Multiplication

Multiplication

Since the early 90s, perhaps earlier, the standard multiplication algorithm and the memorization of single-digit facts (along with long division) have been under attack and marginalized by progressive reform math programs. Teachers, I think, can limit some of the damage. There is no reason to delay the standard multiplication algorithm until the 5th grade as in Common Core reform math, and, in the name of "many ways" fluency, substitute inefficient alternatives, especially nonstandard or invented algorithms, such as lattice, area/array, partial products, for the standard multiplication algorithm. 

Students need to learn the basics first, which means they should master standard [classic] arithmetic starting in 1st grade, not Common Core's brand of "many ways" reform math, which is just a continuation of an ill-advised, academic fad. Multiplication should begin in 1st grade as repeated addition. In 2nd grade, students should switch to memorizing (by rote) half the multiplication table and continually use the single-digit facts to solve word problems. The other half of the times table is memorized at the start of 3rd grade as students begin to use the standard multiplication algorithm and, later, the standard long-division algorithm (not partial quotients, etc.).

At the beginning of 3rd grade, teach/explain/practice the standard multiplication algorithm first (as kids memorize the single-digit multiplication facts), then, later on, when students are very good at the standard algorithm, show some tricks (shortcuts) using the rules of arithmetic and compatible numbers (easier numbers to calculate, such as 25 x 8 = 200) to give students a deeper perspective. 

Third Grade Benchmarks: Multiplication & Long Division
Don't use alternative strategies--such as lattice method, area model, partial products, or complicated [invented] strategies like Rachel's (See Investigations below)--for multiplication. Focus on the memorization of the times table and the standard multiplication algorithm from the get go, then introduce some cool tricks later on for variety. In the second semester of 3rd grade, continue with multiplication and include long division and fractions. Don't forget to do lots of word problems. Lastly, smart kids don't need to practice as much as other kids, but they still need to practice. Smart kids also need more challenging content at a faster pace. Remember, it is drill that leads to skill: Drill for Skill. Understanding does not produce mastery; practice does!

Insert: --------------------
From 5th Grade Investigations (TERC): NCTM Reform Math 
Under “Developing Computation Strategies That Make Sense,” is Rachel’s solution to the multiplication of 59 x 13. I left out some of the words (explanation) to save space, but not the numbers. Rachel's strategy is roughly the essence of reform math, which makes arithmetic more complicated, confusing, and slower than it is. 

Rachel’s Method (Alternative Strategy)
59 x 13 is 50 groups of 13 plus 9 groups of 13.
For 50 x 13: (10 x 13) + (10 x 13) + (10 x 13) + (10 x 13) + (10 x 13)        
For 9 x 13: (9 x 10) + (9 x 3)
Thus: 130 + 130 + 130 + 130 + 130 + 90 + 27
Grouping 100s, etc.: 500 + 150 + 100 + 17 = 767
(LT: Note how 90 + 27 became 100 + 17)

Investigations justifies Rachel's invented method this way: “While Rachel’s method may look more cumbersome at first glance than the historically taught multi-digit multiplication algorithm, it is actually easy to keep track of, results in the numbers that are easy to work with, is not prone to calculation errors, and for someone fluent with the relationships  in the problem, can be carried out fairly quickly. Not all the procedures that students try will be equally manageable.” The Investigations website states that the curriculum focuses on computational fluency, then defines computational fluency as using "many different ways" to solve a problem; hence, the emphasis is on non-standard computational methods.  Rachel's Strategy & Quote From Grade 5, Investigations, TERC, 1998)

Note. Common Core interprets computational fluency as "many ways." Consequently, in Common Core, the standard algorithm for multiplication is ignored and postponed to 5th grade. Common Core claims that "many ways" improves the student's understanding. In reality, the "many ways" fluency idea confuses novices and slows learning.  

Standard Algorithm
12 Seconds

For novices, the breaking down of numbers [Rachel's Method] is cumbersome, convoluted, confusing, and a waste of instructional time. Who would use Rachel's Method to multiply two simple numbers? It is not the way to teach 3rd graders multiplication. Regrettably, the focus in reform math programs is predominantly on nonessentials. On the other hand, according to Tom Loveless (Brookings Institution), the only algorithm specifically mentioned in Common Core is the standard algorithm (right), which is essential arithmetic needed for algebra. (It took me about 12 seconds to do the standard algorithm, but kids can do it faster with practice.)

