in some intellectual form to any child at any stage of development." -Jerome Bruner]

Students Must Learn Patterns for Word Problem Types
What is actually important is that the student recognizes a "multiplication" pattern in the word problem and can differentiate operational patterns among different word problem types, such as addition, subtraction, multiplication, division, multi-step, areas, proportions, etc. Furthermore, the student needs to know and be able to apply "tried and true" algorithms and formulas to get the right answer as quickly as possible. Competency implies speed. Educators should worry less about a child's self-esteem and worry more about the student's competency, says Janine Bempechat (Getting Our Kids Back on Track). 

Pattern recognition comes from experience (practice) by working lots of different word problems, not by drawing pictures. Pattern recognition is the key to learning. [This is a multiplication problem! This is a percent of a number problem! This is an area of a circle problem! etc.] Furthermore, being able to calculate the right answer swiftly using efficient procedural knowledge from long-term memory is just as important. Ray Kurzweil, a graduate of MIT, the director of engineering at Google, and author of How To Create a Mind, says that we have a "deep core of capability of recognizing patterns." However, we need to "train this ability," which is why not everyone can learn calculus. It is one reason that children vary widely in mathematical ability, but there are other factors that limit ability. 

Brain plasticity has limits. There are also genetic, environmental, and IQ components as well as soft skills such as persistence and completing tasks. The premise that all children are geniuses put forth by John Taylor Gatto is nonsense. The late Richard Feynman, Nobel Prize in Physics, writes, "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?" We are squandering a lot of talent because kids who are better in math or science are not supported ($) or challenged with an accelerated curriculum. Common Core cuts off STEM. Its "one size fits all" does not account for the academic variability of children. 
Comment. Incidentally, making a drawing is commonplace in the sciences (e.g., molecular structures, etc.). The periodic table holds an immense amount of information for science students, but students are not asked to draw it. In addition, parts of geometry/trig and graphs and tables of functions are useful: students should be able to construct them. But that's not we are talking about here, which is essential arithmetic. (Incidentally, tables and graphs of linear equations and figuring out function rules can be taught to 1st graders.) 

Mathematician H. Wu says that very young children should use symbolics, not draw visuals, to do arithmetic. The only visual Wu recommends for "organizing the mathematical developments of whole numbers, fractions, and negative numbers is the number line [4]." Kids should be taught symbolics from the beginning. The problem is that many K-8 teachers don't understand symbolics. Math is abstract and uses symbols to convey concepts. 
Dr. Wu's Number Line: whole numbers, fractions, and negative numbers. "All early grade teachers should learn to make effective use of the number line at every opportunity," writes Dr. Wu. Moreover, 1st and 2nd grade teachers should never say "you cannot subtract 5 from 3" because it isn't true.  "A teacher has to say that you will learn how to subtract between any two numbers" [3 - 5],  but for now, we will always subtract the smaller from the larger [5 - 3]."
"Whole numbers and fractions are the cornerstones of the early grades mathematics curriculum," says, Dr. H. Wu, but, in my opinion, Common Core, taught through a reform math prism, has placed a progressive pedagogy ahead of content knowledge. Arithmetic, for example, is not taught for mastery. Dr. Wu states over again that content dictates pedagogy, but educators seem to focus more on questionable methods of instruction than on essential content knowledge. Unfortunately, teachers often are required to teach flawed reforms, fads, and ideas that frequently lack an evidential basis. Reformers say that memorization and practice are old school and outmoded. They are wrong!

I believe American education will bloomfrom the classroom on up, if we get rid of the anchors holding us down, starting with Common Core reform math and its test-based instruction, mandated tests, and accountability. "Reformers have put testing ahead of teaching and pitch bad idea after bad idea [2]." They put progressive ed school ideology ahead of cognitive science. Our kids are mediocre at math because we made them that way. We can reverse-engineer the process by teaching the essentials of arithmetic straight-forward [factual and efficient procedural knowledge] that form the foundation for higher mathematics, especially algebra, precalculus, and calculus, and through eliminating inefficient methods and procedures and fads. There is nothing wrong with traditional arithmetic when it is taught properly and explained well, but progressive reformers trivialize it. 

US educators underestimate what very young children can learn and persist in using popular instructional methods, reform math ideas, and fads that have no basis in cognitive science. The late Jeanne S. Chall (Harvard) pointed out that many popular classroom practices and reforms are not supported by evidence.  

