Educators are wasting valuable instruction time teaching nonessentials rather than standard algorithms, which are vital to advance in math. Instructional time should be used for essentials, not nonessentials. Common Core specifically states that students should learn efficient procedures called standard algorithms for whole number operations--not area models for multiplication, partial quotient methods for division, and so on. Briefly, teachers should explicitly teach classic arithmetic and dump the nonessentials of reform math, along with the misguided policies and practices. Bad ideas, fads, myths, whims, and pseudo-science have corrupted classic arithmetic and, consequently, have impeded US math achievement for decades. To optimize learning, kids need achievable goals, strong teacher guidance--not minimal teacher guidance, more technology, or unwise policies like inclusion. Minimal guidance means minimal learning. Instead of doing arithmetic, kids are drawing pictures. Also, the latest technology has little to do with education and actual achievement, yet school districts are pouring millions into tech and raising property taxes to fund it. Put simply, schools are spending tons of money on things (e.g., technology), ideas, or programs that lack solid evidence and are not needed--all at a time when our students are barely treading water.
Advice.
Make sure kids learn their times tables for auto recall (not just "from memory") no later than the 1st semester of 3rd grade so that you can spend the 2nd semester on long division and basic fraction operations. Put simply, classic arithmetic must start in 1st grade. Kids should practice the standard algorithms for multiplication and long division no later than the 3rd grade to gain competency and to prepare for 4th grade and beyond. None of this requires Common Core reform math or technology, just good teaching. Starting in 1st grade, teach only the standard algorithms, not reform math stuff (such as nonstandard procedures, strategies, drawings, using popular minimal guidance methods, etc.), which slows learning and wastes crucial instructional time on nonessentials and ill-advised practices. Common Core states kids should learn the standard algorithms, not nonstandard procedures or drawings as in reform math.
Dr. Marina Ratner, professor emerita of mathematics at the University of California at Berkeley, writes that Common Core "represents a huge step backward" from the 1997 California math standards. She exclaims, "Who would draw a picture to divide 2/3 by 3/4?" It is nonsense, not "deeper and more rigorous" math. She points out, "Mathematics is not about visual models [drawings] and real-world stories," which her grandson endured almost every day in 6th grade Common Core math class.
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Special Note. Be aware that Common Core is filled with pedagogy, such as instructional strategies, even though it claims to be pedagogy-free. First Grade Common Core: "Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13)." (Common Core 2014 website.)
The strategies approach is pedagogy: counting on, making ten, decomposing, relationships, equivalent sums, etc. None of these should have been stated in a standard. Teachers should make the decisions, not a committee writing standards. Furthermore, nothing in the standards prohibits students from memorizing the addition facts for instant use in problem-solving or practicing the standard algorithm in 1st grade. Note that Common Core asks students to calculate single-digit sums, which is an epic error. Price, Mazzocco, & Ansari (The Journal of Neuroscience) point out that students should not calculate single-digits sums in working memory; they should automate them in long-term memory for instant use in problem-solving. Strategies should not replace, hinder, or disrupt the memorization of basic number facts (factual knowledge) or the practicing of standard algorithms (procedural knowledge) in 1st grade or any grade. Read Outcomes.
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"The ability to communicate (e.g., write a paragraph to explain your answer) is not essential to understanding mathematics," points out mathematician W. Stephen Wilson at Johns Hopkins University. Kids are not little mathematicians, so teach them the standard arithmetic for automaticity and focus on one efficient method to calculate each operation (i.e., the standard algorithms). To learn whole number arithmetic well, students need to memorize single-digit number facts and practice standard algorithms, including long division. If I see another student using the lattice method or area model to multiply, I think I will join Alice down the rabbit hole. Why do educators waste valuable instructional time on such nonsense?