Incidentally, Investigations is still a very popular NCTM reform math program with the 3rd edition, aligned more with Common Core, coming out in 2017, but available only through Pearson.  Investigations won't change much under Common Core. In fact, the Pearson website states that Investigations is an easy transition to Common Core. Furthermore, teachers can use the older editions with patches from Pearson. I do not recommend Investigations or any repackaging of math reform strategies that are often found in typical Common Core programs. Instead of dealing with numbers directly as they are, students are taught to use compensation strategies, that is, changing a calculation to easier (compatible) numbers that will yield the same answer. Rachel changed the original numbers to compatible ones, which is an inefficient method. 

Special Note. Compensation using compatible numbers, such as in Common Core, is nothing new; however, it should be tangential (an add-on) and not the primary method for operating on numbers for beginners. Furthermore, numbers are compatible only for a specific operation. For example, while the expression 3 + 7 makes 10,  3 x 7 does not. This type of inconsistency often confuses novices. Also, instructional time should be reserved for learning and practicing essentials, such as single-digit number facts, standard algorithms, pattern recognition in word problems, equation solving, rules and definitions, etc., all of which require plenty of exposure and practice (i.e., drill for skill). 
End Insert --------------------

Common Core reform math screws up standard arithmetic, doesn't measure up to the Asian level starting in 1st grade or prepare capable students for Algebra 1 in 7th or 8th grade. Our kids have been left behind. This trend of backward thinking emerged in the early 90s. The National Council of Teachers of Mathematics (NCTM), which rejected rote learning in 1989, overstated that its standards for school mathematics, which have dominated math education since the early 1990s, show “impressive accomplishments,” yet, today, most kids stumble over simple arithmetic because the basics have not been automated.

Without solid arithmetic knowledge, both factual and procedural, in long-term memory, kids can't do more complex mathematics well (e.g., algebra, trig, etc.). Unfortunately, Common Core follows the same NCTM reform math trend, that is, children are "taught" to use many alternative strategies (algorithms) to do arithmetic, not the standard algorithms, which are put on the back burner. The problem continues because the Common Core math content has been interpreted through the narrow lens of the eight mathematical practices, making the practices much more important than the content. In short, there is no balance. 

For example, this is evident in how Common Core treats the "tried and true" standard algorithm for multiplication. Under Common Core, standard multiplication, which has been traditionally taught in 3rd grade, is pushed aside to 5th grade, then only with minimal coverage. In many elementary schools, even classic arithmetic is often taught poorly because many elementary school teachers then and now are weak in basic arithmetic. Somehow, perhaps by magic, working with manipulatives or lattice and area strategies will give students a better understanding of the standard algorithm. How? It is what educators have been led to believe.

Comment1. Teaching kids alternative strategies (reform math as in Common Core) is not the same as teaching the real thing, which is standard arithmetic, and it shows as US students stumble badly over basic math. For example, in one southern Arizona school district, 87% of the students who enrolled in community college in 2014 were placed in remedial math. It was 88% in an adjacent school district. The fundamentals of arithmetic and algebra, under NCTM reform math (and now Common Core), have been taught badly. To do well in algebra, students must have solid arithmetic knowledge in long-term memory. They don't. 

Comment2. Teachers often say that kids don’t understand the standard algorithm, so they avoid teaching it. Perhaps, the reason is that elementary teachers, themselves, are weak in basic arithmetic and don’t know how to teach the standard algorithms. Consequently, they substitute alternative strategies (reform math), such as lattice method for multiplication, thinking that kids will somehow understand the standard algorithm. Really? Anyway, this is what they told to do by the reformists.

Comment3. Amber M. Northern, Ph.D., comments on a new study about the effectiveness of instructional practices in first-grade classrooms. She writes, "Youngsters who struggle with math simply need their teachers to show them how to do the math and then practice themselves how to do it—a lot! Why is such instruction so hard for them to come by?" According to the study, Northern writes, "The most effective instructional practice teachers could use with these struggling students was routine practice and drills (that’s right, drill and kill!). Similarly, lots of chalkboard instruction, traditional textbook practice problems, and worksheets that went over math skills and concepts were also effective with them." Furthermore, according to the study's abstract, the teacher-directed techniques that worked for students with mathematical difficulties also worked well for better students. When teaching arithmetic or algebra, I seldom used group work because minimal guidance leads to slow, minimal learning. I have always used strong teacher guidance (explicit instruction, explaining worked examples, lots of practice, and feedback), which has worked well for most students. Unfortunately, for decades, the instruction used most often has been minimal guidance. Classic arithmetic that is taught well works well. [Dr. Northern refers to the study by Paul L. Morgan, George Farkas, and Steve Maczuga, "Which Instructional Practices Most Help First-Grade Students With and Without Mathematics Difficulties?," Education Evaluation and Policy Analysis vol. 37 no. 2 (June 2015)]

