Thursday, December 28, 2017


Our best math students often go unchallenged and fend for themselves in mixed classrooms. To find talented youth requires above-grade-level testing for quantitative and verbal abilities, not grade-level testing.

Note: "Above-grade-level testing identifies students who have the greatest need for advanced-level and fast-paced coursework." (Johns Hopkins Center for Talented Youth) For example, students in grades 2-3 take the Elementary SCAT (School and College Ability Tests) designed for students in grades 4-5.) Many K-8 talented and gifted programs run by school districts are token programs to please parents. Little if any above-grade-level testing is given to elementary students to assess quantitative and verbal abilities.  

Taking Algebra-1 in middle school is for average students who are prepared.  It is not challenging enough for talented youth. Unfortunately, our reform math programs, minimal guidance instructional methods that usually involve group work, and the early use of calculators do not prepare average or above-average students for algebra. H. Wu, a mathematician, points out, "Elementary teachers are generalists and do not know enough math to teach it well. Also, Richard Rusczyk (the Art of Problem Solving) says that AP Calculus is for average high-school students who are prepared. Rusczyk's Introduction to Algebra includes logarithms, complex numbers, and other topics typically found in Algebra-2 and high-level math courses. His 1st-year algebra text has three major chapters on quadratic equations. 

"If you find that the problems are too easy, this means that you should try harder problems. Nobody learns very much by solving problems that are too easy for them." Rusczyk says his text was written for outstanding math students, not average students. Also, it requires excellent arithmetic, prealgebra, and reading skills. A good prealgebra text for average 6th and 7th-grade students contains many topics found in Algebra-1 along with right triangle trig.       

Our K-8 students should do better than average given the amount of money poured into schools. I think U.S. students would be able to compete with the high-achieving Asian nations in math if educators focused more on performance and paid more attention to high-achieving students. Click/Read Focus on Performance.

Note: Don't be misled. If the State stamps your child "proficient" in math, it doesn't mean he is. Be skeptical. 

More is said than done. It is especially true in education. We say we want students to engage in "higher-level" thinking, yet we don't focus on lower-level thinking (i.e., knowing and applying content) that leads to and enables higher-level thinking. Put simply: our actions do not support our goals. We say one thing but do another. 

Unfortunately, the progressive reformers continue to pitch evidence-lacking ideas, theories that don't work (Piaget), and ineffective reform math programs that impede achievement. Special interests, including testing companies (College Board, Common Core, "college readiness" claims, etc.), tech companies like Texas Instruments (calculators), and publishers of textbooks and "learning" software (Pearson, etc.) have infiltrated almost every classroom. Starting in K, children do not need a calculator to learn arithmetic or a graphing calculator to learn algebra. Also, students don't need to watch videos, use social media, or operate tech online (computers, laptops, tablets, smartphones) to learn math well. Over the decades, the latest tech has never been the panacea for our math and science woes.    

Immanuel Kant wrote that thought (e.g., critical thinking, problem-solving, analysis, etc.) without content is empty. To learn something means remembering it from long-term memory such as the single-digit number facts and standard algorithms in arithmetic. Learning requires effort, memorization, drill to develop skill (practice-practice-practice), and review. The primary focus has been on reform-math alternatives, not standard arithmetic or standard algorithms. 

In the self-contained classroom, the focus has been on giving everyone the same. Also, much of the attention is given to low-achieving students with very little attention paid to our high-ability students. Thomas Sowell (Dismantling America) called it a "fallacy of fairness." He points out that "equalizing downward by lowering those at the top" is a "crazy idea taught in schools of education." Our best students should receive accelerated instruction, but this is rare in K-8. Sowell explains, "Fairness as the equal treatment does not produce fairness as equal outcomes." Our best students should be taught a different curriculum in math and other disciplines. They are not. 

Moreover, teachers are often required to teach "items on the test" often using inferior methods of instruction (minimal guidance methods of teaching); consequently, many students never master arithmetic. In effect, the math curriculum is fragmented and below world-class benchmarksThe disparity begins in the 1st grade. Moreover, in many school districts, online practice tests take up an enormous amount of time. Often, the practice tests contain questions on topics not yet taught. Thus, it is not possible to track a student's progress because the test keeps changing. In short, the test does not match the curriculum teachers are told to teach. In December, several 7th-grade students said to me that it took three hours to do a writing test. Why? It is testing run amok. Kids are tested to death. 

In contrast to the American method of developing many different alternative algorithms and strategies (aka reform math), the 3rd-grade focus in Russia is on learning the standard multiplication and division algorithms. Dr. R. James Milgram (Eminent Mathematician, Stanford U.) wrote that "the [3rd Grade] material starts with multiplying multi-digit numbers by one-digit and immediately continues with dividing by one-digit numbers." 

According to the UCSMP translation, the Russian content in 4th grade features 1,500-word problems--many are multistep and demonstrate a "remarkable level of abstract reasoning expected of students." The Russian 4th graders are at least two years ahead in arithmetic. Some of the algebra in the 4th-grade text is beyond the content I present to 4th graders once a week in my Teach Kids Algebra Program, such as solving inequalities and complex equations. American students do not know the arithmetic (e.g., long division or the distributive property) needed to solve the equations below. Compared to students in some other nations, most American students are weak in basic arithmetic.

