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.)
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.
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.
-----
Additional information
This post is a work in progress. Expect frequent changes.
©2018 LT/ThinkAlgebra
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.
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 |
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.
-----
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.
This post is a work in progress. Expect frequent changes.
©2018 LT/ThinkAlgebra