Multiplication is one of the most fundamental operations in arithmetic, algebra, and higher math. It is a prerequisite for learning long division, fractions-decimals-percentages, algebra, etc. Learning the times-table for auto recall and practicing the standard algorithm for proficiency are primary goals of classic arithmetic in the first half of 3rd grade, but not in Common Core reform math, which pontificates an alternative strategies approach. Common Core reformists say that understanding comes only from learning a bunch of alternative strategies, not the standard algorithms. But how can that be?

In contrast, G. Polya (How to Solve It) writes, "To understand mathematics is to be able to do mathematics." It requires factual and procedural knowledge in long-term memory, lots of experience, and skill development through practice. It is not going to change. Fundamentals are put into long-term memory--via memorization and practice--to free the mind for problem-solving. Kids need to learn classic arithmetic, not reform math, to prepare for algebra. The standard multiplication algorithm is needed for kids to advance--not the lattice method or other alternative strategies often found in Common Core reform math.

Educators waste valuable instruction time teaching nonessentials rather than standard algorithms, which are vital to advance in math. In fact, according to Tom Loveless, Common Core specifies that students should learn only standard algorithms for whole number operations--not all those inefficient, alternative procedures (strategies) and minimal guidance methods (collectively called reform math), which have been a waste of valuable instructional time. The strategies approach of reform math is a misinterpretation of Common Core. [Read Standard Algorithms]

3rd Grade Common Core reform math
According to the renamed 3rd grade standards in Arizona (which are Common Core), students should ”know from memory” all products of two one-digit numbers.” But, what does “know from memory” mean? The document states, “Know from memory does not mean focusing only on timed tests and repetitive practice, but ample experience working with manipulatives, pictures, arrays, word problems, and numbers to internalize facts (up to 9 x 9)”. I think not. Kids need an auto recall of single-digit number facts. They must be practiced by rote to stick in the long-term memory for instant use. 

The document lists 9 strategies to do multiplication within 100. It doesn't mention that students should memorize single-digit multiplication facts for auto recall in order to use the standard multiplication algorithm. Moreover, the standard multiplication algorithm is not covered in 3rd grade. It is pushed into 5th grade. “The distributive property is the basis for the standard multiplication algorithm that students can use to multiply fluently multi-digit whole numbers in Grade 5.”

The writers conclude, “Using various strategies to solve different contextual problems that use the same two one-digit whole numbers requiring multiplication allows for students to commit to memory all products of two one-digit numbers.” Really? I think not. 

 5th Grade Common Core reform math
You would think that the standard algorithm would be the primary way to multiply by the 5th grade, but this is not the case in Common Core reform math. Instead, alternative strategies to multiply hog the instructional time. The standard algorithm is merely another strategy among many others and receives scant coverage. It is a mistake.

Common Core says, “Connections between the algorithm for multiplying multi-digit whole numbers and strategies such as partial products or lattice multiplication are necessary for students’ understanding.” No, they are not necessary! It is reform math, not classic arithmetic. Kids need to master the arithmetic that is needed for algebra, but reform math slows their progress and doesn't get them there. Reform math has a shameful track record of sending students straight to remedial math at a community college. 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 complex math just like the students in China, South Korea, and Singapore.

These are examples of some of the alternative multiplication strategies.
5th Grade Common Core State Standards

In Common Core reform math, the justification for alternative strategies is to build understanding before moving to the standard algorithm. But is this happening? The alternative strategies to the standard algorithms seem to build "understanding, if that" only of the alternative strategies, themselves, and hog time away from learning and practicing the standard algorithm.

Indeed, time is better spent on explicit teaching of standard algorithms (not alternative strategies) and practicing them for mastery.  

Source. The two documents I used were the 3rd Grade Common Core State Standards Flip Book and the 5th Grade Common Core State Standards Flip Book, both of which are based on following resources: Common Core, Arizona DOE, Ohio DOE, and North Carolina DOE. Each book is 60 pages long. They were found on a southern Arizona school district website.