Russian 4th Grade Textbook 1980 
Solve the equations: 
1) 69k = 14076;
2) b×74=22348;
3) (7001 + x) × 42 = 441000; 
4) (8001+y)×32=656000. 

Milgram stated in a 2016 interview that the current American system is dysfunctional! "It's hard to imagine any system that works more poorly." He pointed out, "You have certain key topics that you have to carefully teach all the way to real mastery in these early grades. These key subjects include fractions and above all ratios, rates, percentages, and proportions."  They aren't taught well. Indeed, the whole number standard operations are not explained well or mastered either.
Read Bad Math Education 

The Asian system is built on memorization, which forces students to store information in long-term memory where it is ready for use to solve problems (Vohra). American educators don't get it, that is, students need to practice for mastery. Asian children are taught mechanics of operations first with the explanation later, and it works! We do it backward with understanding first, and it doesn't work. Simply, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory. (Note: Asian students have a lot of intrinsic motivation. They try harder in school and learn more.)

There will always be gaps! In real education, you increase differences (Richard Feynman). We have been "equalizing down" to close gaps in the name of fairness (Thomas Sowell) and feeding all students the same math curriculum starting in the 1st grade ("one size fits all" Common Core and state standards). These are counterproductive approaches. The reformers and professors of education say that teachers should differentiate instruction in a mixed ability classroom, but it never works well. Chester Finn, Jr. and Brandon Wright write in EducationNext, "Rare is the teacher who can do right by her ablest pupils at the same time she provides slower learners in her classroom the attention that they need." Also, misinformed reformers and others argue that kids need less mastery of traditional arithmetic and its standard algorithms because they can use calculators. WRONG! A weak math student with a calculator is still a weak math student! 

The only way to improve math achievement is to spend more time on task. The best intervention is one-to-one tutoring (1:1), not a small group such as 1:3 or 1:5. 


"Education should continually be upgraded not continually reinvented and reformed."  The idea that math has changed is nonsense. Still, reformers who have attempted to change basic arithmetic by diminishing traditional arithmetic and its standard algorithms. Reformers seem to ignore valid research evidence. Instead, they cite observational and anecdotal examples. It's not evidence.  

Mathematician Ian Stewart (Significant Figures, 2017) wrote that "math endures." Solving quadratic equations has been around since 2000 BC. The results are not obsolete. "It is still correct today." Indeed,  2 + 5 = 7, and that won't change either. The factual and procedural knowledge of basic arithmetic and algebra has not changed either, but, sadly, it has been distorted, twisted, and reinvented by reform math people. Indeed, reform math has been the dominant force behind stagnation.    

Our education culture [i.e., the status quo] is hard to change. Edward Luce (Time To Start Thinking, 2012, p 75-76) writes that in our schools, sports seem more important than STEM and self-esteem trumps academics. There are exceptions, of course. Also, the bureaucracy in our education culture is incredibly thick, and its lack of flexibility and slow response often ensure "mediocre outcomes" and "stagnation." 

Laurie Rogers (Betrayed, 2010, p. 12) writes, "In reform math, children don't practice skills to mastery." Yet, as Clifford Stoll (High-Tech Heretic, p. 82) explains, memorization and practice are necessary for "competency" in arithmetic and algebra. Indeed, practice and repetition are essential for learning anything well. Furthermore, Daniel Willingham, a cognitive scientist, reminds us that background knowledge, both factual and procedural, in long-term memory frees up working-memory space for problem-solving in mathematics. That is to say, prior knowledge feeds problem-solving.

Don't think the rote learners in Asian nations such as China cannot innovate, says Bill Gates (Source: Thomas L. Friedman, The World Is Flat 2.0, 2006, p. 351). They can, and they are getting better at it because innovation grows out of knowledge. Edward Luce (p. 100) writes, "In 2010, the Chinese built the world's fastest computer." He also points out that our "lead in innovations can no longer be taken for granted."

Knowledge drives the innovative process. Harry Shum (Head of Microsoft Research Asia), "Once you have this foundation [knowledge], being creative can be trainable. China is building that foundation" through rigorous math and science education (Friedman quotes Harry Shum, The World Is Flat V2.0, p. 355). 

We need to do the same, but, unfortunately, the trajectory of Common Core math or state standards, starting in grade 1, is not steep enough to get our kids up to peers in top-performing nations, much less STEM. Moreover, many of our K-8 teachers need to learn more mathematics and use more effective instructional methods. Most teachers, especially those who are weak in mathematics, cannot design lessons from their knowledge of mathematics. Instead, they often rely on the lesson descriptions (scripts) in the Teacher's Manual, along with minimal guidance methods taught in ed school. 

