Mechanics first Explanation later

Asian children are taught mechanics first with an explanation later, and it works!  We do it backward, and it doesn't work. Furthermore, the use of calculators masks weak arithmetic and algebra skills. And, it hurts our kids. Massive numbers of high school graduates are placed in remedial math at community colleges. Many community college administrators want to do away with the "algebra requirement" to increase the graduation rate. Why? Most kids don't pass the remedial algebra courses at Community Colleges either. 

In other words, the teaching of remedial (high school level) algebra courses at community colleges is just as poor as the teaching of algebra in high schools. Apparently, using calculators doesn't help much. Indeed, they are "absolutely unnecessary," writes W. Stephen Wilson.


Keith Devlin says that with all the cheap gadgets available today, the need for mastery of arithmetic has lessened considerably. I think he is wrong. Being able to do arithmetic well--especially fractions--is needed for Algebra. Kids are not little mathematicians or junior scientists. They are novices. They do not think like adults.

Note: If there is no need to learn the mechanics (procedures) of algebra because technology does the procedures faster, as Wagner & Dintersmith (Most Likely to Succeed) argue, then, using the same argument, there is no need to learn the mechanics of arithmetic such as standard algorithms because calculators can do the operations faster. But, oddly, Wagner & Dintersmith don't think that about arithmetic. They claim, arithmetic is useful and algebra and high school math are not. They also claim that "most college courses require only K-8 school mathematics, "especially arithmetic, ratio, proportion, expressions, and simple equations." 

According to Wagner & Dintersmith, arithmetic is needed because it is used in everyday life. Really? When was the last time you did long division, a decimal multiplication problem, fractions, proportions, percentages, arithmetic mean, or a financial calculation without a calculator? 

According to the National Math Panel, "To prepare students for Algebra, the curriculum must simultaneously develop conceptual understanding, computational fluency, and problem-solving skills." 

They are interwoven! You have to teach more than just concepts. Students need to know foundational knowledge, both factual and procedural (mechanics).
  
"By the nature of algebra, the most important foundational skill is proficiency with fractions (including decimals, percent, and negative fractions)." We do not teach fractions well. "The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in Algebra can be expected." (Source: Math Panel, Critical Foundations of Algebra)   


In many K-8 schools, we seem to have a "steady diet of test prep." Is that a good thing? Wagner & Dintersmith (Most Likely to Succeed) think test prep is bad for kids because it hinders creativity and curiosity, but there is little evidence to support the claim. Rather, I think test prep is a poor substitute for a math curriculum and a waste of time. In my view, test prep is not the same as learning core arithmetic fundamentals in long-term memory. 


Wagner & Dintersmith write, "Clearly we want every child by a certain grade level to be an adept and passionate reader, to be proficient (without a calculator) at core math operations (e.g., division, fractions, estimation, and financial literacy), and be able to communicate well. Mastering these foundational building blocks requires repetition and practice." 

I agree. Kids need to master basic arithmetic. But...

Wagner & Dintersmith also conclude that the mechanics of algebra and much of high school math is outdated and unnecessary. They explain, "Don't get us wrong about the need to drill on lower-level math operations. Any young adult needs to be facile with core operations (+, -. x ÷, %, fractions, and decimals," which are K-8 math. But, we also know "they will never have to solve a quadratic equation by hand." 

Today's college entrance tests, they say, "require students to master skills that are obsolete. Indeed, "80% of U.S. adults never use any math beyond decimals, fractions, and percentages." Wagner & Dintersmith assert that "the math needed for college courses does not require high school math (algebra, trig, or calculus)--except for STEM students. Most college courses require only K-8 school mathematics, "especially arithmetic, ratio, proportion, expressions, and simple equations."
 
Comment: The problem I see is that the core basics are not taught well in our K-8 schools, which seem to focus more on test prep than on learning core fundamentals. (By core, I do not mean Common Core.)
Wagner & Dintersmith want to toss out much of high school math as obsolete, especially the paper-pencil techniques, procedures or mechanics of algebra, trig, and calculus, replacing them with calculators, tablets, computers, and smartphones. While calculus concepts are useful, they say, the "mechanics [of calculus] belong to smartphones." They imply the same for algebra and trig. Kids should learn just the concepts, not the mechanics.

I think they are wrong! 

Comment: In the real world, students still need to pass the admission and math placement tests at colleges and universities. Unfortunately, a massive number of high school graduates have been placed in remedial Algebra courses at community colleges because they didn't learn the mechanics and weren't able to apply concepts.
  
