Wednesday, December 1, 2021

MathPage1

Welcome to MathPage1
December 1-2-3-4, 2021

Merry Christmas!

This website is for parents and educators who want their children and students to go to college and earn a degree.

I have a lot of empathy for teachers with everything that is going on in schools today. It was different when I first started teaching. I also have empathy for students who go unchallenged in many of today's classrooms. 

Going Beyond the Curriculum
STEM Math for Grades 1-5: Teach Kids Algebra TKA/Feedback
TKA started in 2011 as a reaction against Common Core reform math and minimal guidance methods. I fused basic algebra ideas with standard arithmetic. I taught algebra ideas by explaining worked examples (models) with practice and feedback in class. The importance of traditional arithmetic was stressed, starting with automating single-digit math facts that supported the standard algorithms. It begins in 1st grade. (Note:
 The pandemic effect put kids behind. Unfortunately, it was a lost year for many students. For example, I was able to teach much more content to 4th graders a few years ago than 4th graders today. Elementary school classroom teachers are struggling to help kids catch up, but I think some arithmetic gaps will remain.)

Note: Giving immediate feedback (below) is important when children are learning algebra fused to traditional arithmetic. It's STEM mathChildren need guidance and encouragement. They also need persistence

Immediate Feedback



It's Ideology!

Today, much of education is not about teaching kids to read, write, do arithmetic, or about achievement, merit, and excellence. Instead, schools have become a bastion of equity, diversity, identity, and inclusion. The liberal ideology is strongly supported by teacher unions, schools of education, federal and state education departments, school boards, and others, including the media. I believe education has gone down the rabbit hole, an abyss. Commonly, there seems to be an inverse relationship between excellent academics and equity. The "equity" reformers have been dumbing down the curriculum for decades. Equity has come to mean the same outcomes. The same outcomes are not possible unless content and standards are lowered substantially. Furthermore, kids are not the same in academic ability. 


Reflections of an Education Contrarian: A Divergent View

I am an education contrarian. My views often diverge from mainstream education "group think" and orthodoxy. For example, I teach 4th graders real algebra. Also, by 1st or 2nd grade, students work with expressions and variables (symbols that represent a number), evaluate expressions, solve addition and subtraction equations (functions) use guess and check, and later inverses (3rd grade). Given an equation, the student builds a table of values (x - y) and extracts the (x,y) number pairs from the table to plot points in Q-I, even in 1st grade.


1st Grade Algebra: Teach Kids Algebra - TKA

In TKA the focus is on variables, order of operations, and functions.

Functions: Equation-Table-Graph y = x + x +2

Functions In the 1st Grade (STEM Math)
Equation, Table Building, & Point Plotting (Graph)
 2011
 

Algebra is about figuring out what you don't know from what you do know. (LT) That's why we study algebra! It's why I teach 1st graders functions (equations-tables-graphs) fused to the arithmetic they are learning. Ian Stewart points out, "Mathematics happens to require rather a lot of basic knowledge and technique." But, first, you have to know something in long-term memory: facts and procedures. Learning arithmetic or algebra well involves memorization, repetition, review, and practice-practice-practice. In short, hard work and persistence!


Note: I am teaching two 4th grade classes at the Title-1 urban school. Almost all students are minorities. I am a volunteer guest algebra teacher and give lessons two hours once a week. 12-1-21


Teach Above the Curriculum!

I am increasingly concerned about academic achievement in schools, especially mathematics and science. Kids can learn more in-depth, key math content earlier and faster. I challenge teachers to stretch beyond state standards and progressive reform math. Teachers of 2nd graders, for example, should teach above the 2nd-grade curriculum and dive into the middle of the 3rd-grade curriculum. Students in 2nd grade should memorize half of the multiplication facts and calculate with the standard algorithm. 


One of the most debilitating trends in many progressive schools that teach reform math is their excessive emphasis on calculators. Many reform math programs expect students to use them starting in the elementary grades. Calculators enable students to do arithmetic without thinking about the numbers involved in a calculation. For this reason, it makes sense to use them, let's say, in a high school science class (chemistry and physics). But for elementary students, even middle school students, the main goal of math education is to get them to think about numbers and learn arithmetic--both factual and procedural knowledge.


In K-7, toss out calculators, most manipulatives, minimal guidance methods, and group work. Instead, explain worked examples (models) and give plenty of practice. Students should practice 30 problems a day, mostly review.


We need to train teachers better. They should be required to take a precalculus course in college, along with college-level chemistry and physics, even future 1st-grade teachers. Most educators will disagree. Still, K-8 teachers must know math and science because they teach math and science.


More-Earlier-Faster

(1) Students can learn more mathematics than the current curriculum.

(2) Students can learn more math earlier than previously thought (e.g., the beginnings of multiplication and algebra in 1st grade). 

(3) Students can learn more math faster.


Note: Teach Kids Algebra (TKA), my algebra program for grades 1 to 5, started in mid-January 2011 with two 1st-grade classes (7 Lessons), two 2nd-grade classes (7 Lessons), and one 3rd grade class (twice-a-week Lessons). The algebraic thinking lessons introduced true/false statements (=), variables, and the algebraic rule for substitution. Then, it quickly expanded to functions: equation-table-graph. TKA is STEM mathematics for very young children. Through the years, most students learned the algebra content with once-a-week, hour-long lessons. TKA in 4th-grade classes started late this school year (November 2021). TKA for 2nd grade will begin early next year. 


