Reform Math

"Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older." WSW

The Calculator Problem

There is no concept in arithmetic that requires a calculator.

Using calculators came straight out of the reform math playbook when, in 1989, the new standards from the National Council of Teachers of Mathematics (NCTM standards) actively promoted the use of calculators starting in K. What a dumb idea! Calculators are not integral to the teaching and learning of basic arithmetic, and graphing calculators are not integral to the teaching and learning of algebra and calculus. There is no concept in arithmetic that requires a calculator. Questions in reform math textbooks were designed for calculators, so time is spent learning to use a calculator, not arithmetic. Moreover, there are questions on state tests and national tests (e.g., GED, SAT, AP) that intentionally require the use of calculators. We used to teach arithmetic and algebra well without calculators and gadgets. 

Note. Calculators are a useful tool for complex calculations in science, higher-level math, such as trig and log functions. There are no concepts in arithmetic or algebra that require calculator use.

For decades, many children have not been learning (mastering) basic arithmetic. Instead, they are taught a version of reform math that downplays memorization, standard algorithms, and traditional instructional methods such as drill-to-develop-skill, all of which can be traced to the 1947 NEA Yearbook and more recently to the 1989 NCTM standards. Incidentally, state math standards based on Common Core are not world-class. 

According to the NCTM, children are no longer expected to master paper-pencil arithmetic, which opens the door for calculator use as early as kindergarten (Charles Sykes, Dumbing Down Our Kids). Moreover, the reformers insist that students should dive right into problem-solving before the basics are taught, which, in my opinion, is an inane strategy. Also, the so-called math educators—straight from schools of education—insist that young students will pick up the arithmetic along the way and invent their own math through the discussion of math problems in small groups. (Sure, and the moon really is cheese.)

Children are not little mathematicians. They are not geniuses like Fermat, Newton, Euler, Gauss, Boole, Hilbert, and so on. They should not be asked to make up their own math in group work or reinvent the wheel. 

W. Stephen Wilson (WSW), a mathematician at Johns Hopkins, wrote that AP calculus is not college Calculus. "[A] very large flaw is the association of AP Calculus with the graphing calculator ... The graphing calculator is not integral to Calculus. It [calculus] can be taught and learned without any technology." Also, "At the college level, many professors do not emphasize (or even allow) the use of graphing calculators since there is no concept in Calculus that requires the technology either to teach or to assess." Wilson observes, "The graphing calculator is used on the [AP] exam to solve completely contrived problems designed so that the graphing calculator is required." Does that sound familiar? Kids must use calculators, we are told, for real-world problems in elementary and middle school. Nonsense!  

Likewise, the graphing calculator is not essential for learning algebra, either, and there is no concept in arithmetic that requires a calculator. Yet, students are weaned on calculators beginning in elementary and middle school. Is it any wonder that our kids stumble over simple arithmetic? 

Furthermore, the curriculum is dumbed down so that "everyone can pass--but no one can excel," which is the essence of reform math in the progressive era. The mastery of fundamentals starting with 1st-grade arithmetic is not a high priority in modern classrooms. Memorization and traditional algorithms are discouraged. Furthermore, the conventional standard algorithms have been replaced with more complicated, alternative algorithms to do arithmetic. It is a sad era for kids.

Steven Strogatz (Mathematician at Cornell) writes, "If we only teach conceptual approaches to math [i.e., reform math] without developing skill at actually solving math problems, [then] students will feel weak. Their mathematical powers will be flimsy. And if they don't memorize anything, if they don't know the basic facts of addition and multiplication or, later, geometry or still later, calculus, it becomes impossible for them to be creative." (Strogatz's quote from The Atlantic in an article by Jessica Lahey)

"Barbara Oakley, an engineering professor at Oakland University in Rochester, Mich., says the key to mathematics expertise is practice, not conceptual understanding as some common-core proponents would have educators believe." 

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. 

