Tuesday, March 30, 2021


 😎Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a Divergent View. 

Happy Birthday, Cindy! 4-4-21

My Impressions
April 2021

What has happened to Math Education? 

Education has been hijacked by extremists who have a utopian vision of the future (Marxism). Traditional arithmetic has been replaced with liberal reform math that puts our kids behind internationally. The curriculum is dumbed-down, "so everyone can pass--but no one can excel." It is feel-good grade inflation (for no good reason) in K-8 and above, sometimes in college. Charles J. Sykes (Dumbing Down Our Kids) explains, "In the name of equity, fairness, inclusiveness, and self-esteem, standards of excellence are being eroded throughout American education." 

I often hear the motto that "All Children Can Learn." Really? What does it mean? Calculus, Advanced Physics? Does it mean that all children have the same capabilities to learn what schools teach? Like so many liberal ideas, it sounds great, but it is wrong!   4/7-13/21

"Facts seem to have become irrelevant for all too many people, who rely instead on visions and rhetoric. The entitlement mentality has eroded the once common belief that you earned things, including respect, instead of being given them. One of the trends that can become part of a perfect storm of disasters for American society has been a decades-long dumbing down of education, producing a citizenry poorly equipped to see through the political rhetoric and even more poorly supplied with facts and the ability to analyze opposing arguments." (Quotes: Thomas Sowell

I see smart kids who don't know much. Memorization and practice for mastery have drifted out of liberal classrooms over the decades. Thus, students don't know much science, history, mathematics, grammar, or vocabulary, and so on. Still, they know how to text, snap selfies, use social media like Instagram and YouTube, and Google. Often, students don't have textbooks to take home to review or study. Parents are left in the dark. What does a "pass" grade mean--a D- or an A+? Who knows? 04-7-8-21

In my opinion, equalizing down and academic rigor are inverses. The more equalization, the less academic progress. (I could be wrong, but I don't think I am.) You can't balance the two. By equalizing down in schooling, you are asking for lower standards. If this is true, then equalizing down is a fallacy of fairness, as Thomas Sowell explains.

"Equal opportunities do not result in equal outcomes." 

Nevertheless, opportunities should be available for anyone who wants them. But, schools should not water down the curriculum to accommodate ill-prepared students, let's say for Algebra. Remedial courses might be needed if students stumble over fractions and long division. The K-12 math curriculum needs to be upgraded and grade inflation erased. "Do you want to build a snowman?"

Can the students figure out these equations that I give to 2nd and 3rd-grade students? No calculators!

12 + 4 +  = 12 
12 x 4 x  = 12

How about 4th-5th Grade? No calculators!
(x + 3)


(2 3/4) ÷ 7/15 

How about 6th Grade? No calculators!
Jill walked at a pace of 3.5 mph for 30 minutes. How far did she walk?

(Answers are at the end of this page.)

Note: Beliefs, fads, practices, and policies in our schools, from Common Core to social-emotional lessons and remote, are often based on anecdotal evidence or ideology rather than scientific evidence. Many so-called "studies" do not account for time-on-task or other variables that are hard to control or ascertain. Too many studies spin correlation as causation, which it is not. Also, just because an association is "statistically significant" doesn't mean it works in the classroom or is meaningful. 

Lastly, any evaluation of an individual, class, group, school, or district "based solely on test scores will present a dangerously inaccurate picture," points out Charles Wheelan. But isn't this what educators and elite leaders do all the time: rank schools, students, programs, etc.? "School spending," "parental education," and "family history" are key variables that are difficult to control. Often a study or research in education cannot be replicated. The acceptance of a belief frequently boils down to judgment, but experts are often untrustworthy with faulty conclusions. Anecdotal evidence is not evidence at all, much less scientific. Often, leaders assert, "Research shows..." What research?

Note:  "A shocking amount of expert research turns out to be wrong." 

Wheelan points out, "We can isolate a strong association between two variables by using statistical analysis [i.,e., regression], but we cannot necessarily explain why that relationship exists, and in some cases, we cannot know for certain that the relationship is causal, meaning that a change in one variable is causing a change in the other. Even in the best circumstances, statistical analysis rarely unveils the truth. There are limits on the data we can gather and the kinds of experiments we can perform." So what's the point? It boils down to the judgment and integrity of experts because statistical analysis is not always reliable. But, then again, experts are often wrong, too. Wheelan  explains, "Statistics cannot prove anything with certainty." (Dear esteemed researcher: It is next to impossible to replicate the study to verify your conclusion.) What is my point? Both expert judgment and statistical analysis in education can often be false. Unless they have a Ph.D. in statistics, journalists (the media) often repeat the conclusions of studies that are likely to be flawed or invalid, but they wouldn't know that, would they? 4-6-21

Comment: Indeed, the misrepresentation of data or a biased interpretation of studies is commonplace, especially in education when a company is trying to sell its software to schools. Common Core is another example. It was "sold" to States as internationally benchmarked, but I showed in 2010 that it wasn't. 4-6-21  

Success Maker [math] software is built on a framework of fun and games. [Clifford Stoll would have a fit!] The company (Pearson Education) says Success Maker software, which is used to supplement a school’s math program, has 40 years of success, but I never heard of it. Furthermore, Pearson backs its claims with faulty research (time-on-task was not controlled, schools volunteered for the study, etc.), unproved education theory, and anecdotal evidence--not hard facts. Its most recent study ("Pearson Success Maker Math Efficacy Study," published 9/15/2010), which is used to promote the math software as effective, was funded by Pearson. 

Moreover, the standardized math test (GMADE) used to make comparisons is from Pearson also. Gee, a Pearson product study (Success Maker), is funded by Pearson. Publishers often fund studies to show that their math programs, textbooks, and software work in the classroom. They are in sales! (Note: I wrote about Success Maker in 2010.) 