The textbook becomes the curriculum. But math topics and lessons are often poorly sequenced or inconsistent. They often lack sufficient depth or explanation. The lack of focus and coherency are only two of the problems with today's textbooks, according to Beverlee Jobrack (Tyranny of the Textbook, 2012). The textbooks are too fat. Too many topics are tossed in to satisfy the NCTM, special interests, and state standards. To me, it makes no sense for a 6th-grade math textbook to be over 800 pages long or for a 3rd grader to take home a 500-page textbook. Jobrack says that there are so many topics at each grade level, which makes it impossible to teach each one well in a school year. Thus, kids get bits and pieces. She says that publishers must produce better instructional materials to guide teachers. It is crucial to get sequence right and to make sure prerequisite skills are in the right order because, in mathematics, one idea builds on another and things fit together logically. Jobrack also reminds us that "learning is work" and that we can "build engagement through student achievement." She observes insightfully, "Education should continually be upgraded not continually reinvented and reformed."

Often, teachers teach math like they teach social studies--bits and pieces--and focus more on pedagogy, such as working in groups, than on content. This doesn't work in math. Put simply, teachers seldom teach math for mastery. 6-16-12 

The National Mathematics Advisory Panel (2008) made it clear that kids need to memorize math facts for auto recall. Teachers should also stress high levels of proficiency in whole number operations, including long division, all fraction operations, along with parts of algebra, measurement, and geometry. In short, kids must master traditional arithmetic and standard algorithms. We need teachers who teach content and who use explicit instructional methods to prepare most kids for algebra by middle school.

Dr. H. Wu (Mathematician, UC-Berkeley) explains, "Computational facility on the numerical level is a prerequisite for facility on the symbolic level." In short, students who know arithmetic number facts and procedures to automaticity (numerical level) do well in algebra (symbolic level). Almost everything I ask kids to do in algebra requires calculations in arithmetic, including fractions and integers. My approach has been to fuse algebra to arithmetic because algebra grows out of arithmetic. No calculators are used. The foundation for algebra is competency in arithmetic, which includes fractions, decimals, percentages, and proportions. Algebra is a process, but it is more than a method; it is a way of thinking. Concepts fit together logically. One idea builds on another idea.

©2018 LT/ThinkAlgebra

Sunday, December 17, 2017

Focus on Performance

Introduction & Perspective

Kids are novices, but we often treat them as little mathematicians, which they are not. They should focus on performance. 

It's not that there are no other ways to approach a problem in math, but novices need to learn one way that always works to move them forward, starting with standard algorithms and memorizing single-digit math facts, some formulas, axioms, etc. At first, the "why" or proof is not always important, while a practical understanding of the "how" gets students moving in the right direction.

Ian Stewart explains, "One of the biggest differences between school math and university math is proof. At school, we learn how to solve equations or find the area of a triangle; at the university, we learn why those methods work and prove that they do."  

Also, the reason students should learn higher-level math is that our understanding of the universe is written in differential equations (calculus). Pulitzer Prize-winning novelist Herman Wouk in The Language GOD Talks recalls his various conversations with Nobel physicist Richard Feynman. Feynman asked novelist Wouk, "Do you know calculus?" I admitted that I didn't. "You had better learn it," he said. "It's the language God talks." Wouk writes that both he and Feynman were mavericks. "Just as I did not know calculus, so Feynman had no knowledge of fiction." Wouk writes that his conversations with Feynman were insightful. When I talk, I learn nothing, but when I listen (to Feynman talk), I learn something extraordinary." The prerequisites for the study of calculus begin with mastering basic arithmetic and algebra with trig.

(Trends: Approximately half the Calculus 101 college instructors do not allow calculators on exams. Many universities no longer accept AP calculus as credit toward a STEM major because AP calculus depends too much on calculators and skips important topics, including proofs. One parent remarked that AP courses were worthless for STEM kids. Her daughter had to take the university's calculus courses because AP was inferior compared to the university course. Indeed, AP (College Board) is a special-interest ruse just like TI calculators, which are not essential for learning arithmetic or algebra well.)

Memorizing the multiplication facts is not always fun,
but it is a necessity for performing math well. The standard algorithms for addition, subtraction, multiplication, and division should be learned no later than 3rd grade.

Students need to practice-practice-practice to get good at math. Without instant recall of multiplication facts, the student cannot do multiplication, long division, fractions, decimals, percentages, ratio/proportions, algebra, geometry, etc. In short, the student cannot move forward.

Focus On Performance
Going Old School to teach basic arithmetic for mastery is forward- thinking because it stresses performance and competency. The ideas of addition and multiplication are not difficult to understand when explained on a number line. The barrier to adequate achievement has been the lack of practice to automate essential factual and efficient procedural knowledge in long-term memory. In short, the arithmetic fundamentals are not taught for mastery.

Instead of focusing so much on understanding, which is difficult to measure and prone to many different interpretations, we should be much more worried about lackluster performance in arithmetic and algebra fundamentals, as measured by both national and international tests. (In reform math, many different alternative strategies are taught. They confuse students, clutter the curriculum, and create cognitive load.) We can measure and evaluate performing math, but we cannot do that with an ambiguous verb "to understand." Moreover, we should stress performing math well beginning in the 1st grade through memorizing single-digit number facts and practicing the standard whole number algorithms.

We should be better than average, given the amount of money poured into schools. Our kids could compete with their peers from high-achieving Asian nations if we focused on performance. We also need to sort students according to math achievement, upgrade teaching, and eliminate test scores as the focus of teaching. 