4th Grade TKA/Title-1, Urban School
Almost all students are minorities.
April 2012

Math is used extensively in science, but science isn't taught that way in our elementary and middle schools. There was one exception: The math needed to do the science was integrated into Science--A Process Approach (1967, SAPA K-6). So, for example, 4 of the 6 processes taught in 1st grade were math or math-related: 

  • 1. Using Numbers, 
  • 2. Measuring, 
  • 3. Communicating (graphs), and 
  • 4. Using Time/Space Relationships (geometry). 

In SAPA, 1st-grade students learned negative numbers. Today, there is nothing like it. However, some SAPA math topics are part of TKA, especially integers, linear functions, and graphing. Much of the math taught in SAPA was above the current curriculum. 


Algebra is about figuring out what you don't know from what you do know. That's why we study algebra! 


What has happened to math education in the US? 
We chase after test scores instead of credible academic achievement and continue a sameness ideology via Common Core and most state standards. Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Also, we expect kids to do critical thinking without sufficient background knowledge in long-term memory, even though, according to cognitive science, factual knowledge in long-term memory must precede higher-thinking skills. 

Minimal teacher-guided instruction, very popular in reform math, is ineffective and inefficient and contributes to substandard achievement, according to researchers Kirschner, Sweller, & Clark. Kids cannot solve math problems on a near-empty math tank (the long-term memory). For example, critical thinking will not help a student solve a trig problem without specific, prerequisite trig knowledge in long-term memory and practical experience working with similar trig problems.


Drills & Lessons

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 recommendations make good sense even today, but, unfortunately, the wisdom of old souls has been replaced by a progressive agenda of reforms and ideology that don't work (e.g., Common Core taught as reform math). 


I read comments in other blogs that are opinions or off the mark. I often exclaimed, "What is this person talking about?" When I write about education, I write from things I have done in the classroom. I was not born when the Committee of 15 laid down its recommendation, but if you want kids to learn stuff, then the Committee of 15 was correct, daily drills in addition to five weekly lessons. When I was a classroom teacher, I often implemented daily practice with daily lessons.


The test of the morality of a society is what it does for its children. --Dietrich Bonhoeffer, Theologian


LT Retired









    

 








Friday, October 15, 2021

Musing2

Welcome to Musing 2
November 27-28-29-30, 2021

Musings of an Education Contrarian: A Divergent View

I am an education contrarian. My views often diverge from mainstream education "group think" and orthodoxy. For example, I teach 4th graders real algebra. By 1st and 2nd grade, students are working with expressions, evaluating expressions, solving addition and subtraction equations (functions) using guess and check, and later inverses (3rd grade). Given an equation, the student builds a table of values and extracts number pairs from the table to plot points in Q-I.

 

Christmas Greetings 2021!😇

Nothing like Charlie Brown's Christmas tree to start the season!

Musing 2

When I was in elementary school, grades 1 to 5, there were no adults on duty before school, during morning recess, or at lunchtime. Students walked to school, and many walked home for lunch. Again, no adult supervision. Lunch was one hour long. Parents did not drive kids to school. No buses. I could run to the school in 5 minutes. The school bell was heard for blocks. Desks were bolted to the floor. Every K-5 elementary school teacher I had could play the piano. Also, for the last half-hour of the school day, the teacher would read to the class. There was a morning milk break. We were graded on deportment (behavior and manners) on the report card. Beginning in the 3rd grade, we wrote in cursive. 

Three Passions
My three passions are teaching kids algebra, researching and writing about education, and doing photography. Click Photography

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To learn arithmetic and algebra well, students need competent teaching, not graphing calculators or manipulatives (e.g., Cuisenaire rods, Algebra Tiles, etc., yet they seem commonplace in some reform math programs. Calculator use was just one of the bad ideas advocated by the National Council of Teachers of Mathematics or NCTM. Another was "minimal guidance" during instruction such as popular discovery/inquiry or project learning, and so on, i.e., group work. 
Over 30 years ago, the NCTM advocated calculator use as early as the 1st grade. No need to memorize math facts that supported the standard algorithms for calculating. It was a terrible idea because good calculating skills are vital for problem-solving.

If you want to teach, don't major in education. Avoid easy APs. 

In high school, take as much mathematics and science as possible and avoid easy AP courses like Human Geography, Environmental Science, Psychology, etc. As one parent put it, these AP courses are a waste of money, including AP calculus for STEM students. Instead of education, major in a regular academic subject such as history, literature, chemistry, mathematics, finance, physics, computer science, etc. In other words, get an excellent liberal arts education first

The presumption is that American teachers know best how to teach math and reading; however, national and international test scores don't support the conjecture. Many of today's teachers are products of liberal education schools. Future teachers should not major in education but a regular academic discipline. Along with the humanities, prospective teachers should pass college-level precalculus, chemistry, physics, history, and literature, even those college students who want to teach 1st grade. A methods course is not the same as a rigorous freshman chemistry course in qualitative analysis. 

Note: "Correlation is not equivalent to cause for one major reason. Correlation is well defined in terms of a mathematical formula. Cause is not well defined," explains David Salsburg, Errors, Blunders, and Lies, 2017. You can't jump from correlation to cause. Yet, that's what many journalists, the media, policymakers, politicians, writers, and so-called experts often do, thinking their opinion is valid, consequential, and factual. 

Even the most careful scientific investigations can be afflicted by uncertainty and bias. Every measurement has a degree of error, a "cloud of uncertainty." Over time, new observations often forge changes in physical laws and scientific theories. For example, the idea that mass is conserved was wrong. Energy, not mass, is conserved in reactions. Moreover, extrapolating beyond the observed data is pseudo-science and reckless practice. Science is not absolute like mathematics. Things can change in science, but in arithmetic, 2 + 3 still is 5 in our regular number system, and it has been the way for centuries.  
  