Note: Summer school is probably not the answer. Parents can help their kids catch up using grade-level workbooks from a bookstore and with lots of practice and review. If possible, enroll your child in Kumon or hire a tutor who knows how to teach standard arithmetic and algebra. Still, how many parents will do this? 4-4-21 

Note: Technology and engagement activities are not the silver bullet. Engagement is not the same as learning content. Also, we have had computers in classrooms for over 30 years, and student achievement has stagnated since Common Core. Michael J. Petrilli at Fordham points out, "It's now been almost a decade since we've seen strong growth in either reading or math." Petrilli calls it the "Lost Decade of educational progress," referring to NAEP results. 4-6-21 

➜ Testing and accountability are still the national educational strategies and more critical than reforming and upgrading curriculum and instruction for all students. (More important, Really?) Test-based accountability has failed. The teacher functions as a facilitator while children are expected to teach one another in group work and find or use multiple ways to solve a problem without a strong background in factual and procedural knowledge. Crazy? Diane Ravitch explains that "kids must have background knowledge, not just strategies." 

Kids who lack strong standard computational skills, one of the five priorities in elementary school mathematics (W. Stephen Wilson), are not prepared to succeed in high school and college math courses. Instead, they end up in remedial high-school-level algebra classes at community colleges. Test-prep games the system. Grades are inflated on report cards. What else? Common Core is not world-class. What a mess! Adding to it is remote/hybrid. (Note: By essential computational skills, I mean the standard algorithms. By instruction, I mean explicit teaching with worked examples--not group work, discovery or project-based learning, etc.)  

Note: Beliefs, fads, practices, and policies in our schools are often based on anecdotal evidence or ideology rather than scientific evidence. Many so-called "studies" do not account for time-on-task or other variables, which are hard to control. Still, many studies hyped correlation as causation, which it is not. Also, just because an association is "statistically significant" doesn't mean it works in the classroom or is meaningful. Lastly, any evaluation of a class, school, or district "based solely on test scores will present a dangerously inaccurate picture," writes Charles Wheelan. But isn't this what educators and leaders do all the time: rank schools, students, programs, etc.? "School spending" and "parental education" are other variables that are difficult to control. Often a study or research in education cannot be replicated.  4-6-21

➜  Not all students have an equal opportunity to learn arithmetic and algebra, reports William Schmidt ("Equality of Educational Opportunity, Myth or Reality" from American Educator/Winter 2010-2011). Schmidt links a student's social-economic status to the opportunity to learn. This is a correlation, but it should not imply causation. Schmidt points out, "Not only do we have great variability [of math content] across districts but by international standards, our eighth-grade students are exposed to sixth-grade content." He is correct. But, I think Schmidt assumes that children of lower social-economic status have lower IQs. I disagree. He also assumes that all students have the same ability to learn content that schools teach when, clearly, ability widely varies

Robert Plomin writes, "Performance on tests of school achievement is 60 percent heritable on average." The other 40% is environmental (nurture). Plomin's percentages are not deterministic; they are propensities. He explains, "The most important point about equality of opportunity from a genetic perspective is that equality of opportunity does not translate to equality of outcome." It's DNA that makes us who we are. (Quotes: Robert Plomin: blueprint, 2018, MIT) 

Equal does not mean identical. (Thomas Sowell

I know average Title-1 minority kids, ages 6 to 10, in my algebra program (TKA) who are stellar in algebra. The difference often boils down to attitude, persistence, and effort (nurture), qualities I stress. Some students listen better, work harder, study more, and don't give up as fast as others.

The radical reformists claim that the only way to level the playing field is to lower the bar so that similar outcomes are possible for all children. But equalizing downward is not equity, writes Thomas Sowell. It's hardly fair. It's a crazy idea! Moreover, chasing after "gap closing" should not be an educational goal, writes Sandra Stotsky (The Roots of Low Achievement, 2019). 

Note: Most students, if not all, should have the opportunity to take a rigorous pre-algebra course in the 7th grade to prepare for Algebra-1 in 8th grade, but many would fail because they were not taught the prerequisites in K-6. Common-Core-based reform math is not world-class math and doesn't get students ready for prealgebra in 7th grade. Equal opportunity to learn is not the same as "equal learning." In short, as Plomin suggests, equal opportunity doesn't produce equal outcomes. 

  Math, starting with counting and standard arithmetic, is hierarchical but often not taught that way. Traditional or standard arithmetic has been replaced by a corrupt version of Common-Core-related reform math, an aberration of standard arithmetic. Indeed, math education has been usurped by liberal extremists and a radical agenda of forcing equality of outcomes (i.e., sameness) on all students. It's not equity! Leading the equalizing crusade is reformist Jo Boaler and her cohort. 

 The reform math idea of "gaining conceptual understanding and learning applications in math [without] mastering procedures" has been promoted through Common Core and influential reformists like Jo Boaler and others. What good is learning about perimeters if you can't calculate them? For example, Boaler brags that she never memoried the multiplication table, yet she acquired a Ph.D. in math education, which is far different from a degree in mathematics. She wants to kick out standard algorithms because they" interfere with the students' progress of discovering the properties of mathematical objects on their own." Memorizing the multiplication table and finding exact answers are not needed, asserts Boaler. Moreover, kids should work in groups, engage in discovery lessons, think critically, and use calculators. Frankly, I am not sure how students can engage in critical thinking when they lack factual and procedural knowledge that enables it.

