This is a work in progress. Please excuse typos and other errors.
Last update: August 17, 2014
Zig berates the distastefulness results from the [Common Core] committee's strange notions of how children learn. "The [Common Core] standards clearly follow the Piagetian myth that children first manipulate then internalize the manipulations, which slowly grows into concrete operations, and later into abstractions or formal operations" (The Dreaded Standards, Zig Engelmann. Zig lambastes the Common Core Mathematical Practices for the early grades).
"It does not work that way, and manipulatives are an instructional nightmare in K [and the early grades]. The products--what children actually learn from manipulation activities--are trivial compared to what could be taught directly in the same amount of time," explains Zig.
Concrete models are often ineffective or detrimental. Symbols-only work better, conclude Mix, Prather, Smith, & Stockton (Mutli-Digit Number Names). Students should not make drawings (visuals) to perform arithmetic.
Every year, state test scores go up slightly, but this should not imply that cognitive abilities (e.g., abstract reasoning) go up too, say MIT neuroscientists. Anne Trafton, MIT News Office, writes, "The researchers found that educational practices designed to raise knowledge and boost test scores do not improve fluid intelligence, such as working memory capacity, speed of information processing, and the ability to solve abstract problems." Our schools, even the best ones (perception), are worse than we think. Furthermore, the highly praised constructivism in reform math and the irrational testing used to compare [judge] states, schools, teachers, and students are all ed talk gobbledygook.
The purpose of learning to do math is to get the right answer. Getting the right answer is just as important being able to figure out the problem types, select an efficient algorithm and calculate the right answer using it. James Shuls, Ph.D. says that "students display deep understanding by getting answers correct." The reformers say getting the right answer is not all that important. Since when?
Constructivism does not work in the classroom!
Kirschner, Sweller, & Clark in 2006 wrote an article for Edcuational Pyschologist called Why Minimal Guidance During Instruction Does Not Work: An analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching. The pedagogy of minimal guidance during instruction comes with different names, but they are all basically constructivism. In math, they are often called reform math. K-S-C write, "The constructivist argument [minimal teacher guidance, such as that in reform math and now Common Core], has attached a significant following; however, the goal of this article is to suggest that, based on our current knowledge of human cognitive architecture, minimally guided instruction is likely to be ineffective." The current research supports direct guidance, which is explicit teaching, something Zig Engelmann emphasized in the 1960s with his symbols-only approach with pre-first-graders.
Traditional, explicit teaching with carefully selected worked-examples, which are expertly explained, is by far the best way to teach children arithmetic. The evidence seems overwhelming. On the other hand, reform math, laced in Piagetian theory (constructivism) and endorsed by ed schools for decades, has screwed up arithmetic instruction. Unfortunately, Common Core follows Piaget's constructivism approach.
The Canadian Study
Evidence that supports Kirschner-Sweller-Clark comes from a Canadian study that started in the early 2000s, when Quebec switched from traditional to constructivist reform math, which was designed after reform math instruction in the US. In the 2014 issue of Economics of Education Review (Haeck, Lefebvre, & Merrigan), the conclusion was that the reform math was a failure in more ways than one.
"The Quebec education program (MELS, 2001, 2003, 2007) relied on a socio-constructivist teaching approach, focused on problem-based and self-directed learning. This approach mainly moved teaching away from the traditional/academic approaches of memorization, repetitions and activity books, to a much more comprehensive approach focused on learning in a contextual setting in which children are expected to find answers for themselves." In short, students are asked to reinvent arithmetic, which is a really dumb idea.
"More specifically, the teaching approach promoted by the Quebec reform is comparable to the reform-oriented teaching approach in the United States. As of 2006, this approach was widely spread across the United States (although more traditional approaches remained dominant) and it was supported by leading organizations such as the National Council of Teachers of Mathematics, the National Research Council, and the American Association for the Advancement of Science."
The researchers (Haeck, Lefebvre, & Merrigan) observed, "We find strong evidence of negative effects of the reform on the development of students’ mathematical abilities." But, there is more to it than this. "We find that the [socio-consructivist] reform had negative effects on students’ scores at all points on the skills distribution and that the effects were larger the longer the exposure to the reform," explain the researchers."
(Note: Excerpts in "quotes" on the Canadian study are found at Kitchen Table Math)
Canada has gone back to direct instruction with excellent results. I think some US teachers see the clear benefits of explicit teaching of arithmetic and disregard the Common Core reform math approach (constructivism), which is inefficient, ineffective, and not backed by evidence. The reinventing of arithmetic in small groups does not work. We have known this for decades, but many teachers continue it. Common Core reform math continues it as well.