Some kids, for whatever reason, will always be better than others in math, but a widespread educational dogma taught in many ed schools is to equalize downward to narrow gaps, that is, let no child get ahead. However, kids are not the same in ability, so why feed them the same curriculum? It's a daft idea! We can never have equal outcomes. 

There will always be achievement gaps because good education creates inequalities, says the late Richard Feynman. Students who study more, practice more, pay attention in class, are more industrious, can delay gratification, and have an optimistic attitude create inequalities, etc. To excel in school mathematics, students need academic ability, realistic goals, and persistence, and conscientiousness to achieve through practice-practice-practice.

We are squandering a lot of talent because kids who are better in math or science are not supported ($) or challenged with an accelerated curriculum. Instead, we give them grade-level math to pass a state test. Common Core cuts off STEM. Its "one size fits all" does not account for the academic variability of children. 

Mathematician H. Wu says that very young children should use symbolics to do arithmetic, not draw visuals. The only visual Wu recommends is the number line for "organizing the mathematical developments of whole numbers, fractions, and negative numbers." Kids should be taught symbolics from the beginning. The problem is that many K-8 teachers are weak in symbolics. 

Math is abstract and uses symbols to convey concepts.

Below is a typical 3rd-Grade Common Core reform math problem [Sandra....], but it is Pre-First-Grade Arithmetic, not 3rd grade.

In the Zig Engelmann 1966 film, this problem would be a pre-first-grade problem. Zig's kids would not draw a picture to find the answer; they would count memorized multiples to find 4 x 6: 6-12-18-24. Singapore first graders would calculate 4 x 6 as repeated addition: 6 + 6 + 6 + 6. They are excellent at adding numbers. By 2nd grade, Singapore students memorize at least half the times tables.

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

Why draw a picture? It is nonsense and a waste of time for basic arithmetic. 

We are told that students "draw a picture" to figure out that 4 groups of 6 are 24. I hope not, but this is what is implied. Indeed, the math fact should have been memorized, and the pattern in the word problem for multiplying should have been recognized immediately.  

Make a drawing, say Common Core reformers, is a vital skill in arithmetic, but mathematicians, such as Dr. H. Wu (UC-Berkeley), disagree. Make a drawing, using manipulatives, keying numbers into calculators, and so on, are not integral for the teaching or learning of arithmetic. 

Note. Astute students would recognize the multiplication pattern and retrieve 4 x 6 = 24 from long-term memory. No need to waste time by drawing a picture or proving that 4 x 6 = 24. Answer: Sandra needs 24 fruits.

Very young children should use mathematical symbols, not draw visuals, to do the math. Putting dots in circles is not essential or practical. How about 12 x 127? 12 circles with 127 dots in each circle, or repeated addition, a 1st-grade idea, or the distributive. You don't draw a picture or do repeated addition or the distributive to do basic arithmetic. You use the standard algorithm for multiplication. Of course, you can't do that unless you have memorized the multiplication table to automation. That's the rub. Reform math downplays the mastery of number facts and standard procedures, putting our students behind.  

In math, symbols convey concepts. 

First graders in Singapore start multiplication as repeated addition and write numerical equations that are solutions to word problems. In 2nd grade, they leave repeated addition behind and memorize half the multiplication facts to write and solve equations, such as 5 • x = 20, etc. 

Writing an equation that shows a solution is "showing your work," but that's not good enough in Common Core. Yet, the equation is the model; it does show your work. In algebra, students need to write and solve equations without graphing calculators. They need to be able to plot all functions (paper-pencil), which squarely conflicts with the Common Core pedagogy of using graphing calculators and technology. 

Major Problems Persist

Thanks to reform math and progressive ideas, "Our kids rank near to, or at the bottom of international tests in math and science." (Quotes from Charles Sykes)  There are many reasons why achievement has stagnated. One is the "religion of self-esteem," now disguised as social-emotional learning. Another is the "attack on excellence." The curriculum in math, for example, is not world-class and does not predict readiness for career/college as claimed. Common Core and state standards, along with insidious, yearly testing, are a political solution to our education woes (NCLB, ESSA), not an education solution. 
Also, teachers major in education, not in regular academic subjects such as physics, mathematics, history, literature, etc. In short, they are poorly trained in academics.  