 Tom Loveless observes that Common Core reform math failed to boost student achievement in math and reading. Math is hierarchical. Each objective has a sequence of prerequisite objectives (Robert Gagne) that must be learned before moving to the next. Gagne (Science--A Process Approach or SAPA, 1967) identified the sequences and applied performance-based learning objectives (i.e., behavioral objectives) with measurable or performance verbs of action. In short, learning objectives must be quantifiable. Gagne did not use standards. He wrote tangible, clear, and specific objectives that were observable and measurable (i.e., performance-based). Note: Understanding, creativity, appreciation, problem-solving, and other vague terms cannot be behavioral objectives because they are difficult to measure accurately. What does it mean to say that a student has acquired problem-solving skills? How do you quantify that? It's not objective; it's subjective. It's a guess! On the other hand, the fundamentals of arithmetic are observable and measurable. The arithmetic objectives follow a sequence of prerequisites and are clear, specific, and quantifiable. 3-31-21

"Robert Gagne was one of the chief architects responsible for developing the systematic structure of Science--A Process Approach [SAPA]. The entire curriculum was based on the theory that any learning act--such as a process of science--can be broken down into component skills. These skills can be arranged and taught in a hierarchical order, from simple to complex." (Sanderson & Kratochvil) 3-30-21

Also, the SAPA math content was more advanced than the grade-level school curriculum. Today, nothing that comes close to matching SAPA and its instructional design. Gagne's conditions for learning: First, determine the learner's objectives that affect the required learning outcomes. Second, figure out the sequence: the learning hierarchies (content maps) that describe the prerequisites necessary for reaching the terminal objectives. Of course, the learner must have the prerequisite knowledge to learn new material. Third, engage in problem-solving using rules (intellectual skills). Learners should be able to state knowledge (verbal information). (3-30-21)

Of the six science processes taught in 1st-grade SAPA, four were math or math-related: Using Numbers (arithmetic), Measuring (metric system), Communicating (graphing), and Using Space/Time Relationships (geometry). SAPA emphasized the importance of math in science. The lessons were thoroughly field-tested and revised often before becoming commercial-ready to schools through Xerox. 3-30-21  

➜  Below: Science--A Process Approach (SAPA K-6, 1967): Part C is 2nd-grade SAPA. I used a similar hierarchical sequence for integers in my 1st-grade algebra program: Teach Kids Algebra, TKA, 2011. It worked well. 3-30-21

➜ LT: If you can't calculate it, then you don't know it. "The teacher's least favorite thing to hear from a student is 'I get the concept, but I couldn't do the problems.' The student doesn't know that this is shorthand for 'I don't get the concept.' The ideas of mathematics can sound abstract, but they make sense only in reference to concrete computations," writes Jordan Ellenberg (The Power of Mathematical Thinking, 2014). When I tutored Algebra2 and precalc, some students would say they understand the idea (concept) but had difficulty calculating the answers. To which I would say, "If you can't calculate it, you don't know it. Let's review the steps." Ellenberg points out, "It is pretty hard to understand mathematics without doing some mathematics." G. Polya (How to Solve It, 1956) explains, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics." The stress is on the doing, the performance! You won't understand perimeters if you can't do sums to calculate them. 3-30-21

 Common Core reform math, often called today's New Math, "is a nightmare for those of us who know full well how to solve most basic math problems, just not the crazy-making, convoluted way it's being taught today," writes Jennifer Jolly (USA Today). Parents are upset because they cannot help their K-8 children with math. Reform math is absurd. Its alternative algorithms taught for equity and understanding at the expense of mastering standard algorithms, which are necessary for problem-solving in math, are a radical approach that doesn't work. The progressive extremists say that math has changed. No, it hasn't. Indeed, 7 + 5 still equals 12, a basic fact that should be memorized along with many others in 1st grade. The standard addition algorithm is organized around place value and supported by single-digit sums. The single-digit sums like 6 + 8 = 14 are easily understood on a simple 0-20 number line. Indeed, the single-digit sums (+) and products (x) should be sanctified, not marginalized. 

Indeed, a fact such as 4 + 7 = 11 is valid and useful in hundreds of concrete situations and allows us to solve real problems. Also, if 67 + 85 = 152, is true, then 152 - 67 = 85 and 152 - 85 = 67 are also true by the basic rerlatiionship between addition and subtraction. Using smaller numbers, if 4 + 7 = 11 is valid and true, then 11 - 7 = 4 and 11 - 4 = 7 are also valid and true.  3-28-21 

The Standard Algorithm Is Primary!

The standard algorithm is built on place value and simplifies regrouping. The idea behind the algorithm is to add ones to ones and tens to tens, etc., because this is how the number system works. The grid above shows that when ones are added to ones, there are 12ones, which is immediately broken into 1ten 2ones. (See the 12.) Record the 2ones in the ones column and "carry" the 1ten over to the tens column (carry mark). Then, add tens to tens for a total of 3tens. The answer is 32, which means 3tens + 2 ones. The standard algorithm is economical and helps automate fundamentals. It is what we want first graders to do. Students are novices; they do not need a perfect understanding of regrouping or place value to use the grid. Understanding grows slowly. At first, only a functional understanding is necessary. The standard algorithm is a focused goal and should be presented at the beginning of the first grade. 3-30-21

 A project-based curriculum doesn't work in math, and the reason should be apparent. Math is highly structured; that is, one idea builds on other ideas, says Ian Stewart. In short, math is hierarchical. There are no shortcuts to grasping arithmetic, just a lot of practice-practice-practice and review. 

Project-based science requires an exceptionally well-written, field-tested curriculum in which academic learning (content) is the focus. Even then, projects often waste educational time as little science is learned. Students would learn much more science by reading a good textbook with some hands-on stuff that supports the content. Moreover, teachers must be highly educated to teach project-based science well. Unfortunately, many K-8 teachers are weak in science and math, yet they must teach these subjects. I don't recommend project-based. 

You can't force-feed project-based learning and expect it to work. It won't scale up, and this is why! In the late 60s and early 70s, with the advent of Science--A Process Approach (SAPA K-6), K-6 teachers had undergone extensive training, workshops, and courses, and it didn't work. Teachers didn't know enough math and science to teach SAPA well. Thus, project-based stuff won't work in most classrooms because teachers lack the prerequisite knowledge and skills to teach it well. And, the knowledge needed is not gained via workshops or PD sessions. That's what we learned. Note: Little has changed since the 1970s. Many K-8 teachers are still weak in both math and science. I blame progressive schools of education that train classroom teachers. But training is not the same as educating. 