1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math), which I call pretend arithmetic, instead of tried and true standard algorithms to do arithmetic.
2. See SingleDigit
Don't calculate single-digit math facts; memorize them.
3. See Memorization & Practice
US schools are a breeding group for mediocrity, not excellence. Early on, parents should provide an environment for achievement and teach their kids how to be successful in school. Early arithmetic at home pays off. Very young children can learn a lot more than we think or are prepared to teach. American children underperform in math!
4. See CommonCore
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education.
To comment, email LarryT: ThinkAlgebra@cox.net
To be continued.
© LT ThinkAlgebra.org
Thursday, August 14, 2014
Wednesday, August 6, 2014
Memorization & Practice
Please excuse typos and errors. Content in this post changes every few days. Last update: 9-17-14
Without memorization and practice to gain knowledge there can be no critical thinking, no bright ideas, no higher level thinking--not in math, science, etc. Everything flows from knowledge. Knowledge is the basis for higher level thinking. Common Core is all about testing, not teaching basic arithmetic.
Kids need achievable objectives, and, because academic ability varies widely, the learning objectives should vary or be flexible from school to school, even from classroom to classroom. This works best at the school building level with individual teachers making the educational decisions, policies, and tests. Yet, top-down Common Core does not allow this. Every student gets the same. One size fits all is idiotic.
A Russian 10-year-old plays Mozart's 12th Piano Concerto with an orchestra. She is better than just good! She has won lots of awards in competitions. In addition to recitals, orchestras invite her to play piano concertos. Varvara doesn't know how good she really is. She is a rare individual in her music ability, nurturing, and passion for practicing. She absolutely loves what she is doing! But, it all started by being in the right environment at the right time. In short, Varvara has been very lucky. Is she an outlier, a prodigy, a genius, or gifted? I don't know. I dislike using these terms to classify children, but she clearly has musical talent way beyond an average child, and that musical ability is being developed. Today, musical ability in very young kids seems to surge, especially in Asian nations and elsewhere. Is it fair that Asian or Russian parents, even poor parents, push their kids into math and music, especially violin and piano, at very early ages, while many American parents think pushing kids is bad parenting? Clearly, there is a clash of cultures. Frankly, I think that many American kids don't study enough, read literature enough, write essays enough, learn math & science enough, or exercise enough. Many students often lack industriousness or do just enough to get by, but that is not the teacher's fault. American parents shell out money for video games, sports, lessons, entertainment, and technology--but not for tutors.
To become better at anything, a student has to memorize and practice and memorize and practice, have passion and persistence to learn, and get expert teaching. The problem in American schools is that most students do not get expert teaching early on; hence, [hidden] abilities in many capable very young children are squandered, never discovered or developed, whether it be in music, art, math, science, literature, writing, etc. Zig Engelmann writes, "The genetic interpretation of giftedness and intelligence is a myth." Yet, it [IQ] seems to govern the education policy in our schools, thanks to the influences of Darwin, Binet, Piaget, Dewey, et al. Even though brain plasticity (that IQ can be increased within limits) has been known for years, yet teachers are told to treat all students the same, that is, in Common Core all students learn the same content. It is the "triumph of mediocrity." The US education system seems to relish mediocrity. Our kids are not dumb. Indeed, better teaching and teacher quality are important, but, according to Will Fitzhugh (Concord Review), "The most important variable in student academic achievement is not teacher quality, but the student's academic work." In short, students should produce high quality academic work, but many capable students do not. Paul Zoch (Doomed to Fail), "Let there be no doubt about it: the United States looks to its teachers and their efforts, but not to its students and their efforts for success in education." It's not the teacher's fault that your kid is lazy. "Your kids are your own fault," writes Larry Winget. The blame should go to the parent, too, not the teacher. Early on, parents should provide the environment for achievement and teach kids how to be successful in school. But, this should not be the message: "If you're not the best, you're a loser." If you're not brilliant, you're worthless (Excellence Sheep, The Miseducation of the American Elite by William Deresiewicz). In short, a child does not have to be perfect or the best to be successful, because success requires a lot of missteps and failures. Failure is part of learning. Typical students can learn arithmetic and algebra with average mathematical ability.
