Upgrade Curriculum & Instruction

Kids Can Learn Harder Stuff--if they work hard & if their teachers work hard! 

Build Mastery Through Practice! 

Some educators say that little kids cannot learn algebra or memorize multiplication facts because it is too hard. I often hear, “Kids are not developmentally ready for that.” Or, “That’s too advanced.” Or, “Our kids cannot compete with Singapore kids.” Or, "Rote learning is bad learning." It’s bunk! It is low expectations! And, it is contrary to experience and cognitive science. Repetition, multiple exposures, effort, and extensive practice are important. Students need strong teacher guidance, explicit instruction, a world-class curriculum, and lots of practice to learn arithmetic and algebra well.  In short, our students need to master a world-class curriculum starting in grade 1 to stay competitive. It takes effort, work, and the development of smart thinking. Students cannot apply arithmetic or algebra that they do not know well. Thus, acquiring background knowledge quickly is critically important. Student achievement, which is graded C- by Quality Counts 2012, should be the highest priority in our classrooms. What has gone wrong? It boils down to the curriculum, which is the content in the textbook or the math program. Beverlee Jobrack (Tyranny of the Textbook), observes that "standards, achievement testing, technology, and professional development [and other reforms] have had little effect on improving student achievement" (p. xx). It also boils down to instructional methods.  

We need a world-class curriculum that is logically sequenced, organized by familiar course names, and focused on achievement. 

Such a math curriculum is not found in many textbooks or popular math programs. The reason students do not do well in arithmetic or algebra is a lack of pivotal background knowledge (Knowing). (The same is true for reading comprehension.) Students cannot think about math ideas, do the math, or solve problems in math if they do not know the key fundamentals of arithmetic and algebra. Unfortunately, most students underperform in math? If teachers do the same things they have done before, then how can they expect major academic improvement? Teachers must analyze what they teach (content), how they teach it (instruction), and how they measure what they teach (assessment). See Common Core on ThinkAlgebra. Updated 1-21-12, 9-2-12

[FYI: Random Thoughts: I do not think Singapore textbooks are sufficient either. Typical students in 1st-grade can learn basic algebra ideas that use, connect to, and reinforce basic arithmetic concepts and skills (see Teach Kids Algebra project in March-May 2010). If I had the technology, every first-grade student would learn to write procedures that make polygons (squares, rectangles, etc.) using the LOGO programming language, which is what my 1st-grade class did in the early 80s. Also, I suggested years ago that teachers break from the textbook and the status quo and explicitly teach content and skills that are above those expected at a given grade level. This takes good instruction and practice. Teach grade level and above because teaching at grade level sets the bar too low. Also, teachers must accept that learning math well requires hard work.] Updated 1-21-12, 9-2-12

Current Textbooks & Popular Math Programs "Perpetuate Mediocrity.”

Textbooks have been the de facto curriculum (what is taught in the classroom), and, according to Beverlee Jobrack (Tyranny of the Textbook), textbooks have changed little over the years and "perpetuate mediocrity," regardless of the many math reforms.  Textbooks sell, not because they are effective in student learning, but because of their "design, labeling, and teacher appeal" (p. xviii). Jobrack writes, "Textbooks sell based on design and superficial features, not because they are based on the latest research on how children learn [cognitive science] and how well they promote student achievement." In my experience, K-8 math textbooks are cluttered with colorful, eye-catching graphics with extra topics included. But they often lack a logical, well-organized sequence of concepts, skills, and applications (scope and sequence) that lead to algebra in middle school. Moreover, textbooks often lack enough challenging content and sufficient practice exercises for skill mastery. After over 40 years in education, I have observed that math textbooks and popular math programs are not focused on achievement; they also marginalize or ignore the basics of cognitive science. Research-based does not mean that the textbook was tested for its effectiveness because effectiveness is not a priority in publishing. Most teachers tenaciously follow the textbook, no matter what, and, therefore, students get only bits and pieces and practice very little. Jobrack writes that the most effective curriculum materials do not appeal to most teachers and are rarely adopted (p. 1). Updated 1-21-12, 1-22-11

Of course, each state has a rough "curriculum framework" or set of standards [Common Core], but, in reality, teachers use state-approved textbooks for curriculum. Moreover, the difference between what teachers teach (the textbook curriculum) and what students actually learn, know, or master is large, while the difference in Singapore, for example, is much smaller. In short, the minimal guidance instructional methodologies that most teachers were taught in ed school and that permeate most math programs and textbooks have not been effective. Many instructional strategies (e.g., group work, differentiated instruction, learning styles, etc.) are especially ineffective. Furthermore, praising students for no good reason and telling students they are smart have backfired [self-esteem movement]. Teachers should stop inflating grades and focus on the child's competency. Also, teachers should make nested connections (i.e., connections within content) but most do not know how because elementary teachers are trained as generalists, not math people, according to Dr. H. Wu. Students need to practice a lot to master content. For example, math facts must be memorized early and used repeatedly.

Curriculum & Instruction

In my view, curriculum and instruction have been the biggest problems we have had in math education. Jobrack says that "educational publishers must play a much greater role in improving education" (p. xii). The textbooks add significantly to the problem of inadequate math learning because they lack content and coherency, are not based on cognitive science and do not "promote student achievement." Jobrack adds, "The [Common Core] standards are not the curriculum. The curriculum is what teachers do every day, introduce a concept, check for understanding, have students practice, and assess it." In short, the math curriculum has been the textbook, and, over the years, the curriculum in classrooms has not changed much. FYI: The Quality Counts 2012 annual report, which grades each state, gives K-12 student achievement in the United States a C-. 