Insert: Have we gone nuts? I don't want boys, thinking they are girls, entering the girl's locker or restrooms, much less participating in girls' sports. However, I believe some girls can play boys' sports, but that's on athletic merit and skill, not a girl pretending to be a boy.

  

Note: My algebra program for grades 1 to 5 is Teach Kids Algebra (TKA), which started in January 2011. The lessons for algebraic thinking introduce true/false statements (=), variables, and the algebraic rule for substitution. Then, it quickly expands to the three representations of a function: equation-table-graph. TKA is STEM mathematics for very young children. Most students learned most of the algebra content. 

TKA 4th Grade: Finding slopes of lines and 
writing the equations in y =mx+b form.
Grade 4 TKA March 2012
 I met the 4th-grade classes twice a week, an hour each time.

Joanne Jacobs education blog: Pre-K doesn't lead to lasting gains in academic skills, but Pre-K grads may be more likely to complete high school and avoid repeating grades. "Most studies suggest a benefit in social-emotional skills such as self-control," writes Claire Cain Miller in the New York Times. "The effects are larger for children whose parents are poor; Black or Hispanic; or did not finish high school" and for boys." (11-16-21)


Note: Academic gains don't last. We have known it since Head Start. What does more likely mean? It is an estimate, and estimates always come with error, a probability distribution. Many estimates or judgments by experts are often shown to be dead wrong. 

 

Pre-K programs do all these other things listed by Miller. Really? It's a miracle! The government spends billions and billions on programs without assessing their quality. The alleged benefits are often anecdotal. Completing high school and doing well in school are far more influenced by family background than the teachers, according to the Coleman Report (1966). Sandra Stotsky (The Roots of Low Achievement, 2019) points out, "Low achievement was, as Coleman and his colleagues concluded, more a reflection of family background than their schools and teachers--then and presumably now."

 

Stotsky observed, "Most of the changes during the twentieth century reduced the content of academic coursework and time spent on this coursework." Many of the initiatives, such as gap closing, national standards approach, math reforms,  diversity, equity (meaning equal outcomes), Zoom learning, etc., didn't work. Educators still don't know how to change low-performing kids into high-performing kids, explains Stotsky. Also, it seems apparent that the Coleman Report findings regarding family background have been ignored well into the 21st century.

 
 
✓  "If you can't explain difficult mathematics to little kids, then you don't know it well enough."  (Or, you don't know how children learn math or both.) H. Wu, a mathematician at UC-Berkeley, has been teaching workshops and courses for K-8 teachers for decades. Wu wrote that many K-8 teachers don't know enough math to teach Common Core math. (Most state math standards are based on 
Common Core.) Also, many teachers are just average and have difficulty explaining complicated math to students. Teachers take my course in the summer, says Wu, but when they return to the classroom, they teach the same old reform math with minimal guidance group-work methods. Learning more content doesn't always alter the way teachers teach, noted Wu, but, I think, it is a step in the right direction

K-8 teachers need to know more about mathematics, at least to the precalculus level, and science, especially chemistry and physics. Unfortunately, for many decades, K-8 teachers have been weak in both math and science. They should be held to higher academic standards, but it's not their fault if they are not. The schools of education are at fault. Education should not be a major. Future teachers should major in a regular academic discipline such as history, English, literature, mathematics, science, philosophy, finance, economics, engineering, classical languages, etc. 

In addition to precalculus, a combination of college algebra and trigonometry, wannabe teachers should take rigorous chemistry and physics courses in college, not watered-down courses. For example, a methods course in science should not be substituted for a college-level physics or chemistry course. Likewise, a math education course should not be substituted for a real math course such as precalculus or calculus. Furthermore, a statistics course should not be substituted for a precalculus course.

The presumption has been that teachers know best how to teach math and reading; however, national and international test scores don't support the conjecture.

Sandra Stotsky (The Roots of Low Achievement, 2019) points out that schools should spend money on improving the curriculum for all students, not always on struggling students or closing gaps. The standards approach has not worked.
✍️ The late Richard Feynman wrote, "I would rather have questions that can't be answered than answers that can't be questioned." It is why I oppose those in education who think they know best how to teach children math, such as Jo Boaler, a so-called math educator. If reformists like Boalerknewn how best to teach math, almost all our kids would learn some algebra skills in the 1st grade and perform the standard long-division algorithm by the 3rd grade. Frankly, when asked how I teach 1st and 2nd graders real algebra, I reply, "I don't know how I explain complex material to children ... But I do know that if I can't explain complex math to young students, then I don't know it well enough." Also, I think in terms of prerequisites (Gagne's Learning Hierarchy), not stages (Piaget), and write coherent "worked-out examples" or models that are performance-based (Mager's behavioral objectives). I often think, is there a more straightforward way to get from A to B? 

**********

Inadequate Calculating Skills & Deficient Problem-Solving Skills. 

In my opinion, the decline in math and reading, as shown by the disappointing data from the NAEP Long-Term Trends (LTT) before the pandemic, roughly matches the advent of Common Core in 2011 through 2020, during which there was a swing toward conceptual understanding using minimal guidance methods, often at the expense of computational fluency. Common Core math and state standards delayed the standard algorithms for no good reason. Students must learn mathematical procedures. Many aren't. They must perform procedures efficiently. Many can't. 


