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| US kids stumble over basic arithmetic. |
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 Jacob Vigdor (Education Next, Winter 2013) writes, that in math, rigor has been replaced by practicality and equity (kids get the same), and counter-intuitively, this has turned students off to math.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}
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