Sunday, February 26, 2017

How do we fix education?

How do we fix education?
LT: I have been in education since 1966. I don't think there is a quick fix. I go into classrooms as a guest teacher to give algebra lessons once a week to urban elementary school students of color. I see very bright students of color, which gives me high hopes for the future.

Sometimes, I think that education is a type of complex problem that has no single solution. Piecemeal solutions without evidence have failed. Top-down solutions from the district, state, and federal governments have fallen flat. The technology hype to improve achievement has never worked. I believe that many educational problems in schools can be solved by adjusting the curriculum to fit the students who walk through the school door. But, this rarely happens in a top-down bureaucratic system that dictates policy and methodology. (Also, kids need books, not laptops or tablets.)

We should upgrade curriculum and instructional methods to fit the students that walk through the school door, but we don't do that.  We should sort math students by achievement starting in early elementary school, but we don't do that because it runs counter to inclusion policies of fairness. We allow our best math students to fend for themselves when they should be accelerated. It is a fallacy of fairness, says Thomas Sowell. Starting in the first grade, you don't place high-achieving math students with low-achieving students in the same math class, but that's what we do. In elementary school, high-achieving math kids need acceleration taught by experienced algebra teachers, not a so-called "math educators" trained in progressive schools of education. 

Policies dumped on teachers by higher-ups should not drive education, but they often do. Common Core and annual testing are two examples. So-called "fairness" policies that are an illusion of fairness are another. Group work and sitting in groups so kids can socialize (oops, I mean "collaborate") are more examples. Furthermore, often teachers are asked to teach a deficient curriculum using ineffective methods. Instead of teaching core (standard) arithmetic, we teach reform math, which crowds the curriculum with many alternative strategies (algorithms) and neglects standard algorithms. 

Basic arithmetic doesn't change. But this has not stopped math reformists from marginalizing standard arithmetic. Reform math does not stress the mastery of standard arithmetic in long-term memory. Reformists believe that "memorization" and "drill to improve skill" are bad teaching methods. They are dead wrong! Consequently, American kids suffer and stumble over simple arithmetic and do not learn algebra well. Moreover, American reformist pedagogy promotes the early use of calculators and online programs via computers, laptops, or tablets to improve math achievement, but it hasn't worked. In fact, such technologies are "conspicuously absent" in high-performing nations. 
Many education policies are toxic, or counterproductive, such as "equalizing downward by lowering those at the top," says Thomas
 SowellYou can't equalize outcomes (Sowell). In fact, education should increases differences (Feynman). Making matters worse for children is that they are taught a convoluted form of math called reform math that hinders progress to algebra by middle school. The state standards, which are grounded in Common Core, are not world-class; consequently, our students start behind and stay behind internationally. Some of these things, I think, are fixable, but it doesn't take more government control or intrusion. Bold, individual teachers can upgrade math content to world-class and implement strong teacher guidance methods that are supported by the cognitive science of learning. Unfortunately, such teachers receive little if any support from the education establishment. 

Dinesh D'Souza (America) points out that "Thomas Jefferson [Federalist No. 10] supported differences that were based on achievement and merit." The role of government is not to remedy the inequality of outcomes, he says. It is not the purpose of education either. Unfortunately, in education, any variation, difference, or disparity is automatically perceived as discrimination by the "equality" zealots, says Thomas Sowell. The idea is incongruous with the facts. Instead of attempting to close gaps by lowering those at the top, a pernicious practice, we should encourage and applaud the achievement of individual students who strive to better themselves no matter the color of their skin and not denigrate their efforts or accomplishment.

Ten years from now, I will echo the same narrative. Instead of pouring money into people and overhauling the teaching profession, we continue to put it in gadgets (technology) to improve achievement in our schools. There is a catch. 
Where's the improvement? Good intentions have never been good enough. The influx of expensive technology ($$$$$) in our schools, which had been promoted by special interest groups, has not resulted in the expected math achievement. Billions are spent on gadgets (tech), but little is learned. 

Michael E. Martinez (Future Bright) writes, "Averaging the ups and downs over the years, academic achievement is slowly edging upward. The real challenge is that the rest of the world has made huge gains. In comparison to other countries, the United States has regressed in effectiveness." What we have been doing in math education has not been effective. 