Note: I shall always remember a quote from Ian Stewart, "Mathematics happens to require rather a lot of basic knowledge and technique." (Letters to a Young Mathematician, Ian Stewart, 2006)


 Understanding does not produce mastery; practice does!


I remember tutoring precalc students who often would say that they understood the concepts but could not work the problems. So, what I did was reteach the procedures. Kids need practice problems, a lot of practice problems, even precalc students, to hammer in the facts and procedures. And, they need review! No matter how good students think they are, they will never come close to matching Richard Feynman's calculating skills. He used to do calculus problems in his head, even at breakfast, which drove his wife crazy. 


Today, poor problem-solving skills can be attributed to inadequate calculating skills, ineffective instructional methods, a dumbed-down curriculum, and just plain poor teaching. For example, students were handed a calculator rather than a pencil. Of course, both concepts and calculating are essential, but the mix became wrong when concepts pushed aside calculating fluency. 


If you can't calculate it, you don't know it. 

How can you solve a problem if you can't calculate the solution on paper like Einstein? How can you solve a trig problem if you don't have adequate knowledge of trig in long-term memory and experience solving trig problems? Knowledge, solving routine problems, and review are needed no matter the level, starting in 1st grade. 


Strong calculating skills are needed to solve problems. Fundamentals begin in 1st grade.

********** 


Measuring is an essential process in Science--A Process Approach (SAPA, 1967). It should also be an essential process in 1st grade. In the early 80s, my students measured mass on old SAPA equal-arm balances in my self-contained 1st-grade class at a Title-1 urban school. Students used centimeter cubes to measure the mass of various objects. Each cm^3 had a mass of 1 gram. Students also learned to "balance the balance" by adjusting the riders before measuring an object. 


🎯 Measuring gives students opportunities to use numbers and learn parts of the metric system.


First-Grade Science Vocabulary

SAPA (Measuring Lessons)

  • centimeter, decimeter, meter, horizontal, vertical, between
  • volume, liter, milliliter, graduated cylinder, pipette, beaker, meniscus
  • area (how much space), comparing, ordering, matching, approximate
  • mass, gram, kilogram, equal-arm balance, cubic centimeter

🎯 Today, how many 1st-grade students are introduced to science vocabulary and ideas like these found in 1st-grade SAPA (1967)?


The main weakness of Science--A Process Approach (SAPA) was process at the expense of content knowledge. The expectation was that students would explore an exciting topic more deeply, prompted in a process lesson by finding books and materials at the library. Most children won't do that! There were no SAPA textbooks for students to read and gain deeper information about a topic. 


In short, science is not only what scientists do (process); it also is an accumulation of knowledge or facts. Reading helps kids acquire more profound content knowledge. Memorizing facts and definitions in science is essential, too. Watching TV science programs can boost a child's interest in science, especially chemistry and physics, topics that are often ignored in elementary school. 


Mathematics is a key part of SAPA. Indeed, "competence in using numbers is essential in science." The K-6 SAPA science program gives using numbers proper value. SAPA included the Using Numbers process "first, to make pupils realize that the ability to use numbers is an essential and fundamental process of science; second, to give pupils an opportunity to use numbers in finding answers to scientific questions in real problem situations." (SAPA Commentary for Teachers, AAAS, 1970)

In writing specific performance (behavioral) objectives, the action words are: identify, name, order, describe, distinguish, construct, demonstrate, state, and apply a rule.


Equations act like a balance.

The SAPA balances are long gone, but today, I use a commercial equal-arm balance (below) to demonstrate the idea of balance or equal (=) in equations, that in all equations, the left side equals the right side or visa-versa. In my algebra lessons for grades 1 to 5, students are asked to "Think, Like a Balance."

Jayne, a 1st grader, measures mass in units called cubes.
Note: These cubes are larger than centimeter cubes and have a mass of about 1.7g each. However, you can round up and assign them a mass of 2 grams (2g) each until kids get to decimals. Thus, for 1st graders, 6 cubes would approximately equal 12 grams in mass ( 12g), etc. The symbol means approximately equal to. Every measurement has an error component. Another symbol I use a lot is , which means not equal to. Given the equation 3 + 2 = 2 ● 3, is it true or false? It's false:
3 + 2 2 ● 3 because 5 6. 
  

Note: Measurement = truth + error.  

Expression = Expression: 2 + 5 = 10 - 3 is a true statement because 7 = 7. In math, we work from true statements and make equations true given a variable, such as x + 12 = 10 + 7. (Finding x is called solving an equation.) If the right side is 17, then the left side must be 17, too. What number x do we add to 12 to make a true statement, 17 = 17? {5} An equation tells that two calculations have the same answer. Stress: A variable like x is a number 


Note: SAPA was the best science curriculum available because it integrated mathematics into science. Its downfall was that ordinary teachers were unable to teach SAPA well, not even after extensive training. This is because they didn't have a strong science or math background, to begin with. In addition, SAPA lessons required lots of materials, and when government money ran out, kits were not replenished in the late 70s. As a result, much of the equipment was stored, lost, or discarded.    


🍎 "Children differ a lot in how well they do at school." --Robert Plomin
"When it comes to mathematical ability, we're not all born equal."
Too many educators believe that "anyone can do anything: all they need is training and lots of practice." But it doesn't work that way. "Most of us can be trained in mathematics up to high school level, but few go beyond." (Ian Stewart

Robert Plomin explains, "Schools matter in that they teach basic skills such as literacy and numeracy, and they dispense fundamental information about history, science, maths, and culture." Ian Stewart points out, "That's why education is worth the effort."