Note: Children are not asked to memorize without understanding. Asking students to memorize (automate) 7 + 5 = 12 is not without some level of  "understanding" of numbers, addition, magnitude, and place value, that 12 is 1ten+2ones), etc. The number line shows that 7 + 5 is 12. No other explanation is required. The single-digit number facts need to be automated in long-term memory, which involves drill-to-develop-skill. Memorizing factual knowledge and practicing standard algorithms are not obsolete. They are essential. 

7 + 5 = 12
Understanding math requires factual and procedural knowledge in long-term memory. Performing math measures it. "You don't know anything until practiced," says, Richard Feynman. Unfortunately, according to national and international tests, most American kids lack competency in basic arithmetic and algebra. But, it is our fault for not teaching the basics to mastery. It boils down to a curriculum that is not world-class and minimal guidance instructional methods (group work) that leads to minimal learning. (Minimal Guidance = Minimal Learning.) There is no magic pill. 

A performance-learning objective "describes the specific act students should be able to perform if they have successfully completed a particular learning experience," writes, Vincent O'Keeffe. Verbs such as understand, know, be aware of, comprehend, appreciate, and others are vague and not easily measured. For example, the verb "to understand" should be avoided because it is vague and open to many interpretations. 

Performance is Understanding.
If you cannot do addition, then you do not understand addition. Performing math well is what novices need. Understanding is a vague idea, open to interpretation, and difficult to measure, but applying as a specific performance is measurable. In short, think performance.

Understanding is hidden in the doing. 
Understanding is in the "doing" or performance of math to solve problems. G. Poyla (How to Solve It) pointed out, "Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics [i.e., performing math]. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." 

Emphasizing the performing of math first with explanations later is an essential leap for changing the lackluster math achievement of American school children.  

Specific Performance is MeasurableThe learner will be able to do (something) that is measurable. Either the student can perform long-division correctly, solve fraction problems, calculate the area of a triangle, demonstrate a percentage problem, solve a proportion problem, or write a linear equation given two points on the line, and so on, or the student can't. Progress is measurable. Note the action verbs: perform, solve, calculate, demonstrate, write. 

Knowing is the foundation for applying. 
Young students are novices and are a lot like engineers in that they learn to apply and execute the right procedures (i.e., algorithms) to solve a problem.  It requires extensive factual and procedural knowledge, pattern recognition, and experience (practice) solving problems. Moreover, novice students must be able to do the standard procedures (algorithms) quickly and correctly, so they should be practiced for mastery. Calculating is vital to solving problems in math. 

Unfortunately, most school math programs are hung up on "understanding," which has been one of the hallmarks of reform math and opened to many interpretations. Also, over the years, reform math shifted from the standard algorithms to many different, alternative "strategies." For example, instead of teaching the mechanics of the standard algorithm for multiplication first with the explanation later, students are presented 5 or 6 multiplication strategies that clutter the curriculum and diminish working memory space needed for problem-solving and learning. 

Reform Math Multiplication Strategies
Cluttering The Curriculum & Increasing The Cognitive Load
Repeated Addition
Partial Products
Make A Drawing
Write a paragraph
Use a Calculator

Indeed, mastery (i.e., performance or competency) of essential arithmetic has not been the primary goal of reform math. Memorization and repetition for mastery are sidelined as obsolete and poor teaching by the reform math zealots. What is necessary, they say, is not memorization but to think critically and deeply. Really? The stumbling block is that it is not possible to think critically and deeply about math (i.e., problem-solving) without sufficient knowledge in long-term memory. You cannot work a trig problem without knowing some trig. Also, so-called "fairness" policies, using technology, and other innovations and policies have not leapfrogged math achievement. The reform math mindset of the 21st century must change. 

Note: Engineers are not mathematicians. They do not prove the algorithms or equations they apply. They know they work. Proof (i.e., showing why something works) is what mathematicians do. Children are novices, not little mathematicians. In other words, first-grade students do not need to prove or show with the different strategies of reform math (e.g., drawings, dots, etc.) that 2 + 3 = 5 or that the standard algorithms always work. Novices need to know "how" to do and apply the math, not "why" it works. We should refocus on performing math that high-achieving Asian nations have done for decades. 

Ian Stewart explains, "One of the biggest differences between school math and university math is proof. At school, we learn how to solve equations or find the area of a triangle; at the university, we learn why those methods work and prove that they do." Thus, understanding for school children can be defined as knowing when to apply the right algorithm and be able to do it quickly to get the correct answer. In short, it is performing mathematics. 

Additional information

  • According to Mark Seidenberg, the U.S. culture of education has produced "chronic underachievement" in both math and reading. 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 well) while U.S. educators stress higher-level thinking. Our approach is backward. We should first highlight lower-level thinking skills (i.e., knowing and applying) to build a strong foundation for higher-level thinking skills. 
  • Mathematician Richard Askey in American Educator points out that student understanding is a function of teacher understanding. Mathematician H. Wu acknowledges that most elementary teachers are trained as generalists and don't know enough math to teach Common Core (i.e., state standards) well. Furthermore, the state standards are not world-class, so our kids start behind beginning in 1st grade and stay behind through the grades.  
  • Mathematician W. Stephen Wilson points out that calculators are "absolutely unnecessary." He writes, "The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them.” 
  • The idea that elementary students must know the "why" of everything rather than the "how" is nonsense. Students should practice standard procedures until they are automatic, which is what kids in other nations do, especially the East Asian nations that trounce American students in factual and procedural knowledge and creative problem solving on international tests (TIMSS, PISA). Memorization and repetition are keys to learning because learning is remembering from long-term memory. 
Note: Discipline reform has caused a school-climate catastrophe, according to researchers Steinberg & Lance in Education Next

This post is a work in progress. Expect frequent changes. 