In my mind, well-trained competent teachers in math and science, starting in early elementary school, can make a huge difference in math achievement--not technology. Also, it seems that high-achieving university students don't want to become teachers because of poor working conditions and low pay. Another reason: teachers are not well respected.
Also, I want to state that the teachers are not to blame for the bureaucratic mess in education. They didn't create the problems. The teachers I have known are dedicated and do the best they can given the context in which they work. Instead, we should go after the progressive elites who train and certify the teachers, and we should hold accountable for the policymakers and the special interests. The public schools are expected to solve societal problems in our communities, which often sideline their primary function which is to teach kids to read, write, and do arithmetic.
To Be Revised 

There will always be disparities in education and life, but, too often, the differences are automatically misconstrued as discrimination. "Differences in geography, demography, culture, and other factors can make economic and other prospects or outcomes unequal for different individuals and groups, even if particular institutions or society were to treat everyone the same," explains Thomas Sowell (Wealth, Poverty, and Politics, 2016). 

It is unfortunate that education has been overrun by the progressive elites, reformists, ideologues, politicians, philanthropists, special interests, and lobbyists. It is called 21st-century education. (How has it worked? Achievement is flat.) For example, a particular TI calculator (30-SX MultiView) is required on the new GED. The College Board SAT is now an exit test in many high schools. Algebra textbooks often boil down to using TI graphing calculators. Online instruction and testing have become commonplace. Money has poured from our schools to special interests, e.g. technology and online "learning" and testing. Moreover, schools of education keep promoting flawed theories, unpragmatic Utopian ideas, progressive ideologies, and evidence-lacking fads or trends that don't work in the classrooms. 

Twenty-first-century reformers demand more technology in classrooms to improve achievement, but the evidence of tech effectiveness has been scant. Indeed, we have had computers in classrooms since the 1980s, but recent national and international test 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.
© 2017 LT/ThinkAlgebra

Wednesday, February 22, 2017

Core Arithmetic

Core Math
Students need to memorize single-digit number facts and learn standard algorithms from the get-go (1st grade), not reform math pedagogy. The aim of learning core math is competency, i.e.,  the automaticity of foundational factual and procedural knowledge in long-term memory. Kids need to drill to improve skills. W. Stephen Wilson, a mathematics professor who holds a Ph.D. from MIT, writes, "The foundation for K-12 mathematics is laid in the early years of elementary school. [For students] to succeed in college, this foundation must be solid." It isn't, which is the reason that a massive number of high school graduates end up in remedial math at a community college. The message from Dr. Wilson is that if you want to get to college and do well, then you must learn core arithmetic in elementary school. Therefore, teachers need to teach core math (the fundamentals) beginning in 1st grade. Reform math, which still dominates the teaching of math today especially in many elementary schools, gets you to remedial math at a community college. It is a shock to students who were weaned on calculators, received A's and B's in K-8 reform math, and took wrongly named college prep courses in algebra. Common Core or state standards have not corrected the major pedagogical flaw in American math education. Ask any pianist or gymnast about the critical importance of excellent instruction and energetic practice (drill) to improve skill. The same is true for math. 

Common Core mentions but does not define standard algorithms, so I am defining standard algorithms as found in Kaplan GED Test.* The single-digit number facts are not hard to memorize in first grade, and the standard algorithms are easy to learn. "Core elementary school mathematics content is straightforward and focused. The catch is that it must be learned well to progress," writes Professor Wilson. In short, children need to automate core arithmetic. And, by core arithmetic, I do not mean substandard Common Core, which includes most state standards or inferior reform math pedagogy. K-5 core arithmetic includes whole numbers, fractions-decimals-percentages, exponents, and proportions. In elementary school, the core math students must also learn parts of algebra, measurement, and geometry. (Let me make clear that the partial quotient method is not the standard algorithm for long division.)