According to the document, "The goal of every teacher should be to guide students in understanding and making sense of mathematics." This interpretation of Common Core comes straight out of NCTM reform math dogma, which failed in the past because the focus was not on practicing and mastering content knowledge.

It is clear that the document is based on "practices" and reform math going back to the NCTM math standards that stressed understanding and sense-making rather than learning content knowledge and skills kids must know to do mathematics. The critical problem is that the Common Core math content has been interpreted through the narrow lens of the eight dubious mathematical practices. Content knowledge, according to Common Core reform math, is inferior to the practices, which many mathematicians say are nonsense. 



(Note on Addition. Common Core stresses “making 10” to add, not the standard algorithm, so 8 + 6 is 8 + 2 to make 10 (add 2), then take those 2 away from the 6 (6 - 2) to make 4, so the equation becomes: 8 + 6 = (8 + 2) + (6 - 2), which is 10 + 4 or 14. That’s a total of three calculations, enough to clog up working memory. Gee, why not memorize 8 + 6 = 14 in the first place? The strategy is called compensation, which makes numbers easier to add, that is, the numbers are more compatible. Here is an example of  "make 10" compensation with 4 + 3: add 6 to 4 to make 10, then subtract 6 from 3 to make -3 (oops!). It doesn't work in this context. Gee, don't you think it is easier just to memorize 4 + 3 = 7? Numbers are compatible only on a specific operation. FYI: Singapore kids do not learn operations with negative numbers until the 7th grade.)

Common Core: EngageNY (K-5) Curriculum
In 3rd grade, there are 33 math standards on the Checklist, but only two require fluency, and of the two, only one is listed as a Major Emphasis, which is 3.OA-7 (Multiply and divide within 100 using strategies).  

“3.OA.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.” In Common Core, "from memory" is not defined as instant recall from long-term memory. The standard multiplication and long division algorithms are not found in Common Core 3rd grade. They are found in 5th and 6th grade respectively, which puts our kids at least two years behind.

In contrast, Singapore 3rd grade students learn standard algorithms and do multiplication and division up to 3 digits by 1 digit. Singapore students figure out and memorize by rote the single-digit multiplication facts so they stick in the long-term memory for use in problem-solving, and they practice the standard algorithms from the get go. We should do the same in the US, but we don't. Well, we used to until NCTM reform math came along, stressing the use of calculators starting in grade K and screwing up arithmetic. 

Students must have auto recall of multiplication facts to work the standard multiplication and division algorithms. But it is not a catastrophic if the student has a weak understanding the algorithm, itself. What is much more important is that the student understands the operation, itself, and can do the algorithm quickly as needed. Multiplication and division undo each other. It is also important that students can interpret remainders, know the rules for multiplication, and can apply the relationship between multiplication and division. [Examples: 7 x 9 = 63 and 9 x 7 = 63 (Commutative rule of multiplication); 7 × 5 × 2 is 7 x 10 by compatible numbers or 70 (Associative rule of multiplication); 4 x 354 = (4 x 300 + 4 x 50 + 4 x 4), which is 1200 + 200 + 16 or 1416 (distributive rule of multiplication). The standard multiplication algorithm is based on the distributive rule.

Memorize: Fact Order (Times Table)
2nd Grade
n x 0
n x 1
2s
perfect squares (2 x 2 , 3 x 3 , 4 x 4 ... 10 x 10)
3s
4s
5s
3rd Grade
6s
7s
8s
9s

First Draft. Please excuse typos and other errors. 
Comments: ThinkAlgebra@cox.net

© 2015 LT/ThinkAlgebra.org

Look to the past

Minimal Guidance = Minimal Learning
Perhaps, we should look to the past to get to the future. 
Let me dust off my slide rule! 
Drill For Skill Mastery

I have been in education and the classroom for over 45 years, and I do not think the "achievement gap" can be easily solved, not from a top-down accountability culture that dominates our schools. The main goals of education should be excellence and equality, but they are stalled and miles apart. I have often stated that the kids who walk through the school door vary widely in background knowledge, industriousness, persistence, attitude, home-support, and academic ability. The students are different from every school door in every district; consequently, most of the educational decisions for children in a particular school or classroom should be made by the teachers in the classroom and not imposed by district, state, or federal policies and mandates. In fact, for many times in the past, I have written that teachers should make up and correct their achievement tests. 

