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!
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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)
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| Standard Algorithm 12 Seconds |
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).
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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.
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| These are examples of some of the alternative multiplication strategies. 5th Grade Common Core State Standards |
Indeed, time is better spent on explicit teaching of standard algorithms (not alternative strategies) and practicing them for mastery.
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.
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