To make matters worse, education is loaded with junk science. Many acceptable classroom practices are not evidence-based. Most programs (a shocking 82% of them) funded by the U.S. Department of Education did not improve reading or math achievement. Still, they lurk in our schools.  

In math education, there has been an over-reliance on manipulates, calculators,  and alternative algorithms that lead to nowhere. Moreover, there is a lack of vocabulary study in reading programs. 

The training of teachers is dumbed down, too. Sykes writes, "Enthusiasm for cooperative learning (now called collaboration and group work), distrust of competition (students are the same), a suspicion of grades and tests, and an aversion to traditional methods of teaching (including phonics) are almost universal in the schools of education." The teacher's role in the classroom has switched from academic leader to facilitator of learning. In short, the teacher gives minimal guidance during instruction, which equates to minimal learning. Individual excellence is no longer revered. Furthermore, the rise of "self-esteem has supplanted grades and genuine achievement." Almost all students get good grades and are passed to the next grade level regardless of achievement (i.e., rampant grade inflation).  

Elementary teachers don't know enough mathematics to teach arithmetic well, reports Hung-Hsi Wu, a mathematician at UC-Berkeley.

Common Core is not near the Asian level. "The curriculum is dumbed down so that "everyone can pass--but no one can excel." 

Not much had changed since Sykes wrote his book nearly 25 years ago.
Quotes are from Charles Sykes (Dumbing Down Our Kids, 1995)

The Math Wars

WSW: The end of the math wars!  You must be joking. There will always be people who think that calculators work just fine.

There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is bad for a student, perhaps believing that it means students can no longer understand them. Of course, this permanently slows students down, plus it requires students to think about 3rd-grade mathematics when they are trying to solve a college-level problem. 

There will always be the standard algorithm deniers.

Avoiding hard mathematics with young students does not prepare them for hard mathematics when they are older. 

©2019 - 2020 LT/ThinkAlgebra
Comments: ThinkAlgebra@cox.net
Updated: 5-28-19, 5-30-19






  

Teach Kids Algebra

My Teach Kids Algebra Program (TKA) started in the winter of 2011 and has been taught off and on in grades 1 to 5 at a Title-1 K-8 urban school. This school year (2018-2019), I gave lessons to two 4th-grade classes and a 2nd-grade class. TKA started as a reaction against progressive teaching methods and Common Core reform math, which was not world-class. 

Teach Kids Algebra (TKA) is a singular, innovative algebra project for typical students grades 1 to 4. Students use arithmetic to learn basic algebra ideas such as variables, equations, functions, negative numbers, graphs, etc.  The TKA curriculum is designed to challenge students at a much deeper level than traditional elementary school mathematics. TKA is more content and less enrichment. Still, most students can learn it given prerequisites and the proper instruction.

Teach Kids Algebra or TKA helps build a mathematical background for future studies in math, science, engineering, computer science, and so on. TKA is STEM math for elementary school students. Even 1st graders dive right into numerical equations and equations in one variable.

TKA 3rd Grade 2011-2012 School Year

Progressive methods of teaching (minimal guidance, discovery, group work, test prep, etc.) encouraged reasoning or thinking without content knowledge, not memorization and mastery, even though "thought without content knowledge" in long-term memory is empty (I. Kant). (In short, thinking without content sounds great, but it doesn't work that way. It is domain-specific. There is no generalized thinking skill. Thinking in math is different from thinking in science.) Also, because of working memory limitations, all arithmetic basics should be memorized and over-learned so that they can be recalled instantly from long-term memory to help solve problems in working memory. Such is the science of learning. Learning means the student can recall information from long-term memory. For example, if you can't instantly recall multiplication facts such as 8 x 7 = 56, then you haven't learned them. Also, if you can't calculate it (e.g., perimeter), then you don't know it. In math, knowing something means being able to perform it. Understanding cannot be measured directly, but it can be implied by the student's performance of arithmetic and algebra on paper.   