1. Reform math screws up arithmetic!  Unfortunately, many teachers [and parents] have bought into reform math dogma, which has been preached in ed schools for decades.  
2. Teachers should teach arithmetic content explicitly for mastery, not multiple models [strategies] and inquiry-based group work that lead nowhere. "The importance of early arithmetic skills for math competence" is critical. 
3. There was a time when arithmetic was arithmetic, and kids automated it in long-term memory through practice and memorization for use in problem solving. If I were in the classroom today, I would disregard Common Core, constructivist reforms, fads, and test prep and teach kids traditional arithmetic, as I always have, because it advances kids to algebra by middle school--even if I get lousy reviews or lose my job. Test scores are not equivalent to achievement. 
4. Dr. H. Wu knows that regular teachers are generalists and are not trained to teach math properly. [I also think they are not trained to teach reading properly too.] Thus, Wu suggests that most K-5 teachers end up teaching math from books that is "usually not learnable." [Our textbooks stink!] Kids in K-5 need real math teachers to fix the problem, writes Dr. Wu ("Americans stink at math but we can fix that," 8-22-14 SF Gate), but, in my view, if we train the 'math teachers' to teach constructivist reform math we have not made progress. In contrast, Wu wants intensive content-based training, but I am skeptical that we can change elementary teachers into math teachers. Wu also writes, "Mathematics being hierarchical, students cannot learn upper-grade math if their foundation K-5 math is shaky. We need to first improve the teaching of math in elementary school." But, I have misgivings about Wu's solution. Wu wants to start by selecting a few districts and "require [a representative number of elementary teachers to become] the 'math teachers' of these districts by going through intensive content-based training," so they can help other teachers, etc. In my view, math teachers in K-5 should already know content and know how to teach it well. The training model Wu suggests may not help much. The idea didn't work for Science--A Process Approach (60s), the New Math (70s), and it probably won't work for Common Core reform math, which will be lucky to have a life spam of 2 to 3 years, if that. Dr. Wu was on the Common Core math committee, but he is a university mathematics professor and has never taught elementary school children; however, he is right about several things. Teachers don't know enough content to teach math. The problems kids have in math stem from poor instruction in K-5 because elementary school teachers have not received the proper math content training from schools of education. Most elementary teachers are weak in both math and science content. Little has changed in 50 years. 

[Note. Dr. Wu has some good ideas, and I often quote him. I have read most of his papers over the years, and his writings have been helpful in my teaching. Having a few teachers go through intensive content-based math training will help, but its not enough. He gives workshops to elementary and middle school teachers, and I applaud his effort, but Dr. Wu cannot change the powerful progressive American culture in which "decline has become a policy objective" (Dinesh D'SouzaAmerica).]

[Note. I don't have a solution for the decline of American education unless we junk fads (often called innovations), weak instructional methods (group work, inquiry-based activities, etc.), and constructivist reform math (Common Core and its accountability testing). Content should dictate pedagogy, says Dr. Wu, not the other way around. We don't want kids reinventing arithmetic, which is basic idea in reform math. In addition, we need outstanding textbooks that are content-based (traditional arithmetic), stress the practice of key factual and procedural knowledge to automaticity, and adhere to world-class standards, such as the Core Knowledge K-8 math sequence, but we don't have them. In my opinion, the conundrum is that most elementary teachers don't know enough content to teach it well. We also need to arrange students into homogeneous math sections according to their mathematical knowledge. Putting high achievers and low achievers in the same math class, starting in 1st grade, has been a recipe for underachievement and mediocrity for all kids. For example, Singapore has a pull-out math class for incoming 1st graders weak in math skills with a strong teacher. It is called tracking for math. The program lasts up to two years to catch kids up. The tracking continues in the 4th, 5th, and 6th grades when the math content becomes much more complex.] 

The standard algorithms [traditional arithmetic] should be taught first and be the primary methods for calculating. This requires auto recall of math facts through drill, practice, and usage (applications). Unfortunately, Common Core has been interpreted in terms of reform math; hence, kids are taught bad ideas and inferior or weak methods that are not useful or practical. Some of the bad ideas promoted by progressive reformers have different names, such as constructivist, discovery, problem-based, experiential and inquiry based teaching (Kirschner, Sweller & Clark, 2006). These "minimal teacher guidance" instructional methods (i.e., reform math), according to the research from Kirschner, Sweller & Clark, are inefficient and inferior to traditional arithmetic methods that stress explicit instruction, memorization, and practice of fundamentals for mastery. 