Cognitive Science: Practice, The Key to Competency
Clifford Stoll (Berkeley astronomer and physicist) says that explicit practice has fallen out of favor, "yet it ['repeated drill'] is the core of mathematical competency." For example, in 3rd grade, the multiplication algorithm, finding areas (squares, rectangles, triangles), word problems, and counting problems are excellent ways to practice or review multiplication facts, but Common Core delays the mastery of the standard algorithm until 5th grade for no good reason. [Note. Common Core 5th Grade (p. 35): "Fluently multiply multi-digit whole numbers using the standard algorithm."In Common Core, "fluency" does not necessarily imply "automaticity." In my view, the explicit practice has been replaced by manipulatives, calculators, group work, and strategies (e.g., Everyday Mathematics).] Added 1-16-12, Revised 1-17-12

Cognitive Science: Thinking [Learning] Is Hard Work!
Dr. Art Markman (Smart Thinking) writes that students who do not know content and who are not persistent will have difficulty solving math problems (“Smart Thinking"), such as word problems. Markman, a cognitive scientist, says that, like chess, smart thinking is a skill that can be learned.  In short, for students to solve math problems, they need to learn lots of basic content to build a tool kit for problem solving. It should start in 1st grade with the mastery of math facts (auto recall), procedures (algorithms), place value, equations, and properties of numbers. Moreover, Markman says that students learn new stuff better when it is linked to stuff they have already learned and practiced. Thus, coherent sequencing (Bruner), attention to prerequisites (Gagne), and nested connections (Markman) in arithmetic and algebra are important because math is highly structured and hierarchical; i.e., one idea builds on or links to another. In short, you cannot teach math like you teach social studies. (Note. The TIMSS definition of "Applying" in its advanced math framework means routine problems found in textbooks. Students must be good at routine problems before they proceed to non-routine problems.) 

Cognitive Science: Math Ability Grows!

Understanding Grows Over Time. 
If we want our children to excel in mathematics like kids in other nations, then our kids must know arithmetic and algebra flat out. Most don't. There are no short cuts. It takes consistent practice, hard work, and smart thinking (Markman) to learn anything well, especially mathematics. To establish good math habits, students need to build mastery through practice. Thinking, says Markman, is hard work because "your mind is designed to think as little as possible." Learning something new (i.e., content) in arithmetic or algebra takes effort and smart thinking. Repeatedly, cognitive scientists have told educators that math ability is not fixed; it grows through a child's persistence, effort, and hard work to learn new stuff, which grows brain cells. Students don't learn anything unless it is practiced well. Again, this takes effort, persistence, and hard work. Learning something new in arithmetic or algebra can be frustrating at times, but students must learn to persevere and get through it to foster cognitive growth. Thus, students need to develop persistence and an ethic of hard work to learn challenging content. In short, brain cells must grow to achieve in mathematics. It is about time we educators listen.

We need to train children for the future, but they won’t have much of a future [very limited postsecondary choices] unless they master mathematics (arithmetic, algebra, and precalculus), which takes discipline, concentration, effort, and hard work. Most children grossly underachieve in mathematics and science. For nearly half a century, we have known that very young children can learn math faster and at a much higher level, but this is not the way math is taught in most schools. I ask educators to shake off the role of "guide on the side" and put aside group work and manipulatives. Teachers should explicitly teach challenging content and help students attain mastery through practice.

From ThinkAlgebra

©2012 LT/ThinkAlgebra

US kids stumble over arithmetic

US kids stumble over basic arithmetic.

Stuck in the middle!
Many educators assume that the US reform math methods are better than methods in other nations, but, on international tests, US 4th and 8th-grade scores are just average because both our curriculum content and instructional methods are off-target. Moreover, for decades, we have jumped from fad to fad, from one sweeping idea to another. Teachers are taught to favor group work, inquiry methods, manipulatives, technology (calculators, software, videos, etc.), differentiated instruction, strategies, etc. Teachers are told that students should jump to collaboration and critical thinking without a focus on knowledge mastery. It makes no sense! Kids are novices, so teachers should focus on content mastery--the content that leads to algebra in middle school. 

US math scores are not shooting up; they are stuck in the middle. Unfortunately, the American character seems focused on utility. What is the utility of Algebra? What is meant by utility? The reason to study algebra is to cultivate the mind so kids learn to think, reason, and plan. In addition, kids need a good grasp of algebra to get into college. Furthermore, many college majors require calculus, not just STEM majors (e.g., business, finance, economics, etc.). Richard Hofstadter (Anti-Intellectualism in American Life) points out that there has been a bedrock of anti-intellectualism in America, and algebra is just one subject of some attack. Jacob Vigdor (Education Next, Winter 2013) writes, that in math, rigor has been replaced by practicality and equity (kids get the same), and counter-intuitively, this has turned students off to math. He observes, "In the 21st-century workplace, mathematical capability is a key determinant of productivity." 

The great majority of innovators of tomorrow are the STEM and academically gifted students of today, but our school policies and programs critically shortchange our top students! Jacob Vigdor concludes, "Evidence suggests that the dramatic watering down of curricular standards . . . has made our top performers worse-off." Our best students are just average when compared to their cohorts in top-performing nations. In short, our best students underachieve. Eric A. Hanushek points out, "We like to talk about American innovation, but many of the people doing the innovating here [e.g., Silicon Valley] were in fact born elsewhere." The amount of money and assistance spent on the "most able and motivated kids" is a drop in the bucket compared to the huge amount of money, resources, and action spent on the "least able, or the disinterested, or the unmotivated." Sarah Lichtenwalter writes, "Funding for gifted education programs today is inadequate and upsetting. The National Association for Gifted Children (2009) notes that the federal government provides only two cents of every hundred dollars spent on education to gifted children (National Association for Gifted Children, 2009, p.2). Yet, as Charles Murray (Real Education) aptly points out, "America's future depends on how we educate the academically gifted." They will be the leaders and innovators of tomorrow. But, Murray cautions that it is not good enough for leaders to be smart; leaders must also be wise. We have a lot of smart people in education, but I don't see a lot of wisdom.