©2018 LT/ThinkAlgebra

Monday, December 11, 2017

Children Are Novices

I am a novice, not an expert.

Children are novices not experts. Their academic learning is rule-based at first, which is the way arithmetic should be taught but often isn't. Breznitz & Hemingway (Maximum Brainpower)  write, "We are quite good at rule-based thinking, [which] has led to the development of fields such as mathematics, geometry, physics, and, of course, computer science." 

Children "master a skill initially by following a set of rules." Learning the standard algorithms is rule-based and mechanical.  Proficiency in arithmetic requires a place-value system, the automation of single-digit number facts, knowing the behavior of numbers (axioms), and applying factual and procedural knowledge to solve a problem. Being proficient in math does not make you an expert--far from it. Kids don't think like adults because they have not had a lifetime of experience to supplement rule-thinking. Real experts cannot explain what they do. 

Ordinary kids can learn arithmetic if they learn a place value system, the standard algorithms, the single-digit number facts, and practice for mastery. Mathematician Steven Strogatz (The Joy of x), writes, "Any calculation involving a pair of numbers, no matter how big, can be performed by applying the same sets of facts, over and over again, recursively. It sounds mechanical, and that's the point." It is mechanical. Arithmetic is rule-based. 

R. Barker Bausell (Too Simple To Fail) writes, "Children who are given more instruction learn more than those who are given less. Too much time is squandered in the classroom. Time on task is essential, but too many educators do not maximize time in academic learning. In short, teachers should use efficient instructional methods, but many do not. 

We have state standards, mostly Common Core, but in math, for example, the standards are not broken down to a hierarchy of learning objectives that are specific, discrete, and measurable (Robert Mager: Preparing Instructional Objectives). Moreover, teachers often use time-consuming, minimal guidance methods such as hands-on or discovery. They are inefficient compared to explicit teaching.

Bausell explains, "Using discovery learning, in which children are guided to uncover principles that took some of our best minds centuries to come up with, is also contraindicated (and borders upon the ridiculous.) It would make a lot more sense to give students the principles they need to begin with, then teach them how those principles are applied."

Note: Do not confuse cleverness with giftedness or expertise.

© 2017 LT/ThinkAlgebra

Tuesday, November 28, 2017

Bad Math Education

Parents, educators, and citizens don't realize how ineptly math has been taught in our K-12 public schools, even highly rated schools, compared to schools in high achieving countries. The high school graduation rate of high achieving nations is 90%, and half of those students have had calculus reports, Dr. R. James Milgram, a researcher and mathematician at Stanford. The math taught in our K-12 public schools is inferior. It is not world-class. Milgram says that "our current system is dysfunctional." We don't have the teachers, the textbooks, the programs, or the resolve to achieve such a high level in mathematics. Will we ever get to the point at which 1/3 to 1/2 of our students can be successful in a real college-level calculus course in high school (not AP)? 

Note: For years, U.S. high schools have inflated graduation rates via bogus credit recovery, grade inflation, and substandard courses. 

Richard Rusczyk (the Art of Problem Solving) says that there is no reason we can't. He explains that calculus is for average high school students who are prepared. The conundrum is that our students are poorly prepared even in 1st grade. Students are novices, not little mathematicians. They need to learn content that is world-class to support problem-solving, but they don't under reform math.

(Note: Singapore 1st-grade students learn much more key content than American 1st-grade students. For example, Singapore students memorize addition facts, write equations from word problems in three operations (+ - x), drill to develop skill, learn formal algorithms, practice multiplication as repeated addition, and much more.) We don't do any of these in most 1st-grade classrooms. 

The major textbook companies such as Pearson dictate the math curriculum, which is reform math. Instead of standard or traditional arithmetic and its standard algorithms, students are introduced to a hodgepodge of inefficient, alternative algorithms (aka reform math). Rather than teaching content for mastery (i.e., competency), the grade 3-8 teachers are told to teach to "items on the state test," which is a fragmented curriculum. Professor Milgram stated in a 2016 interview that the reform math textbooks, programs, and methods are "a total waste of time for your average, above average, and accelerated students. Just a complete waste." After reading parts of a 1st-grade enVision textbook and other textbooks from Pearson, I think he is right. 

Most of the math class time is misdirected into group work, discovery/inquiry or other minimal guidance methods. The content is lean. Kids are encouraged to use calculators. Also, little time is given for practice, review, and feedback. Students do not memorize or drill-to-develop-skill because the mastery of fundamentals in long-term memory is not the primary goal of reform math. Consequently, in the real world, 54% of Singapore 8th-grade students score at the Advanced Level compare to only 10% of U. S. 8th-grade students (TIMSS). The great majority of students who want to go to community college will likely end up in remedial math because they have not mastered basic arithmetic and algebra. (Note: This has been the case for at least a decade or two, probably longer. Sufficient content is lacking in many so-called college-prep algebra courses in high school.)