Wagner & Dintersmith think that students should drill the procedures of arithmetic, but there is no need to drill the procedures of algebra, trig, or calculus because they are old-fashioned and nonessential. They argue that high school kids should use the latest technology such as WolframAlpha on their smartphone, tablet, or computer. Their main argument has been that no one will ever need to solve a quadratic equation or an integral on paper. 

According to Wagner & Dintersmith, arithmetic is needed because it is used in everyday life. Really? When was the last time you did long division, a multiplication problem, or a financial calculation without a calculator? Algebra and Calculus are not needed because they are not used, but neither is arithmetic. Notice the inconsistency! The lack of numeracy among students and adults has been a perplexing problem for decades. Today, only 3% of adults can ace a math quiz without reaching for a calculator. The questions on the quiz featured multiplication, subtraction, percentages, and division.

 
Comment: What do Wagner & Dintersmith and many others have against algebra mechanics and quadratic equations? Physics is loaded with quadratic equations. To understand physics well requires a calculus background, but most beginning physics courses in high school and college are algebra-based. Kids need to know algebra, but many don't know it well enough. So, they struggle in the hard sciences, if they get that far.  


Also, I guess no one will quote Shakespeare either, so his plays should be eliminated from the high school curriculum. Latin is gone too. Gee, I thought the mechanics of algebra were embedded in the understanding of algebra concepts. "If you can't do it, then you don't understand it," says mathematician W. Stephen Wilson.

If there is no need to learn the mechanics (procedures) of algebra because technology does the procedures faster, then, by the same argument, there is no need to learn the mechanics of arithmetic such as standard algorithms because calculators can do them faster. 

I don't buy it!!!

In the real world, kids need to learn the concepts, but they also need to learn the mechanics (e.g., standard algorithms, distributive property, etc.) and solving routine problems. Most of the concepts in arithmetic and algebra are easy. Even 1st-grade students can learn some of the fundamental concepts of algebra via arithmetic. By 3rd grade, well-prepared students can learn the mechanics (technique) for solving simple addition or subtraction equations using inverses.
 
© 2017 LT/ThinkAlgebra

Memorizing & Learning

Much is taught, but little is learned.

Students need to automate factual and procedural knowledge (the basics) to do well in mathematics. Long-term memory is the key component in learning math or any academic discipline. Also, drill-to-improve-skill (repeated practice) is necessary. "To learn something is to remember it." If you can't instantly recall 6 x 7 = 42, then you haven't learned it. If you can't write and solve a proportion with an unknown, then you haven't learned it. All learning makes changes in long-term memory. Indeed, competency in math or any discipline is built on practice-practice-practice. Also, "Our ability to think depends on memory," explains William R. Klemm in a journal article, "What Good Is Learning I You Don't Remember It?"). Knowledge and skills are gained through memory.

William R. Klemm (Texas A&M University) points out that kids lack memorization skills. He writes, "Teachers should emphasize the educational importance of understanding, but not at the expense of overlooking the importance of memorization skills. Currently, mainstream educational theory embraces such attributes as insight, creativity, inquiry learning, and self-expression. But such emphases lead to bias and under-appreciation of the role of memory in learning. Students cannot apply what they understand if they don’t remember it. In the process of educational reform, the reformers discount the importance of memory." They are dead wrong! "Society needs people with knowledge and skills, which are acquired through memory." Indeed, "Our ability to think depends on memory." You can't think about something that you do not remember.]


In math, the fundamentals should be memorized. You don't want to figure out (i.e., calculate) 7 x 4 in [limited] working memory every time you come across it. Also, attention and focus are critical in class. Writing notes is important. Furthermore, teachers should present content that is well organized. Moreover, students can learn something better when it is associated with something they already know. 

The progressive idea of learning in groups without an organized, hierarchical curriculum (e.g., Piagetian learning) is vacuous. Klemm observes, "Kids don't appreciate how extraordinary attentiveness is." We teach kids not to pay attention when they are seated facing each other in small groups of four. Apparently, in the progressive classrooms of today, "collaboration" supersedes attention and learning. 

Often, teachers are required to teach to the test. Unfortunately, they miss the idea that learning should be cumulative and permanent, not merely test to test. Abstract symbols and structures learned in math, (e.g.,  y = mx + b, 3 + 8 = 11, a + 0 = a, or y = x^2, etc.) are the visual images. Even if students know that an equation is like a balance (visual), they often do not apply the idea. An equation can be visualized by the equation, itself, a table of values, and a graph.   