Do all students come to 1st grade with the same background knowledge, the same vocabulary, etc.? So, what do we do in our elementary schools? To paraphrase Horace Secrist, a professor of statistics, "Pupils of different mentality, and of training, grouped together in a a single room were to be education, which resulted in inefficiency and regression to mediocrity." When I first started teaching, we tracked students in elementary school and junior high school--to put superior math students with other superior math students with an expert math teacher, etc., but, for decades, this has been taboo in public elementary and middle schools and often called racism. Elementary school classrooms are similar to the old one-room schools. They are the breeding ground for mediocrity. It is called regression to the mean. Mike Petrilli says, "The greatest challenge facing America's schools today isn't the budget crisis, or standardized testing, or teacher quality. It's the enormous variation in the academic level of students coming into any given classroom." Before it was condemned in the US [as racism], tracking actually worked well for almost all students. It isn't a perfect system, but it is far superior than what we have today (inclusion and differentiated instruction). "Academic ability varies widely," observes Charles Murray (Real Education). [Note. For math class, Singapore pulls out weak math students at the beginning of 1st grade and places them with an expert math teacher to catch them up. The program lasts through 2nd grade for kids who need it.] Indeed tracking, when done right, is an effective strategy to reduce variation in the classroom, which means that students [by achievement] go to different teachers for math and reading starting in 1st grade.
Do most infants have the potential necessary to become gifted? Does intensive early training and education work? Should we push children? Varvara started piano at age 4. She was "lucky" to grow up in an environment that developed her outstanding musical ability. Luck plays a much larger role in how much children achieve. Even so, there are no short cuts. Real achievement requires a lot of memorization, practice, good teaching, and luck.
For example, Varvara has a passion for Mozart because her teacher has a passion for Mozart. She understands Mozart because her teacher understands Mozart. She grew up in an environment that fostered, nurtured, and developed her abilities. She was lucky! Varvara gets many standing ovations, and when she plays, hundreds and hundreds and hundreds of notes flow effortlessly, without thinking, from her long-term memory to her fingers. It is a remarkable process. She has mastered not only the piano parts and how to play them, but she also knows the orchestra parts. I saw a performance, and she was having fun playing Mozart! She started piano at age 4. She has developed music ability far beyond that of most kids, but without early intensive practice and expert teaching (nurture), she would be average, if that. And, without the right environment, she may have never started piano in the first place. I want to make clear that practice improves performance, but it does not create talent. There has to be something there to begin with (nature).
Here is my point. Start Young! Very young children are good at memorizing stuff. But, in my view, American educators don't take advantage of this, not in math. Young children are novices; they are not experts; and they are not little mathematicians, but most average children can master arithmetic and algebra when taught and practiced well.
The world was lucky that Newton decided not to become a farmer like his mother wanted.
Most likely your child will not become an expert [genius] in mathematics, such as Newton, Leibniz, Euclid, Gauss, Descartes, etc., or go on to take calculus; however, your child, at the least, can become competent in arithmetic and algebra, even higher, with practice and expert teaching in the right environment. Some American children [we need more] will advance to calculus and beyond and even to STEM careers. We have a shortage and have been importing STEM talent for decades. The K-12 American education system has not been developing the future talent we need in science, mathematics and other STEM fields. Our system produces mediocrity at a time we need excellence. Everyone in education talks about excellence, but the progressive reforms haven't worked. Gee, there must be something wrong with the reforms!
But lurking in some children could be another Newton, an old soul. Newton learned Latin but not mathematics at the King's School until age 17, when his mother decided to make Isaac into a farmer, but he hated farming. So later, he returned to King's School to complete his education, then went to Trinity College in Cambridge, where he started as a subsizar--a student who did chores on campus to pay for his education, room and board, etc. Thank goodness Newton was no good at farming and that the headmaster at King's School convinced his mother to send Isaac back school.
Very young children can learn a lot more than we think or are prepared to teach. American children consistently underperform in math! Kids often say they want to go into STEM, but when they get to college, they don't.