To me, it makes little sense to disconnect what is taught in the classroom, that is, the behavioral learning objectives, and the testing of what has been taught. (Some pundits think that teachers cannot be objective in grading. These non-educators say that kids need an independent test. I disagree.) In my opinion, ever since NCLB, classroom instruction and the "imposed" state or federal standardized testing have been uncoupled when the testing should have been intrinsically connected at the classroom level.

We are entrenched in a culture of accountability, metrics, benchmarks, and performance indicators, says Jerry Z. Muller, a history professor at the Catholic University of America. This might be okay for business, but schools are not a business. He writes, "Under NCLB [No Child Left Behind], scores on standardized tests are the numerical metric by which success and failure are judged." In my opinion, "the progressive powers that be" should not evoke accountability or transparency as an excuse for blatant misuse of standardized test scores to judge, rank, threaten, or punish schools, teachers, and pupils. Have these people gone mad? It is anathema to publish test scores that should have been kept and used internally to better instruction. Also, the progressive idea that all kids are the same and can learn the same math content describes a utopia, not reality. For example, in the real world, there have always been students more knowledgable and skillful in math than others, etc.

The upshot of progressive math reforms is that students stumble over basic arithmetic because, for decades, many students have not routinized fundamentals to free mental space (working memory) for problem-solving. For example, in the largest southern Arizona school district, 87% of the students who enrolled at a community college in 2014 were placed in remedial math. In an adjacent district, the percentage was even higher at 88%. The lack of knowledge and skills is clear. The shocking truth is that we truly teach math badly. 

Daniel Willingham, a cognitive scientist, writes, "Schooling makes students smarter largely by increasing what they know, both factual knowledge and specific mental skills ... like learning procedures in mathematics." But, the 21st Century progressive reformists say that "technology has rendered memory unnecessary; what matters is learning to think," which is a flawed argument, says Willingham. Nothing beats knowledge and skills in long-term memory. Schooling makes you smart predominately because of gains in crystallized knowledge (acquired knowledge and skills) that can increase scores in math and reading). 

"Going to school boosts IQ," asserts Willingham.  Thus, practice (drill for skill) is important in schooling. For instance, once times tables or steps for a standard algorithm are memorized, they must be used again and again and again to stick in the long-term memory for instant use in working memory as needed. Willingham also points out, "Research on human memory indicates that academic content and the way it is sequenced--i.e., curriculum--are vital determinants of educational outcomes." For decades, our math content has been weak and curriculum has been disconnected, and incoherent. Math is hierarchical, that is, one idea builds on another, but it is seldom taught this way. Everything fits together logically, says mathematician Ian Stewart.

Regrettably, Common Core does not
easily accommodate fast learners and ignores STEM.

Academic Faddism = Stagnating Achievement
The Common Core math standards were not written for fast learners or to accommodate acceleration. Everyone gets the same, which implies no child gets ahead. Common Core also ignores STEM--a really bad decision. In my opinion, students must learn the basics first, which means they need to master standard [classic] arithmetic starting in 1st grade, not Common Core's brand of reform math, which is just another ill-advised, top-down academic fad. The greatest risk in blindly embracing the latest education fixes, whether they be Common Core, accountability, or technology--just to name a few--is wanting to believe that they will work, but they never do.

Common Core doesn't focus on standard arithmetic--not the way it is taught. Rather, it typically substitutes "reform math" strategies based on doubtful, arguable "mathematical practices." Consequently, students never seem to become adept at standard arithmetic. Regrettably, it is often taboo for kids to memorize, practice for proficiency (drill for skill), or even use straightforward standard algorithms.

Dr. W. Stephen Wilson, a mathematician at Johns Hopkins University, after reviewing a popular 5th-grade reform math program (Investigations), explains that kids never get to or learn standard arithmetic. Standard arithmetic, not reform math, is the foundation for algebra.

Mediocre Outcomes
The New Math (the 70s), NCTM reform math, popular group work & discovery/inquiry learning, No Child Left Behind, Common Core standards and the government's Race to the Top are "academic fads," writes Mark Levin (Plunder and Deceit), "for which trillions of dollars have been and are being wasted on inferior educational outcomes." 

The ideas, rules, skills, and uses of standard arithmetic to prepare for algebra have not changed. But progressive reformers, who are clearly in charge, have tampered with and radically altered standard arithmetic and its teaching by teaching kids alternative strategies (i.e., remnants of reform math as in NCTM math and now in Common Core ), which are not the same as teaching the real thing (i.e., standard arithmetic). Furthermore, the teacher's role has radically changed to that of being a "facilitator" of learning rather than that of being the academic leader.