K-8 teachers are trained for classroom teaching but not well educated in the subjects they are asked to teach, explains Diane Ravitch. Indeed, little has changed since the 70s. Also, the idea that testing would spur K-12 school achievement didn't work, either. Only 24% of the 12th-grade students were proficient in math by 2019 NAEP standards.

 Common Core (CC) reform math has failed, concludes Tom Loveless at Brookings. He writes, "No convincing evidence exists that the standards had a significant, positive impact on student achievement." In my opinion, CC has had a negative impact. Math and reading achievement scores (NAEP 2019) have stagnated for over a decade, and achievement is poor with remote and hybrid models. Zoom is not school! After 8 years of CC reform math, only 24% of 12th graders were proficient in math (NAEP 2019), a sad truth. Loveless points out, "Replacing Common Core with a different set of standards will solve very little. The United States cannot regulate its way to educational excellence. 3-23-21

 In my view, if you can't do it (i.e., calculate it), then you don't know it. You cannot apply something you don't know well. Perimeter is an example. If a student cannot do sums, then the student cannot calculate the perimeters of polygons. Knowing the idea of perimeter and calculating perimeters correctly are not the same. Finding the perimeters of squares, rectangles, and other shapes is a 1st-grade application of sums. The addends must be the same units, such as centimeters.

1st-Grade Plotting Points and Finding Perimeters of Shapes
Teach Kids Algebra (TKA) Lesson 7, May 2011
Title-1 City K-8 school.

 Calculating skill is one of the five basic building blocks of elementary school arithmetic, explains W. Stephen Wilson (Elementary School Mathematics Priorities). He writes, "You cannot teach mathematics without the place value system, standard algorithms, and our other building blocks." In short, the basics are for everyone. Tom Loveless notes, "The real problem is that the research evidence that these reforms boost reading or math achievement is spotty at best." 

 The problem has been deeply rooted in our curriculum and instructional methods for decades. It will be hard for teachers to change from discovery/inquiry or project-based group work to knowledge-based schooling and explicit explaining of carefully selected worked examples. Ian Stewart, a mathematician, points out, "Mathematics happens to require rather a lot of basic knowledge and technique." Marginalizing the standard procedures (i.e., technique) has been the wrong approach. And implementing constructivist "minimal guidance methods" of instruction has paved the way for minimal learning, explain Kirschner, Sweller, and Clark ("Why Minimal Guidance During Instruction Does Not Work...", 2006). 3-23-21 

➜ Minimal Guidance = Minimal Learning!

My algebra program for grades 1 to 4 is called 

Teach Kids Algebra. 

  • Understanding does not produce mastery; practice does!
  • You cannot apply something you don't know well.   
  • If you can't calculate it, then you don't know it. 
  • You know nothing until you have practiced. 


2nd or 3rd-grade TKA

12 + 4 +  = 12

What is box to make a true equation? 

-4 (negative 4)

4 and -4 are opposites (inverses) and add to zero, thus 12 + 0 = 12.

Note: Adding negative 4 is the same as subtracting 4. To subtract 4, add its opposite. The idea of inverse is fundamental in algebra.

0 and 1 are identities for addition and multiplication, respectively. 

12 + 0 = 12 and 

12 x 1 = 12.

In 1st Grade Arithmetic

Add 1 to get the next natural number. (Building Numbers)

6 + 1 = 7 

7 + 1 = 8

11 + 1 = 12

Add 0 (Inverse in Addition)

6 + 0 = 6

7 + 0 = 7

11 + 0 = 11

Can the students figure out these equations that I give to 2nd and 3rd-grade students? No calculators!

12 + 4 +  = 12 
12 x 4 x  = 12

How about 4th-5th Grade? No calculators!
(x + 3) 

(169)^½ = 

(2 3/4) ÷ 7/15 

How about 6th Grade? No calculators!

Jill walked at a pace of 3.5 mph for 30 minutes. How far did she walk?

Answers to the equations: 

-4, 1/4, 1, 13, 165/28, 1.75 miles


STEM Math for Elementary School Students by LT: Teach Kids Algebra Program

  • Addition: It must be the integer -4 to make 0, the identity. 
  • Multiplication: It must be a fraction 1/4 to make 1, the identity.
  • Any expression to the zero power is 1 by definition.
  • The exponent 1/2 means the positive square root of 169 or 13.
  • The fraction problem works out to be 165/28, which is in the lowest terms. Students must have lots of fraction practice and know that division is multiplying by the reciprocal. 
  • Units must be consistent, so 30 minutes is 1/2 of an hour, thus d = rt or d = 3.5 mph x 30/60 or 1/2 of an hour. The multiplication is simple: half of 3.5  is 1.75 miles as hours cancel.  

© 2021 ThinkAlgebra.org/LT

Monday, March 8, 2021


😎Inklings2: Observations, Ideas, and Opinions on Math Education by a Contrarian in 2021, a Divergent View. Inklings3 will open soon. 

My Impressions
March 2021

See the latest: Click Inkings3 
Some content from this page is now on Inkings3, plus more.

 Calculating skill is one of the five basic building blocks of elementary school arithmetic, explains W. Stephen Wilson (Elementary School Mathematics Priorities). He writes, "You cannot teach mathematics without the place value system, standard algorithms, and our other building blocks." In short, the basics are for everyone. 

Tom Loveless notes, "The real problem is that the research evidence that latest reforms boost reading or math achievement is spotty at best." 