There is no reason why typical kids at very early ages cannot memorize multiples, just as Zig's disadvantaged inner-city pre-first-graders (4 and 5 year olds) did in the 60s, or the addition and subtraction tables for auto recall that Singapore 1st graders master (6 year olds), along with the standard algorithms. But this takes daily practice, oral drills, persistence, etc.! It also takes expert teaching by teachers who know traditional math and are passionate about math and its power, which is the conundrum in America. Most American teachers are weak in both math and science. Some elementary teachers hate math or push it aside. For decades, memorization and practice have fallen out of favor in progressive classrooms. Schools of education push progressive ideas and reforms. This is irresponsible. We don't push kids because "experts" tell us that this is bad for them. Wrong! Parents should drill math facts at home before kids are school age. It is easy to show 4-year-olds addition on a number line. Early on, parents should provide the environment for achievement and teach kids how to be successful in school. Early arithmetic practice at home pays off. Cognitive science has been catching up to anecdotal evidence, which is correct.
Unfortunately, there are idiots in education who say long division is either useless or too hard for kids to learn. But, 3rd graders can easily learn it, iff they have mastered the times tables in long-term memory. [iff means if and only if ⇔]. There are idiots in education who say that memorization of math facts and learning standard algorithms early on are not that important. The progressive reformers are dead wrong! Their policies have led to stagnation in math and reading achievement; impoverishment of public schools, thanks to the costs of Common Core reform and testing; national indebtedness--17 trillion; and a decline in education, rephrasing Dinesh D'Souza (America). The progressive reformers, who see themselves as disruptive innovators, are responsible for the decline and stagnation in US education because of bad ideas.
One unique feature of America has been as a "wealth creator" because of its sound innovations and enterprise, says D'Souza, but our K-12 education system no longer produces enough well-educated and informed citizens. Instead, our education system produces many people who don't work and who are dependent on government. Many college students don't belong in college, and if they pick the wrong major, there is no way they can pay back student loans. College students, even at elite universities [e.g., Harvard], are not taught to think for themselves, says William Deresiewicz. They graduate with "no sense of purpose."
Knowledge is the key to innovation.
It's not that our kids can't do these things--memorize math facts, learn standard algorithms, and apply arithmetic in science to solve problems early on. Our kids are not stupid; they are not taught. Some are lazy! The education establishment doesn't or won't do the right thing. Reform math has screwed up arithmetic for decades; consequently, most kids learn very little substantive arithmetic that prepares them for algebra by middle school. They cannot move forward or excel. Our kids are novices, and novices need to memorize and practice, starting with traditional arithmetic. They also need to be challenged. Our system does not find talent (STEM), and when it is found, it does not support it ($) or offer a different and accelerated curriculum, not in elementary school or middle school. Taking algebra-one in 8th grade is not an accelerated curriculum. Taking AP Calculus is for average kids who are prepared. [AP calculus relies on graphing calculators, has no proofs, and skips important topics, unlike university courses for STEM kids.] US education policies squander talent.
Let me reassure you that there is nothing wrong with classic arithmetic when it is taught properly and explained well, but progressive reformers, such as Common Core reform math crusaders, trivialize it.
The standard algorithms [traditional arithmetic] should be taught first and be the primary methods for calculating. Students must memorize number facts and use standard algorithms at the same time. They should encounter a wide range of routine word problems. First graders should also learn perimeter and area, parts of measurement, geometry, and algebra. I know; I taught it.
1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math) instead of tried and true standard algorithms to do arithmetic.
2. See SingleDigit
Don't calculate single-digit math facts; memorize them.
3. See PiagetianMyth
Constructivism does not work in the classroom.
3. See CommonCore
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education.
©2014 LT ThinkAlgebra.org
Without memorization and practice to gain knowledge there can be no critical thinking, no bright ideas, no higher level thinking--not in math, science, etc. Everything flows from knowledge. Knowledge is the basis for higher level thinking. Common Core is all about testing, not teaching basic arithmetic.
Kids need achievable objectives, and, because academic ability varies widely, the learning objectives should vary or be flexible from school to school, even from classroom to classroom. This works best at the school building level with individual teachers making the educational decisions, policies, and tests. Yet, top-down Common Core does not allow this. Every student gets the same. One size fits all is idiotic.