Many reformists like Jo Boaler (Stanford Graduate School of Education) pretend that math education is the same as math, but it is not. It’s no joke that she titles her book What’s Math Got To Do With It?. The "it" is "math education." Reformists like Boaler believe that the eight Standards for Mathematical Practices (SAPs) in Common Core are much more important than math content, which they often deride as "rote," says Dr. W. Stephen Wilson, a mathematician. They are wrong! The reformists think children should be little mathematicians, which they are not. They are novices who need to learn standard arithmetic content through memorization and practice to advance to more abstract and complex math just like the students in China or South Korea.
Don't worry about a child's self-esteem; worry about the child's competency in performing straightforward arithmetic. Professor W. Stephen Wilson writes, "Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way." Also, placing kids of different achievement levels (such as mixing high math achievers with low math achievers) in the same math classroom (called inclusion) is like the one-room school house, which was an inefficient model to educate children and has led to "regression to mediocrity," explains mathematician Jordan Ellenberg. Inclusion is an implausible idea, yet it is considered a good "education practice" among reformers.
W. Stephen Wilson Sounds Off: There will always be ...
Mathematician W. Stephen Wilson ("The Common Core Math Standards," Education Next, Summer 2012) writes, “The end of the math wars! You must be joking!
Common Core
Common Core specifies that students should learn only standard algorithms for whole number operations--not all those inefficient, alternative procedures, or strategies, or minimal guidance methods (collectively called reform math), which have been a waste of valuable instructional time.
I don't want students using the area model to do multiplication or the partial quotient method to do long division. And, I don't want them making drawings, etc. That's reform math and not essential. I want students to use the standard algorithms. Ideologically-driven reforms or policy fixes, such as NCTM reform math, NCLB, now Common Core (reform math), etc, are the reasons that, in the largest school district in southern Arizona, 87% of high school graduates entering community college in 2014 needed remedial math. It is shocking. Reform math and /or math poorly taught prepares students for remedial math at a community college.
Briefly, we are terrible at teaching basic arithmetic and algebra to prepare students for college-level work. Most K-6 teachers are weak in arithmetic, but this is nothing new. Over the decades, this has not been adequately addressed, much less corrected. Often, teachers are not teaching classic arithmetic; they are teaching reform math using group work. Moreover, based on my analysis in 2010-2011, the Common Core math standards, themselves, are not at the Asian level, so our kids start behind beginning in 1st grade.
Instructional time is limited, says Tom Loveless, so, in my opinion, it should be used for what is required, not for extra stuff that isn't required. Barry Garelick comments that under the reform math interpretation of Common Core, "Students are required to use inefficient, cumbersome methods for two years...which confuse more than enlighten." It puts kids behind 2 years, but reformers like Jo Boaler don't seem to acknowledge that reform math has been ineffective. Furthermore, Jason Zimba, one of the writers of the math standards, writes, "The standards also allow for approaches in which the standard algorithm is introduced in grade 1, and in which only a single algorithm is taught for each operation." Thus, Zimba opens up Common Core to all types of interpretations and misinterpretations. FYI: The standard algorithm for addition can be taught in 1st grade in the first month of school. I know; I did it in the early 80s when I had a self-contained 1st-grade class in a city school.
Why waste time with non-standard algorithms that lead to remedial math? In contrast, the standard algorithms (old school) and the memorization of single-digit number facts (old school) get the job done when taught well. In elementary school, kids need to master the arithmetic that is needed for algebra, but reform math slows their progress and doesn't get them there.
The Third Grade Rule
The old California standards, which were adopted in December of 1997, got it right. By the end of 3rd grade, students should know well (be able to apply and perform) the standard algorithms for whole numbers (addition, subtraction, multiplication, and long division). Many schools on the east coast used to do this--public, Catholic, and independent--before NCTM math standards came along in 1989 and screwed up arithmetic. Today, arithmetic is still screwed up. Instead of learning standard algorithms, kids learn a distorted version of arithmetic called reform math, which is a waste of precious instructional time.
The standard algorithms for whole numbers are based on single-digit number facts, which must be memorized for auto recall; place value; properties of operations; and relationships between addition & subtraction and between multiplication & division. They are critically important for the child's future. Without memorizing single-digit number facts for instant recall (factual knowledge) and gaining proficiency in standard algorithms (procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics," writes Professor W. Stephen Wilson.