TKA does not follow Common-Core-aligned textbooks or the state's curriculum. I make a list of skills students need to learn to be successful in Algebra-1, then I figure out the most efficient methods to teach those skills to young beginners. The lessons are not student-centered. The goal of the lessons is not to support the self-esteem of children or to boost their creativity. The main goal is learning content (i.e., factual and procedural knowledge) to prepare for harder math later on. Students must use arithmetic stored in long-term memory to do problem-solving in working memory. 

I started from scratch, fusing algebra ideas into basic arithmetic. 
When I started giving algebra lessons to 1st through 3rd grades, I made a topic list. Then, I developed specific behavioral objectives (i.e., performance-based, Mager) and sample problems (i.e., worked examples) to explain how to perform. I often asked the students questions and gave them a couple of problems to work on their own (guided practice). Finally, I handed out the lesson's practice sheet (independent practice), which included both current and problems from previous lessons (review). For the remaining 30 minutes, I walked around the room to give encouragement and corrective feedback to individual students. I called my algebra program Teach Kids Algebra or TKA. Sessions were weekly for an hour. 

In short, I constructed my curriculum, but I used Gagne's idea of building knowledge hierarchically. I identified the prerequisites needed for the student to perform the objective. One idea builds on another. It's called background knowledge. 

Algebra is symbolic arithmetic. In my algebra program (TKA), algebra ideas build on basic arithmetic students know or should know (i.e., factual and standard procedural knowledge). TKA is an enrichment program given once a week for an hour. For 1st and 2nd graders, the lessons lasted 7 weeks (i.e., 7 hours). TKA reinforces basic arithmetic and points out weaknesses that teachers can correct. TKA helps students get ready for algebra. 

TKA is not progressive teaching. It is the opposite. It is explicit teaching with feedback. In short, I teach children algebra, not reform math. I do not follow the district's or state's curriculum. I explain content the first half-hour with worked examples and give "guided practice," then, as students work independently on a practice sheet, I roam around the room, answering questions, giving hints, and offering encouragement and feedback—no group work, no calculators, no manipulatives, no Common Core, and so on. 

Many allege that algebra in the early grades is not developmentally appropriate. How would they know? The same old progressive assumptions and theories that put our kids behind in the first place continue to haunt us. 

Like Singapore, I embrace the ideas of Bruner, not Piaget.

Bruner, not Piaget!

The rules of arithmetic are the same as the rules in algebra, so I fused algebra to basic arithmetic.

The 7 lessons of TKA for 1st and 2nd grade. 
The lessons were extended for older students in grades 3 to 5 (20-25 Lessons) with more emphasis on writing equations (English to math symbols), solving a wide range of equations using algebraic techniques (inverses), y = mx+b, integers (4 operations), fractions (4 operations), evaluating expressions, simplifying expressions, geometry, working with square roots, equal ratios and solving proportions, etc.  

➡ 1st-Grade Teach Kids Algebra (TKA), Spring 2011

Fusing Algebra to Standard Arithmetic.

• True/False & Equality (=) "Think Like A Balance."
• Equations in One Variable, Guess & Check
• Equations in Two Variables (Input-Output Model)
• Function Rules & Building x-y Tables
• Plotting Points in Q-I & Finding Perimeters
• Graphing Linear Equations in Quadrant I
• Given y, Find x (Reverse, Undo) & Steepness of a Line (Slope)

David G. Bonagura Jr. writes in the Wall Street Journal, "Contrary to today's education theories; memorization is critical in the classroom and life." Memorizing math facts and standard procedures is good for kids. It enables them to solve math problems.

"Extracting a math problem from a word problem requires a high level of critical thinking," writes Professor W. Stephen Wilson, Johns Hopkins. It is called mathematizing (English words to math symbols and vice versa). Students should extract numbers and write an equation to symbolize and model the word problem. 

