Tuesday, July 14, 2015

Standard Algorithms

Tom Loveless says that instructional time is limited, so it needs to be used wisely, but too often it isn't.  He writes, "The standard algorithm is the only algorithm identified [in Common Core] as required for students to learn." The standard algorithm, which is classic arithmetic, should be the primary method for calculating from the get go, that is, starting in the 1st semester of 1st grade. On a school district's website I read, "The best learning often happens in teams." It is bunk. (Quote: Tom Loveless, Implementing Common Core: The Problem of Instructional Time)

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. 

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 backwardfrom 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.

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

"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 EllenbergInclusion 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!
  • 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 GarelickTeaching 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...)

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."

"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!

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.
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

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Monday, July 6, 2015


I have written many times about repackaging and recycling failed ideas then and now in education, but so have Larry Cuban (Stanford) and others. The reformists keep bringing back stuff that didn’t work in the past and won't work now. They don’t learn from the mistakes of the past. The current Common Core/NCLB maneuver is left over from the standards+testing+(government sanctions) hatched under No Child Left Behind (NCLB) in 2001. The policy fix [test-driven reforms] has led to “many standardized tests and coercive accountability,” along with substantial “federal overreach,” says Cuban. It has also prompted a harsh backlash and opt-out movements against Common Core and its testing. 

Dr. Cuban also categorizes the policy fixes: (1) Fix the students (e.g, early childhood education, etc (2) Fix the schools (e.g., school choice, higher standards, etc.), and (3) Fix the teachers (e.g., teacher education, child-centered, etc.). He writes, “School reform over the past century has skipped from one big polity fix to another without a backward look at what happened the first time around.” In fact, "Public and policymaker affections have hopscotched from one solution to another then and now and in some instances, combined different fixes (e.g., extending the school day, raising standards and increasing accountability for schools and teachers, promoting universal pre-school, pushing problem-based learning)."

We keep “fixing” but nothing seems to get fixed properly because “reform-driven policies are (and have been) hardly researched-based, observes Cuban. Put simply, reform-driven policies or fixes are ideologically driven, not evidence driven. In my opinion, the liberal agenda of reform fixes has corrupted traditional education and has wasted billions upon billions of dollars on ineffective government programs, policy fixes and reforms that don't work, and bad ideas. Consequently, kids do not get the education they need.

Ideologically-driven reforms or fixes are the reason that, in the largest school district in southern Arizona, 87% of high school graduates entering community college in 2014 needed remedial math. And, in an adjacent school district, it is 88%. So much for the policy fixes that corrupt and don't fix! They have hurt students, especially minority students. But, its not only math. Many of the intro courses at Community College today are actually equivalent to top-notch high school courses decades ago. Community Colleges should start with precalculus, not with remedial basic math, or pre-algebra, or high school algebra courses.

[This is the situation: Steven Strogatz, Professor of Applied Mathematics at Cornell University, writes, "Dad, can you show me how to do these multiplication problems?" Sure. "No Dad, that's not how we're supposed to do it. That's the old school method. Don't you know the lattice method?" No? Well, what about partial products?" Strogatz's experience uncovers the failings of reform fixes. Students never learn standard arithmetic for automaticity. Instead, they are taught reform math, which is bunch of inefficient, alternative algorithms. Kids raised on reform math, minimal guidance approaches, group work, and calculators often end up in remedial math at community college.] 

Lastly, Cuban quotes Andre Gide, a French novelist, “Everything has been said before, but since nobody listens we have to keep going back and beginning all over again.” Such has been the state of education reform for at least a century. 

[Note. This post features Larry Cuban analysis about recycling past failures, along with several relevant comments I wrote in 2010.]

I believe in public education, but I agonize over what has been happening in schools across America for 50 years.
 We could teach children actual arithmetic--which includes, standard algorithms, long division, fractions, percentages  proportions, and equations--straightforwardly, but, instead, we teach nonstandard reform math in "inquiry/discovery or problem-based" group work, which does not prepare typical kids for algebra by 8th grade. Kids don't routinize arithmetic because memorization, hard work, and practice for automation of fundamentals, such as times tables and standard algorithms, have fallen out of favor in child-centered classrooms. 

Kids need to memorize times tables and master long division, fractions, percentages, proportions, and equations in elementary school to do algebra in middle school, which is what typical kids in Asian nations do. The problem begins in 1st grade. Furthermore, teachers are often led to believe they are doing the right thing, that is, teaching reform math using minimal teacher guidance methods, because this is what they are continually told to do in professional development or taught in ed school. Common Core is typically misinterpreted by the powers that be as reform math.