We seem stuck in a testing mentality rather than an achievement culture. Instead of soaring to the top, we barely make it to the middle. Diana Senechal points out that standardized tests have limitations and that test scores are not "precise measures of teaching quality, school quality, or student achievement." No one listens! Furthermore, our idea of equity focuses instruction on struggling or least able students, so our best students, even our above-average students, go unchallenged in many of today’s classrooms because instruction is not tailored to their needs. Teachers are often faced with mixed groups and told to differentiate instruction within the classroom. As a consequence, the self-contained teacher may have up to 4 or 5 different math groups, which makes absolutely no sense. Yet, educators seem to come up with sweeping ideas they claim are transformative. For example, the solution to differentiated instruction has been adaptive math software or ideas such as online learning, but software or videos cannot replace a talented classroom teacher. Mixed groupings should be abandoned. Instead, kids should be sorted by math knowledge with instruction tailored to their needs. The sorting should begin in primary school. Note. The main premise of the common core is that one size fits all (uniformity), but how is this possible? The common core premise is already falling apart because common core kindergarten kids move into 1st grade without knowing many of the K common core standards. Moreover, the common core writers tell us that the new math standards are not as high as top-achieving nations, so our kids will continue to lag behind the rest of the world. Furthermore, it seems that state tests, under NCLB, are often the main focus of instruction in the classroom. Teachers are pressured to focus more on items found on the state tests. Thus, many kids don't get a coherent math program that propels them to algebra in middle school. The new math standards won't change classroom practice much, which is to focus on test items.   

TIMSS 2011 [International Tests]
"The most striking contrast comes in the 8th grade, where nearly half of all students tested in South Korea, Singapore, and Chinese Taipei (Taiwan) reached the "advanced" level in math compared with only 7% of American test-takers." (Quote: Education Week, 12-11-12, on TIMSS 2011 results. US 8th graders averaged 509 in math, which is above the international average (500), an improvement, but lag significantly behind top-achieving nations; e.g., South Korea 613.

Will common core math catch our kids up to other nations? 
{No! Common Core is not at the Asian level.} 
Has the common core been tested in real classrooms? 
(Surely, you're joking! The hypothesis is that common core math will correct our math problem and properly prepare kids for college-level math. But, common core is untested and, in my opinion, is nothing more than junk science when it is put into classrooms. The hypothesis has no supporting evidence, but plenty of special interest support, including the US government.) 


How long? (In many states, the common core standards are better than the old standards, but it may take years for typical kids to meet the new math standards, which, incidentally, frequently fall below international benchmarks. How long? Who knows? Many 1st graders do not know the K standards, and K is where the new standards begin. You don't phase in K standards! Deficiencies in the early grades, like before, will march up the grades and grow wider. It is the same old problem. Somehow, 3rd, 4th, and 5th-grade teachers are charged with the task of catching kids up, perhaps by a miracle. This hasn't worked well in the past and won't work now. You don't start intensive intervention programs in upper elementary or middle school, you start them at the beginning of 1st grade where the problems arise. That's what Singapore does!)

"It's not just urban kids who are struggling. Even wealthy suburbs are lagging behind countries like Singapore and Finland," writes Jennie Rothenberg Gritz, The Atlantic, 12-10-12. Also, students who struggle early in math are unlikely to catch up. Math intervention programs should start no later than 1st grade, not 4th grade, which is too late for most kids according to a study done by ACT.  

Note. Everyone talks about higher-level thinking, but, as Robert Pondiscio (The Core Knowledge Blog) points out, "In reality, without the basic skills and knowledge firmly in place, there’s no such thing as higher-order anything and never will be."  

Instant Recall of Math Facts should be a 3rd-grade benchmark!
W. Stephen Wilson points out, "Arithmetic is the foundation. Arithmetic has to be a priority, and it has to be done right." But, things have gone wrong in elementary school arithmetic, such as the use of calculators, the lack of instant recall of multiplication facts, the lack of fluency in standard algorithms, the overuse of manipulates, a steady diet of discovery activities or group work, and so on. Professor Wilson observes, "Avoiding hard mathematics with younger students does not prepare them for hard mathematics when they are older." Indeed, he argues correctly that the lack of instant recall of multiplication "permanently slows students down." But, many educators, administrators, and policymakers do not seem to comprehend this. Over the years, I have appealed to educators and administrators that multiplication facts for instant recall and the multiplication algorithm should be the top benchmarks in 3rd grade. (They used to be.) Also, students should start the long division algorithm in 3rd grade. In particular, I am not in favor of asking novices to find multiple ways of doing procedures, when the standard algorithm is put on the back burner. In my view, this is exactly what has happened with NCTM reform math over the past couple of decades and, now, it is part of the common core. Nothing really changes that much. 

Standardized Testing Problem
We seem to be stuck in a "testing culture" rather than an "achievement culture." Standards are general, often vague statements; they are not instructional [behavioral] objectives, which are specific and actionable. Dr. R. Baker Bausell (Too Simple To Fail) writes, "Our standards-based testing system does not (a) assess school-based learning, (b) reflect any school's actual curriculum, or (c) has any known implications for instruction." The state math tests, at least the ones in Arizona, are not actual content achievement tests. The state tests measure how students score on a test. I do not expect anything different from the new common core math tests. Often, teachers are pressured to spend more instructional time on state reading and math test items. In my view, this practice narrows the curriculum. Frequently, PE, art, music, science, or history/geography are squeezed thin or not available in many elementary schools because of time or staff shortages. While some achievement testing, which is based on very specific instructional [behavioral] objectives, is needed to give teachers feedback. I think standards-based testing is not the answer. 

Change Is Hard; Ability Varies
Changing our culture, or education ideology, or its entrenched bureaucracy is hard. Progressive ideas seem to dominate K-12 education. Progressives often put "ideological value [e.g., fairness, equity, social justice, etc.] above excellence" and "social engineering ahead of ability." Common Core is within the progressive frame in that every student gets the same. Yet, kids are not the same. "Many students spend their days in school unchallenged--relearning materials they have already mastered," write Subotnik, Olszewski-Kubillius, & Warrell (Scientific American Mind 2012). Academic ability, motivation, and persistence vary widely among the school population. Thus, outcomes can never be the same. Teachers are not the same, either. The progressives' answer to better public education has been more money, more teachers, more technology, smaller classes, etc. Sound familiar? Some progressives are often driven by ideology and frequently dismiss scientific findings as "just another opinion." Results don't matter much, just intentions. In short, over the years, academic excellence has eroded in our classrooms. 