Mathematician Jordan Ellenberg (How Not To Be Wrong: The Power of Mathematical Thinking, 2014) defends the classic algorithms that kids must know to advance. He writes, "Some reformists go so far as to say that the classical algorithms (like add two multi-digit numbers by stacking one atop the other and carrying the one when necessary) should be taken out of the classroom, lest they interfere with the students' process of discovering the properties of mathematical objects on their own. That seems like a terrible idea to me: these algorithms are useful tools that people worked hard to make, and there's no reason we should have to start completely from scratch." Professor Ellenberg also points out, "It is pretty hard to understand mathematics without doing some mathematics."
(Sources: Tom Loveless, "Implementing Common Core: The Problem of Instructional Time;" the 1997 California math standards (Ze've Wurman); Barry Garelick, Teaching in the 21st Century and other writings; the NCTM 1989 math standards; Jason Zimba, "When the Standard Algorithm Is the Only Algorithm Taught"; Jo Boaler, What's Math Got To Do With It?; Larry Cuban, "Fixing Schools Again & Again; and Pima Community College"; Jordan Ellenberg, How Not To Be Wrong...)
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Comment.
G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." It requires sufficient knowledge in long-term memory, experience, and skill development through practice. It implies that if a student cannot do the math, then the student doesn't understand it or know it. It relates to Richard Feynman's insight: "You don't know anything until you have practiced."
Aim
"The aim of mathematics," writes mathematician Eugenia Cheng (How to Bake Pi). "is to make things easier. Math is hard, but it makes hard things easier." Should we accept the reform template that makes arithmetic harder than it is? It is the reason we should teach kids the standard algorithms as the primary way to calculate from the get go. But we don't, not in reform math and not in Common Core, which is often interpreted as reform math. Common Core (reform math) implies that memorization of single-digit multiplication facts and the practice to automation of the multiplication and long division standard algorithms are not a priority and not good teaching. They are dead wrong!
- There will always be people who think that calculators work just fine, and there is no need to teach much arithmetic, thus making career decisions for 4th graders that the students should make for themselves in college. Downplaying the development of pencil and paper number sense might work for future shoppers, but doesn’t work for students headed for Science, Technology, Engineering, and Mathematics (STEM) fields.
- There will always be the anti-memorization crowd who think that learning the multiplication facts to the point of instant recall is not good for a student, perhaps believing that it means students can no longer understand them. Of course this permanently slows students down, plus it requires students to think about 3rd- grade mathematics when they are trying to solve a college-level problem.
- There will always be the standard algorithm deniers, the first line of defense for those who are anti-standard algorithms being just deny they exist. Some seem to believe it is easier to teach “high-level critical thinking” than it is to teach the standard algorithms with understanding. The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared.
- There will always be people who believe that you do not understand mathematics if you cannot write a coherent essay about how you solved a problem, thus driving future STEM students away from mathematics at an early age. A fairness doctrine would require English language arts (ELA) students to write essays about the standard algorithms, thus also driving students away from ELA at an early age. The ability to communicate is NOT essential to understanding mathematics.
- There will always be people who think that you must be able to solve problems in multiple ways. This is probably similar to thinking that it is important to teach creativity in mathematics in elementary school as if such a thing were possible. Forget creativity; the truly rare student is the one who can solve straightforward problems in a straightforward way.
- There will always be people who think that statistics and probability are more important than arithmetic and algebra, despite the fact that you can’t do statistics and probability without arithmetic and algebra and that you will never see a question about statistics or probability on a college placement exam, thus making statistics and probability irrelevant for college preparation.
- There will always be people who think that teaching kids to “think like a mathematician,” whether they have met a mathematician or not, can be done independently of content. At present, it seems that the majority of people in power think the three pages of Mathematical Practices in Common Core, which they sometimes think is the “real” mathematics, are more important than the 75 pages of content standards, which they sometimes refer to as the “rote” mathematics. They are wrong. You learn Mathematical Practices just like the name implies; you practice mathematics with content.