 The problem has been deeply rooted in our curriculum and instructional methods for decades. It will be hard for teachers to change from discovery/inquiry or project-based group work to knowledge-based schooling and explicit explaining of carefully selected worked examples. Ian Stewart, a mathematician, points out, "Mathematics happens to require rather a lot of basic knowledge and technique." Marginalizing the standard procedures (i.e., technique) has been the wrong approach. And implementing constructivist "minimal guidance methods" of instruction has paved the way for minimal learning, explain Kirschner, Sweller, and Clark ("Why Minimal Guidance During Instruction Does Not Work...", 2006). 3-23-21 

➜ Minimal Guidance = Minimal Learning!

➜ Delay, Delay, Delay ...

➜ Common Core (CC) and state-standards-built-on-CC delay the standard algorithms of whole numbers. It's called reform math. Also, CC is not world-class math. We were misled into thinking that CC math was internationally benchmarked. It wasn't. I pointed it out in 2010 when I compared CC's 1st-grade math draft to Singapore's 1st-grade math syllabus. In 2011, my Teach Kids Algebra (TKA) 1st and 2nd-grade curriculum (7 Lessons) was a strong reaction against Common Core reform math. No matter the grade level, my best students in algebra had a good command of single-digit math facts, place value, standard algorithms. By 4th grade, TKA students studied a more structured approach to solving equations (unpacking/inverse), fractions/decimals, and percentages.  

➜ Even though my algebra program (Teach Kids Algebra) is considered acceleration, I teach in mixed classrooms and expect most students to learn at least 70% of the pre-algebra skills. But the expectation is not enough. I also give students help and encouragement during each session. TKA students do not work in groups or use calculators. They are explicitly taught via worked examples. TKA is STEM math for young elementary school students and builds on traditional arithmetic. There are rules to learn, including the "substitution rule" for variables. In the 3rd and 4th grades, TKA students also learn inverses and unpacking to solve equations, although I often start the undo or inverse idea in 2nd grade (e.g., 12 + 4 - 4 = x).

I do not Zoom. Zoom is not school. 3-26-21

Students are not learning enough basic arithmetic early enough. For example, they don't master multiplication in the 2nd and 3rd grades like kids in top-performing nations. The problem will persist as long as teachers focus on strategies and alternative algorithms (aka reform math) and use inefficient minimal-guidance methods (e.g., group work, discovery, project-based, etc.) and test prep instead of the explicit teaching of traditional arithmetic from the get-go (1st grade on up). Often, standard algorithms are delayed by Common Core or State standards mainly built on Common Core.

By 4th grade, students should focus on fractions and fraction operations--the international benchmarks, but, instead, they are still on addition and subtraction. Unfortunately, too many students come to 4th grade or 5th grade without knowing the multiplication facts, a 2nd-3rd grade benchmark. Over the decades, I have seen this in many classrooms. 3-19-21

From Jill Barshay at the Hechinger Report (3-22-21) about reading comprehension:

  • Generally, children posted higher comprehension scores after reading a print version of a picture book compared with a digital version.
  • Digital enhancements, such as games, pop-ups, and sounds, can distract children from the narrative storyline. Built-in dictionaries were bad for comprehension but good for vocabulary development. 

➜ LT: For vocabulary development, which is the missing link in reading comprehension, students would benefit from a workbook written specifically for that purpose, such as Vocabulary Workshop from Sadler. Many Catholic schools and other independent schools use Sadler. Also, content-rich history lessons from Core Knowledge (CK) with vocabulary in the sidebars (e.g., Ancient Greece and Rome) are excellent, too. Vocabulary lessons are in the CK Teacher Guides. And it is all free via pdf. (Note: Core Knowledge or CK should not be confused with Common Core or CC. Also, CK focuses on content-rich history and geography, not social studies.) 3-22-21


"Teaching is very easy if you don't care about doing it right and very hard if you do." Thomas Sowell 

"I don't believe I can really do without teaching." [When new ideas are not coming to mind, I must have something to do.] If I am teaching, "at least I am doing something and making some contribution." It never hurts my teaching, says Feynman, to review the fundamentals in my mind! "Is there a better way to present them?" Richard Feynman

"Fairness as equal treatment does not produce fairness as equal outcomes." Thomas Sowell

"But in recent times, virtually any disparity in outcomes is almost automatically blamed on discrimination, despite the incredible range of other reasons for disparities between individuals and groups." Thomas Sowell (Discrimination and Disparities, 2019)

The idea of equal outcomes is nonsense, yet it is the goal of radical education leaders whose strategy has been to "equalize downward, by lowering those at the top. It is a crazy idea taught in schools of education," writes Thomas Sowell. It is biased.



"Working memory is limited. So "whenever something clogs up working memory, your all-around problem-solving abilities take a hit ... Learning via worked examples, instead of solving problems for yourself, is one potential way past such working-memory roadblocks," writes Sanjay Sarma, MIT (Grasp, 2020). One major roadblock is not having auto recall of multiplication tables, which should be "overlearned" to stick in long-term memory and be instantly available in the problem-solving process in the working memory.  3-21-21

Furthermore, in my opinion, remote or hybrid is "pretend" education. "71% of parents think their kids learned less than they would have had schools remained open ... [while] 72% of parents say they are satisfied with the instruction and activities provided by schools during the closure," according to surveys conducted by EducationNext (Winter 2021). (Really? Are parents satisfied with their kids learning less?) 3-21-21


Give teachers a break. It is not their fault the State adopted Common Core reform math knowing it was not internationally benchmarked (i.e., not world-class) or shut the schools for inferior remote and hybrid instructional methods. 3-16-21 

"We don't have to understand the mechanics of the universe to go about our daily lives," suggests physicist Carlo Rovelli. And we don't have to understand calculus either, even though it unlocks the secrets of the universe. After all, Issac Newton, who invented the calculus, didn't understand its underpinning other than solving physics problems. So, let's think of mathematics as a tool to solve problems, which requires knowing calculating skills, ideas, and uses. Richard Feynman pointed out that there are not enough word problems or questions in elementary school math textbooks. 3-21-21

Standard arithmetic is not taught for mastery. This must change!

