Monday, November 19, 2018


Let's face it. We teach math badly, starting in the 1st grade!

In the real world, pouring billions into programs without substantial evidence, because they are such good ideas, does not work. Indeed, reducing class size and upgrading tech in the classroom (both, very expensive), and so on, have not boosted achievement or made America into a math superpower. We have been going in the wrong direction. We are told that our kids are doing well in math when, in fact, compared to international achievement, they are not. Top-performing nations are years ahead. Common Core, state standards, and state testing have not been the answer. Math is taught as a version of NCTM reform math that failed in the past. Furthermore, Common Core and state standards based on Common Core are not benchmarked to international math standards.  

We should make sure that kids have the calculating skills to do the applications and solve problems. If you cannot calculate it, then you don't know it. Calculating skills should come before applications. We give beginners word problems before they know 5 + 7 = 12. We should first require kids to memorize facts like 5 + 7 = 12 for instant recall before asking them to solve word problems that involve the addition and subtraction facts (i.e., standard algorithms). The same is true for long-division and the other operations with whole numbers, fractions, decimals, and percentages.   

NAEP Changes Won't Solve the Problem
In November 2018, the NAEP committee made name changes: NAEP Basic, NAEP Proficient, and NAEP Advanced. The committee stated that the NAEP Proficient achievement level does not mean grade level, even though the tests are given at the  4th-,  8th-, and 12th-grades. The definitions of these levels were altered somewhat to make them more accessible (equitable), but at what cost? Less achievement? The commission now says that NAEP-BASIC is grade level. "See, our kids are doing okay in math! Most kids reach the NAEP-Basic level." I don't buy the committee's argument. Our kids are lousy at math, and everyone seems to know that. But, for decades, American educators and leaders have glossed over the problem and made excuses. The committee, in my mind, has done the same by making the NAEP-Basic grade level. 

Another red flag is in your community and elsewhere. For example, in the Tucson area, from the nine school districts that feed into a local community college, 74 to 88% of the students who had applied at Pima Community College were placed in remedial math (PCC 2014).

U.S. math is going down (across the spectrum), not rising, and the state standards, which were strongly influenced by Common Core and reform math enthusiasts, add to the decline.


1. NAEP (National Assessment of Educational Progress) is The Nation's Report Card. 
2. TIMSS (Trends in International Mathematics and Science Study)

I don't think I am misusing the NAEP results because they indicate major problems with U.S. math instruction. Achievement is not getting better. It is flat. The same is true for international math tests such as TIMSS. 

NAEP 2017 (Nations Report Card): How well we educate children in math can be inferred by examining the Advanced levels of NAEP and TIMSS. For example, only 8% of 4th graders, 10% of 8th graders, and 3% of 12th graders scored at the Advanced level of math in NAEP government tests. 

Latest TIMSS: In TIMSS, only 14% of U.S. 4th graders reached the Advanced Level in math, but Singapore had 50%, Hong-Kong 45%, S. Korea 41%. We are not in the same ball park. The trend continues. Only 10% or U.S. 8th graders reached the Advanced Level of achievement, while 54% of Singapore 8th-grade students reached that level, and so on. The international difference in achievement is stark. 

ACT Math Scores Are at a 20-Year Low.
NAEP fails to mention that the most recent ACT math scores are at a 20-year low and that the most recent NAEP math scores are no better than they were a decade ago. On the other hand, students from Asian nations leave U.S. students in the dust. U.S. state math standards are not world class, and it shows up early when only 8% of 4th graders, 10% of 8th graders, and 3% of 12th graders scored at the Advanced level of math in government tests (NAEP).  

International: Asian countries dominated the 2015 TIMSS Math results. 

U.S. students were not in the same ballpark. The more "rote" East Asian learners, who memorize and drill-to-improve-skill, soared far above U.S. students not only in content knowledge and ability to perform mathematics correctly but also problem-solving at the Advanced levels. WOW!  At the 4th-grade TIMSS level, for example, Singapore's scale score was 618 compared to the U.S. scale score of 539, which, incidentally, is slightly better than Finland's 535. 

Reform Math
Reform math, which focuses on reasoning and multiple strategies often at the expense of learning content knowledge and standard algorithms, permeates math teaching today. Kids are novices, not experts. They need to memorize stuff for mastery. Proficiency on the state test is the goal, not the mastery of essential content.