✔︎ Here are a few interesting ideas I teach in TKA. 
1. Equation: Expression = Expression
If 3 + 4 = 7 and if 8 - 1 = 7, then 3 + 4 = 8 - 1 or 7 = 7 by the Transitive Property of Equality, but most teachers never heard of Transitivity. Thus, an equation tells that two calculations have the same answer. Two things equal to the same thing are equal to each other (Transitivity). This is not meant to be formal proof, but it is interesting. I give my 1st graders more difficult problems, such as 5 + 7 = x + 5. Find x. If the left side is 12, then the right side must be 12 as well. Well, that is easy for 1st graders who know the commutative property of addition; how about 5 + 7 = x - 12? {24} Students at this age use guess and check to solve equations. In 3rd grade, they use inverses. 

2. Division by zero is not an acceptable operation because it produces a false statement. 0 x 1 = 0 x 2 (True 0 = 0). Now, divide the original equation (0 x 1 = 0 x 2) by zero on both sides, and cancel the zeros. You end up with 1 = 2a false statement. This is not meant to be formal proof. (Sure, I can divide by zero, but  it doesn't work and is not acceptable in regular arithmetic or mathematics.) Yet, it is interesting. An ordinary fraction is a quotient of two integers: a/b, but b cannot be zero.

3. -3 x -3 must equal either 9 or -9, but which one is it? 
Let's test -9 because we learned 3 x -3 = -9 on the number line. 
-3  x -3 = 3 x -3   (Does -3 x -3 equal -9?)
Okay, let's divide both sides of the equation by -3. Cancel.
We end up with -3 = 3 (False), so it cannot be -9. It's a counterexample. Thus, the answer to -3 x -3 must be 9, not -9, which suggests that the product of two negatives is always positive in Real Numbers. This is not meant to be formal proof, but it is interesting.   

Where did I get these ideas? 
My main source was Ian Stewart, a mathematician who wrote many books. Stewart also shows that every number has 2 square roots except 0. It's the Pythagorean Theorem, of course. Thus 3^2 + 4^2 = x^2, But x can either be 5 or (-5). (Just to toss something out for middle school math teachers: solve x^2 = -9. It's complex.)  

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The False Narrative of Progressives: & The Fallacy of Fairness
If we provide the right environment in school and at home, students will do well academically (equity). "What we need are government preschools," we are told. Really? Sounds great, but it doesn't work that way. As it turns out, school achievement is more genetics than nurture (Robert Plomin, blueprint, how DNA makes us who we are, 2018). Also, poverty is not the root cause of low academic achievement. The root causes have been government policies and substandard teaching of basics. A major factor of declining achievement in government schools is dictating equal outcomes by intentionally lowering standards and expectations. No child gets ahead. 

Equity as equal outcomes is a "fallacy of fairness," writes Thomas Sowell. Equity cannot produce equal outcomesKids get better through practice, but practice and memorization have fallen out of favor in progressive schools. Drill is considered old-fashioned, old-school, and poor pedagogy. It's not. It works! 

Robert Plomin explains, "Children differ a lot in how well they do at school ... How much do differences in children's school achievement depend on which school they go to? "The answer is not much ... Our focus is on individual differences [not schools] ... Schools matter in that they teach basic skills such as literacy and numeracy, and they dispense fundamental information about history, science, maths, and culture. Schools also matter because children spend half of their childhood in school ... This does not mean that the quality of teaching and support offered by schools is unimportant. It matters a lot for the quality of life of students, but it doesn't make a difference in their educational achievement ... Some children in the worst schools outperform most children in the best schools." 

Plomin points out, " Inherited DNA differences account for more than half of the differences between children in their school achievement. Thus, genetics is by far the major source of individual differences in school achievement, even though genetics is rarely mentioned in relation to education." 

Environment (nurture) accounts for the rest. Plomin makes clear, "Equality of opportunity does not translate to equality of outcome." Thomas Sowell has been saying the same thing for decades.
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Stalled Achievement: It's the Teaching!
Kids are not learning the fundamentals of arithmetic or algebra well enough. It's not the teaching we are told, but what else could it be? Frankly, many teachers aren't teaching fundamentals effectively. Either the curriculum is wrong, or the instructional methods are wrong, or both. A steady diet of group work, discovery learning, project learning, or other minimal guidance methods doesn't work in math. There is little explicit instruction using examples (models) in today's classrooms. Kids must master facts and learn procedural knowledge to advance. Sadly, memorizing facts and practicing standard algorithms have been pushed aside as poor pedagogy. Really? 

Furthermore, equity has come to mean equal outcomes. Not possible, a liberal pipe dream! The only way to close gaps is to significantly lower standards and expectations, a dumbing down of curriculum, which Thomas Sowell calls a "fallacy of fairness." Common Core or state math standards were shaped so that all students get the same math curriculum. No student gets ahead. The idea of sameness in the name of equity doesn't fit the real world or the science of learning. Some kids are better at math than others. That's DNA, instruction, and practice. Some kids are better at the violin. That's DNA, instruction, and practice. 

Almost all kids can get better in math through practice-practice-practice and better teaching. Nearly all students can learn arithmetic at an acceptable level to prepare for a rigorous pre-algebra course in 6th or 7th grade that prepares students for Algebra-1 in 8th grade. It happens when expectations are high. But, the teaching must get significantly better in the lower grades. Therefore, the pathway is conditional on K-6 teachers teaching arithmetic well starting in 1st grade. Hence, part of the answer is for schools to train teachers better in arithmetic and teaching math. Reform math and minimal guidance methods have not worked.

Math Facts: Start with a simple, 0-20 number line, then use flashcards. 
Math facts must be automated in long-term memory! 