If "learning is remembering" from long-term memory, then as Zig Engelmann points out, "You learn only through mastery" (i.e., practice-practice-practice). And, he is right! While other nations focus on mastery of fundamentals, many American educators complain that the content is developmentally inappropriate. Why is the content inappropriate here and not in the high achieving countries? The U. S. followed Piaget, even though much of his developmental theory had been refuted. Many other nations, including East Asians, did not follow Piaget. 

Note:  R. Barker BausellToo Simple To Fail, wrote that the work of Jean Piaget would ultimately wind up having no recognizable application to classroom instruction. Unfortunately, many teachers still hold to Piaget's claims that children grow into math and abstraction. The reason young children don't know much math isn't a matter of age or development but a matter of not being exposed to it (National Math Panel 2008).

The crux is that under reform math, which dominates American classrooms, "children do not practice math skills to mastery" (Laurie Rogers, Betrayed). Simply, reform math with its different strategies (i.e., inefficient alternative algorithms) does not work. Also, children might enjoy discovery activities, group work, and other minimal guidance methods, which are time-consuming, but they aren't learning enough math. Skills should come first, but not in reform math. 

In contrast to American elementary schools, students in other nations such as Russia learn the standard algorithms for multiplication (e.g., 4987 x 6) and long division (e.g., 4987 ÷ 8) no later than the 3rd grade through practice-practice-practice.  In Singapore, multiplication starts in the 1st grade, half of the multiplication table is memorized in the 2nd grade, and the rest in 3rd grade. Unfortunately, we have a barrage of math educators, teachers, professors of education, administrators, reformers, and so-called experts who denigrate standard arithmetic and want to abolish algebra as a requirement for college. 

Parents don't seem concerned that their kids are 2 or 3 years behind in learning math content and problem-solving. The bottom line is that many students do not master basic arithmetic or algebra. Calculators disrupt mastery and camouflage weak math students. Parents say that education is a priority, but it isn't in practice. They gladly put out money for the latest gadgets, video games, smartphones, kids' sports programs, lessons, TV service, and so on but seldom for Kumon math lessons or a private math tutor. 

Peg Tyre (The Good School) writes that (in the 60s) Singapore rejected Piaget's notion of kids growing into math and abstraction, but American educators eagerly adopted Piaget's progressive theory, which was a colossal mistake. In contrast to Piaget's notions, East Asian countries and other nations embraced the views of Jerome Bruner "who argued that kids are capable of learning nearly any material so long as it is organized, sequenced, and represented in a way they can understand." (Note: Bruner's quote is from Tyre's book.)

Moreover, the National Math Advisory Panel (2008) rejected the claims of Piaget. Kids do not grow into abstract thinking. The reason our "children often don't know math at an early age is not that the content is developmentally inappropriate but that they haven't been exposed to it." 

In short, U.S. kids are not taught math they should learn. They underachieve compared to their peers in some other nations. Many primary teachers de-emphasize traditional arithmetic and its standard algorithms and, instead, teach reform math. The elementary teachers, themselves, are weak in arithmetic and algebra. Also, teachers try to make math fun, but learning math is hard work. Students need to memorize and drill to develop skill. American educators and parents need to wake up about what it takes to improve math performance.  

©2017 LT/ThinkAlgebra

Thursday, November 16, 2017

Thoughts on First Grade Math

Random Thoughts on First Grade Arithmetic

Number lines are seldom found in primary school math materials. One exception (?) is the 11" by 16.5", 645-page 1st grade enVision Math textbook from Pearson 2011, but the number line is hardly used--just one lesson: "You can use a number line to find missing numbers: 2 _ 4 with "before" and "after" as vocabulary. A ten-frame is used extensively to model numbers and figure out single-digit number facts, not a number line. I think this is a mistake. Also, memorization and drill are not part of the enVision reform math program. Why not? 

11" by 16.5" enVision Math 1st-Grade textbook. It is 645 pages.

The reform math idea is that students should calculate the single-digit number facts using a variety of methods, not memorize them, which is a shift from the Old School ideas of memorization and drill to develop skill in arithmetic. Frankly, the ten-frame method and counters are inferior to a number line, which shows basic arithmetic. In my self-contained 1st-grade class (the early 80s), I stopped using counters (manipulatives) after the first couple of weeks of school. The number line is essential mathematics, not the ten-frame, counters, or calculators. The 0-20 number line is an extension of the number line kids come to school with. 