"If we want high-skill, high-wage jobs, then that is impossible to do with the education and job training system we have." (Tucker's Lens, 2-24-21)

American educators have not faced the reality that our educational and vocational structure (i.e., K-12 to Community College) is broken and, in my opinion, often lacking in quality and rigor. For example, Common Core math standards, now State math standards, are below international benchmarks, starting in 1st grade. So why did almost every state adopt Common Core in the first place? Many radical progressive leaders continue to claim that America is not a nation at risk. But, the facts don't support the claim.

After eight years of Common Core, only 24% of our 12th graders were proficient in math (2019 NAEP). Surprise! No great leap in achievement. Many AP courses are not equivalent to real college-level courses, primarily in the STEM areas. Surprise! For example, some colleges and universities, such as the University of Texas, do not accept AP calculus for STEM credit. (STEM students are required to take UT's calculus courses and for a good reason.) Frankly, I don't trust the quality or rigor of numerous so-called online college courses, much less online degrees. It is too easy to cheat. For example, you can get or hire someone to help you with an online test. (I know. I was asked to help with the final exam in a required statistics course.)  

Note: The superintendent of a local school district had this to say about returning to in-person classes on March 22nd for the last two months of school. I paraphrase: We don't expect students to make gains or make up lost learning. We expect a return to normalcy. In fairness, the super has been very supportive of teachers. 3-15-21

Nearly 80% of K-12 parents favor in-person schooling in their communities, writes Megan Brenan (Gallup) 3-11-21. I wonder if it is in-person, hybrid, or a combination of the two?

Scores: Reading and math scores should not be going down or stagnating. We are spending all this money on education, and the results are mediocre if that. Most kids are not proficient in math or reading at the state or international levels. What we have been doing in the classroom has not worked well. Richard Feynman was critical of elementary school math and science textbooks. He once observed, "So we really ought to look into theories that don't work, and science that isn't science." Also, "None of the textbooks said anything about using arithmetic in science." Nothing has changed!

Richard Feynman

Dr. Richard Feynman ("Surely You're Joking, Mr. Feynman!" 1985) is a Nobel Winner in Physics. He writes, "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences or inequalities. So if education increases inequality, is this ethical?"

➜ "Factual knowledge must precede skill," writes cognitive researcher Daniel T. Willingham. By factual knowledge, he means math knowledge such as concepts, facts, and standard algorithms (often called calculating skills). By skill, he refers to higher-level thinking skills often called "critical thinking," which is at the top of Bloom's cognitive taxonomy. Therefore, it must be more important than knowledge--a gross misinterpretation of Bloom. In short, a student cannot do problem-solving in math without first knowing the math (i.e., knowledge). Why are teachers glossing over, even skipping the knowledge foundation and jumping into critical thinking? It's taught in ed schools, but it's backward and doesn't work!
Start with Knowledge!
is needed for applying and reasoning.

[Note. The idea that all children have the same ability to learn is nonsense. "Children differ [widely] in their ability to learn academic materials," yet schools and educators often set aside this fact and treat all children the same, according to Charles Murray (Real Education). Consequently, we end up with mixed-ability classrooms (equity-diversity-inclusion policies) in which many children go unchallenged or shortchanged in mathematics. Children don't have the same ability but get the same opportunities to learn, not identical opportunities.

If students of similar math achievement are put in the same math class, they can excel and learn the math content. Also, I do not think we can catch kids up who have low academic ability because these students need a specialized curriculum that matches their abilities); however, effective intervention programs for incoming 1st-grade students with weak number skills, such as the safety net programs in Singapore, can help. It's called tracking!

Tracking in Singapore
Singapore has a pull-out math class with an expert teacher for incoming 1st graders weak in math skills. The program lasts up to two years to catch students up and continues in the 5th and 6th grades when the math content becomes much more complex. Singapore tracks students to remediate observed weaknesses. The 6th-grade exit test tracks students by math achievement to specific middle schools.]

➜ In my opinion, "equal opportunity to learn" does not mean learning equal content. Also, according to William H. Schmidt, "By international standards, our eighth-grade students are exposed to sixth-grade mathematics content." All students should have the opportunity to upgrade their content knowledge. But why are we teaching 6th-grade math to most of our 8th graders who should be capable of much more? The reason is simple. American students are not learning nearly enough basic arithmetic early enough.

Students lack good calculating skills. 

Teachers are often asked to teach math programs that do not stress the automaticity of math facts and standard procedures, i.e., arithmetic fundamentals. The lack of fluency in basic arithmetic often starts in 1st grade with addition and subtraction. Many educators endorse a misguided notion that kids will learn facts and procedures when they are ready. In reality, students without essential background knowledge will struggle in math up the grades. In short, students lack standard calculating skills, one of the five most important priorities of elementary school arithmetic, explains W. Stephen Wilson, a mathematician. Competence in Computational Skills is the backbone of a good math program, explains Robert B. Davis, The Madison Project: skills, ideas, and uses). If you can't calculate, then you can't solve math problems. 

Traditional Arithmetic trains a young child's mind for higher math if the fundamentals are automated, especially the early memorization of math facts and the learning of standard algorithms, starting in 1st grade with Addition,  which I think all students can learn. 

1st-grade Arithmetic: 67 + 85 can be rewritten as columns to stress place value, putting ones under ones, tens under tens, etc. The 7 and 5 make 12, which is 1 ten plus 2 ones or (t + 2). Record the 2 in the ones place as shown, carry the ten to the tens column (carry mark), and then add the tens. The standard algorithm is built on place value. 
1st Grade by Christmas


Competence in Computational Skills (i.e., standard algorithms) is essential for problem-solving. It requires memorization and practice-practice-practice. 