First graders should memorize math facts from the start, not count on their fingers to figure out a fact each time needed. Bad habits are hard to break. Also, they should not use calculators! Still, I see exercises in 1st-grade math textbooks that use calculators. 

Elementary School Mathematics Priorities  ( Dr. W. Stephen Wilson)

The five building blocks for higher mathematics: 

1. Numbers

2. Place value system

3. Whole number operations (i.e., The Standard Algorithms)

4. Fractions and decimals

5. Problem solving 


Most teachers are unaware of the content very young children can learn given the proper instruction.


Note1: 2nd-Grade kids learned to do sums of integers (-10 to 10), plotted (x, y) points, and identified ordered pairs of numbers, all were part of the 2nd Grade Science A Process Approach (SAPA), the Using Numbers process (SAPA 1967). Moreover, 1st-grade students were introduced to integers (-10 to 10) in SAPA. These exercises were thoroughly tested in classrooms and not optional. In short, SAPA taught the mathematics needed to understand and do the science.  


Note2: In my algebra curriculum (Teach Kids Algebra, TKA), 40 1st graders at an urban Title-1 school, after 7 hours of instruction, used a linear equation (e.g., y = x + x + 1), made an x-y table of values given the x values, and plotted the corresponding number pairs (x, y) in Quadrant I (TKA, 2011). The TKA lessons were for all students--almost all were minority students. Below is one of my 1st-Grade TKA Students (TKA, Spring 2011).

TKA was a stark response against Common Core reform math, minimal guidance instructional methods, calculator use, the dumbing down of math, grade inflation, and other inane policies. In 1989, for example, the National Council of Teachers of Mathematics (NCTM) reforms included calculators in 1st grade. Is it any wonder that achievement in arithmetic and mathematics, in general, went downhill? In my opinion, the NCTM lost all credibility. Higher-level thinking skills, without substantial knowledge in long-term memory supporting them, are at best superficial if that.  


Declining Achievement

Bummer: Declining Achievement!

Pre-pandemic Data (The Nations Report Card)

NAEP Long-Term Trends (2012 - 2020)

Since 2012, our kids are not getting better in math or reading, according to pre-pandemic data (NAEP Long-Term Trends LTT2020). Betsy DeVos writes, "Not even high-performing students recorded any measurable achievement gains. There wasn't a single bright spot to be found anywhere in the data. No student, of any age, of any subgroup, saw their performance improve since 2012. Most saw declines." (Note: NAEP Long Term assessments are different from the regular NAEP assessments.)

  • "One of the biggest differences between school math and university math is proof. At school, we learn how to solve equations or find the area of a triangle; at the university, we learn why those methods work and prove that they do." -- Ian Stewart (Letters to a Young Mathematician, 2006) 
  • Thus, teaching children arithmetic and algebra should show how to do and apply things, not why they work. Factual and procedural knowledge is essential to learning math well. Kids need a solid grasp of basics--factual knowledge and technique, says Ian Stewart. They don't need to make drawings or write explanations to prove they are correct. 
  • "In my opinion, the implementation of Common Core reform math and the resurgence of minimal guidance instructional methods seem to coincide with the stagnation of achievement. The stagnation is the result of the poor teaching of fundamentals, starting in grade 1." -- LT
  • "Thoughts without content are empty"  -- Immanuel Kant
  • "It's called remote learning because the chance of a child learning anything is remote." (Cartoon)
  • "As for the Biden education agenda, I couldn’t be more disappointed by their emphasis on 'inputs' reminiscent of a pre-Coleman era, when schools were judged by their resources and promises rather than by whether kids learned anything in them." -- Chester E. Finn Jr.
  • Most gifted programs do not boost academic achievement because they are enrichment programs or project-based, not acceleration. The fact that knowledge sparks critical thinking and problem-solving seems lost in many gifted education programs. Thinking without sufficient knowledge in long-term memory seems shallow and superficial. Most gifted programs are not based on the academic standards (math and verbal) established by the Johns Hopkins Center for Talented Youth (CTY), which requires the School and College Ability Test (SCAT) two grade levels ahead to identify students eligible for its programs. For example, a 2nd-grade student would take the 4th-grade SCAT. Also, 7th-grade students should score at least 500 on the College Board SAT math or 510 on the verbal part to qualify for CTY courses in math or the humanities. 
  • After Brown v. Board of Education, Thomas Sowell explains, “There was a lot of turmoil, racial polarization, and bitter backlashes, but no general educational improvement from seating black school children next to white school children.” Chief Justice Warren, who claimed that schools were “inherently unequal,” was wrong! There were many excellent all-black schools, too, that did the same. But, forced integration ruined these black schools, such as the all-black Dunbar High School in Washington DC, says Thomas Sowell, a graduate of Dunbar. (Thomas Sowell: Discrimination and Disparities, 2019) 
  • When you learn something, teach it to someone else. 
  • I often hear, Follow the Science; however, I think most people don't grasp science. Science does not prove something correct. Rather, it is a process that eliminates wrong ideas. With new data, scientific ideas often change. It is not absolute like mathematics. Data interpretations can often result in divergent inferences

✔✔✔✔✔ Declining Achievement!

In my opinion, the implementation of Common Core reform math and the resurgence of minimal guidance instructional methods seem to coincide with the stagnation of achievement, which, I believe, is primarily the result of the poor teaching of fundamentals, starting in grade 1.  