I think reform math, which is favored by many educators and often championed in schools of education, has kept American math education off course for decades, especially in terms of curriculum content and instructional methods. And, policies, such as mixed groups (inclusion) don't help either. Most students never develop the math skills they need to move forward (NAEP). They stumble over arithmetic skills, even in middle school--especially fractions, decimals, percentages, and proportions. Moreover, many educators often interpret common core in terms of reform math methods, which, unfortunately, downplay memorization and practice, and emphasize group work rather than individual achievement and excellence. Kids do not learn much math by group or inquiry methodology.

To get good at arithmetic requires lots of practice with corrective feedback so things stick in long-term memory. The process should start in 1st grade with the learning of addition facts for auto recall, the use of the standard algorithm for quick calculations, the introduction to multiplication as repeated addition to solve word problems, the writing of equations that model word problems in addition, subtraction, and multiplication. Students should also learn how new knowledge builds on existing knowledge and how things fit together logically in arithmetic and algebra. Algebra grows out of arithmetic. In short, there is a lot of reasoning and inferential leaps students should learn in the study of arithmetic and algebra. It is not easy.

Note. To catch other nations, educators should think in terms of getting at least 60% [my figure] of students ready for Algebra I by 8th grade. The goal of 60% is flexible, of course, but it is attainable only if the K-7 curriculum is upgraded to world-class and only if teachers improve instruction substantially. Also, common core sidelines the idea by placing critical Algebra I standards in high school, not 8th grade. In my view, the common core is off course. 

Algebra is for kids who are prepared, so let's prepare more kids! Unfortunately, there has been a trend in the US to water down math. Indeed, under common core, every student gets the same. But, equal treatment (content) should not imply equal outcomes. Some kids learn math faster, study more, are more persistent, work harder, have pushier parents, etc. Even though academic ability varies widely, I believe most kids, but not all, can learn algebra when they are properly prepared in earlier grades. It is important to start teaching algebra ideas in 1st grade. Let's be straightforward and focus on individual excellence. Tell students what math content they are expected to learn. Teach it. Practice it. Test it. Use it. Furthermore, kids do not learn math by group; they learn math individually through persistence and practice. We have lost sight of the importance of the individual and of excellence. In many schools, "sameness" is extolled and individual excellence trivialized. Note. Some students might take half of Algebra 1 in 8th grade and the other half in 9th grade, but coordination between the middle schools and high schools in the same district is often nonexistent, which is another problem in math education. 

Automation
The automation of  fundamentals is found in the 1st-grade curricula from top-performing nations (e.g., Singapore, etc.), but automation is not specifically stated in common core 1st grade. (In Asian nations, kids memorize and practice for automaticity.) Instead of automation, common core says that kids should use strategies to add and subtract. Furthermore, I think the common core is similar to NCTM reform math in that it encourages students not to memorize basic number combinations, etc. Students need to automate fundamentals to move forward and to engage in higher-order thinking. Kids must build content knowledge in long-term memory!

Note. Some argue that common core requires kids to think, but students won't do much thinking without a lot of background knowledge, both factual and procedural, in long-term memory. You can't solve a mathematical problem without "prior knowledge," writes G. Polya (How to Solve It). Poyla also writes that students often need a lot of practice with routine math problems before advancing to more complex problems. Content-less thinking isn't thinking at all. 

Enable Higher Order Thinking
You don't get to higher-order thinking unless you know a lot in long term memory! Knowledge is the most important treasure we can give kids. 

Burger & Starbird (The 5 Elements of Effective Thinking) write, "Commonly held opinions [or assumptions] are frequently just plain false." Often, drill or rote learning is seen as the enemy of higher-order thinking says Doug Lemov (Practice Perfect), but it's not true. He writes, "It's all but impossible to have higher-order thinking without strongly established skills and lots of knowledge of facts" in long-term memory. Indeed, Lemov says that many types of higher-order thinking are a function of rote learning. Drill or rote learning is not the enemy of higher-order thinking; it enables it.  For decades math education has favored progressive [ideological] assumptions that have no basis in cognitive science. Thus, memorization and practice, which are needed to automate core skills and procedures, have fallen out of favor in our schools. This ideological path is a reason that most of our kids grossly underachieve compared to their peers from other nations.  

Some of the strategies can be useful, of course, but they should not replace auto recall of basic number facts or the standard algorithm. A student should not have to figure out a simple number fact using a strategy every time the combination pops up. Thus, it seems that most US students will continue to stumble over arithmetic early on, never mastering the essential factual and procedural background knowledge needed to move forward to algebra by middle school. US students are always behind the curve. Being proficient at something is not the same as mastering it until it's automatic. In school, students should automate the fundamentals (Lemov). Students should not have to figure out that  5 + 7 is 5 + 5 + 2 or 12, etc. The number facts should be automatic, not figured out every time the fact pops up. 

Common core pushes most of the algebra standards to high school and delays "fluency" in standard algorithms for addition and subtraction of whole numbers to 4th grade. Unfortunately, US students lag behind their peers in other nations. Doug Lemov (Practice Perfect, p. 204) writes, "When students don’t do something a lot, they never get very good at it." This describes the dilemma we have had for decades in math education. The progressive ideology of many education elites squeezes out memorization and practice in our math classrooms. This is nothing new. I think there are close parallels between the NCTM reform math standards, which failed to turn around math education in the US, and their common core replacement, which is untested in the classroom. 