Working memory is limited. so "whenever something clogs up working memory, your all-around problem-solving abilities take a hit ... Learning via worked examples, instead of solving problems for yourself, is one potential way past such working-memory roadblocks." One major roadblock is not having auto recall of the multiplication table, which should be overlearned to stick in long-term memory and instantly be available for the problem-solving process in working memory. The same is true with other single-digit math facts, standard algorithms, and fraction operations.

Note: Under a remote or hybrid system, "My child is learning less." 
It's "pretend" education, which is incongruous with the EducationNext survey of parents who say they are happy with their child's education. Then again, surveys are opinions, which can vary from month to month. 
😎 Basics are not enough. Teach content that prepares more kids for algebra in middle school by restoring real education--not group work with minimal guidance methods of ineffective instruction. Moreover, students don't need to reinvent arithmetic. They are novices, not experts or little mathematicians. Algebra should start in 1st grade. Also, you don't make drawings to do arithmetic. Who does that? "It's not that Asian kids overachieve. It's that American kids grossly underachieve," explains Richard NesbettIntelligence and How to Get It

The Asian culture of working hard and persistence is something we could and should emulate. Intelligence, in Confucian terms, is a matter of hard work, not IQ, notes Nesbett (Schooling Makes You Smarter)For decades, the achievement bar for math has been set too low for American students. Most State math standards are below international benchmarks. For example, many kids struggle with whole number operations, such as addition (Common Core), when they should be learning all four fraction operations in 4th grade, an international benchmark. 

Instead of Piaget, we should have followed Jerome Bruner
like top-performing nations.

Professor W. Stephen Wilson points out, "Avoiding hard mathematics with younger students does not prepare them for hard mathematics when they are older." Indeed, he argues correctly that the lack of instant recall of multiplication facts permanently slows students down."

Note: If we were good at teaching arithmetic, then our kids would be among the best in the world, based on the billions and billions ($) we spend on schooling. They aren't. We have been on the wrong path for decades! Knowledge is the foundation for higher-order thinking in mathematics, so the first priority should be mastering fundamental mathematical knowledge (conceptual, factual, and procedural), which is what top-performing nations do. Often, weaknesses in elementary math don't show up right away but suddenly pop up in middle school or high school when students hit a wall and struggle with higher-level maths.

➜ Teaching kids to score high on a state test is not the same as teaching content that prepares students for algebra by middle school and beyond. We are merely teaching the minimum, just enough to get by if that. Common Core, which embraces reform math, won't change this. Preparing more kids for algebra by middle school was the highlight of the National Mathematics Advisory Panel report (2008). Still, Common Core and state standards did not embrace the premise by postponing most algebra standards to high school. It is a huge mistake to ignore standard arithmetic in the early grades. 

Issac Newton
Math is a tool to solve problems. But, we don't teach math as a tool. Think, Issac Newton! Newton invented calculations [calculus] that solved physics problems without a deep conceptual understanding of the procedures he used to solve the problems. The conceptual basis for calculus didn't come for another 250 years (limits). WOW! An incomplete understanding of, let's say, the long division algorithm should not a big deal for novices. Likewise, an incomplete understanding of calculus did not diminish Newton's worth or prestige as a super mathematician and physicist.

4th Grade Algebra (TKA)

The 4th graders in my Teach Kids Algebra (TKA) class find how the height of the "tennis ball bounce" varies according to the height the ball is dropped. [Discussion: The height of bounce is directly proportional to the height of the tennis ball drop.] After calculating the averages of the trials of the ball bounce from several heights, students plotted the results, drew the best fit line, figured out the slope of the line, wrote a linear equation that models the relationship (y = kx), and made predictions. Calculators are not used in TKA. The equation is a direct proportion and starts at (0,0)--without a bounce, there isn't a measurable height. When a prediction was made, the team set out to confirm it. Sometimes, the prediction (made by extrapolating beyond the known data) didn't match the experimental result. If the prediction does not agree with experiment, then the equation is wrong (Richard Feynman). [Discussion: Name the factors that could have interfered with accurate measurements or slope calculation.] 

4th-Grade TKA: Ball Bounce

In my Teach Kids Algebra (TKA) program, 4th graders did a ball bounce experiment from 5 different heights, 3 trials per height, recorded data, calculated averages, graphed it, figured out the best fit line, the slope, and wrote a linear equation in (y= mx + b) form. There was a lot of discussion about measurement error and why the equations differed from group to group. Then the kids tried interpolating and extrapolating values in their equations. They tested the new values to see if the equations worked. When the new observations didn't match the predictions made from their graphs, they knew something was wrong. Either the measurements and calculations were wrong, or the equation of the best fit line was wrong, or both. More class discussion, etc. (2012-2013)
Teach Kids Algebra (TKA) introduces advanced content to very young children, from integers to equations [functions] in two variables (y = mx + b). Educators should be teaching more than just basics. Starting in 1st grade, elementary school teachers should be preparing students for algebra by middle school. There is no set curriculum for this, so I made it up and wrote my own classroom lessons [curriculum] and taught it. I fused algebra to arithmetic to make algebra accessible to very young children. The students were in Title 1 urban schools.

Children don’t need to reinvent arithmetic; they are not little mathematicians. They are novicesUnderstanding does not produce mastery; practice does.

➜ Problem-solving is domain-specific.
Polya poses a problem.
The length of the perimeter of a right triangle is 60 inches, and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides?

A typical student cannot solve this problem without substantial knowledge of high school mathematics (algebra and geometry). But, isn't prerequisite knowledge necessary for any math problem, at any level, even for routine problems? Yes, knowledge precedes thinking, says Willingham, a cognitive scientist. Knowledge first! Why are teachers glossing over the knowledge foundation and jumping into critical thinking? It is backward!