Also, there is no reason to interpret Common Core as reform math rather than traditional arithmetic, but, unfortunately, Common Core is seen through the reform math prism. Also, there are many code words that refer to constructivist reform math in Common Core. Traditional arithmetic and explicit instruction need to be restored. Students should memorize math facts and use standard algorithms from the start of 1st grade. Students need straightforward, clear instruction and lots of practice. [Note. Sweller's cognitive load theory (1988) predicts the worked-example effect, which is a positive learning effect when used as part of instruction for new content, especially in mathematics. I used key worked examples when I taught traditional algebra to 1st through 5th graders as a guest teacher several times a month. Teachers must be knowledgeable and select a sequence of examples carefully and be able to explain the examples to students, even to 1st graders.]

What does college and career "readiness" mean? It is vague, yet it is the main claim of Common Core. The extravagant claims that were made to "sell" Common Core to states have been bogus from the start, and the costs ($$$$$) for CC and its testing mentality have been enormous burdens on ailing school budgets. What has happened to critical thinking? Apparently, state education officials, boards, and politicians shut out critical thinking by adopting or supporting Common Core and test-based accountability for teachers. Many organizations, including teacher unions, were paid to promote Common Core. It is well known that most of Common Core has been financed by Bill Gates.

Dana Goldstein (The Teacher Wars) writes, "When American policy makers require every public school to use the same strategies [e.g., Common Core reform math--without evidence], they reduce the discretion of the most motivated teachers. This is an age-old problem in American education. Our system is highly decentralized in terms of curriculum, organization, function, and student demographics and needs, yet we have expected local schools to implement a one-size-fit-all reforms or agendas imposed from above." 9-3-14

"We have set our standards too low in all all fields for far too long," writes Richard Askey, Professor of Mathematics. Common Core is an attempt to raise standards, but it is not good enough because Common Core, itself, is flawed, below world-class benchmarks, and laced with constructivist reform math ideas that don't work. 

 Note. My 1st grade students (80s) were taught equality and transitivity: 3 + 4 + 1 = 5 + 4 - 1, that is, both expressions name the same point on the number line {8} and therefore are equal to each other (8 = 8). It is also the reason that 2 + 3 = 5. Both expressions name the same point on the number line {5}, and therefore are equal to each other (5 = 5), which is the idea of transitivity (roughly stated: two things equal to the same thing are equal). These ideas are essential to the algebra. But, you won't find this reasoning in elementary school textbooks. Also, addition and subtraction are similar. Subtraction undoes addition; however, subtraction is actually the addition of negative number. Kids should use integer number line to start this key idea, but you won't find it in Common Core K-5. In the early 80s, my 1st graders discovered that 5 - 3 = 2 and 5 + -3 = 2 (number line) give the same answer. Therefore, 5 - 3 = 5 + -3 because they name the same point on the number line {2}. This sets them up for a fundamental idea: a - b = a + -b, that subtraction is nothing more than addition. To subtract a number, add its opposite. Subtraction is not commutative, but addition is. To subtract an integer, add its opposite is a concept I usually introduce in 3rd and 4th grade algebra. (Actually, a negative number, such as -2 [by itself] means 0 - 2, which is 0 + -2).  

Common Core Reform Math is a Mismatch
Costly reforms, often called innovations, have not made things much better in education. We have had reform after reform with little change in achievement over the years. We keep repeating the same mistakes and recycling bad ideas, and Common Core, which is the latest version of reform math, is one of them. Common Core reform math marginalizes time-honored arithmetic, such as standard algorithms, and tried and true instructional methods, such as explicit teaching, memorization, repetition, and practice-practice-practice for automaticity. Common Core, which is taught as reform math, is a mismatch to the needs of most students, according to a study from the University of California at Irvine and Penn State, headed by Professor George Farkas. Daniel Willingham (Why Don't Students ...), a cognitive scientist, says that beginning in 1st grade, kids need to automate background (prerequisite) knowledge, both factual and procedural, to solve problems in mathematics. They don't get that in Common Core when taught as reform math, which downgrades early memorization, practice, and standard algorithms. Under Common Core, kids don't have to be "fluent" using the standard algorithm for whole number addition until the end of 4th grade. Instead, students are asked to reinvent arithmetic through inquiry/discovery lessons in groups and "learn" inefficient, alternative strategies to add. No wonder our kids stink at arithmetic! Barry Garelick (A Common Sense Approach....) writes, "One has to accept that procedural fluency is essential for students to achieve and that ultimately it leads to understanding." The late Robert B. Davis (Madison Project) writes that "understanding grows [and changes] gradually [over time]" and takes lots of practice. "The way we understand any particular things will necessarily change over time," explains Davis. Understanding is indeterminate. Thus, it is normal for kids to have an imperfect understanding rather than "the correct idea." For example, a 1st grade understanding of place value is different from a 4th grade understanding. Procedural knowledge is a key part of understanding mathematics and important in problem solving. The purpose for learning math is to do mathematics efficiently to solve problems.  G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." This requires knowledge, experience, and skill through lots of practice. American students are weak in procedural knowledge. They cannot apply arithmetic or algebra they don't know well or weren't taught. 