The standard algorithm starts in 1st grade with the memorization of single-digit math facts. Below (12 + 25) shows the practicing of n+2 facts without carrying. The standard algorithm organizes the place value for students. Addition with carrying comes next. The place value idea is to add ones to ones, tens to tens, etc. Also, ten ones make one ten (10), and ten tens make one hundred (100), etc. Numbers are broken down by place value.  Dr. Wilson writes, "The place value system is, indeed, mathematics! You cannot teach mathematics without the place value system, standard algorithms, and our other building blocks." Note: In 24 + 8, the 4 and 8 make 12ones, but the "12ones" break down to 1ten+2ones (place value system). The 1ten is carried to the tens column to add tens to tens. See 12.
The Standard Algorithm
First Grade: 1st Marking Period
Without Carrying --> With Carrying
Single-digit number facts must be memorized.
The meaning of numbers is through place value: 

37 is 3tens+7ones.
Students need to drill to improve skill.

The problem I see is that students lack competence in core arithmetic, starting with single-digit number facts, which must be memorized (automated), used continually, and reviewed regularly. The standard algorithm is one of the best ways for students to practice number facts. Indeed, students need to drill to improve skill. Also, the standard algorithm organizes the place value system. Indeed, standard algorithms are the best models for place value. Even mathematicians as children drilled to improve skills in the core arithmetic such as number facts and standard algorithms. 

The alternative strategies (algorithms) of reform math, which I call pretend arithmetic and mathematician W. Stephen Wilson calls pre-arithmetic, have been ineffective in the teaching of core arithmeticIn my view, the alternative strategies are "extras" that crowd the curriculum and overload students’ working memory. They are not needed and do not lead smoothly to the standard algorithms, which children must master to get to algebra by middle school.  

"Mathematics developed incrementally over millennia by geniuses," writes Dr. Wayne Bishop. Children are novices, not geniuses. Asking children to discover math or invent algorithms (e.g., constructivist pedagogy of reform math) is an "absurd pedagogy." In my mind, the use of calculators in American classrooms has hindered achievement. In stark contrast to American teaching, the students in the East Asian (high achieving) nations seldom use technology in their classrooms. It is "conspicuously absent." 

Even though reform math's constructivist pedagogy has been promoted by schools of education and supported by organizations such as the National Council of Teachers of Mathematics (NCTM) and the National Science Foundation (NSF), which selected reform math programs as exemplary, it has failed and has led to remedial math at a community college--not college readiness. Not only is constructivist pedagogy inferior, but it also conflicts with the cognitive science of learning. Moreover, researchers Kirschner, Sweller and Clark (2006)** point out that discovery learning and other popular minimal guidance teaching methods are not the best way for students to learn basic math. Over the years, millions of kids have suffered the consequences of constructivist pedagogy. 

©2017 LT/ThinkAlgebra

* Kaplan GED Test, 2015 Mathematical Reasoning Prep
Note: I do not approve of using calculators on the GED, SAT, ACT, or any standardized or state test.

** Kirschner, Sweller and Clark (2006): Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching 

Saturday, February 18, 2017

Our best students left behind

We need to focus on the best students rather than always on the worst students. 
We should let our best students go as fast as they can, but we don't. Instead, our best students are dumped into "inclusive" elementary school classrooms in the name of fairness. Charles Murray (Real Education) writes, "Performing poorly in the classroom is not a big deal socially. Performing conspicuously well is often a social liability. ... When it comes to athletic and musical ability no one considers withholding training that could realize those gifts. It is just as senseless, and as ethically warped, to withhold training that can realize academic ability. ... America's future depends on how we educate the academically gifted."

How well? For decades, we have done a crummy job in that 
our best students are grossly underfunded and left behind on their own. The purpose of math education is to make students competent in arithmetic and algebra; however, our best math and science students need acceleration early on. It means hiring algebra teachers for grades 1 to 5 and science teachers who can create and implement legitimate chemistry and physics courses in elementary school. Astute 1st-grade students can study functions and equation solving (math) and also learn the difference between an observation and an inference and study atomic theory (science). We don't have good textbooks for advanced elementary students, not in math or in science.

Note: "The insistence that we can dramatically improve the academic performance of low-ability students has an almost religious tenacity," writes Murray. While there are always exceptions, it hasn't worked well in real education. There are limits.