The main goals in education should be excellence and equality (Michael E. Martinez), but for progressive reformists, the core issues in education are "fairness and equality--not excellence," explain Berezow & Campbell, who trace today's quandary in education back to "unrealistic faith [in] the magical solutions" of John Dewey, "the father of progressive education." They write, "But rather than keep what worked and improve what did not, Dewey set out to reshape education from the ground up." 

Today's progressive reformists have done just that to standard arithmetic and instruction with "bizarre education reforms." Their "ideology [utopia is truly possible] disconnects with reality." Regrettably, education is no longer "a matter of promoting excellence; it is a matter of pursuing political priorities," especially "social engineering," astutely observe Berezow & Campbell. Thomas Sowell (Dismantling America) calls "equalizing downward" the fallacy of fairness.   

The upshot of progressive math reforms is that students stumble over basic arithmetic because, for decades, many students have not routinized fundamentals to free mental space (working memory) for problem-solving. For example, in the largest southern Arizona school district, 87% of the students who enrolled in community college in 2014 were placed in remedial math. In an adjacent district, the percentage was even higher at 88%. The shocking truth is that we truly teach math badly, and faddish reform math is just another example.  The outcome has been that kids do not master fundamentals, which require memorization and practice, starting in 1st grade.  The progressive reforms in math content and the teaching of math (i.e., curriculum and instruction) totally failed these kids. Common Core as reform math, more technology, and NCLB "test & punish" are not the answers to our woes in education. The consequence of popular minimal guidance methods during instruction has been minimal learning. The progressive reformists say that group work and inclusion policies level the playing field, which is not reality as the kids coming through the school door vary widely in background knowledge, industriousness, and academic ability. Kids no longer drill for skill mastery, but they should!

Adding to the Common Core confusion, frustration, faddism, and extra baggage is a fallacy that students should be able to explain math to demonstrate understanding, but explaining something and understanding something are not the same. Even experts often have difficulty explaining themselves, so why should we expect little kids to explain themselves?  David Ruelle (The Mathematician's Brian) says that part of mathematical thinking takes place unconsciously and part nonverbally, so I am not sure how explaining an answer, which burdens and overloads the working memory of novices, points exactly to understanding, which is difficult to measure or quantify. In sum, explaining something should not imply understanding, but not according to Common Core. 

On the other hand, math is rule-based, which is marginalized by progressive reformists. Students must know and follow the rules to do correct mathematics. If they write proper steps and show standard calculations to get an answer, then those should be sufficient to imply some level of understanding, yet Common Core progressive reformists say that's not enough. Well, yes it is!

Also, when working a word problem, adept students should be able to extract the relevant numbers and operate on them abstractly to obtain a solution. It suggests that students must be given ample experience in recognizing patterns to figure out when to add, subtract, multiply, divide, apply a formula, write an equation, etc. To sum up, students need to be competent in basic arithmetic, but most are not. 

Note. This post is in first draft form. It is a collection of random thoughts, both new and old, in no particular order. Please excuse typos and other errors. Please ignore the reputation. 

Dr. Amber Northern, who is senior vice president for research at the Thomas B. Fordham Institute, comments on a new study about the effectiveness of instructional practices. She summarizes the report, "The most effective instructional practice teachers could use with these struggling students was routine practice and drills (that’s right, drill and kill!). Similarly, lots of chalkboard instruction, traditional textbook practice problems, and worksheets that went over math skills and concepts were also effective with them." Furthermore, according to the study's abstract I read, the teacher-directed techniques that worked for students with mathematical difficulties also worked okay for better students. But, I would add a caveat. High achieving students should be tracked into a fast paced class (acceleration) that is challenging, in my opinion.  Ze’ve Wurman jokes, “This [study] can’t be true! We all KNOW that working in groups and playing with strategies, and invented algorithms are the way to know math. And we KNOW that teachers should guide from the side rather than teach.” In summary, teachers don't teach standard arithmetic straightforward anymore. Many elementary teachers, who have been instructed and indoctrinated with reform math and progressive dogma and ideology, either in ed school and in professional development, don't know how to teach standard arithmetic, which has always worked well in the past when taught well.  They have been led to believe by the powers that be that teaching classic arithmetic is bad teaching. Well, it is good teaching. 