Professor George Farkas (head of a study from the University of California at Irvine and Penn State) points out“However, activities such as routine practice or drill, math worksheets, problems from textbooks and math on the chalkboard appear to be most effective, probably because they increase the automaticity of arithmetic. Foundation skills need to be routinized so that the mind is free to think.” In short, Common Core reform math screws up arithmetic. Furthermore, Dr. H. Wu (mathematician UC/Berkeley) observes, "We have not been teaching learnable math to students--only what passes as math [i.e., reform math] in standard school textbooks." Dr. Wu writes, "K-5 math classes [should] be taught only by math teachers"--not elementary school teachers who are generalists or by so called math educators coming out of progressive schools of education. Kids need teachers who know content. Teachers need content-intensive math courses in ed school. Unfortunately, he says, "the implementing of this idea is not yet forthcoming." Frankly, I don't see this happening. Math textbooks are lousy, and regular classroom teachers follow them. Wu writes, "There is no quick fix here."   

In 1895 the "Committee of 15," sanctioned by the National Education Association (NEA), made recommendations for elementary schools (grades 1-8). Arithmetic should be taught beginning in 1st grade with 60 minutes of oral drill daily plus five textbook lessons a week to prepare students for geometry and physics in 7th grade and algebra in 8th grade.  

These 1895 recommendations make good sense even today, but, unfortunately the wisdom of old souls has been dumped in favor of progressive reforms that don't work (e.g., Common Core taught as reform math). [Note. Kids who come into 3rd grade can't add numbers, never learned the standard algorithms, and don't know place value. In short, under Common Core reform math, the students weren't taught to automate basics, that is, essential factual and procedural knowledge of traditional arithmetic.]

Marc Tucker writes, (Fixing Our National Accountability System), "High stakes tests in the top performing countries are used to hold students, not teachers, accountable, the obverse of what happens in the United States." We have it backwards. Also, our teachers have not been treated or paid as professionals, but teachers need to teach--not facilitate--for students to learn. Furthermore, schools of education must select and prepare the best candidates to be future teachers. [The don't.] Indeed, better teaching and teacher quality are important, but, according to Will Fitzhugh (Concord Review), "The most important variable in student academic achievement is not teacher quality, but the student's academic work." In short, students should produce high quality academic work, but many capable students do not. We are not holding students accountable in the American system. That's wrong! For decades, teachers have been scapegoats and treated poorly.  

Marc Tucker (Fixing our National Accountability) points out, "Test-based accountability and teacher evaluation systems do not simply fail to improve student performance. Their pernicious effect is to create an environment that could not be better calculated to drive the best practitioners out of teaching and to prevent the most promising young people from entering it." Tucker states that "only 52% of the teaching force from any given year in the US is teaching 6 years later." The failure of schooling should not be blamed only on teachers. The blame is the entire American system, says Tucker, including "the public, school boards, and the Congress." In short, our kids stink at math because WE made them that way. Common Core, with its accountability testing, is just another bad idea that adds to the trend. 

Proton Quark Structure
Kids need achievable objectives, and, because academic ability varies widely, the learning objectives should vary or be flexible from school to school. The policymakers should be the classroom teachers. Teachers should write different levels of curriculum (different flavors, like quarks) that better fit the needs of students who walk through the school door. The curriculum should be flexible because kids vary widely in academic ability. The one-size-fits-all, Common Core reform math is the wrong approach. That said, it is highly unlikely that states or the federal government would give up control of the public schools to the teachers in those schools. 