"It's the teaching,the late Zig Engelmann would retort. Teaching involves both the curriculum and methods of instruction. For example, you can't teach arithmetic the way you teach social studies because arithmetic is symbolic, abstract, and hierarchically based, so that "new mathematical ideas build on older ones" (Ian Stewart, Letters to a Young Mathematician), so the sequence is critical. Students who do not master the multiplication table or practice the multiplication and long-division standard algorithms by the 3rd grade will struggle with fractions, proportions, percentages, algebra, higher-level math, etc. Indeed, the alternative algorithms advocated by many reputed experts, math educators, reformers, and policymakers should never replace the standard algorithms, which are primary.  


Chester E. Finn Jr. on Excellence 
"I generally view “excellence” in education as a relentless push to maximize student learning—for all kids, including the very bright and those who move at a slower pace, certainly including kids from every sort of background. There’s a meritocratic element to it, but there’s also a strong push for accelerating everyone to the max. Today’s push for “equity” is surely at war with meritocracy, especially if it leads to things like eliminating “gifted education” in the name of equality rather than augmenting the scope of gifted ed. to serve many more kids. The “war on testing” is ultimately a “shoot the messenger” project that would blind us to the failings, gaps, and shortfalls that beset both equity and excellence. And the distraction of nonacademic mandates for schools, such as SEL and other “softer” school qualities and pupil attributes, can only deflect us from the pursuit of excellence. Do please bear in mind that “Excellence in Education” was the name of the commission that issued A Nation at Risk!"  (Finn's Interview with Rick Hess, Education Week)

Failed Educational Goal: Gap Closing
Sandra Stotsky explains that the primary educational goal should be improving the curriculum and teaching for all students, not closing gaps. After dumping billions and billions into schools over the years, the gaps aren't closing and, in many cases, increasing. Over the years, the curriculum has been dumbed down, and teachers have been using minimal guidance methods that are ineffective. Gap closing has been a failed educational goal. It has also been the wrong goal, articulates Stotsky (The Roots of Low Achievement, 2019). 

"How exactly does a classroom teacher close gaps if she also is trying to improve academic achievement in all students in the class? She can't easily do both at the same time." Educators are obsessed with gap closing. But, Stotsky suggests that educators don't know how to change "massive numbers of low achievers into high achievers," not in math or reading. She points out, "So far, studies have produced no useful insights for K-12 educators to address low achievement." 

✔✔✔✔✔ In short, what we have tried hasn't worked; e.g., reform math, minimal guidance methods of instruction, mixing achievement groups in the same classroom, group projects, block scheduling, more money, etc. 

Not Ready for College
A new report from ACT states that only 1/4 of the students who took the exam were ready for college. (https://leadershipblog.act.org/2021/10/2021-ACT-Achievement-Data.html)


ACT CEO Janet Godwin observes, "We are seeing a number of year-over-year trends that suggest the emergence of a 'lost generation' that is less likely to succeed academically and in the workplace ... These trends have all been worsened by the COVID-19 pandemic, but it is not the single cause nor excuse for them." The decline started years ago. In my view, it started with NCTM reform math and 1st-grade arithmetic. Common Core or state standards have exacerbated the decline. 


Godwin points out, "This is the fourth consecutive year of declining achievement of high school seniors, and too many of our seniors are simply not prepared for college-level work."


The decline didn't start in high school or the Zoom year. We have not been teaching the foundations of arithmetic well enough beginning in elementary school, so a decline is inevitable. On average, our students cannot match their peers in top-performing nations, not in mathematics.  


Justice-Orientated Science

I read parts of the Science and Engineering in Preschool Through Elementary Grades (2021), a Consensus Study Report that focuses on "justice-oriented science and engineering instruction." Really? It is just as bad as equity math. 


We need to go back to move forward.


In the 1960s, Science--A Process Approach (SAPA K-6) integrated mathematics as an essential process. In fact, of the six science processes taught in 1st-grade SAPA, four were math (Using Numbers, e.g., Integers, etc.) or math-related, such as Measuring, Communicating (graphs), Using Space/Time Relationships (geometry). Robert M. Gagne laid the foundation for SAPA's Learning HierarchyBelow is a glimpse of the 2nd-grade Using Numbers. The sequence is important, linking a new idea to an old idea. Prerequisites!


Part C is 2nd Grade

In SAPA, the stress was on science processes applied to biology, chemistry, and
physics--not social justice, identity, or diversity. The math needed to do and grasp the science was taught as an essential part of SAPA, such as Using Numbers 8 above.


Finnish Schools (are not that great!)

When Finnish schools were proposed as a model the U. S. should follow a few years ago, Dr. Olli Martio, University of Helsinki, found conspicuous math deficiencies in Finnish schools. I gave my two 5th-grade classes at a Title-1 Urban school some of the 9th-grade questions out of curiosity. Martio concluded that the use of calculators and a spotty curriculum have kept many students from learning basic arithmetic. (Note: 9th graders are ages 15-16 in Finland; 1st graders start at age 7.)



Note: I also found that Finnish 4th and 8th-grade students ranked slightly below the US in international testing (TIMSS). I concluded that Finnish math education was not superior after all. In the most recent TIMSS cycle, I also observed that the results of Finland's 8th graders were conspicuously absent. Therefore, we should not imitate Finland. 

When our math standards are below international benchmarks, we tell our kids they are not smart enough to learn the same math (or science) that kids in many other nations routinely learn. So, for example, the fluency of standard algorithms is delayed for no good reason.