Students should start with basic arithmetic and routine problems in mathematics to build a storehouse of knowledge and experience in long-term memory before moving to more complex problems that require more insight. The idea that students can do problem-solving (i.e., critical thinking in math) without fundamentals in place is illogical and backward, yet this is what many teachers think or are taught to think in education schools. In contrast to reform math, elementary school students should focus on mastering basic arithmetic and routine word problems, grade by grade. It requires a lot of practice and review. In short, solving problems requires domain-specific knowledge. The same is true for science, but the thinking in math (deductive) is different from the thinking in science (observation-inference), and so on. While math is absolute, science is subject to change as new observations become available.

I pose a chemistry problem.
Calculate the grams of hydrogen required to produce 82.000 grams of ammonia from nitrogen and hydrogen gasses.

Would you attempt to solve this routine chemistry problem without knowing the fundamentals of high school chemistry and algebra? Of course not! Solving chem problems requires domain-specific knowledge. Moreover, learning to solve routine problems, whether in chemistry or elementary school arithmetic, presupposes both knowledge and practice.   

I pose a Latin problem.
Ego vos hortor ut amicitiam ombibus rebus humanis anteponatis. Sentio equidem, excepta sapientia, nihil melius homini a deis immortablibus datum esse. 

Would you attempt to translate Latin without knowing the fundamentals of Latin? Of course not. Translating Latin requires domain-specific knowledge. 

What do I think? Teachers should teach, not facilitate. To teach math well, K-8 teachers must know mathematics through precalculus. Moreover, they should know college-level chemistry and algebra-based physics to grasp how arithmetic and algebra are used. The methods of instruction must be effective, not faddish or minimal guidance. We need smart and wise teachers in the classroom to produce smart and wise students. Schools of education have been producing mediocre teachers if that. Academic coursework for wannabe teachers, especially in higher mathematics and science, is weak. Blame education schools with low standards.  

In 2009, the Albert Shanker Institute wrote, "Advances in cognitive science make it clear that very young children are capable of much more academically than was previously imagined." This is not a new idea! We figured this out in the 50s and 60s with real instruction in the classroom--not from fads, ed school theory, progressive ideology, reforms without evidence, or beliefs without reason. 

The "many levels" in the same math class and an emphasis on group work and minimal guidance methods of instruction, which are very common in elementary schools and many middle schools, have led to widespread underachievement and other unintended consequences. We expect too little from students. 

Elementary teachers of self-contained K-6 classrooms spend about an hour a day teaching math, yet, over the years, there has been little [flat] progress in US math achievement. One might ask, what are teachers doing in the math hour? Instruction has not been as effective as it needs to be. Teachers work hard, but there seems to be little payoff. The grouping of students for math class (heterogeneity) is part of the problem. The instructional time consumed by test prep is another. The "group work" way of thinking by many teachers is still another. A dumbed-down math curriculum and minimal guidance methods contribute greatly to poor student achievement. Teacher training is inadequate in math and science. 

Not That Good ...
The Finnish school system has been lauded as one of the best in the world. It's not. In TIMSS 2011, Finnish 4th and 8th graders scored below US students in mathematics. In 2009, Professor Olli Martio, University of Helsinki, pointed out in a mathematics journal that less than 30% of Finnish 9th graders (15-16-year-olds) can multiply 1/6 x 1/2 correctly. He attributes the decline in competency on curriculum changes that sidelined basic arithmetic in favor of calculators. In Finland, the national standards have eroded to a lax framework that allows individual teachers to decide what they will teach and how. Thus, many Finnish teachers no longer teach some of the arithmetic skills that can be done on a calculator. 

I tested typical Finnish questions with my two 5th grade TKA classes. No calculators were used by Finnish students. Also, TKA students never use calculators. The Finish students didn't do well. Maybe, American educators should think twice before emulating Finland. Not much has changed! 

Straight Talk

1. Confirmation Bias

In Being WrongKathryn Schulz points out that "we believe things based on meager evidence." For example, experts tell us that technology in schools boosts math achievement. But is this guess true or false? We have had computers in classrooms for over 30 years. Where is the evidence? Is this a false claim? Schulz also defines confirmation bias as the "tendency to give more weight to evidence that confirms our beliefs than to evidence that challenges them." She explains that we fail to look for evidence that would contradict our beliefs. 

2. Integrity in Education

Richard Feynman explains the essence of good science: "If it disagrees with experiment, then it is wrong." Can this science idea be applied to education? I think so, but education is not physics. We have been relying too much on opinions rather than experiments. We in education have lots of guesses and programs built on guesses and assumptions; however, we do not conduct randomized sound experiments to test ideas. Indeed, many popular classroom practices have no basis in evidence. Feynman often talks about integrity in science. He writes, "If you're doing an experiment, you should report everything that you think might make it invalid." We never talk about this kind of integrity in education. Many education studies are flawed. It isn't easy to find a control group. Also, time-on-task is often ignored in educational studies, which makes them suspect. 

3. Minimal-Guided Instruction = Minimal Learning

Clark, Kirschner, & Sweller state that minimally-guided constructivist instruction (e.g., discovery, inquiry, project-based, problem-based, etc.), which is revered in schools of education and prevalent in today's classrooms, can "increase the achievement gap" and that the "failure to provide strong instructional support produced a measurable loss of learning." 

The researchers explain that the "aim of all instruction is to add knowledge and skills to long-term memory. If nothing has been added to long-term memory, nothing has been learned." If nothing has been remembered (retrieval), then nothing has been learned. In short, students must have the proper prerequisite knowledge in long-term memory to solve problems in math. Prior knowledge is essential. 

 The parent of a 3rd grader asked the teacher why the multiplication table isn't being taught. "We don't do that anymore; we teach critical thinking." The teacher's reply is the essence of reform math. And, it's wrong! One way for parents to fight woke teachers is to withdraw their children, which is what the parent did.  3-27-21

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