[Comment. Larry Cuban points out that current reformers are on thin ice because the research doesn't support them. He writes, "The pumped up language accompanying “personalized learning” resonates like the slap of high-fives between earlier Progressive educators and current reformers. Rhetoric aside, however, issues of research and accountability continue to bedevil those clanging the cymbals for “student-centered” instruction and learning. The research supporting “personalized” or “blended learning” is, at best thin. Then again, few innovators, past or present, seldom invoked research support for their initiatives."]
Please excuse typos, errors, and embedded Notes and Comments. Draft 1.
Reform Math Fails. Over the years, multiple reforms have been imposed on schools. Indeed, educators have been "reformed to death," says Diane Ravitch. None of the math reforms have worked well and for good reason. Believing an idea will work because it sounds good and seems reasonable is not the same as having solid evidence. It appears that facts and scientific evidence don't matter much. Indeed, a lot of "studies" in education are junk science and often promote disinformation. In education, studies are rarely duplicated to verify results. Indeed, education is filled with ideas, common notions, and popular practices that have not been tested scientifically. Thus, we have had bad idea after bad idea introduced into our classrooms. The consequences of bad ideas are often calamitous. In one urban school district, 87% of the students who had enrolled in community college in 2014 needed remedial courses in mathematics. [Another school district is 88%; Another district is 83%; etc.--all in the Tucson area.] These kids are products of the K-12 reform math movement, which took hold in the 90s with NCTM standards. The new reformers say that Common Core will fix rampant remediation, yet Common Core is typically interpreted through the lens of reform math. Repeating or repackaging the mistakes from the past does not move math achievement forward. Also, test-prep, technology, curriculum narrowing, threats to withhold funding, minimal guidance methods, group work, and so on, will not erase the achievement woes.
Jordan Ellenberg (How Not To Be Wrong: The Power of Mathematical Thinking, 2014) writes, "Some reformist go so far as to say that the classical algorithms (like add two multi-digit numbers by stacking one atop the other and carrying the one when necessary) should be taken out of the classroom, lest they interfere with the students' process of discovering the properties of mathematical objects on their own. That seems like a terrible idea to me: these algorithm are useful tools that people worked hard to make, and there's no reason we should have to start completely from scratch." Professor Ellenberg also points out, "It is pretty hard to understand mathematics without doing some mathematics."
Higher Math is Needed. There are a lot of smart kids, but the Common Core reforms do not prepare enough of them for college--not with watered-down math and science courses. There are exceptions of course. We do have pre-college kids who hold their own with the best high school math students from other nations in international contests. That said, many smart kids are not prepared for college level STEM courses, especially in mathematics and science. Indeed, Common Core ignores STEM, even though "high math" is needed markedly more today and in the future than in previous generations. The reforms are illogical and flawed. "The ability to create algorithms that imitate, better, and eventually replace humans is a paramount skill of the next one hundred years," explains Christopher Steiner (Automate This: How Algorithms Came to Rule Our World). The bots (algorithms) are coming for your job.
Kids Are Ill-Prepared. Steiner says the two economic-growth-drivers over the next 50 years will be health care and tech, so how well are we preparing kids in chemistry, physics, and mathematics? We aren’t! A "dearth of engineers" is the reason the Silicon Valley imports talent. “The problem,” writes, Steiner, “is that not enough US kids get that foundation of upper-level math before arriving at college.” It is unfortunate that many smart kids can’t hack the STEM college courses in chemistry, physics, math, and computer science—the real stuff--because they lack sufficient preparation in high school. [Comment. The problem of inadequate preparation does not suddenly pop up in high school. It begins early in elementary school with the way arithmetic has been taught (as reform math). Today, Common Core is often interpreted in terms of reform math by the "progressive" powers that be. Indeed, memorizing single-digit number facts, learning standard algorithms, and practicing essential math to mastery have been intentionally lessened, even ignored, in reform math approaches.]
[Note. The reform math people have exploited the eight Standards for Mathematical Practice in Common Core, which are vague and open to various interpretations, as a means to devalue the traditional teaching of mathematics and promote their own constructivist pedagogy and progressive agenda. In short, the people in charge impose their favored progressive reforms.]
Teach Standard Arithmetic Straightforward. Many teachers get hung up on understanding, critical thinking, and group work, which are some of the elements of the reform math movement. Consequently, they miss the target. Kids learn very little math in small-group-centered reform math "discovery" activities. Instead of reform math, teachers should teach standard arithmetic content straightforward and require that students practice it for mastery beginning in 1st grade. Indeed, the standard algorithm for addition can be taught to 1st graders in the 1st month of school when children begin to study place value, that 25 is 10+10 + 5, or 2t + 5, or 2tens+5ones, and memorize the addition facts, such as add 2 number facts, which students easily figure out on the number line. Soon, students can add 23 + 42 in columns using memorized single-digit addition facts [i.e., the standard addition algorithm]. Later, well-taught 1st graders can add 40 + 129 + 24 in columns. They also learn subtraction and much more.
[Note. Reform math has had many different names over the years. One feature of reform math is minimal teacher guidance, that is, the teacher's role is passive and diminished to being a facilitator so that students work in groups. Examples of reform math include constructivist, discovery, problem-based, experiential, inquiry-based, etc. Kirschner-Sweller-Clark say that minimal guidance is inferior and has failed, yet reform math elements are still championed in schools of education and extend to popular interpretations of Common Core. According to cognitive science, problem solving in math is not possible without prerequisite factual and procedural knowledge in long-term memory. For example, you can't solve a trig problem unless you know trig, and knowing trig requires practice-practice-practice. (Likewise, you can't translate Latin without knowing some Latin.) In mathematics, the emphasis should be on students acquiring factual and procedural knowledge in order to do and/or apply mathematics to solve problems. You don't get good at math without substantial practice.]