Advanced middle school and high school students need to take the initiative because very few schools can meet the actual accelerative needs of advanced math and science students. They need to study on their own. Self-motivated students can watch "open courseware" from MIT in chemistry, physics, math, and other topics. More expensive options include private tutors and courses such as those offered by The Art of Problem Solving. Our best math and science students in elementary school don't need enrichment programs; they need acceleration and to be with other excellent students who are also inspired.

Children are not the same. "Ability varies," says Charles MurraySo, let's stop pretending that all kids are the same and can learn the same math. Children need different math curricula based on their cognitive abilities and achievement--not the same curriculum as in Common Core and state standards. In sharp contrast to our current policies, the late Nobel-prize winning Physicist Richard Feynman points out, "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities." Education increases inequality

You don't fix achievement gaps by lowering those at the top or by equalizing outcomes.
Our best students should soar, but many do not. Furthermore, we do not live in Lake Wobegon where all kids are above average. There will always be gaps and levels. We are not all equally creative. Indeed, academic ability widely varies as do other abilities such as musical ability. Some kids will be better at math than others. The outcomes of instruction will not be the same. You cannot equalize or legislate outcomes. However, what we can do is to upgrade curriculum and instruction to world-class levels for the majority of students, focus on getting more students ready for Algebra by middle school, and accelerate high-achieving math students starting in early elementary school. We can accomplish those things by teaching K-5 kids the fundamentals of standard arithmetic rather than reform math. Kids are passing without knowing grade-level content. The deficiencies compound up the grades. Also, focusing instruction to improve test scores is an inadequate curriculum and not the same as learning critical content in math. American K-8 students are weak in math and science, but so are many of their teachers. Also, many students have difficulty comprehending complex text (reading). The burden of academic deficiencies carries over to the high school. The problems begin in the 1st grade, not middle school or high school. Students should master the fundamentals of standard arithmetic first. They don't. A typical elementary school classroom is much like the old one-room schoolhouse. (Comment: You cannot solve a math problem without knowing in long-term memory the prerequisite math.)  LT 

©2017 LT/ThinkAlgebra

Thursday, February 9, 2017

American math curriculum and instruction do not work well.

The Failure of a Reform Math Curriculum and its Pedagogy

Content standards are not the same as curriculum standards. 
The curriculum is someone's interpretation or translation of content and how it should be taught. Common Core state standards have been understood as reform math that flunked in the past. Therefore, instead of learning one way to multiply (i.e., the standard algorithm), reform math students are fed a multitude of alternatives or unconventional methods; e.g., arrays, lattice model, area model, partial products, etc. The standard algorithm that is needed for algebra is pushed aside, a big mistake. 

No one uses the area model, the lattice model, and so on to multiply numbers, yet these alternatives are the priority in most elementary schools. In contrast to reform math, students should use the standard algorithm from the get-go (3rd grade) and start long-division in the 2nd semester or 3rd grade (1997 California State Math Standards). The standard algorithm for multiplication is built on a place value system, the distributive property, and the instant recall of single-digit number facts (the multiplication table). Yes, students need to memorize, drill-to-improve-skill, and regularly review the math facts, but they should also practice the standard multiplication algorithm until it is automatic.  

(1) A typical 3rd-grade multiplication is 3476 x 8. A typical 3rd-grade long division is 768 ÷ 8. The standard algorithms are used, of course. 
(2) First-grade students in Singapore start multiplication as repeated addition. Thus, 3 x 4 means 3 groups of 4: 4 + 4 + 4 = 12. Half the times table is memorized in 2nd grade, the rest in 3rd grade. The formal algorithm is 3rd-grade content.
The content standards are different from the curriculum standards, explain Marzano and Kendall (1998). The curriculum reflects how kids are taught to achieve the standards (e.g., linking specific content to specific methods of instruction). In my view, the problem is that the Common Core curriculum stresses the alternative strategies of reform math (low value) and downplays the critical importance of mastering the standard algorithms needed for algebra in middle school (high value). 