In my view, students underachieve in math for these reasons: the way students are grouped for math class, the math content taught (weak curriculum), the methods of instruction (ineffective), teacher training (inadequate in math and science), and the lack of effort and study (industriousness) from many students.

You see, in education, real evidence doesn't matter, and this has been a stumbling block in improving math education for as long as I can remember: progressive ideology, preached in schools of education, trumps cognitive science. Our kids are mediocre in math because we made them that way.  9-1-14

US policies squander talent.
The US neglects its best math and science students, starting in elementary school. We are squandering talent, but this is nothing new. Kids who are better in math or science are not supported ($) or challenged by a totally different accelerated curriculum.

Common Core reform math follows the same path. It has no STEM and teaches basic arithmetic poorly so that many capable kids, especially minority kids, aren't prepared for algebra-one by middle school; consequently, our kids can't compete with peers from other nations. Moreover, the K-7 science textbooks are practically math-less, and the new science standards lack sufficient content in physics, chemistry, and math to do college level work.

"I'm looking at all these [elementary school math] books and none of them has said anything about using arithmetic in science," observed the late Richard Feynman, Nobel Prize Winner, Physics. They are lousy. [He actually read the textbooks.] Not much has changed over 50 years. Textbooks today are still lousy. There are not enough applications or word problems. Some of the math is wrong or off track, and instructional methods and content are inferior. [Note. In 1964, Feynman, an eminent physicist, was a member of the State of California's Curriculum Commission to choose math and science textbooks for elementary grades.]

Sandra Stotsky recently wrote on Breibart, "Despite the billions of dollars showered on our schools, American public education is poor to mediocre and likely to remain so." Indeed, classroom teachers have not been in charge of curriculum, instruction, and testing for decades. I fear they have lost control of their profession. 

Reformers want to transform math education through the US Department of Education using Common Core, more testing, and more technology, which benefits large publishing, testing, and technology companies--not students. No, we do not need to transform math education; we need to restore it, starting with traditional arithmetic in 1st grade. I think Common Core reform math is based more on "ideology and herd thinking than on fact and logic [5]." 

US education policies squander young talent. Common Core reform math has no STEM.      

Our kids are mediocre in math because we made them that way. Instead of teaching "tried and true" traditional arithmetic, educators have taught trendy, popular reform math because this is what teachers were taught in schools of education. Common Core reform math is a distorted version of arithmetic.  Ben Carson (One Nation) writes that in the 1800s "there was an 6th grade exit test, a test you had to pass to get your sixth-grade certificate. I doubt most college graduates today could pass that test. We have dumbed things down to that level."

Note. If the math reforms and progressive instructional methods [i.e., constructivist ideas] of the past 25 years actually worked well, then our kids would be at the top in international math testing. They aren't even close. The new Common Core has been implemented or taught as a version of reform math, which won't get our kids to where they need to be, yet it is being forced upon the public schools. Classroom teachers have almost no impact on top-down education policymaking.  "Arizona Children deserve high quality standards that prepare them to succeed," proclaims Expect More Arizona, a pro-Common Core political organization, but I think the slogan has little merit because standards written on paper mean very little when they are deficient, or when teachers are weak in math, or when the curriculum is taught as reform math, which uses multiple models (strategies) for operations and inquiry/discovery group work for instruction. Teachers have little say. They have to follow Common Core, which is envisioned by most as the resurrection of reform math based on Piaget's constructivist ideas.  In short, children are asked to reinvent arithmetic through inquiry/discovery activities in group work. It's nonsense and a waste of time. 

Note. Under Common Core reform math, "making a drawing" is considered one of the most important skills. It isn't.  Mathematicians, such as Dr. H. Wu (UC-Berkeley), disagree.  Very young children should use mathematical symbols, not draw visuals, to do math.  Zeev Wurman points out another major problem, "I heard Prof. Wu express doubt that our elementary and middle school teachers can prepare our students for Algebra-1 in grade 8 with my own ears, and that he used it as his explanation to me why he now approves of pushing algebra to the high school." In short, kids who are prepared should take Algebra 1 in 8th grade. The problem is that our K-8 teachers don't know how to prepare them, and Common Core reform math does not help much in this regard, in my opinion. Dr. Wu writes that "sound content knowledge is the foundation of teaching." It is also the foundation for higher math. Teachers must teach content. Our schools of education have failed. 