There has been a glut of “opinions” to improve achievement in our schools, but very few of innovations, policy fixes, or government solutions seem to work well because they are not based on solid evidence; consequently, improvements in achievement have been very slow in coming. I lament that I am not sure an easy remedy exists for our schools. However, I think there is a math nucleus from which to start. I do know that in the past kids who mastered traditional arithmetic, algebra, and trig through strong teacher guidance, memorization, and practice for mastery were well prepared for college-level courses in math and science. (These high school students also took traditional courses in chemistry and physics.) So, maybe we should look at the past to get students to the future. Briefly, students need traditional math and science courses. Einstein used the past to reach the future by echoing Isaac Newton who wrote, “If I have seen further than others, it is by standing on the shoulders of giants.” At the right scale, Newton’s laws of motion work well. You don't see scientists tossing them out because they are old school or hundreds of years old. 

In education, perhaps we are asking the wrong question. Perhaps, the problem is not the schools, but the students they teach, according to Robert Weissberg (Bad Students, No Bad Schools). No matter, too often educators and the powers that be follow the herd or proceed from consensus (conventional wisdom) or the opinion of so-called experts rather than from real evidence. Unfortunately, "conventional wisdom" and "experts" are often wrong. I question whether it is possible to close an achievement gap when students vary widely in background knowledge, industriousness, and academic or cognitive ability as they come through the school door. The main goals of education should be excellence and equality, but, even though there has been limited progress, they are miles apart. I question whether standardized test scores should dictate what teachers do in the classroom. And, I wonder whether the "best from the past" should have been abandoned by today's progressive reformists.

For example, classic arithmetic works well when taught well. It has had an excellent track record preparing students for algebra. Even though classic arithmetic works well, it is often dismissed or marginalized by progressive ideologues (the powers that be) because it doesn’t fit their dogma. In contrast to the ideologues, I believe that when standard arithmetic (ideas, rules, skills, and uses) is taught explicitly—with memorization and practice of fundamentals for automation built into the instruction—then students would learn standard arithmetic that advances them to algebra by middle school. It is drill that leads to skill: drill --> skill. Indeed, math fundamentals don't change; 8 x 125 is still 1000.  

I don’t understand much about the inner workings of a computer, but a limited or incomplete understanding doesn’t prevent me from using one. Perhaps the same should be said about long division or other standard algorithms. Why substitute “partial quotients” strategy for the real thing? Teachers often say they gloss over the standard division algorithm because kids don’t understand it, but is this a good enough reason not to teach something as fundamental as long division? Isn’t it more important that students know how to do it, use it, and apply it to problems, even if their understanding is weak, incomplete, or merely functional? The standard algorithms are beautiful in that they have unity and symmetry, that is, they all use the single-digit number facts and the structure of place value.

Few people truly understand Einstein’s counterintuitive ideas: objects shrink, time slows, and mass increases (as objects move very fast). These ideas are backed by experiment. “Nature truly is absurd,” says Richard Feynman. We teach them at some level. I take that back, the new science standards (Next Generation) leave out the quantum theory and what happens when objects move very fast.

Teach and explain the standard algorithm first, and then, when students are good at it, teach the some tricks or shortcuts using the rules of arithmetic to give students a deeper perspective. Don't use alternative strategies--such as lattice method, area model, or partial products--to teach multiplication.  Focus on the memorization of times tables and practicing the standard multiplication algorithm from the get go, and then introduce some cool tricks. Lastly, smart kids don't need to practice as much as other kids. Smart kids will need something different, more difficult and challenging.

Standard Algorithm (Classic Arithmetic) and Breaking Down Numbers (Common Core)
Kids need to master the arithmetic that is needed for algebra, but reform math slows their progress and doesn't get them there. Standard algorithms are based on place value and single-digit number facts.

Here is a 1st grade (first semester) addition example. 

Addition: 1st Grade

Standard Algorithm
It took me 3 seconds to add 65 and 37 using the standard algorithm, which always works and is the standard by which we should judge other methods or strategies. Note. The addition algorithm with carry can be taught 2 or 3 months into 1st grade (during the 1st semester).   I know; I did it. 

The standard algorithm works well when students memorize the addition facts for auto recall. I introduced the standard algorithm (at first, without carry) in the first couple weeks of school when students began to learn place value (23 is 2tens+3ones or 2t + 3) and to memorize, systematically, the single-digit addition facts. The problems I gave students were based on the facts they were memorizing. It is important that students drill for skill to put fundamental factual and procedural knowledge into long term memory. [See First Grade.]