One result has been that American students stumble over basic arithmetic, even in middle school, which has led to a massive influx of students in remedial math at community colleges. In short, our math instruction has been ineffective for decades upon decades. It seems clear, at least to me, that the American "reform math" curriculum, progressive methods of teaching and ideology have not been the best way to teach students standard arithmetic. Beginners don't need a bunch of alternative strategies; they need one efficient method (standard algorithm) to do each operation, which requires the memorization of single-digit number facts starting in 1st grade.  

In 1938 William Bagley (Columbia University) pointed out that "American elementary and secondary students fail to meet standards of achievement set by students in other countries," even though "more money has been spent on education than ever before." (Quote from Ellis and Bond, Research on Educational Innovations, 2016) Sound familiar?

In the real world, the Common Core content standards are not world-class. For example, Singapore 1st-grade students learn significantly more standard arithmetic than American students, according to my analysis. Singapore content standards are straightforward, and the instructional methods are largely supported by the cognitive science of learning. Starting in the 1st grade, Singapore students memorize, drill to improve skill, and regularly review arithmetic basics to cement them to long-term memory for mathematical thinking, learning new content, and problem-solving.

Ellis and Bond question the "linking the achievement of certain content goals with certain methods of instruction in the absence of clear and convincing evidence." American students are not meeting the standards set by students in other countries because of the reform math curriculum and methods of instruction along with progressive ideology. “The more things change, the more things stay the same.”

Al Manne (Stanford) admonishes, "To get a large model to work, you must start with a small model that works, not a large model that doesn't work." In American schools, shoddy models abound. Note: Teachers don't have much choice in a top-down bureaucratic education system. They are required to teach a substandard curriculum with inferior instructional methods. 

It is 2017, and our goal of being the best in the world in math and science has failed repeatedly. American students are not meeting the achievement levels set by students in the East Asian nations, such as the Advanced Levels (TIMSS). Unfortunately, some education leaders rationalize the lackluster performance in math by saying American students have never done well on international tests. It's no big deal; after all, there are incremental improvements. No, it is a big deal! 

February 9, 2017

©2017 LT/ThinkAlgebra

Wednesday, February 1, 2017

Standard Algorithm

The Standard Algorithms Should Be Primary, But They Are Not.
Reform math shortchanges them.

In reform math and now Common Core state standards, which are often interpreted as reform math, students have been taught many alternative strategies for different numbers or situations. Consequently, many ways make simple arithmetic more complicated, mystifying, and confusing. Core arithmetic is straightforward and focused, but not reform math, which marginalizes or delays traditional arithmetic. Often, students are taught 3 or 4 strategies (procedures) to calculate one operation. The alternative approaches overload a beginner's working memory and confuse students. More importantly, they do not prepare students for algebra. In contrast to reform math, the standard algorithms for the four whole number operations do prepare students for algebra and should be primary and taught first, but they are not. The multiple strategies of reform math shortchange or push aside the critical importance of the standard algorithms.  

In fact, the standard algorithms solve the reform math dilemma of many alternative strategies and cognitive load. Each of the four whole number operations has one standard way for calculating called the standard algorithm, which always works no matter the numbers. Think of all the time wasted on alternative strategies when the instructional time should have been spent on learning the standard algorithms (core arithmetic), along with practicing and reviewing.  Students should practice 20 to 30 problems a day, five days a week, even 1st-grade students.
The number 37 means 3tens+3ones.

In 1st-grade whole number addition, the place value system is adding ones to ones, tens to tens, and so on. Students need to memorize the single-digit math facts and break down numbers by place value to make the standard algorithms work. 

In short, the standard algorithm boils down to single-digit number facts in a place value system. It is the quintessential model of place value, and "the case for the importance of the standard algorithms for whole number operations cannot be overstated," explains W. Stephen Wilson, Ph.D., Professor of Mathematics, Johns Hopkins University. Moreover, the study of the place value system should begin in 1st grade with the meaning of numbers. Numbers should be taught as place value, e.g., 12 is 1ten+2ones or t + 2, and so on. Place value is core mathematics, and it leads directly to the standard algorithms. 

Wilson points out, "The place value system, fractions, and the standard algorithms all contribute greatly to algebra readiness."

To Be Continued
Excuse typos and other errors. 

©2017 LT/ThinkAlgebra