Number lines show key math concepts. Students learn magnitudes, number relationships, patterns, whole number operations, fractions, decimals, etc. The "one more" idea shows children how numbers are built: add one

First Grade Number, Line 0 to 20: Equal Units. 
I started with a 0-20 number line, then added a -10 to 10 number line. 
Equal Units
First-grade students come to school with a built-in logarithmic number line (unequal units), but in arithmetic, the number scale is linear (equal unitsand needs to be taught that way. The distance between 4 and 5 is ONE unit or just plain 1. The distance between 10 and 11 is ONE unit, etc. Students can see this on a number line. To get the next number (a'), add 1 to a: a + 1 = a'. (FYI: a' is read a-prime, which is the next number) Thus, 2 + 1 = 3,  4 + 1 = 5, 18 + 1 = 19, 56 + 1 = 57, etc. Linear number lines should be used from day-one in teaching arithmetic to first graders, but I seldom see them in primary math textbooks. Counting starts at 1, but number lines and rulers start at 0.  Magnitude is an important idea, too. Students can see that 4 is greater than 3 (4 > 3) or that 3 is less than 4 (3 < 4). These are inequalities, and the symbol always points to the smaller of the two numbers when they are compared. When comparing numbers m and n, only one of these is true: m = n, m > n, or m < n. 

Okay, how do we get from 4 to 5?
ADD ONE to get the next integer.
The number line shows how (and why). We add 1, which, on the number line, means you move one unit to the right of 4 to get to 5: + 1 = 5, that is, to get 5, add 1 to 4. This is the successor rule for integers: a + 1 = a', in which a is an integer {4} and a' (a-prime) is its successor {5}. The idea that you get the next integer by adding one to the previous integer is very important, but, unfortunately, it is seldom taught this way in 1st grade. The successor rule works for all integers. For example, -6 + 1 = -5 or -1 + 1 = 0. You don't need to call it the "successor" rule. For whole numbers, you can call it the "add 1" rule. Furthermore, what seems obvious to adults, is not always clear to 5 and 6-year-olds. Remember, kids don't think like adults, which is a basic premise of cognitive science. Hence, presentation and explanation are important in teaching young children arithmetic as are memorization, "drill for skill," pattern recognition, place value, magnitude, etc. 

Adding 3 and 4
Once children see the basic relationship of 3, 4 and 7, then they should memorize the fact (3 + 4 = 7) and solve missing addend problems such as 7 = x  + 4. Also, note that subtraction is defined in terms of addition: 7 - 3 = 4 if and only if (iff) 3 + 4 = 7. And, 7 - 4 = 3 iff 4 + 3 = 7. It makes good mathematical sense that addition and subraction should be taught together. 

The concepts in arithmetic are elementary, and the number line makes them accessible to 1st-grade students. Children are novices, not experts or little mathematicians. They need to memorize and practice to learn. Learning something is remembring it from long-term memory.

Also, the standard algorithm is not found in the enVision 1st-grade text. The textbook is too big to take home, so students tear out the lessons. Moreover, there are numerous calculator activities called Going Digital to get kids on calculators. Using calculators starting in kindergarten was an imprudent guideline introduced by the National Council of Teachers of Mathematics (NCTM 1989). You would think the NCTM would know better. 

W. Stephen Wilson, a mathematics professor at Johns Hopkins University, sets the record straight by pointing out that using calculators is "absolutely unnecessary" in arithmetic and algebra. He writes, "The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."

enVision Math - 1st Grade

You see, where reform math programs such as enVision Math are headed, which is to do away with memorization of facts for auto recall and the standard algorithms (Old School). Calculators (tech) are substituted for paper-pencil standard arithmetic. It is already happening. The reform idea is that computations can or should be done on calculators, even easy ones. Therefore, the student can focus on critical thinking or problem-solving. But, the math doesn't work that way. You cannot apply something you don't know well in long-term memory. Students should master the fundamentals, but many don't. 

The content in 1st-grade enVision Math is significantly below the benchmarks of some Asian nations. American kids start behind and stay behind their Asian peers. Often, in reform math, kids are asked to draw something to get an answer or as proof. A drawing is not math. Counting is not a property of numbers. Using a calculator is pressing keys; it's not math.

Arithmetic is mostly addition, subtraction, multiplication, division, fractions-decimals-percentages, and ratio/proportion. In the early grades, parts of arithmetic are unexciting. Also, it is not much fun memorizing the addition and multiplication tables or practicing the standard algorithms. Reform math and minimal guidance constructivist methods seem to dominate K-8 classrooms. Consequently, many students do not master traditional arithmetic or standard algorithms. They are ill-prepared for Algebra. Indeed, basic arithmetic has not been taught well for years. The problem starts in 1st grade and jumps up the grades.  

Kids are novices and need to memorize and drill for developing skills. Learning arithmetic means remembering arithmetic from long-term memory. Learning arithmetic is work, but it is essential because it forms the cognitive architecture of mathematics in your long-term memory and helps students develop number sense. But, later, students will likely use calculators for more complex calculations, especially in chemistry and physics. I used a slide rule. Today, unfortunately, calculators often cover up weak arithmetic skills even in elementary school. 

Liberal reformers (the progressives) tossed out the Old School stuff even if it worked well when it was taught well. Tossing out the good things of the past is hardly a reasoned strategy. 

The doing of mathematics and the understanding of mathematics are rooted in the symbolic language of mathematics (i.e., abstraction). But, teachers don’t know symbolics or properties (rules) of numbers. They do not know that equality is reflexive, symmetric, and transitive (Peano axioms), much less the ideas that subtraction is defined in terms of addition and division is defined in terms of multiplication. In algebra, subtractions are changed to additions, and divisions are changed to multiplications. I am not sure that teachers know that the sum of identical addends is called multiplication. Counting is not a property of numbers.   