Kids Should Master Standard Arithmetic

In contrast to reform math, the traditional approach focused on the mastery of content knowledge and basic calculating skills such as the standard algorithms. It worked for the Asian students who are leaps ahead of U.S. students. Common Core and state standards are not benchmarked to international standards. They are not world-class. To do arithmetic well means to know facts and procedures. It requires memorization and practice-practice-practice. Unfortunately, the trend in U.S. education has been to eschew memorization and the practice of standard (traditional) arithmetic fundamentals. 

Kids are not mastering simple arithmetic. 


If you had learned arithmetic in school but can’t remember it, it means you never really learned it in the first place. Learning something is recalling it from long-term memory. It involves mastery, which, in turn, requires practice-practice-practice. 

To apply mathematics to solve problems, first, you need to know the building blocks. You don't start with an application; you start with basic arithmetic--numbers and how they behave and relate to each other, the single-digit number facts, and the standard algorithms.  These are the building blocks of arithmetic, along with patterns (i.e., rules) such as the Commutative rule.

©2018 LT/ThinkAlgebra

Tuesday, November 13, 2018

such good ideas

In education, we have embraced many beliefs, fads, innovations, trends, reforms, and theory without substantial evidence because they are such good ideas. A lot of good ideas turn out to be duds such as Common Core and personalized learning through tech, blended learning, group work, and so on. According to a recent government report, a whopping 82% of the innovations funded by the U.S. Department of Education did not improve student achievement. In short, many of the so-called innovations often hyped in our schools don't work, even the popular ones, such as using technology and software to tailor instruction to each child. 

Many educators believe educational technology can personalize learning. It sounds like another good idea! But, innovations such as using technology and software to tailor instruction to each child have failed to post gains in achievement. The U.S. Deparment of Education keeps funding innovations, but almost all of them fail.

Still, we keep hearing that the way to improve achievement is only through innovation. It's bunk. I think, the best way to improve math education is to look to the past and resurrect the well-established ideas that worked for almost all students, such as memorizing the math facts and learning the standard algorithms as the primary method of calculating. Kids need to "drill to develop math skills." In short, they need to practice-practice-practice because they are novices

Also, students need to learn content knowledge, lots of it, because critical thinking in math (i.e., problem-solving) is empty without specific content in long-term memory. Reform math has left students with weak calculating skills and deficient content knowledge. The ACT Math scores have sunk to a 20-year low. In short, many students don't have strong math skills and the content knowledge that are essential for STEM jobs to power the economy. 

The popular problem-solving approach might work for a lesson now and then provided children have strong, calculating skills and sufficient content knowledge. However, often, this is not the case. If a student cannot calculate it, then they do not know it. We should make sure that kids have the calculating skills to do the applications and solve problems. 

The use of software and tech may not be the best way to teach multiplication.
The teacher can demonstrate this on the chalkboard and ask students questions to clarify that multiplication is a sum. Software is not a substitute for the teacher. Software can't ask follow up questions, explain points, or talk to students.
Multiplication is a 1st-grade level concept, not a third-grade level. Students should use a number line on their desks. Multiplication is a sum and should be taught that way. First graders in Singapore are good at adding, so 4 x 7 means the sum of four sevens: 7 + 7 + 7 + 7= 28

To multiply, Zig Engelmann asked disadvantaged, urban pre-1st-grade students to memorize multiples, such as the 7s: 7, 14, 21, 28, and so on, to do multiplication and find areas of rectangles. In other words, students should be able to do the calculations before an application is introduced.    

Robert Holland writes about Common Core and state standards and quotes Mark Bauerlein: “These standards rely on process-and skills-based ideas [thinking skills], which brush aside knowledge of Western and English literary traditions,” remarked one of the Pioneer analysts, Emory University English professor Mark Bauerlein."

Holland writes, "Instead, the new-age standards offer a heaping serving of multiculturalism, plus the preposterous notion of 21st-century mathematics that renders obsolete principles validated centuries ago. This is how the dumbing-down so beloved by social levelers proceeds." Traditional or standard arithmetic has been bumped by reform math and Common Core, which is one-size-fits-all. Indeed, Common Core and state standards based on Common Core are below world-class math standards.

Teaching kids to fish is a waste of time if there are no fish! 

H. D. Hirsch, Jr. writes, "American educators have been pronouncing ideas about all-purpose critical-thinking skills for more than a century. But we now know from cognitive science that the idea is essentially mistaken. Critical thinking does not exist as an independent skill." It is specific to a domain. 