No one bothers to ask critical questions? Here are four of many. Will the new standards work in our classrooms? (No one seems to know.) Where is the evidence of effectiveness? (You must be kidding. There is no evidence. The standards have never been tested.) Will the standards get kids into college? (No one knows!) Will common core math catch our kids up to other nations? (No! It is not at the Asian level) What is the cost ($$$$) of implementation to the States and school districts? (I have no idea, just rough estimates.) 
10-4-12, 10-15-12, 10-22-12

"Neglect of mathematics works injury to all knowledge since one who is ignorant of it cannot know the other sciences or the things of this world. And what is worst, those who are thus ignorant are unable to perceive their own ignorance, and so do not seek a remedy.” {Roger Bacon}

Return to ThinkAlgebra
© 2012-2013 LT/ThinkAlgebra/MathNotes

Math Grouping

Sort kids into homogeneous math sections by achievement, starting in 1st grade.

Arrange kids into homogeneous math sections by achievement. Don't worry about their self-esteem. Worry about their competency! 

Equal coverage of core math content is often confused with equal coverage of math content. While most kids can learn arithmetic basics and some algebra, many kids will not go on to learn trig or calculus. Let's face it; some kids are much better at math than others, and many are not being taught math they are capable of learning. 

Putting high achievers and low achievers in the same math class have been a recipe for underachievement and mediocrity. We have been mainstreaming kids for as long as I can remember. In a typical classroom, there is often a wide range of abilities or achievements in math. It means that the kids who learn math faster get bored and the kids who struggle stay behind. In my view, mainstreaming [inclusion] for math classes has led to underperformance at all levels. In short, the traditional system of heterogeneous classes for math is deeply flawed. Putting low achievers and high achievers in the same math class have hurt all our students because they are not challenged to learn the content they are capable of learning. Moreover, the "one-size-fits-all" common core reform math is not the answer. In addition, the popular instructional methods (minimal teacher guidance, inquiry, group work, etc.) are often less effective in teaching arithmetic and algebra. In my view, students underachieve for these reasons: the way students are grouped in math class, the math content taught (weak curriculum), the methods of instruction (ineffective), and teacher training (inadequate in math and science). [Note. In my view, common core is often driven by pedagogy leftover from the NCTM reform math it has replaced. More, later.]

Education creates inequalities. "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?"  (Surely You're Joking, Mr. Feynman! by Richard P. Feynman.)  (First Draft. Please excuse errors. There are last-minute inserts.) 


Good education, says Feynman, should increase differences. The most realistic way to meet the diverse needs of most students in mathematics is homogeneous sectioning for weak students, average students, and advanced students. It is the most pragmatic means for upgrading all students to significantly higher levels of achievement. The sorting of students should be flexible and start early in the 1st grade. (Note. Kids enter school with differences, but US schools tend to ignore differences. In Singapore, differences are addressed, not ignored. In 1st grade, kids with weak number skills are pulled out for math periods and taught by a high-quality teacher. They are expected to learn core arithmetic just like the regular kids. The difference is these pull out kids are in a smaller class, instead of a class of 30, use different materials, and are assigned the best teacher. Most students catch up within a year because that is the goal, although the safety net program lasts through the 2nd grade.) 

The "many levels" in the same math class, an overemphasis on group work and inquiry learning, and minimal guidance methods have led to widespread underachievement and unintended consequences. In my view, the implementation of "many levels" in the same math class and a steady diet of group work do not define quality learning or excellence. Furthermore, educators seem caught up in a self-esteem mode rather than in a competency and achievement mode. Many students who get As and Bs in so-called "college prep" courses end up in remedial math courses at community college. We are told that common core math should eliminate the mushrooming remedial math problem, yet there is no evidence that it can. 

[Insert. Education is off the track because academic excellence has been replaced by a "fairness" mindset that equalizes downward and leads to widespread underachievement for all students. In addition, we need to switch from a "test culture" to an "achievement culture." The curriculum, which is taught in the classroom, is test-focused and narrow. In my view, the common core is an outcome of the fairness mindset and testing culture, which are interrelated. Common core math is not at the Asian level. Thus, US students start at a lower level and never catch up. Beginning in the 1st grade, math is not taught well. Nothing new. The math curriculum is weak, poorly thought out, and poorly taught. Standards (e.g., common core) are not the curriculum. The curriculum is what is actually taught in the classroom. Katherine Baird (Trapped in Mediocrity) says that too many teachers do not make good use of classroom time, which is one of the main themes of this post. One reason is that kids of uneven skill levels are in the same math class. She also states that most states define "proficiency" at low levels of competency. Professor Baird wonders why there are so few elementary schools that teach algebra and geometry? 

On ThinkAlgebra, I conclude that states should opt-out of the common core, adopt the Core Knowledge K-8 content and skills sequence, which is better than the common core, and allow individual schools to compose their own rigorous curriculum and achievement tests based on the students they serve. Note. Core Knowledge, not common core, sets up a coherent K-8 math sequence that prepares more students for Algebra 1 in 8th grade, which is the fundamental tenet of the National Mathematics Advisory Panel (2008) ]  

In reality, not all students will be high achievers. Kids vary widely in academic ability, motivation, persistence, effort, self-control, numbers, vocabulary, etc., yet we educators pretend that each student is the same, which is nonsense. We need to tackle the real world, not play with Utopian models. We create idealized models of reality (e.g., equality) then think they are a reality [William Byers], but every teacher knows some kids make little effort to learn things, many kids do enough to get by, and other kids just do not have the smarts.  Nevertheless, we bend over backward with money, time, and resources to make equality work. It just isn't going to work--it's not reality. We need a different recipe, one that works in the real world, not a fantasy.

The late Professor Feynman is right. In a sound education system, there will always be inequalities because education creates differences; however, the mixed group approach in place today is an unacceptable model because it has led to underachievement at all levels. And that's what many educators do not want to acknowledge. In short, our present system symbolizes [is code for] low-quality schooling. Kids need strong teacher guidance, a world-class math curriculum, a grouping that matches their achievement level, lots of practice to master math, and persistence. Instead, most kids get "minimal guidance" instructional methods (e.g., discovery, inquiry, group work, etc.), a weak math curriculum, mixed grouping in math class, and insufficient practice to automate fundamentals.