The main reason children have difficulty with a math topic is that they have not learned the prerequisites. Cognitive Scientist Daniel Willingham (American Educator | Summer 2008) wrote: "Recognize that no content is inherently developmentally inappropriate. If we accept that students' failure to understand is not a matter of content, but either of presentation or a lack of background knowledge, then the natural extension is that no content should be off-limits for school-age children." Willingham sides with Bruner. 

For additional information, click Teach Kids Algebra (1-6-18)


Updated 5-25-19, 5-28-19, 1-10-2020

©2019-2020 LT/ThinkAlgebra.org

Digital Age

You don't get good at reading by not reading. Kids in the Digital Age read less. Likewise, you don't get good at math by not practicing the fundamentals--facts and procedures--so they stick in long-term memory for instant use in problem-solving. 
(Model: Hannah)

"Hard work brings out talent to its fullest extent," says Chloe Chua, 12, Violinist.  "If I don't have time, I practice 1 to 3 hours a day. If I have time, I practice 4 to 6 hours." Chloe has musical ability way above the norm. She learns techniques and memorizes notes by practicing. It is incredible. 

My Thoughts

In school, kids don't practice math facts and standard procedures enough. Hence, many students stumble over simple arithmetic. Mastering arithmetic has not been a primary goal in modern classrooms, but it should have been. And it used to be. Check out 19th Century Arithmetic (3rd/4th Grade): 
Find the interest of $60 for 4 months at 5%. 

Sometimes, I think we have harmed education more by trying to fix it. And, the latest, trendy fixes have been digital. Many fixes have not been supported by scientific evidence. They did not improve achievement in math or reading. Also, a lot of classroom instruction has not been based on the cognitive science of learning. 

Tech Hype (The Folly of Technological Solutionism)
The aggressive marketing of tech by lobbyists, philanthropists, and foundations that want to transform, radically change, or deschool education their way has failed to improve achievement. Billions and billions have been invested in education by donors with unremarkable achievement. Unlike students in Singapore, U.S. 1st-grade students aren't required to memorize math facts and learn other basics (e.g., standard algorithms). Also, students fall behind because the curriculum is not world-class. Furthermore, the minimal guidance methods of instruction have been ineffective. 

Larry Cuban (Stanford) writes, "There are donors who push for certain kinds of reforms that undermine the public good." Consequently, schools spend a massive amount of tech and ineffective programs, which leaves less money for textbooks, classroom supplies, and teachers. 

Algorithms are not the real world. 
Google thinks its algorithms never get things wrong. Really? Algorithms are not perfect. They are not the real world. Since when should algorithms dictate solutions to complex education issues or define humanity? Evgeny Morozov (The Folly of Technological Solutionism) writes, "Not every problem is amenable to a technological fix," especially education. 

Digital Kids know less math. Reading books has declined. For a decade, both math and reading achievement has stagnated. Mark Bauerlein (The Dumbest Generation) says that "leisure reading is a significant factor in academic progress. The more kids read out of school and in school, the higher their scores." The problem is that kids are reading less, not more. Martin Heidegger (1954 essay) says that "technology is now the center of our being. It has replaced us." It is much worse now, says Christian Madsbjerg. "We have stopped thinking. Machines do it for us." Our cognitive abilities are slowly eroding because they are no longer used. "We stop seeing numbers and models as a representation of the world, and we start seeing them as the truth--the only truth. We often make really poor decisions just because it is so uncomfortable to do the hard work of thinking." (Note: Heidegger statement is from Christian Madsbjerg (Sensemaking). 

Silicon Valley
The ideology of Silicon Valley is "the promise that technology will solve it--whatever it is. And the solution is sure to be revolutionary." In education, it replaces the old with the new. Everything "old school" has been tossed out. Anything digital is good. Many of today's digital kids don't want to work hard, think, read, or do math. For students, learning stuff (knowledge) is not that important. Bauerlein points out that there were "little or no gains in achievement once the schools went digital." In today's schools, effort trumps excellence. Grades are no longer based on performance, accomplishment, or merit. All students pass regardless of achievement. 