- There will always be people who think that teaching kids about geometric slides, flips, and turns is just as important as teaching them arithmetic. It isn’t. Ask any college math teacher.
- The end of the math wars! You must be joking."
Common Core
Common Core specifies that students should learn only standard algorithms for whole number operations--not all those inefficient, alternative procedures, or strategies, or minimal guidance methods (collectively called reform math), which have been a waste of valuable instructional time.
I don't want students using the area model to do multiplication or the partial quotient method to do long division. And, I don't want them making drawings, etc. That's reform math and not essential. I want students to use the standard algorithms. Ideologically-driven reforms or policy fixes, such as NCTM reform math, NCLB, now Common Core (reform math), etc, are the reasons that, in the largest school district in southern Arizona, 87% of high school graduates entering community college in 2014 needed remedial math. It is shocking. Reform math and /or math poorly taught prepares students for remedial math at a community college.
Briefly, we are terrible at teaching basic arithmetic and algebra to prepare students for college-level work. Most K-6 teachers are weak in arithmetic, but this is nothing new. Over the decades, this has not been adequately addressed, much less corrected. Often, teachers are not teaching classic arithmetic; they are teaching reform math using group work. Moreover, based on my analysis in 2010-2011, the Common Core math standards, themselves, are not at the Asian level, so our kids start behind beginning in 1st grade.
Instructional time is limited, says Tom Loveless, so, in my opinion, it should be used for what is required, not for extra stuff that isn't required. Barry Garelick comments that under the reform math interpretation of Common Core, "Students are required to use inefficient, cumbersome methods for two years...which confuse more than enlighten." It puts kids behind 2 years, but reformers like Jo Boaler don't seem to acknowledge that reform math has been ineffective. Furthermore, Jason Zimba, one of the writers of the math standards, writes, "The standards also allow for approaches in which the standard algorithm is introduced in grade 1, and in which only a single algorithm is taught for each operation." Thus, Zimba opens up Common Core to all types of interpretations and misinterpretations. FYI: The standard algorithm for addition can be taught in 1st grade in the first month of school. I know; I did it in the early 80s when I had a self-contained 1st-grade class in a city school.
Why waste time with non-standard algorithms that lead to remedial math? In contrast, the standard algorithms (old school) and the memorization of single-digit number facts (old school) get the job done when taught well. In elementary school, kids need to master the arithmetic that is needed for algebra, but reform math slows their progress and doesn't get them there.
The Third Grade Rule
The old California standards, which were adopted in December of 1997, got it right. By the end of 3rd grade, students should know well (be able to apply and perform) the standard algorithms for whole numbers (addition, subtraction, multiplication, and long division). Many schools on the east coast used to do this--public, Catholic, and independent--before NCTM math standards came along in 1989 and screwed up arithmetic. Today, arithmetic is still screwed up. Instead of learning standard algorithms, kids learn a distorted version of arithmetic called reform math, which is a waste of precious instructional time.
The standard algorithms for whole numbers are based on single-digit number facts, which must be memorized for auto recall; place value; properties of operations; and relationships between addition & subtraction and between multiplication & division. They are critically important for the child's future. Without memorizing single-digit number facts for instant recall (factual knowledge) and gaining proficiency in standard algorithms (procedural knowledge), "students are severely handicapped as [they] attempt to pursue the next levels of mathematics," writes Professor W. Stephen Wilson.
Mathematician Jordan Ellenberg (How Not To Be Wrong: The Power of Mathematical Thinking, 2014) defends the classic algorithms that kids must know to advance. He writes, "Some reformists go so far as to say that the classical algorithms (like add two multi-digit numbers by stacking one atop the other and carrying the one when necessary) should be taken out of the classroom, lest they interfere with the students' process of discovering the properties of mathematical objects on their own. That seems like a terrible idea to me: these algorithms are useful tools that people worked hard to make, and there's no reason we should have to start completely from scratch." Professor Ellenberg also points out, "It is pretty hard to understand mathematics without doing some mathematics."