 |
| Varvara started piano at 4. She is 10 in the photo. |
To become better at anything, a student has to memorize and practice and memorize and practice, have passion and persistence to learn, and get expert teaching. The problem in American schools is that most students do not get expert teaching early on; hence, [hidden] abilities in many capable very young children are squandered, never discovered or developed, whether it be in music, art, math, science, literature, writing, etc. Zig Engelmann writes, "The genetic interpretation of giftedness and intelligence is a myth." Yet, it [IQ] seems to govern the education policy in our schools, thanks to the influences of Darwin, Binet, Piaget, Dewey, et al. Even though brain plasticity (that IQ can be increased within limits) has been known for years, yet teachers are told to treat all students the same, that is, in Common Core all students learn the same content. It is the "triumph of mediocrity." The US education system seems to relish mediocrity. Our kids are not dumb. Indeed, better teaching and teacher quality are important, but, according to Will Fitzhugh (Concord Review), "The most important variable in student academic achievement is not teacher quality, but the student's academic work." In short, students should produce high quality academic work, but many capable students do not. Paul Zoch (Doomed to Fail), "Let there be no doubt about it: the United States looks to its teachers and their efforts, but not to its students and their efforts for success in education." It's not the teacher's fault that your kid is lazy. "Your kids are your own fault," writes Larry Winget. The blame should go to the parent, too, not the teacher. Early on, parents should provide the environment for achievement and teach kids how to be successful in school. But, this should not be the message: "If you're not the best, you're a loser." If you're not brilliant, you're worthless (
Do all students come to 1st grade with the same background knowledge, the same vocabulary, etc.? So, what do we do in our elementary schools? To paraphrase Horace Secrist, a professor of statistics, "Pupils of different mentality, and of training, grouped together in a a single room were to be education, which resulted in inefficiency and regression to mediocrity." When I first started teaching, we tracked students in elementary school and junior high school--to put superior math students with other superior math students with an expert math teacher, etc., but, for decades, this has been taboo in public elementary and middle schools and often called racism. Elementary school classrooms are similar to the old one-room schools. They are the breeding ground for mediocrity. It is called regression to the mean. Mike Petrilli says, "The greatest challenge facing America's schools today isn't the budget crisis, or standardized testing, or teacher quality. It's the enormous variation in the academic level of students coming into any given classroom." Before it was condemned in the US [as racism], tracking actually worked well for almost all students. It isn't a perfect system, but it is far superior than what we have today (inclusion and differentiated instruction). "Academic ability varies widely," observes Charles Murray (Real Education). [Note. For math class, Singapore pulls out weak math students at the beginning of 1st grade and places them with an expert math teacher to catch them up. The program lasts through 2nd grade for kids who need it.] Indeed tracking, when done right, is an effective strategy to reduce variation in the classroom, which means that students [by achievement] go to different teachers for math and reading starting in 1st grade.
Do most infants have the potential necessary to become gifted? Does intensive early training and education work? Should we push children? Varvara started piano at age 4. She was "lucky" to grow up in an environment that developed her outstanding musical ability. Luck plays a much larger role in how much children achieve. Even so, there are no short cuts. Real achievement requires a lot of memorization, practice, good teaching, and luck.
For example, Varvara has a passion for Mozart because her teacher has a passion for Mozart. She understands Mozart because her teacher understands Mozart. She grew up in an environment that fostered, nurtured, and developed her abilities. She was lucky! Varvara gets many standing ovations, and when she plays, hundreds and hundreds and hundreds of notes flow effortlessly, without thinking, from her long-term memory to her fingers. It is a remarkable process. She has mastered not only the piano parts and how to play them, but she also knows the orchestra parts. I saw a performance, and she was having fun playing Mozart! She started piano at age 4. She has developed music ability far beyond that of most kids, but without early intensive practice and expert teaching (nurture), she would be average, if that. And, without the right environment, she may have never started piano in the first place. I want to make clear that practice improves performance, but it does not create talent. There has to be something there to begin with (nature).
Here is my point. Start Young! Very young children are good at memorizing stuff. But, in my view, American educators don't take advantage of this, not in math. Young children are novices; they are not experts; and they are not little mathematicians, but most average children can master arithmetic and algebra when taught and practiced well.
 |
| Your child may not be another Newton, but, at the least, she can learn arithmetic, algebra, and probably precalculus well. |
Most likely your child will not become an expert [genius] in mathematics, such as Newton, Leibniz, Euclid, Gauss, Descartes, etc., or go on to take calculus; however, your child, at the least, can become competent in arithmetic and algebra, even higher, with practice and expert teaching in the right environment. Some American children [we need more] will advance to calculus and beyond and even to STEM careers. We have a shortage and have been importing STEM talent for decades. The K-12 American education system has not been developing the future talent we need in science, mathematics and other STEM fields. Our system produces mediocrity at a time we need excellence. Everyone in education talks about excellence, but the progressive reforms haven't worked. Gee, there must be something wrong with the reforms!