"The domain specificity of skills is one of the most important scientific findings of our era for teachers and parents to know about." Still, it is widely unknown. Critical thinking in math is called problem-solving. You have to know some trig in long-term memory in order to solve trig problems. In short, the "basis of thinking skills is specific domain knowledge." 

Evidence to the contrary has been ignored by the liberal education establishment. Jill Barshay points out, "Experts have long known that the research evidence doesn’t consistently support the notion that smaller classes increase how much students learn." 

For math, no benefits were found in reduced class size. 

Personal Note. In the early 80s, I had 25 students in a self-contained 1st-grade class at a city Title-1 K-3 school. Except for 2, the students learned to read, write, and do arithmetic at or above grade level. The math was above the level found in the Singapore syllabus in that students worked with negative numbers, fractions, and algebra ideas. A few years ago, Singapore lowered the number of students for 1st grade to 30. But, the problem isn't the number of students. Zig Engelmann would say, "It's the teaching in the classroom." 

Kids aren't learning enough factual and procedural knowledge in long-term memory because reform math instruction doesn't stress "mastery." Kids need to drill to develop skill in arithmetic. Also, reform math, with its many cumbersome, alternative, nonstandard algorithms (strategies) to do straightforward arithmetic, induces cognitive overload. Moreover, E. D. Hirsch, Jr. writes, "Teaching strategies [in both math and reading] instead of knowledge has only yielded an enormous waste of school time." 

Furthermore, in arithmetic and algebra, the algorithms should be efficient, not cumbersome or complicated.  

Andreas Schleicher, director of the education and skills unit at the Organization for Economic Cooperation and Development, has long been arguing that the U.S. overemphasizes small classes at the expense of good teaching. In his 2018 Book, “World Class,” Schleicher debunks the “myth” that “smaller classes always mean better results.”

I think it is okay to have large classes with special tutoring (pull out) for those students who are behind. For example, Singapore places incoming 1st-grade students who have weak number skills with a separate teacher for math class to catch them up within two years. In short, Singapore sorts kids for math class. We don't do that. We mix low-achieving math students with high-achieving math students in K-8 under the pretense of equity and inclusion. U.S. achievement in math has been lackluster, flat, and below expectation. Our policies often hinder achievement.     
Robert Slavin, director of Johns Hopkins’ Center for Research and Reform in Education, told Jill Barshay, “But…no studies of high quality have ever found substantial positive effects of reducing class size.” Despite popularity with parents and teachers, review of research finds small benefits to small classes (by Jill Barshay, The Hechinger Report).

The liberal Education Establishment promotes unconfirmed ideas and fads because they are "such good ideas." No need to test, they say. Scientific evidence of effectiveness isn't required. But, too many allegedly "good ideas" have turned out to be duds. They failed in the classroom and hurt the students. It is called reform! Unfortunately, there are many education practices, beliefs, fads, innovations, reforms, and theory that are not supported by evidence. No wonder the U.S. has major education woes. In fact, progressive reformers say there is no need for proof because the ideas and theories are just common sense and accepted by everyone (e.g., self-esteem, learning styles, small classes). Really?  

Teaching kids to fish is empty if there are no fish! 
The way we teach basic arithmetic and reading has produced lackluster results. East Asian nations focus on performance in math procedures (doing and applying math knowledge well) while the U.S. educators stress higher level thinking strategies, fads, and small classes. Our approach has failed.

E. D. Hirsch, Jr. points out that the education establishment is "captive to bad ideas." Thus, over the decades, "learning has declined." Sadly, the progressive education establishment insists that "students should 'learn how to learn' to develop abstract thinking skills that they could apply fruitfully to various facts, events, and conditions." Sounds great! But, it is contrary to the cognitive science of learning and domain specificity of skills. Indeed, a thinking-skills-orientated approach has radically shaped the curriculum. It is entrenched and hard to change. Hirsch explains that it's the wrong approach because "abstract thinking skills falter without a foundation of content supporting them."  Kids need background knowledge--not an "anti-core knowledge curriculum favored by the educational establishment."  Thinking without content is empty. (E. Kant)

We should first emphasize lower level thinking (i.e., knowing and applying) to build a strong foundation for higher level thinking. Kids need background knowledge.  

According to Mark Seidenberg, the U.S. culture of education has produced "chronic underachievement" in both math and reading.

This post is updated frequently. 11-16-18, 12-6-18, 12-8-18

©2018 LT/ThinkAlgebra