The road to equality is paved with good intentions, but it is easy to get stuck in the mud with good intentions because good intentions are not the same as good ideas that actually work. In fact, many of the ideas put into the classroom turn out to be bad ideas; e.g., kids must first have high self-esteem before they can learn. Wrong! Intellectual leaders in education have been wrong again and again. In education, inputs (differences like self-control, etc.) do affect outputs (learning). Kids come to school with sizable differences in academic ability and vast differences in vocabulary. You cannot equalize huge differences. The idea of "equalizing downward by lowering those at the top" [Thomas Sowell] is a "prevailing ideology" in education. It hurts kids. Unfortunately, says Sowell, high achievement is often equated with "privilege." Privilege, some say, is not fair! Sowell characterizes the progressive point of view this way: "Tests in school discriminate against students who did not study." Let's abolish tests and homework because some students, apparently the ones who study, delay gratification, and work hard, have an unfair advantage.] Note. Progressives, which have influenced education policies for decades, actually believe that utopia of equality is possible [Berezow & Campbell]. Get real.

Instead of giving each student the same, which is a fundamental premise of Common Core math [uniformity], we should bring students up to the level of mathematics they need to move forward and be successful in life. We are obligated as educators to give children opportunities as equal as possible and encourage students, regardless of background, to work hard to achieve and excel. We need more college-educated minorities and women, especially in the STEM fields. 

When content is taught explicitly, average kids can learn arithmetic and algebra at an acceptable level; however, weak-performing students should be in a math section that receives a double dose of instruction; e.g., KIPP 5th graders get 2 hours of math daily. Indeed, KIPP students spend more relevant time-on-task in mathematics in one day than some elementary students spend in one week. KIPP kids have longer school days, school on Saturdays, longer school year, and math homework. Advanced math students should be placed in a section that stresses depth, content acceleration, and rapid pace. Indeed, to move rapidly forward, the best math students, which often languish in the regular classroom, should be grouped together for math class. This is homogeneous sectioning for regular grades in elementary school and by course in middle school, e.g., pre-algebra or Algebra 1. It is time to bring back the "old school." Hey, I miss the chalkboards.

Homogeneous math sections should be taught by high-quality math teachers (not NCLB definition of highly qualified teachers). We don't have nearly enough high-quality math teachers. Furthermore, the grouping (low, average, high) should start in 1st grade. Weak kids in 1st grade should be pulled out for math class at the beginning of the school year. The best kids should be pulled out for math class, too. My algebra program helps identify young, mathematically able students.

Group the best 1st-grade math kids
for a daily pull-out class taught by
a high-quality math teacher.

Homogeneous Sectioning for ELEM Arithmetic; MS Pre-Algebra, Algebra 1 

1. Low performing students: (2C) double dose of core 

2. Average performing students: Core + 

3. High performing students: Core +++

Core denotes the knowledge and skills learned by average students in top-performing nations, e.g., Singapore. A good curriculum for US kids is Core Knowledge content and skills sequence, which is world-class and puts Algebra 1 in 8th grade. It does not refer to the common core. In short, Core means you don't dumb down the math. Elementary teachers must get better at teaching basic arithmetic. Moreover, the organizing of students should begin in 1st grade and be fluid up and down. Switching to homogeneous groupings and explicit teaching would be important progress, but it is not perfect. There is no perfect system, but there are systems that work much better than others. 

Students in three tracks end up at the same goal, which is to learn core. This would end "content incoherence," a term used by E.D. Hirsch, Jr. Kids need to learn core. The lower track learns core. The middle path moves faster and learns more than the core, and the advanced group soars way above grade level core. This is not equal coverage of math content; it is equal coverage of core.  

[Insert. Many middle schools offer some form of homogeneous groupings, such as advanced or honors-level classes in mathematics; however, this idea (honors math class) is rare in elementary schools because parents seem satisfied with math enrichment or with talented and gifted programs. The problem is that math enrichment does not move kids forward. Also, math enrichment is rarely taught by a high-quality math teacher. Consequently, some elementary school parents hire a private tutor or enroll their child in online courses from EPGY or Art of Problem Solving. In high school, students sort themselves; i.e., they can select from a range of math courses, including AP Calculus.] 

Decades ago, educators replaced the "old school" homogeneous sections (BAD) with a theory of equality (GOOD) that advocates mixed-level groups, self-esteem, group and project work activities, inquiry learning, grade inflation, less rigor, etc. Subsequently, there has been a steady decline in academic rigor in math, science, and other academic subjects. The outcome has been "massive underachievement" [Janine Bempechat], an epidemic of grade inflation, a glorification of "group work" that downplays individual achievement, and an explosion of remedial math classes at community colleges. In short, math achievement has stagnated over the past 30 years. What's more, memorization and practice, which are needed for the mastery of fundamentals of arithmetic and algebra in long-term memory, have fallen out of favor in many classrooms.

[Insert. There is a direct link between knowledge in long-term memory and the child's ability to solve math problems. Mathematicians have pointed out repeatedly that there is an intrinsic fusion between knowing the basics of mathematics in long-term memory (arithmetic, algebra, trig, etc.) and the quantitative reasoning skills needed to solve problems in mathematics. Prior knowledge is essential for problem solving and insight. Indeed, as Dr. Art Markman (Smart Thinking) points out, "Memory is all about connections." In mathematics, connections are vital because one idea builds on another. Everything fits together logically. To free "mental space" for problem-solving, math facts and efficient procedures need to be practiced, so they become automatic (mastery). There are no shortcuts. In spite of this, many US educators think the mastery of math facts, procedures, and skills are not that important.]

The replacement game plan of putting kids of mixed knowledge and skills in the same math classroom (inclusion policy) has not worked well. It does not make sense for a teacher to have several math levels in her elementary classroom. The teacher barely has enough time to plan for one good math lesson a day, much less several different math lessons, plus the reading groups and everything else. Consequently, quality instructional time-on-task at each math level has been limited and leads to underachievement at all levels. A lot of instructional time is wasted.