Below is the standard algorithm I taught 1st graders in the early 80s. It is efficient and the best model for place value. I used the word carry, not renaming or regrouping. Carry makes good sense to beginners: 4 and 8 are 12 ones, or one ten and 2 ones. See the 12. The one ten is carried to the tens position: adding ones to ones, tens to tens, etc.  
Olli Martio pointed out, "Although the changes to the mathematics curriculum were made to help people to use mathematics in everyday life, this aim has badly failed ... Problem-solving has been overestimated in all levels of the mathematics curriculum," and calculators have been misused. "The use of calculators is overemphasized since nowadays their use is extremely limited in everyday life." You never hear, "Let me pull out my calculator" in everyday life. The bank calculates your balances, loans, interest, mortgages, credit card payments, etc. Stores compute your purchases and tax. The same, for bills, such as your electric or gas bill, insurance, finance, etc. Almost everything is calculated for us. In many cases, your calculator collects dust after schooling is finished. (I forgot where I put my slide rule.)

Here is another problem in Martio's test: 0 x 8436 = 0 x 0.536. In 1981, 79% of Finnish students were correct, but by 2003, the percentage of students who got it correct dropped to 65.6%. Of course, multiplying by zero gives zero, a 3rd-grade concept, even earlier.  

Another problem: (-3)^2, 67.8% of Finnish students had "(-3)^2 = 9" correct in 1981, but by 2003 only 47.5% of Finnish students were correct. Not good! What's went wrong? It's the teaching, a spotty curriculum, and the misuse of calculators concluded Dr. Martio

For comparison, in my two 5th grade classes a few years ago, 74% of the students in one class gave the correct answer (9) and 54% in the second class. Calculators were not allowed in my Teach Kids Algebra (TKA) lessons. Group work,  discovery/inquiry, and project-based learning were not used in TKA. Instead, carefully selected examples were explained and linked to old ideas (explicit teaching) along with practice worksheets where more than half the questions were review questions. (Note: I met students in grades 3 to 5 once a week for 60 minutes and, depending on the scheduling, just twice a month for an hour for the school year. I met with 1st and 2nd graders for 7 hours.) 

Another problem: (1/6) x (1/2). In 1981, 56.5% of Finnish 9th graders were correct (1/12), but in 2003 only 28.3% gave the correct answer, a significant drop. Remember, these are 9th graders. It's shocking! In contrast, 74% and 100% of students were correct in my two fifth grade classes, respectively.

I do not think my 5th graders were better students than the Finnish 9th graders. I think the Finnish students were not taught fundamentals, starting from the 1st grade. It's the teaching! Incidentally, I teach my algebra program at a Title-1 city school with about 85% minority population.

(Reference: Long Term Effects in Learning Mathematics In Finland--Curriculum Changes and Calculators, Olli Martio)

The False Narrative of Progressives 
If we provide the right environment in school and at home, students will do well academically (equity). "What we need are government preschools." Really? Sounds great, but it doesn't work that way. As it turns out, school achievement is more genetics than nurture (Plomin). Also, poverty is not the root cause of low academic achievement. The root causes have been DNA, government policies, and substandard teaching of basics. A major factor of declining achievement in government schools is dictating equal outcomes by intentionally lowering standards and expectations. No child gets ahead. Equity as equal outcomes is a "fallacy of fairness." It cannot produce equal outcomes (Thomas Sowell). Kids get better through practice, but practice has fallen out of favor in progressive schools. Drill is considered old-fashioned, old-school, and poor pedagogy. It's not. It works!

A implies B, but B does not imply A.
Successful people practice and practice, but there is no guarantee that I will succeed if I practice and practice. I will improve with good teaching and plenty of practice, of course. "That's why education is worth the effort." In the real world, I will never play the violin as well as 9-year-old Himari, who nails Paganini, no matter how much I practice. I don't have the musical ability she has. (Google "Himari violinist")
Himari from Japan has the soul of Paganini.
She's not a prodigy; she's way above that.
"She is one of a kind, in a class of her own." 
I was astonished when I watched her play.
Paganini lives!

But, you should see her at age 7:
https://www.youtube.com/watch?v=YipD8Npugvg&t=346s

She is so much better at age 9 with an orchestra.
https://www.youtube.com/watch?v=7x2rmL9_JOY 
Start at 3:30 when she walks on stage (and soon dances to Paganini).


Food for Thought
When I was a kid, I walked several blocks to school, even in Kindergarten, in the bitter cold, and, later, played after school until dark, all without adult supervision. I rode my bicycle everywhere, no helmet. I survived lead paint, asbestos dust from an old furnace, coal dust, and cigarette smoke at home. My chemistry set was filled with dangerous chemicals, and I lived. I used encyclopedias (World Book) to help with school work. I read high-school chemistry textbooks in 7th grade and did most of the experiments. Note: Science was one semester in 7th grade and one semester in 8th grade. 

After school, I watched the Mickey Mouse Club. Once a week, I ran home from choir practice to catch part of Milton Berle. Then, on Saturdays, I watched Mr. Wizard, a science show for kids, on NBC. Today, there is nothing like it. 

We would hide under school desks in practice drills because of a perceived Soviet threat; polio was also a concern. But, I never gave either much thought. Contrarily, today, the media turns almost everything into a crisis. Kids are blasted with misinformation on TV and social media. The news isn't favorable. It's depressing. Also, their eyes are glued to a cell phone screen. As a result, students often suffer from anxiety. Circumstances are much different now than in the 1950s.

The latest is climate change hype.
If we don't do something NOW, the world will come to an end. What rubbish! The radicals and so-called experts told us that over 50 years ago. The sun significantly controls our weather. We can't change the sun or sun spots, etc.

©2021 ThinkAlgebra/LT