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 Measurable: The 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
Lattice
Repeated Addition
Scaling
Array
Area
Partial Products
Make A Drawing
Write a paragraph
Use a Calculator
Distributive

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. 
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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

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

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 Bausell, Too 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