The idea of homogeneous sections is not perfect, but it is far better than what we have today, which are mixed-level classrooms and an almost impossible task of differentiating instruction in those classrooms. Consequently, many students, especially our best kids, go unchallenged and underachieve when compared to their peers in other nations. In a mixed group of students, while the teacher is working with one small group of students for 15 to 20 minutes (groups rotate), the teacher also has to classroom manage the other students, who are often distracted (talking, off-task behaviors, etc.) because they sit at desks in groups. In reality, students don't learn as much as they could or should and have less relevant time-on-task. FYI: In many of today's classrooms, the emphasis is more on improving group scores on the state math tests than on individual achievement. We are off-target.

[Insert. All children need challenging content, especially in math. For example, the content I introduce to little kids in Teach Kids Algebra (TKA) is difficult before it becomes easy. It is more difficult for some than for others. The "difficult" becomes easier a little at a time through memory, persistence, and practice--not group work. Indeed, success is a function of perseverance and hard work.] 

An attempt to make math classes [sections] more homogeneous is often met with harsh opposition because homogeneous grouping conflicts with [progressive] equality dogma. To "boost low-performing students," content has been weakened by subtracting core rigor.  States have lowered "proficiency" cut scores on NCLB math tests so that more students pass. Consequently, many students have been labeled "proficient" in state NCLB math tests, yet they do not meet the proficiency level in NAEP tests. 

Many kids, especially academically gifted students, go unchallenged in elementary and middle school and underachieve. It is caused not only by a weak math curriculum but also by mixed math classes. This "unthinking pursuit of equality" hurts all kids, explains Jacob Vigdor (Education Next, Winter 2013). The paradigm of subtracting rigor does not move students forward toward algebra in middle school. Indeed, it delays the math development of all students suggests Vigdor. My observation is the same. For example, 3rd-grade students who are not required to memorize multiplication facts for auto recall or practice the standard algorithm (x) for fluency are stalled. They cannot do long division, fractions, pre-algebra, or algebra. 

Subotnik, Olszewski-Kubilius, & Worrell (Scientific American Mind, November/December 2012) point out, "Today researchers, policymakers, and teachers pay little to no attention to high-achieving students ... Many such students spend their days in schools unchallenged--relearning materials they have already mastered." Students who are behind never catch up because that is not the goal. Our lower-skilled kids might get better in mixed classes because the focus is on them (NCLB), but most kids who are above average, especially our best students, according to Jacob Vigdor (Education Next, Winter 2013), are not challenged and underachieve because "instruction is not tailored to their varying needs." Like me, Vigdor wants to reorganize math classes via homogeneous sections because the sectioning works for most kids: weak, average, high.  

Homogeneous grouping across grade levels or by courses is not a new idea; it is not necessarily innovative, but it meets the needs of the vast majority of students substantially better than the mixed-group classes often found in our elementary and middle schools today. The practice of homogeneous grouping across grade levels was banished because it didn't fit the progressive concept of equality. On the other hand, Jacob Vigdor argues that mixed math classes hold all kids back, and he is right. Like me, Vigdor advocates differentiation via homogeneous groups, not "many groups" within the same math classroom, which impedes all kids.  

[Insert. Elementary teachers do not hesitate to place kids into several groups by ability for reading or for math within their individual classrooms, yet many balk at splitting all the kids at a particular grade level into math sections based on student knowledge and skills (homogeneous groups). The idea of establishing homogeneous sections for math, for example, conflicts with a progressive ideology of equality and self-esteem.]

[Insert. Dr. Janine Bempechat (Getting Our Kids Back on Track) says our children grossly underachieve. She asserts, "We need to worry less about self-esteem and more about competence ... We need to expect much more from our children and challenge our children to confront difficulty [and work hard]." Dr. Bempechat points out, "We have become so consumed with worry over our children's self-esteem that we take pains to manufacture it." Moreover, Bempechat insists, "We need to stop protecting children from hard work and sacrifice in the name of happiness and self-esteem." 

Also, Bempechat says that we need to teach children "critical academic and life skills, which are the ability to persist in the face of challenges, to delay gratification, and to endure boredom." It is indeed unfortunate that "many in our society [including elite educators] look down on academic excellence."]

Technology has been cast as the new panacea because, according to ardent supporters, kids can learn at their own pace. Sounds great, but it has never worked. For example, Individually Prescribed Instruction (IPI) in the early 70s was a total flop! Often in education, grand ideas that failed in the past are repackaged and pushed onto schools as innovative and transformative. They are not. Furthermore, adaptive software, such as Success Maker, is no match for a high-quality teacher, no matter the grade level. The equality dogmatists would have you think that the solution to our math woes is kids sitting at a computer learning math at their rates, which would be the ultimate in differentiation. Sounds great! But, it has not worked in the past. Unfortunately, some good ideas were banished by the "equality" dogmatists. One good idea was homogeneous grouping (weak, average, advanced).

I think the sorting of students should begin in early elementary school. For example, in Singapore, weak math students are pulled out for math class at the beginning of 1st grade and placed with a high-quality math teacher to catch them up. We don't do this. Moreover, weak math students in Singapore are expected to learn the same core arithmetic that is taught in the regular class. 

Notes1. Jacob L. Vigdor (Education Next, Winter 2013) makes a comparable proposal about grouping students in math, and Doug Lemov (Teach Like a Champion) thinks teachers should put desks in rows, so students face the board to enable attention during explicit teaching. Both ideas conflict with current practice and conventional thinking. Kids don't learn math by group; kids learn math [by inference and counterexample] from teachers who know math and use explicit teaching methods. 

Dr. Eric Hunushek says, "Schools do have a big influence on achievement." But, as Hanushek observes, "[Low income] schools...aren't geared to making sure that these kids [mostly minorities] get really high-quality teaching. They get average teachers, which, on average, doesn't make up for a family background [vocabulary gaps, etc.]." I am retired, but, as a guest teacher. I go to a Title 1 elementary school and teach algebra to little kids (grades 3 to 5), all minorities, to show teachers that many kids can learn content that leads to Algebra 1 in middle school. The explicit teaching of content is often a mismatch to school district policies of desks-in-groups, group work, and collaboration. Kids are novices, not experts. In the real world, experts collaborate, often by email. Frequently, schools do not live in the real world; they live in a fantasy world. If we want kids to become future innovators, then they must first "become an expert" [in a discipline] writes Evangelia G. Chrysikou (Scientific American Mind, July/August 2012). "A solid knowledge base will allow you to connect remote ideas and see their relevance to a problem." She points out, "Working alone is usually the best way to come up with creative solutions."  