But lurking in some children could be another Newton, an old soul. Newton learned Latin but not mathematics at the King's School until age 17, when his mother decided to make Isaac into a farmer, but he hated farming. So later, he returned to King's School to complete his education, then went to Trinity College in Cambridge, where he started as a subsizar--a student who did chores on campus to pay for his education, room and board, etc. Thank goodness Newton was no good at farming and that the headmaster at King's School convinced his mother to send Isaac back school.
Very young children can learn a lot more than we think or are prepared to teach. American children consistently underperform in math! Kids often say they want to go into STEM, but when they get to college, they don't.
There is no reason why typical kids at very early ages cannot memorize multiples, just as Zig's disadvantaged inner-city pre-first-graders (4 and 5 year olds) did in the 60s, or the addition and subtraction tables for auto recall that Singapore 1st graders master (6 year olds), along with the standard algorithms. But this takes daily practice, oral drills, persistence, etc.! It also takes expert teaching by teachers who know traditional math and are passionate about math and its power, which is the conundrum in America. Most American teachers are weak in both math and science. Some elementary teachers hate math or push it aside. For decades, memorization and practice have fallen out of favor in progressive classrooms. Schools of education push progressive ideas and reforms. This is irresponsible. We don't push kids because "experts" tell us that this is bad for them. Wrong! Parents should drill math facts at home before kids are school age. It is easy to show 4-year-olds addition on a number line. Early on, parents should provide the environment for achievement and teach kids how to be successful in school. Early arithmetic practice at home pays off. Cognitive science has been catching up to anecdotal evidence, which is correct.
Unfortunately, there are idiots in education who say long division is either useless or too hard for kids to learn. But, 3rd graders can easily learn it, iff they have mastered the times tables in long-term memory. [iff means if and only if ⇔]. There are idiots in education who say that memorization of math facts and learning standard algorithms early on are not that important. The progressive reformers are dead wrong! Their policies have led to stagnation in math and reading achievement; impoverishment of public schools, thanks to the costs of Common Core reform and testing; national indebtedness--17 trillion; and a decline in education, rephrasing Dinesh D'Souza (America). The progressive reformers, who see themselves as disruptive innovators, are responsible for the decline and stagnation in US education because of bad ideas.
One unique feature of America has been as a "wealth creator" because of its sound innovations and enterprise, says D'Souza, but our K-12 education system no longer produces enough well-educated and informed citizens. Instead, our education system produces many people who don't work and who are dependent on government. Many college students don't belong in college, and if they pick the wrong major, there is no way they can pay back student loans. College students, even at elite universities [e.g., Harvard], are not taught to think for themselves, says William Deresiewicz. They graduate with "no sense of purpose."
Knowledge is the key to innovation.
It's not that our kids can't do these things--memorize math facts, learn standard algorithms, and apply arithmetic in science to solve problems early on. Our kids are not stupid; they are not taught. Some are lazy! The education establishment doesn't or won't do the right thing. Reform math has screwed up arithmetic for decades; consequently, most kids learn very little substantive arithmetic that prepares them for algebra by middle school. They cannot move forward or excel. Our kids are novices, and novices need to memorize and practice, starting with traditional arithmetic. They also need to be challenged. Our system does not find talent (STEM), and when it is found, it does not support it ($) or offer a different and accelerated curriculum, not in elementary school or middle school. Taking algebra-one in 8th grade is not an accelerated curriculum. Taking AP Calculus is for average kids who are prepared. [AP calculus relies on graphing calculators, has no proofs, and skips important topics, unlike university courses for STEM kids.] US education policies squander talent.
Let me reassure you that there is nothing wrong with classic arithmetic when it is taught properly and explained well, but progressive reformers, such as Common Core reform math crusaders, trivialize it.
The standard algorithms [traditional arithmetic] should be taught first and be the primary methods for calculating. Students must memorize number facts and use standard algorithms at the same time. They should encounter a wide range of routine word problems. First graders should also learn perimeter and area, parts of measurement, geometry, and algebra. I know; I taught it.
1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math) instead of tried and true standard algorithms to do arithmetic.
2. See SingleDigit
Don't calculate single-digit math facts; memorize them.
3. See PiagetianMyth
Constructivism does not work in the classroom.
3. See CommonCore
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education.