Note: The two primary operations in the set of real numbers are addition and multiplication, which make a Field. A number line is all that is needed to explain addition, subtraction, multiplication, and division of whole numbers and regular fractions. 

The idea that elementary students must know the why of everything and show proof rather than the "how" is nonsense. Indeed, students need to practice procedures until they are automatic, which is what kids in many other nations do, including the East Asian countries that trounce American students in factual and procedural knowledge and creative problem solving on international tests (TIMSS, PISA). Memorization and repetition are keys to learning because learning is remembering from long-term memory. 

Asian children are taught mechanics first with the explanation later, and it works! We do it backward with understanding first, and it doesn't work. In short, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory. 

As I had said many times: There is no substitute for knowledge in long-term memory and the practice that gets it there.

The reality is that the more "rote learners" of the East Asian nations have excelled in factual and procedural knowledge and creative problem-solving (TIMSS, PISA), leaving most American students in the dust.

Symbolic Arithmetic
Standard arithmetic is the basis for higher-level mathematics, such as classical algebra. Algebra is symbolic arithmetic. Newton called elementary algebra the universal arithmetic because the calculation of numbers (arithmetic) and of symbols for unknowns (algebra) were the same. The rules that govern the calculations of arithmetic and algebra are called the field axioms. Simply, elementary algebra obeys the rules of arithmetic. A focus should be on learning of field axioms (rules) that govern arithmetic and elementary algebra calculations. It impacts 1st-grade arithmetic and algebra. 

Should we teach algebra concepts in 1st grade? 
YES! 3 + 7 - x = 5 - 2 
(x = 7 to make a true statement. The inverse idea is often used in solving equations. 
Left side = Right side: 3 = 3)

John Stillwell (Elements of Mathematics From Euclid to Godel) said that the point of doing arithmetic is not to do millions of calculations but to learn the axioms that govern them along with efficient calculation methods using single-digit number facts. Also, He said that the algorithms (step-by-step recipes) to calculate numbers should be fast and efficient. For novices, one algorithm per operation is sufficient to start. The operations can be understood using a number line, and when the numbers are larger, the standard algorithms should be used. This idea goes against the "many methods" of reform math that clutter the math curriculum and create cognitive load. Also, children should learn the mechanics first with an explanation later. Learning the mechanics of an algorithm requires practice-practice-practice and auto recall of single-digit number facts.  

In the first week of school, first-grade students should learn at least two field axioms (rules) via the number line: a + 0 = a and a + b = b + a. Also, students should learn an important "common notion" from Euclid: Things that are equal to the same thing are also equal (i.e., the transitive axiom of equality). 

The meaning of the equal sign is important and often overlooked. If 2 + 3 = 5 and 12 - 7 = 5, then it follows that 2 + 3 = 12 - 7 by the transitive axiom (rule) of equality. The left side of the equal sign {5} is equivalent in value to the right side {5}. Simply, students should "Think Like a Balance." Mathematics is built on true statements.

True/False Statements
3 + 4 = 7 TRUE because both sides are 7 (7 = 7)
3 + 4 = 5 + 6 FALSE because 7 ≠ 11

Here are the nine "field axioms." 
Elementary Algebra (i.e., symbolic arithmetic ) obeys the rules of arithmetic.
I have given an example in arithmetic and the grade level of introduction.
1. a + 0 = a 
5 + 0 = 5 (1st, zero property of addition)
2. a ⋅ 0 = 0
5 x 0 = 0 (2nd, zero property of multiplication)
3. a + b = b + a
2 + 3 = 3 + 2 (1st, commutative property of addition)
4. ab = ba
2 x 3 = 3 x 2 (2nd/3rd, commutiative property of multiplicaiton)
5. a + (b + c) = (a + b ) + c 
3 + (5 + 2) = (3 + 5) + 2 (1st, associative property of addition)
6. a(bc) = (ab)c 
4 x (3 x 2) = (4 x 3) x 2 (2nd/3rd, associative property of multipication)
7. a + -a = 0
2 + -2 = 0 (1st/2nd, the addition of opposites is zero; the inverse property of addition. Note 2 + -2 the same as 2 - 2.)
8. a ⋅ a^-1 = 1 (for a ≠ 0)
3 x 1/3 = 1 (2nd/3rd, the product of reciprocals is one; the inverse property of multiplicaiton)
9. a(b +c) = ab + ac 
3 (4 + 7) = 2 x 4 + 3 x 7 (2nd/3rd, distributive property)

Note: Subtraction can be changed to addition, and division can be changed to multiplication. Thus 5 - 3 = 5 + -3, and 6 ÷ 5 = 6 x 1/5, or 6 times the unit fraction 1/5, which is 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5 or 6/5 by repeated addition. Why is this important?  Both Addition and Multiplication are commutative. (Subtraction and Divison are not commutative) You can add or multiply numbers in any order.

Rules in math are essential. 
I often hear negative remarks about rules in math. Math is governed by rules such as the properties of numbers and equality. The rules should be learned in the early grades. Simply, young students must know the rules and be able to apply them.

Last update: 11-19-17, 11-21-17, 11-23-17, 12-25-17

©2017 LT/ThinkAlgebra