[Insert. In Teach Kids Algebra (TKA), I want kids to master content to form a solid knowledge base because the kids who know stuff (facts, procedures, axioms, apps, and ideas) in long-term memory will be successful. Also, TKA is independent of the district's gifted programs; however, it does help identify students who learn math faster, have more insights, and handle complex, in-depth material. To stretch and inspire able students, I formed 4th grade and 5th grade Honors groups that met once a week. It isn't enough time, but it is a start. Moreover, "In academics, so far only in mathematics do we have reliable ways to detect potential talent early on," writes Subotnik, Olszewski-Kubilius & Worrell (Scientific American Mind, November/December 2012).]

[Insert. The content I introduce in elementary school classrooms is mostly algebra (variables, equality, true/false statements, writing and solving equations, x-y table building, graphing, functions, etc.) and pre-algebra stuff (integers, fractions, formulas, etc.). I do not teach specific items on the state NCLB math tests, although, at times, there is some overlap. By blending algebra ideas with arithmetic, students are more likely to make the conceptual leap from the specific to the general.]

Notes2. In the past couple of decades, [NCTM] reform math disciples have substituted "cooperative group work" for explicit teaching, and school districts, under NCLB, have focused instruction and resources on average and below-average students, often leaving the academically gifted, advanced, and even above-average students unchallenged. Reform math has downplayed the auto recall of number facts and the practice of standard procedures for mastery in long-term memory--both stall student achievement. Even students who are below average are frequently not challenged in this system. It is the wrong approach. We know that a lot of practice solidifies essential factual and procedural knowledge in long-term memory for use in problem-solving [prior knowledge is needed for problem-solving and critical thinking in math and science]. Educators need to reevaluate and challenge their assumptions, but they often don't. For example, kids who are not required to memorize multiplication facts in 3rd grade and work with the standard algorithm for fluency cannot do long division, fractions, or algebra. They are stalled. Group work and collaboration are championed in our schools, not individual achievement and academic excellence. We are in a test mode and not in an achievement mode. We are off-target. Under common core, this will not change much. 

The US math curriculum (i.e., content taught in the classroom) and instruction (i.e., methods of teaching math) are not a good model. If our curriculum and methods of instruction were an exemplary model, then most of our kids would score substantially higher on NAEP government tests and be near or at the top internationally rather than in the middle (TIMSS). Incidentally, Singapore teachers do not put kids on computers to learn basic math. I think much of the technology and software used in US classrooms by students is a distraction, not a viable solution.

Also, nearly 50% of the 8th graders in several Asian nations, including Singapore, score at the "advanced" level on international tests compared to only 7% of US 8th graders (TIMSS). This indicates that classroom teachers and cram school teachers in Asian nations not only teach core but way above the core for able students.  

Thinking Out Loud. Even though there has been some improvement, especially among less-skilled minority students, rapid growth, such as that found in many other nations, escapes us. We remain stuck in the mud [of mediocrity]. There are exceptions. For example, Massachusetts came in 6th with a score of 561 in 8th-grade math. South Korea was 1st at 613 on TIMSS. And, while the curriculum (e.g., algebra) has become more accessible to average students to promote equality, many students are not prepared academically to handle it because they are products of a weak elementary and middle school curriculum and inadequate instruction. Students should not take algebra if they are not prepared, yet many schools push kids into algebra, ready or not. It's an epic mistake. 

Jacob Vigdor writes, "America's lagging mathematics performance reflects a basic failure to understand the benefits of adapting the curriculum to meet the varying instructional needs of students." And, the adaptation Vigdor strongly suggests is differentiating via homogeneous math sections, starting early in elementary school, not the "many levels" in the same math class that we have endured for decades. Differentiating via homogeneous math sections is old school, and it works for almost all kids--weak, average, advanced.   

FYI. Regardless of the rhetoric from common core defenders, the hidden intention of the common core is to homogenize math content (equalize downward). In short, common core math is not designed to catch our kids up to international math benchmarks, which is a reason I classify it in the framework of equality/self-esteem [progressive] ideology. In my view, the common core is the latest manifestation of a "once size fits all" progressive dogma. Furthermore, our math textbooks and instructional methods often unduly focus on understanding, which is difficult to measure, at the expense of competency, which is easy to measure. (Peter Hanley, redefinED 12-31-12, writes, "Common Core standards seek to prepare students to achieve 1200 on the SAT.... [Nevertheless], the average score for America's teachers has been about 1000.") Schools of education lack academic standards. I think there are a lot of good teachers out there; we don’t have enough of them. Schools of education have not been graduating high-quality elementary and middle school math teachers.  

[Insert. Math, such as memorizing times tables, isn't much fun until you get good at it, which requires effort, study, and practice. Once you get good at something, you like it better. Parts of math can be hard and frustrating. It is a giant step to go from the specific (using numbers) to the general (using variables). 

In the US, math is often taught badly. We know that learning fractions and long division well in early elementary schools prepare students for algebra in middle school. Researcher Robert Siegler, Carnegie Mellon University, writes, "Early knowledge of fractions and long division predicts long-term math success." ]  

This document is an abridged version of the original (12-25-12) with additions. It is frequently updated, revised, and tweaked almost daily. 1-27-13

The document is not an essay. Please overlook disjoint parts, awkward sentence structure, incorrect grammar, spelling, and many inserts.

Comments may be addressed to ThinkAlgebra@cox.net.
Model Credit (top of page): Remi, 5th grade
Note. In this document, the term "progressive" is used as defined by Alex B. Berezow and Hank Campbell in their book Science Left Behind. 

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