Also click the following:
1. See SingleDigit 
Don't calculate single-digit math facts; memorize them.
US schools are a breeding ground for mediocrity, not excellence. Early on, parents should provide an environment for achievement and teach their kids how to be successful in school. Early arithmetic at home pays off.  Very young children can learn a lot more than we think or are prepared to teach. American children underperform in math!   
3. See PiagetianMyth
Piaget's constructivist reform as applied to math does not work in the classroom. In short, constructivist reform math screws up arithmetic. The researchers (Haeck, Lefebvre, & Merrigan) observed, "We find strong evidence of negative effects of the reform on the development of students’ mathematical abilities." 
4. See CommonCore 
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education. I am not sure they ever were.


More Thoughts.
I am thankful to Kaplan for distilling what Common Core reform math is all about and how it is often interpreted and implemented (taught). Since 2009, my conclusion has been that Common Core is not about mastering traditional arithmetic that is needed to prepare for algebra by middle school. It is about reform math strategies. 

Who would use an area model to compute the product of two decimals?  In short, inefficient, alternative models in Common Core reform math lead nowhere. They are not economical, practical, or grounded in evidence. They are often an obstacle to learning arithmetic. 

Starting in 1st grade, kids need to learn and automate traditional arithmetic. They don't.

Dr. James Milgram, a mathematician at Stanford, points out that there are "severe problems" in Common Core math with the way kids are taught to add, subtract, multiply, and divide. He continues, "There are also severe problems with the way Common Core handles percents, ratios, rates, and proportions." 

We need to restore arithmetic.  If 12 peaches are worth 84 applies, and 8 apples are worth 24 plums, how many plums shall I give for 5 peaches. (Grades 3 & 4, Ray's New Intellectual Arithmetic, 1877 by Joseph Ray)

Reformers put testing over teaching and pitch bad idea after bad idea, says John Stocks, NEA speech. Common Core reform math, the way it has been interpreted or implemented, is a bad idea. Test-based accountability linked to Common Core is also a bad idea. Evgeny Morozov (To Save Everything, Click) writes, "Schools concentrate all their efforts on improving test scores, even if children learn much less as a result."

Our kids are mediocre in math because we made them that way. Instead of teaching "tried and true" traditional arithmetic, educators have taught trendy, popular reform math because this is what they were taught in schools of education. Common Core, which is often interpreted as reform math, is just more of the same.

If teachers are forced by state and federal mandates to teach Common Core reform math, which is linked to test-based accountability, then they have lost control of their profession. Indeed, classroom teachers have not been in charge of curriculum, instruction, and testing for decades.

In Common Core, calculating single-digit number facts in working memory is stressed. But, it is no substitute for the memorization of single-digit number facts for instant recall from long-term memory. Instant recall frees mental space (working memory) for problem solving. 

T: What is 6 + 7?
S: I don't know.
T: What is 6 + 6?
S: I don't know. But I might be able to figure out 6 + 7. [Calculating.] Let me see. I do know that 6 is 3 and 3. Okay, so I can put together the 3 and the 7 to make 10. but then I have to add the other 3 to the 10 to get 13.

This is an example of calculating a basic number fact, which takes up a lot of mental space and time, but some kids get really good at it. If the student has instant recall of 3 + 7, then why not 6 + 6 or 6 + 7, which is one more than 6 + 6? In short, the student calculates the single digit number fact, which takes up mental space, rather than memorizing the fact for instant recall.

Instant Recall

T: What is 6 + 7?
S: 13  (Instant Response)
Kids with instant recall, rather than kids who calculate single-digit math facts, are the students who do much better when the math becomes more complex. Students should not calculate single-digit number facts, they should memorize them. And, practice for auto recall starts in 1st grade with addition and subtraction facts, which is what has been done in Singapore other Asian nations for decades. We used to do it in the US. Early on, Common Core reform math does not focus on memorization, practice, and standard algorithms.    

[1] Kaplan: A Parent's Guide to the Common Core, Grade 5
[2] From a speech to the NEA by John Stocks
[3] Kaplan: A Parent's Guide to the Common Core, Grade 3
[4] H. Wu: The Mathematics Early Grade Teachers Need to Know
[5] "ideology and herd thinking than on fact and logic" idea from Think Like a Freak (Levitt & Dubner)

Model Credit: HannahE
Model Credit: RemiB