"Students should come to school to learn, not to text," write Friedman and Mandelbaum (That Used to Be Us). Students don't want to study to master the content; they want to use social media and gadgets. 

Arithmetic is not an opinion. 
The problem in education with generous contracts is that school districts cannot afford them. Likewise, States cannot afford "huge pension benefits for union workers who no longer work." State politicians and district leaders need to learn arithmetic and live within their means.

Practice does not cause talent.
I agree with mathematician Ian Stewart that practice does not cause talent. There has to be something there to begin with: an innate ability that can expand and flourish (brain plasticity) given excellent instruction with practice-practice-practice. Like any ability (athletic, musical, logical-mathematical, linguistic, academic, etc.), it varies widely from person to person. It is in your DNA. But, practice alone does not cause talent; it improves an ability one already has. In short, practice improves performance. You don't get good at something unless you practice-practice-practice, then practice some more. Oh, did I say you need to practice? In short, practice activates talent, improves performance, and develops innate ability, but it does not cause talent. 
  
Many innovations have failed in our schools.
If you want to innovate, then first become an expert. Unfortunately, "different" is often perceived as better, but is it? Indeed, 82% of the innovations funded by the U.S. Department of Education had failed to improve reading or math achievement. WOW!!!!! In short, well-intended changes in the classroom (aka innovations) are not practical because they don't work well. Many programs and practices in today's classrooms, including those linked to tech, have not improved achievement. Indeed, so-called technology-driven innovation has been expensive ($$$) with little benefit in achievement. Tech is not a panacea.  

Literacy (?)
The digital age has not redefined literacy, it has replaced it. Fred Siegel calls it digital miseducation. Consequently, today's students know less math, science, geography/history, and literature than students of 20 years ago. Students don't read much, not even textbooks. Summer reading has declined. Also, students lack critical reading skills and cannot tell the difference between fiction and non-fiction, writes David Joliffe, an English professor. Likewise, students have difficulty distinguishing fact from inference. They are often mixed together. So, do adults. 

Knowledge has been shortchanged.
Knowledge is the basis of critical thinking, creativity, and innovation. Why are schools not concentrating on students gaining fundamentals of factual and procedural knowledge in long-term memory? Teachers should focus on the mastery of fundamentals like Singapore, not state test proficiencies! The bottom line is that kids must know facts in long-term memory to perform math, but many don't. 

Knowing something is better than not knowing something. 
Whoever said that "ignorance is bliss" was a fool. Your financial future is built on living below your means and knowing how exponential growth works (math). Even 4th graders in the 1800s calculated simple interest, percentages, and ratios, while many of today's 4th graders stumble of simple arithmetic. Indeed, academic expectations and performances today are much lower than they used to be.

19th Century 4th Grade Arithmetic
Find the interest of $60 for 4 months at 5%. 
(Ray's New Intellectual Arithmetic for 3rd & 4th Grade, 1877)

No repeatability means bad ideas persist in education. 
A lot of people think that you make a bunch of observations, then infer a rule or theory, but that's not the way science works. If you are an expert in a field, you guess a rule, then carefully craft an experiment to make observations (measurements). Intrinsically, real science weeds out bad ideas. Also, experiments need to be replicated by other scientists and peer-reviewed. On the other hand, research in education doesn't do that. It suffers from confirmation bias, so-called anecdotal evidence, or statistical gibberish. There is no repeatability. Consequently, many unsupported ideas or practices in education thrive. Progressive educators (aka liberals) believe that memorizing facts and procedures is bad for kids. It's old school. Really?

Ideas in this post are subject to change. 

Updated: 5-17-19, 5-18-19, 5-19-19, 5-21-19
©2019 - 2020 LT/ThinkAlgebra
Comments: ThinkAlgebra@cox.net