Breaking Down Numbers (Make 100)
While 1st graders can do arithmetic well using the standard algorithm with larger numbers, very few 1st-grade students will be able to deal with larger numbers using the "breaking down numbers" approach, such as "make 100." In other words, when using larger numbers, the standard algorithm always works and moves students forward, but the "breaking down numbers" strategy (and many other reform math strategies) slows up and inhibits mastering the arithmetic necessary for algebra. Substituting the "many ways" of reform math for the classic one way has been the wrong approach. 

 In the second example, which involves breaking down numbers, I first have to think how to "make 100," which will likely take me more time because, as a first grader, I am a novice. If I figure out that 65 and 35 make 100, then I can proceed as shown above (65 +35 +2). But, there are many different ways to break down numbers, and the "many ways" often confuse and baffle 6 and 7-year-olds, which is the reason I don't recommend this approach for novices. For example, I can break up the numbers this way: 60 + 5 + 30 + 7. Add the 60 and 30 first, which is 90, and then add  5 + 7, which is 12, then add the 90 and 12, which is 102. OR, I could break down the numbers this way: 65 + 30 + 7, which is 95 + 7, but 7 is 5 + 2, so 95 + 5 is 100, then add it to 2 to get 102, and so on. Is any of this "breaking down numbers" stuff essential for adding 65 and 37? Of course not. It is merely "toying with strategies," which are not better or faster, says Ze've Wurman. 

From my experience and observations teaching 1st graders classic arithmetic, my best inference is that they can add larger numbers using the standard algorithm, armed with some understanding of place value, rules, and memorized single-digit number facts. However, typical 1st graders would have great difficulty with larger numbers using the "many ways" strategy of breaking down numbers as shown above, which, in my opinion, needlessly complicates classic arithmetic and confuses little kids. Kids are not adults. They are not little mathematicians. They are novices, not experts, but reformists keep forgetting this. Novices need consistency in method and structure to learn arithmetic well, such as the standard algorithms in classic arithmetic. The "many ways" of reform math (alternative strategies) often depend on nice numbers. They are inefficient and often waste valuable instructional time. Tom Loveless says that only the standard algorithm is specially mentioned in Common Core.  [Note. Students should not calculate single-digit math facts. It eats up working memory space.  They should memorize them for instant use in problem-solving.] 

Here is a 3rd grade (first semester) multiplication example: 16 x 125.
The standard multiplication algorithm is needed for kids to advance--not the lattice method or other alternative strategies often portrayed in Common Core reform math.

Multiplication: 3rd Grade

The standard algorithm took me 10 seconds.  It needs no extensive explanation. The 750 is 6 x 125 and 1250 is 10 x 125, and the reason you add these products is the distributive property of multiplication over addition: 125(6 + 10).

"Breaking Down Numbers" Strategy
In the "breaking down numbers" strategy, I have to know in advance that 8 x 125 is 1000. Suppose I forget the fact? Suppose I didn't know this trick? Like most strategies, the numbers have to be just right for this to work well. Furthermore, the lack of consistency confuses and baffles kids who are novices. Indeed, it is very fast, if you know the trick. Try 32 x 125. Quick. "It is 4000!" WOW! You must be a math genius just like Gauss! 

Instead of tricks and other alternative strategies, stick to the standard algorithm, which always works. When students are fluent in the standard algorithm, then it's okay to toss in a few interesting tricks as enrichment, but remember, tricks are nonessential. Do not introduce the multiplication algorithm by first working with partial products, lattice method, or the area model, etc. They are nonessential. Instead, teach and explain the standard multiplication algorithm and practice-practice-practice, that is, drill to skill.

Remember, kids are not little mathematicians, and they are not elementary school math prodigies like Carl Gauss. They are novices and need explicit instruction and lots of practice. Be sure to drill for skill.  

Barry Garelick argues that kids are below grade level mainly because they weren't required to master the basics. 
Amy Chau says that rote repetition is underrated in America. She explains that nothing is fun until you get good at it. Chau writes, "To get good at anything you have to work, and children on their own never want to work."
David G. Bonagura Jr. writes in the Wall Street Journal, "Contrary to today's education theories, memorization is critical in the classroom and life."

It is hard to find classic arithmetic taught straightforward in our schools.

©2015 LT/ThinkAlgebra.org
Comments: ThinkAlgebra@cox.net
Model Credit: HannahE, 7th Grade