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