John von Neumann is alleged to have said, "In mathematics, you don't understand things. You just get used to them." Indeed, children need to learn to deal with and get used to abstractions because "whole numbers, fractions, and the various operations with whole number and fractions are abstractions," writes mathematician Morris Kline (Mathematics for the Nonmathematician, 1967). Dr. Hung-Hsi Wu writes that kids should use symbols to convey mathematical ideas and operations.
Professor Kline writes about a man who walks into a store to buy three pairs of shoes at $10 each. "The difficulty," says Kline," is that you can't multiply shoes and dollars," so you have to abstract from the particulars and "multiply the number 3 by the number 10 to get the number 30." Kline explains, "One must distinguish between the purely mathematical operation of multiplying 3 by 10 and the physical objects with which these numbers may be associated." Starting in 1st grade, children need to learn to pull out the numbers and operation needed to find the solution to a word problem. Translating a word problem into abstract symbols that conveys numbers and operations in the form of an equation, which shows a solution, requires plenty of practice. Singapore kids write equations in 1st grade.
At the basic level, the fact 4 + 7 = 11 is an abstraction that can be applied to hundreds of situations, says Kline. Abstraction is a powerful idea in mathematics. The fraction 3/4 is an abstraction. It can be represented as a point on the number line just like a whole number. But operating on fractions seems "arbitrary and mysterious." Kline explains, "Operations with fractions are formulated [made up] to fit experience."
We make up algorithms to fit our experience in the real world, but they must be efficient for practical use. (All the standard algorithms for whole number operations are based on single-digit number facts.) If we take 10 oranges and split each orange in half, then we would we have 20 halves. That is, 10 ÷ 1/2 = 20 must give the same answer as 10 x 2 = 20; therefore, 10 ÷ 1/2 = 10 x 2. From this abstract idea and others, such as the idea of reciprocals (1/2 and 2 are reciprocals because their product is ONE), we should be able to make up an efficient algorithm for division of fractions that always works in the real world, and we have: To divide by a fraction, multiply by its reciprocal, which is the standard algorithm. It fits our experience. The proof, however, is left to mathematicians, not to novice kids.
G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." This requires sufficient knowledge in long-term memory, experience, and skill development through practice. It implies that if a student cannot do the math, then the student doesn't really understand it or know it. This is related to Richard Feynman's insight: "You don't know anything until you have practiced."
I think it is difficult to discuss understanding with any consensus because the term is vague and hard to measure and analyze, and, in my opinion, overrated in US math programs. I think teachers should talk in terms of doing or applying mathematics, which is observable and measurable. In order to do or apply arithmetic, kids must know some arithmetic--key factual and efficient procedural knowledge. The idea is to master the important stuff, which includes number facts and standard algorithms.
David Ruelle (The Mathematician’s Brain) explains mathematical intuition, “When we study a mathematical topic, we develop an intuition for it. We put in our [long-term] memory a large number of facts that we can access readily and even unconsciously. Since part of our mathematical thinking is unconscious and part nonverbal, it is convenient to say that we proceed intuitively. This means that processes of mathematical thought are difficult to analyze.” Dr. Ruelle also observes, "Mathematics is a matter of knowledge, not of opinion.” (I think understanding falls under opinion.)
Ruelle writes that “mathematicians put a lot of facts in their long-term memory through long days of study.” In this sense, I want kids to be more like mathematicians (or musicians), not little mathematicians, that is, I want kids practicing the essentials so they stick in long-term memory and become automatic. Basic knowledge of arithmetic (ideas, skills, and uses) enables mathematical thought. I don’t want novices trying to discover or reinvent arithmetic in group work or waste time with non-standard algorithms at the expense of standard algorithms. As Hersh & John-Steiner would say, learning mathematics well takes drill and practice, lots of it. There is no workaround. If students can apply the math they have learned, then this cognitive outcome likely implies that they have some understanding of it, but I cannot measure it.
Perhaps, the closest we can come to a student’s actual mathematical thought process, which is often unconscious and nonverbal (Ruelle), is when the student actually performs math that leads to a solution or correct answer, whether it be solving an equation or executing the standard multiplication algorithm, etc. In my opinion, written explanations or drawings are nonessential. Showing steps is sufficient to demonstrate a child's knowledge.
While it is difficult to figure out a student's actual understanding, we can, I think, confirm some level of performance or competency by looking at the steps the student writes on paper to get to a solution. Also, speed is an important variable in competence. As teachers, we should depend more on observable [measurable] behavior and not so much on understanding, which is uncertain, equivocal, and difficult to measure. Here is an example of observable behavior. A 3rd grade student has 4 of 5 long division problems correct within 5 minutes.
In short, teachers should focus more on specific, measurable cognitive outcomes, as described by behavioral objectives, rather than on inexact, hard to measure qualities such as understanding, appreciation, soft cognitive skills, and so on.
First Draft. Please excuse typos and other errors. To Be Revised. June 13, 2015
Comments: ThinkAlgebra@cox.net
© 2015 LT/ThinkAlgebra