In 2010, I wrote: Teachers are taught to be facilitators who use “minimal guidance” instructional methods (discovery, problem-based, inquiry-based, etc.); however, these methods are often ineffective. Sweller, Clark, and Kirschner point out in their 2006 analysis (Why Minimal Guidance During Instruction Does Not Work), “Minimal guidance in mathematics leads to minimal learning,” which is exactly what we have. Inadequate math achievement starts in 1st grade and spirals up the grades through high school. Reformers often use popular ideology, not evidence, to justify or impose their policies. [2010]

Common Core [Math], initiated by state governors for uniformity, advised and written by special interest groups, supported by politicians and the federal department of education, and adopted by most states before the final draft was written is not a solid framework for upgrading math curriculum and instruction to world-class calibre. We can do better. In fact, the 2010 K-8 Core Knowledge Math Sequence has done just that. [2010]

Knowledge Enables Thinking and Learning New Stuff²
We need a pedagogy that liberates and empowers teachers¹ to teach basic knowledge and that enables efficient, “long-term memory” learning.² This  requires meaningful, explicit instruction, substantial daily practice, and recurrent review. Unfortunately, the popular trend in mainstream education is to implement “unguided or minimally guided instructional approaches”³ (e.g., discovery learning and its recycled configurations), which not only limit rapid acquisition of basic knowledge in long-term memory, but also are time-consuming and largely ineffective. Indeed, acquisition of background knowledge in long-term memory is essential for thinking and problem solving in academic disciplines like math and science. Moreover, as Daniel Willingham points out, the more stuff you know in long-term memory, the more stuff you can learn because new ideas are built on old ideas.² Thinking in math, at the very least, requires automatic recall of facts and procedures (background knowledge) from long-term memory into working memory.⁶ Even though “critical thinking processes are tied to background knowledge,”² many educators use pedagogies that do not optimize putting key background knowledge in long-term memory. As Kirschner, Sweller, and Clark argue, "The aim of all instruction is to alter long-term memory. If nothing has changed in long-term memory, nothing has been learned."³ [2010]

The fast acquisition of knowledge in mathematics optimizes thinking and problem solving processes. It is knowledge in long-term memory that enables thinking and learning new stuff.² 

Sources & Information
1. Peter Wood, “Where Do We start? Reforming American Education,” National Association of Scholars, May 21, 2009. Some wording is similar to Wood’s, so I want to credit him.  
2. Daniel Willingham, Why Don’t Students Like School, 2009. Willingham, a cognitive scientist, discusses of the intrinsic value of basic background knowledge in long-term memory to enable thinking and learning. Every parent who teaches their child and every educator should read this book. 
3. Kirschner, Sweller, and Clark, “Why Minimal Guidance During Instruction Does Not
Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching,” Educational Psychologist, 41(2), 2006. "The aim of all instruction is to alter long-term memory. If nothing has changed in long-term memory, nothing has been learned." The method that works best is explicit, strong teacher-guided instruction. 
6.Working memory is where thinking takes place.

Parents should not expect world-class math instruction in elementary school. Sian Beilock (Choke, p. 124) points out that elementary education majors have the “highest levels of math anxiety of any college major in the United States.” Moreover, elementary school teacher preparation programs “include very little math.” Teachers are trained to be generalists. They take "math education" courses, but not math courses. Yet, teaching arithmetic and algebra requires domain-specific knowledge and domain specific methods that optimize rapid mastery of fundamentals in long-term memory. This requires memorization, substantial practice, and solving a wide range of routine word problems. Like parents in Singapore and Korea, U.S. parents should supplement school math with “outside of school” tutoring and instruction. Parents can use the revised 2010 math sequence from Core Knowledge as a guide. In contrast to Common Core, the Core Knowledge math sequence is world-class and leads, grade-by-grade, to Algebra 1 in 8th grade. [Tuesday, December 21, 2010] 

Other comments. 

  • Kids are not fluent in arithmetic needed for algebra. 
  • Algebra courses are watered down. 
  • The new Common Core math standards are not world class. 
  • In NCLB, “mediocre” performance is labeled “proficient.” 
  • Kids lag behind their peers in high-achieving nations. 
  • Society says it is okay to be bad at math. 
  • In school, it’s not ‘cool’ to be smart in math.

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