(Sources: Tom Loveless, "Implementing Common Core: The Problem of Instructional Time;" the 1997 California math standards (Ze've Wurman); Barry Garelick, Teaching in the 21st Century and other writings; the NCTM 1989 math standards; Jason Zimba, "When the Standard Algorithm Is the Only Algorithm Taught"; Jo Boaler, What's Math Got To Do With It?; Larry Cuban, "Fixing Schools Again & Again; and Pima Community College"; Jordan Ellenberg, How Not To Be Wrong...)
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Comment.
G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." It requires sufficient knowledge in long-term memory, experience, and skill development through practice. It implies that if a student cannot do the math, then the student doesn't understand it or know it. It relates to Richard Feynman's insight: "You don't know anything until you have practiced."
Aim
"The aim of mathematics," writes mathematician Eugenia Cheng (How to Bake Pi). "is to make things easier. Math is hard, but it makes hard things easier." Should we accept the reform template that makes arithmetic harder than it is? It is the reason we should teach kids the standard algorithms as the primary way to calculate from the get go. But we don't, not in reform math and not in Common Core, which is often interpreted as reform math. Common Core (reform math) implies that memorization of single-digit multiplication facts and the practice to automation of the multiplication and long division standard algorithms are not a priority and not good teaching. They are dead wrong!
Understanding
Let's talk about understanding, which seems to be a big deal in Common Core reform math, especially deep understanding. But what is deep understanding? Frankly, I don't know. It originated from NCTM math standards. Understanding is a matter of opinion because it is difficult to quantify. But, arithmetic and algebra are a matter of knowledge, not opinion, and are quantifiable.
David Ruelle (The Mathematician’s Brain) explains mathematical intuition, “When we study a mathematical topic, we develop an intuition for it. We put in our [long-term] memory a large number of facts that we can access readily and even unconsciously. Since part of our mathematical thinking is unconscious and part nonverbal, it is convenient to say that we proceed intuitively. It means that processes of mathematical thought are difficult to analyze.” Dr. Ruelle also observes, "Mathematics is a matter of knowledge, not of opinion.” (I think understanding falls under opinion.)
Ruelle also writes that “mathematicians put a lot of facts in their long-term memory through long days of study.” In this sense, I want kids to be more like mathematicians (or musicians), that is, I want kids practicing the essentials, so they stick in the long-term memory and become automatic. Basic knowledge of arithmetic (ideas, skills, and uses) enables mathematical thought. I don’t want novices trying to discover arithmetic in group work or playing with non-standard algorithms at the expense of standard algorithms. As Hersh & John-Steiner would say, learning mathematics well takes drill and practice, lots of it. There is no workaround.
If students can apply the math they have learned; and then this implies that they have some understanding of it, but I cannot measure it except for students showing correct mathematics. Often, I hear reformers claim that novices need "deep understanding." What is that? How do you measure it? I think the closest we can come to a student’s actual mathematical thought process, which is often unconscious and nonverbal, is when the student performs mathematics by writing steps that lead to a solution or correct answer, whether it be solving an equation or executing the standard multiplication algorithm, etc. Written explanations or drawings are nonessential. Of course, there are many reformists who disagree.
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| 4th Grade Smarter Balanced |
Here is a 4th-grade level problem from Smarter Balanced via Katharine Beals' blog Out in Left Field. The addition is something all 2nd graders should know. Indeed, 1st graders who know place value past hundreds can do this. Oddly, there is no carry in this 4th-grade problem. I guess carry concept is too difficult for kids.
Note. 1st Draft. Please excuse typos, errors, and changes.
Comments: ThinkAlgebra@cox.net
Last changes: July 17, 2015, July 22, 2015, July 31st, August 20, 2015
© 2015 LT/ThinkAlgebra.org
Note. 1st Draft. Please excuse typos, errors, and changes.
Comments: ThinkAlgebra@cox.net
Last changes: July 17, 2015, July 22, 2015, July 31st, August 20, 2015
© 2015 LT/ThinkAlgebra.org