Lastly, the attitude of the student plays a critical role in learning. "Too often students have a negative attitude about the academic subject matter, and this attitude is self-defeating," writes Klemm. In short a negative attitude toward math, for example, interferes with learning math. The negative attitude is learned from teachers who hate math and from parents who tell their kids that they were not good at math. I do not think most students dislike math; they just find it harder to learn than other subjects. 

April 8, 2017, LT/ThinkAlgebra

Age of the Algorithm

Human judgment and experience are no longer considered relevant, just spreadsheets, big data, and algorithms, according to the elite progressives.

"We have stopped thinking. Machines do it for us," explains Christian Madsbjerg (Sensemaking) In short, Madsbjerg says that technology has replaced us. Thinking is hard work, and we don't want to make an effort. In fact, we try our best not to think. This is the age of information (Internet), but Beau Lotto (Deviate) suggests that "all information in and of itself is meaningless, and it doesn't come with instructions." Lotto explains, "One of the most challenging things we can do is to step into uncertainty." We are constantly bombarded by uncertain situations, and our brains try to make certainty out of uncertainty says Lotto. Our brains have evolved to do this because our perception of the world is not the real world. "You don't see the reality--only your mind's version of reality." 

In education, we drive for efficiency via "metrics, accountability, and standardized testing" says Madsbjerg. Standardized testing compares kids and crunches them as data. These tests were not designed to assess specific knowledge (what subject matter kids know or don't know) so why are they given in the first place? 

The "Age of the Algorithm" is the wrong approach argues Madsbjerg. In a nutshell, computer algorithms are making decisions for us. Indeed, we have been swept away by Silicon Valley tech ideology: "The promise is that technology will solve it--whatever it is." He states, "Everything is a disruption: a clean break from the past leaning far forward into the future." In other words, out with the old, even if it worked well, and in with the new (e.g., technology and its algorithms), even if it lacks evidence. After all, it's the 21st Century! 

Also, the algorithmically driven culture of Big Data, part of the Silicon Valley state of mind, has "upended the way we educate children" and everything else. The rush to put computers and tech in the classroom ($$$$$) has, indeed, disrupted education often at the expense of the arts and humanities, which are seen as valueless extras. Nicholas Carr (Utopia Is Creepy) asks, "Have we been seduced by a [Silicon Valley] lie?" Carr suggests that while some aspects of technology have enriched us, others have imprisoned us. Also, the idea that tech will magically solve our math achievement problems lacks credibility. 

The "Silicon Valley state of mind" reaches into education. "In education, everyone has trained in the same methods by the same institutions," writes Madsbjerg. Teachers have become interchangeable in an interchangeable system, i.e., they are like cogs in a machine: one teacher is just as good as another. Technology has been revered above humans. It makes everyone the same and everything equivalent.
   

Disruption in education, such as out with the old (traditional arithmetic, knowledge) and in with the new (critical thinking, tech) started in the late 1980s with NCTM reform math, which stressed calculator use in K-12. The progressive idea was that children could learn math naturally and quickly without a curriculum (Piagetian learning). An example was Mathland, a utopian idea outlined by Seymour Papert (MIT) in Mindstorms. Also, memorization and practice were declared obsolete (poor) teaching methods by the progressive elite. Calculators would often replace basic arithmetic. Furthermore, the idea that an algorithm could build a Global community utopia is one of the latest liberal ideas--a fantasy, of course, but it comes from the elite of Silicon Valley. The progressives believe utopia is possible, and they are doing everything to make it happen by placing technology above humans. 

[Comment: Paulo Blikstein (Graduate School of Education, Stanford University) writes in 2013, “The essence of Piaget was how much learning occurs without being planned or organized by teachers or schools. His whole point was that children develop intellectually without being taught!” Seymour Papert latched on to Piaget’s constructivist theory. The fundamental flaw of both Piaget and Papert was that kids don’t learn arithmetic and algebra that way, i.e., without a curriculum. They need to be taught. And, they need to intentionally push factual and procedural knowledge from working memory into long-term memory through repetition. Learning is remembering. If you don’t remember something, then you haven’t learned it.]   