©2014 LT ThinkAlgebra.org
Wednesday, July 30, 2014
SingleDigit
Single-Digit Math Facts Retrieval v.s. Calculating
According to Price, Mazzocco,& Ansari (The Journal of Neuroscience), students who memorize by rote single-digit math facts starting in 1st grade, rather than always calculating them, become good math students, and the benefits reach far into high school with much higher PSAT scores. Being able to retrieve 5 + 6 = 11 automatically from long-term memory is different from [and better than] calculating it using a strategy (e.g., 5 + 6 = 5 + 5 + 1 = 10 + 1 = 11), which is a different mental process that clutters working memory, leaving less space for problem solving. Even though mathematicians repeatedly have stated that kids must learn facts by rote, which requires a lot of memorization, practice, and constant use, many US educators have gone a different road, that of calculating by using strategies (e.g., NCTM reform math, now Common Core reform math).
Why are Asisan students superior in math?
What about problem solving?
In TIMSS (4th & 8th grade students), students who scored at the Advanced Level are good problem solvers. Asian nations consistently have 40 to 50% of their 4th and 8th grade students scoring at the Advanced Level in TIMSS compared to only 7% of US students and 4% of Finland students. In fact, in TIMSS math, the US and Finnish students scored about the same [1].
The "rote-learners" from Asian nations are not only tops in international math tests (TIMSS, PISA-15 year olds), but also tops in "creative problem solving," a new test from PISA. (The results were released earlier in 2014.)
Children who have auto recall will not be inclined to calculate basic facts; they will automatically retrieve them as needed. No matter, the strategy shown above (reasoning from doubles) won't work well without instant recall of 5 + 5. Kids often use strategies to remember 3 or 4 demon facts, which, at first, is okay.
Too often, in US math programs the cart (Whys) is put ahead of the horse (Hows). US students are weak in procedures, such as standard algorithms, while Asian kids excel in them and in problem solving.
Standard Algorithms are Key
Dr. H. Wu states, "The multiplication algorithm reduces the multiplication of all numbers to the multiplication of single-digit numbers, which is part of a general pattern. If you can add, subtract, multiply, and divide using single-digit numbers, then you can add, subtract, multiply, and divide all numbers no matter how big." This is the importance of standard algorithms: They are powerful and fast, and they are based on auto recall of math facts. Unfortunately, Common Core tends to ignore or delay standard algorithms, which is another way of saying that in Common Core reform math the memorization and practice of single-digit math facts early on are not that important.
[1] If you analyze the TIMSS results (2011), the Finnish and US 4th and 8th grade students, on average, are almost the same with Finland having a very slight edge. However, 7% of US 8th grade students reach the TIMSS Advanced Benchmarks in math, compared to only 4% of Finland's 8th graders. [Problem solving is thought to be one of Finland's strengths, but this may not be the case.] In 4th grade, the percentages for the TIMSS Advanced Benchmarks are about the same: US 13%, Finland 12%. For perspective, 40% to 50% of 4th and 8th grade students in the top performing Asian nations reach the Advanced Benchmarks. See Finland.
Recently, Pasi Sahlberg (How American Innovation Improved Finnish Education) wrote that Finland turned around its education system based on innovations in the US, which include progressive ideas (Deweyism: social interaction and group work), cooperative learning (group work), multiple intelligences, alternative assessments, and peer coaching. What turn around? Finnish 15-year-olds dropped from 1st to 12th in the matter of a decade (PISA). Their 4h and 8th graders are not any better than US kids (TIMSS). And, the innovations Dr. Sahlberg talks about were failures in US classrooms, which is a reason that our 4th and 8th graders are about the same in math achievement and problem solving.
1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math) instead of tried and true standard algorithms to do arithmetic.
2. See PiagetianMyth
Constructivism does not work in the classroom.
3. See Memorization & Practice
Very young children can learn a lot more than we think or are prepared to teach. American children underperform in math! Parents should drill math facts at home before kids are school age. It is easy to show 4-year-olds addition using a number line. Early on, parents should provide the environment for achievement and teach their kids how to be successful in school. Early arithmetic at home pays off.
4. See CommonCore
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education.
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To Be Revised
7-30-14, 8-1-14
© 2004 LT/ThinkAlgebra
According to Price, Mazzocco,& Ansari (The Journal of Neuroscience), students who memorize by rote single-digit math facts starting in 1st grade, rather than always calculating them, become good math students, and the benefits reach far into high school with much higher PSAT scores. Being able to retrieve 5 + 6 = 11 automatically from long-term memory is different from [and better than] calculating it using a strategy (e.g., 5 + 6 = 5 + 5 + 1 = 10 + 1 = 11), which is a different mental process that clutters working memory, leaving less space for problem solving. Even though mathematicians repeatedly have stated that kids must learn facts by rote, which requires a lot of memorization, practice, and constant use, many US educators have gone a different road, that of calculating by using strategies (e.g., NCTM reform math, now Common Core reform math).