Mark Zuckerberg's utopian vision of a "post-fact world" driven by "pattern recognition," that is, "the fittest message is the message that wins," is a pipe dream, argues Nicholas Carr on Rough Type. If an idea is repeated enough, let's say on mainstream media, social media, blogs, etc., we soon begin to believe it is true, right, or moral. Indeed, in Zuckerberg's "self-serving world," the “fittest message” always wins, but the "fittest" can be good, bad, or fake. Indeed, a lie when repeated often enough is perceived as true. Still, to Zuckerberg and others, "Facebook [i.e., the algorithm] is never the problem; it is always the solution!" But, Facebook is not perfect code and cannot create the utopian Global community that Zuckerberg envisions. It is "lousy as a news medium; it is terrible as a forum for political discourse," observes Carr. Moreover, Facebook can remove opinions or content that conflicts with its vision via filters. In short, the algorithm and the people behind the algorithm control the content.  

The Silicon Valley elitist mindset has driven the arts and humanities into the ground. The disruptive "out and in" ideology has kicked out chalkboards and traditional mathematics (Old School). Moreover, in progressive-leaning classrooms, history, geography, English, and literature have been downgraded and technology upgraded. Science, visual arts, and music have suffered, too. Also, instead of teaching basic arithmetic, we teach reform math, which crowds the curriculum with a hodgepodge of alternative strategies (algorithms) and neglects standard algorithms. Indeed, standard (traditional) arithmetic has been out of favor for decades.

[Note: Similarly, college students are discouraged from majoring in the arts and humanities. Many students who follow their passion can't find good-paying jobs and end up with large student loans with no means to pay them.]     

Today, in education, most everything is STEM, technology, big data, algorithmic (machine) thinking, and "critical thinking" without content. As a result, we have lost common sense in education, observes Nicholas Carr (Utopia Is Creepy). Also, Madsbjerg finds that human judgment and experience (practical wisdom) are no longer considered relevant in the Age of the Algorithm. Human wisdom and knowledge are out (obsolete); big data and algorithms are in. "We stop seeing value in things like poetry, sculpture, novels, and music," writes Madsbjerg. It is an unfortunate change in our education culture. 

[Comment: An example of technology upending practical wisdom is the 3rd-8th-grade federally-mandated (yearly) accountability testing. In one K-8 school, the standardized testing via computer disrupted education for 4 weeks, while a paper-pencil version would have taken 3 days. Also, much instructional time was wasted on helping kids learn to use the computerized testing program. In short, implementing machine testing created a significant disruption in education at this school. Indeed, "out with the old and in with the new" is a vacuous idea. Still, we are told that anything tech must be good. Really? What has happened to common sense? The focus in our classrooms has been the state test and "teaching to the test," which limits the curriculum. Indeed, teaching-to-the-test fragments the math curriculum. Also, reform math (via Common Core or state standards) taught in many progressive-leaning classrooms is not the standard arithmetic that kids must know to prepare for Algebra in middle school. In short, instead of the standard algorithms, teachers focus on "reform math methods" of calculating that students will never use.]

A liberal arts education, which requires math, science, literature, philosophy, foreign language, economics, the arts, and humanities, is more important today than ever, argues Fareed Zakaria (In Defense of a Liberal Education). Zakaria says that students are warned that if you want to have a good life (the American Dream) in today's high-tech world, then don’t major in the liberal arts in college. “In America, liberal education is out of favor.” College students are advised to “prefer job training [especially STEM] to the liberal arts” to prepare for jobs with good starting salaries to pay off student loans. Okay, but students can have a STEM major and at the same time take courses in the arts and the humanities even though they are not required. Many wise students have double majors--one to follow a passion and one to put food on the table and get out of debt.

The elite of Silicon Valley makes "grandiose claims." They peddle mathematical models (algorithms) as a clear picture or understanding of the real world, which they are not. A model is not the real thing. Most models are wrong; some are useful but imperfect and limited. Furthermore, "we fervently believe that more data will lead to more insights," explains Madsbjerg, but we would be wrong because "algorithmic thinking offers us the illusion of objectivity." We have sophisticated weather models that frequently make wrong predictions. In the real world, facts are never absolute, neither are the algorithms that claim to understand the world. Likewise, Facebook or Google algorithms are far from perfect. Indeed, they can be deeply biased. 

Twenty-first-century reformers demand more technology in classrooms to improve school achievement, even though the evidence of tech effectiveness has been scant. Indeed, we have had computers in classrooms since the 1980s, but national and international tests scores have been flat (2015 NAEP, TIMSS, PISA). In short, putting more costly technology ($$$$) in classrooms has not and probably will not change the narrative. No one considers the cost-benefit. In short, tech is not a silver bullet, not in education.

(See: http://thinkalgebra.blogspot.com/2017/02/how-do-we-fix-education.html) 

©2017 LT/ThinkAlgebra