Why are Asisan students superior in math?
A January 2013 article in The Journal of Neuroscience by Price, Mazzocco, & Ansari confirms the rote learners. The reason that Asian kids excel in math, while US kids struggle, is that Asian kids memorize by rote the single digit math facts early on (1st grade, even earlier) and become proficient in using the standard algorithms at the same time. Strategies (US approach) and calculators (Finland's approach) should not replace or delay the instant recall of single-digit math facts. Instant recall of single-digit math facts is necessary for proficiency in standard algorithms.
What about problem solving?
In TIMSS (4th & 8th grade students), students who scored at the Advanced Level are good problem solvers. Asian nations consistently have 40 to 50% of their 4th and 8th grade students scoring at the Advanced Level in TIMSS compared to only 7% of US students and 4% of Finland students. In fact, in TIMSS math, the US and Finnish students scored about the same [1].
The "rote-learners" from Asian nations are not only tops in international math tests (TIMSS, PISA-15 year olds), but also tops in "creative problem solving," a new test from PISA. (The results were released earlier in 2014.)
Children who have auto recall will not be inclined to calculate basic facts; they will automatically retrieve them as needed. No matter, the strategy shown above (reasoning from doubles) won't work well without instant recall of 5 + 5. Kids often use strategies to remember 3 or 4 demon facts, which, at first, is okay.
Too often, in US math programs the cart (Whys) is put ahead of the horse (Hows). US students are weak in procedures, such as standard algorithms, while Asian kids excel in them and in problem solving.
Standard Algorithms are Key
Dr. H. Wu states, "The multiplication algorithm reduces the multiplication of all numbers to the multiplication of single-digit numbers, which is part of a general pattern. If you can add, subtract, multiply, and divide using single-digit numbers, then you can add, subtract, multiply, and divide all numbers no matter how big." This is the importance of standard algorithms: They are powerful and fast, and they are based on auto recall of math facts. Unfortunately, Common Core tends to ignore or delay standard algorithms, which is another way of saying that in Common Core reform math the memorization and practice of single-digit math facts early on are not that important.
[1] If you analyze the TIMSS results (2011), the Finnish and US 4th and 8th grade students, on average, are almost the same with Finland having a very slight edge. However, 7% of US 8th grade students reach the TIMSS Advanced Benchmarks in math, compared to only 4% of Finland's 8th graders. [Problem solving is thought to be one of Finland's strengths, but this may not be the case.] In 4th grade, the percentages for the TIMSS Advanced Benchmarks are about the same: US 13%, Finland 12%. For perspective, 40% to 50% of 4th and 8th grade students in the top performing Asian nations reach the Advanced Benchmarks. See Finland.
Recently, Pasi Sahlberg (How American Innovation Improved Finnish Education) wrote that Finland turned around its education system based on innovations in the US, which include progressive ideas (Deweyism: social interaction and group work), cooperative learning (group work), multiple intelligences, alternative assessments, and peer coaching. What turn around? Finnish 15-year-olds dropped from 1st to 12th in the matter of a decade (PISA). Their 4h and 8th graders are not any better than US kids (TIMSS). And, the innovations Dr. Sahlberg talks about were failures in US classrooms, which is a reason that our 4th and 8th graders are about the same in math achievement and problem solving.
1. See Multiple Models (Different Strategies)
Students are taught inefficient, alternative strategies (reform math) instead of tried and true standard algorithms to do arithmetic.
2. See PiagetianMyth
Constructivism does not work in the classroom.
3. See Memorization & Practice
Very young children can learn a lot more than we think or are prepared to teach. American children underperform in math! Parents should drill math facts at home before kids are school age. It is easy to show 4-year-olds addition using a number line. Early on, parents should provide the environment for achievement and teach their kids how to be successful in school. Early arithmetic at home pays off.
4. See CommonCore
Common Core is part of the progressive agenda to downgrade American education. Every student gets the same. The government has taken over and controls education. Teachers are no longer in charge of education.
----------
To Be Revised
7-30-14, 8-1-14
© 2004 LT/ThinkAlgebra