Garelick gets it right

Barry Garelick Gets It Right! 

To Parents & Teachers

Barry Garelick (Teaching Math in the 21^st Century), who embarked on a second career as a secondary math teacher in California, disagrees with the way math is taught in our schools. Mediocre math achievement comes from bad policies and bad teaching starting in early elementary school. Garelick writes, “I believe strongly in how math should be taught and even more strongly in how math should not be taught" (p. 21). He points out that practice is the key to learning math well. He writes, "I believe practice is essential in mathematics; it results in automaticity which ultimately allows students to take on increasingly complex tasks” (p. 27). He disputes the value of block scheduling, group work and collaboration, inquiry/discovery activities, teachers as facilitators, far-fetched/nonstandard word problems, and much more.

Garelick observes that the “education establishment mischaracterizes traditionally taught math as being devoid of thinking and solving problems.” For multiple decades, traditionally taught math has been attacked by reformers, calling it obsolete and old school and branding it as poor teaching. Garelick, who has a degree in mathematics, says the reformers are dead wrong! He explains that the traditional teaching of math is not poor teaching. Indeed, traditional instruction demands mathematical reasoning, memorization, and practice to automate essential factual and efficient procedural knowledge in long-term memory for instant use in problem solving. Furthermore, traditional instruction is supported by the fundamental ideas of cognitive science and works well when taught well.

Garelick also finds fault with the eight Standards for Mathematical Practices (SMPs), which are the core of Common Core. He says they are unrealistic and have been strongly criticized by several mathematicians. The SMPs describe “expertise,” but kids are novices, not experts or little mathematicians. 

The reason some kids are below grade level by middle school is that they weren’t taught to "think" or "understand" in elementary school, which is the typical narrative among Common Core and 21st century reformers. But Garelick argues that kids are below grade level mainly because they weren’t required to master the basics. Mastering math requires memorization and practice. 

At a meeting discussing a struggling algebra student, the team contended that under Common Core, which is "more about understanding," the student "wouldn't be burdened with memorization of procedures," such as the quadratic formula. But, this line of thinking baffles Garelick: “How a student could be deemed to understand the quadratic formula without knowing it was puzzling” (p. 146).  Indeed, how can a student understand mathematics that he or she does not know well (i.e., able to do and apply successfully)? The idea that understanding is all students need is misguided! Understanding does not produce mastery; practice does!

Apparently, under Common Core, being able to do and apply the mathematics and get the right answer are not that important. But, they are important, critically important! Those who say they know or understand the math, but can’t do it or apply it, don’t know the math at all. Nobel Prize physicist Richard Feynman wrote, “You do not know anything until you have practiced.” Math is hard to learn compared to other academic subjects because it is abstract, cumulative, and requires focus, effort, perseverance, and practice to learn well. Barry Garelick thinks there is no substitute for practice. Without a solid background in basics, higher math becomes a struggle. Practicing math should be a daily habit.

There is much more to his experiences teaching math as a semester-long substitute teacher at a California middle school. Parents will be astounded at what he was told about Common Core at teacher meetings.

LarryT, ThinkAlgebra.org
Send Comments To: ThinkAlgebra@cox.net

Knowledge

Important Ideas from ThinkAlgebra 
1. Math Fundamentals Don't Change! 
2. Knowledge, both factual and procedural, in long-term memory fires up problem solving, innovation, and creativity.
3. Assault on Knowledge and Liberal Education.
4. Standards for Mathematical Practice (SMPs), the essence of Common Core, are unrealistic because they represent mathematical expertise. Kids are novices, not experts.

Comment. This post is in first draft form and was updated on 5-12-15. It is a collection of random thoughts, both new and old, in no particular order. Please excuse typos and other errors. Sometimes, I repeat and rephrase ideas. Repetition is a good thing! Quotes are from W. Stephen Wilson, Jason Zimba, David G. Bonagura Jr., Robert B. Davis, Ze've Wurman, Will Fitzhugh,  Lisa Hansel, Bruyckere-Kirschner-Hulshof, Steven Strogatz, Leslie Valiant, Barry Garelick, Ian Stewart, Zig Engelmann....  Reform math, including Common Core, which typically is implemented as reform math, has screwed up standard arithmetic by focusing on inefficient, alternative, non-standard procedures. It makes little sense to make standard arithmetic more complicated or difficult than it actually is, but this is what reform math does.  Plausibility, popular beliefs, and common opinion--without scientific evidence--often forge education policies. Comments can be made directly to ThinkAlgebra@cox.net.

::::: Teach basic arithmetic to automaticity and focus on one efficient method to calculate each operation (the standard algorithms). To learn arithmetic well, students need to memorize single-digit number facts and practice standard algorithms, including long division. Essentially, this is what we used to do back in the 60s! Elementary school students also need to know place value, master fractions-decimals, use formulas and write/solve linear equations, grasp ratio-proportion, be able to work many types of word problems (problem solving), and learn parts of geometry, measurement, and algebra. The best way for children to learn these things is through explicit instruction by explaining carefully chosen worked examples and giving feedback to students. The teacher is not a facilitator but the academic leader in the classroom. 

Mathematics Professor W. Stephen Wilson, Johns Hopkins University, writes that 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.”  This is exactly the situation. The memorization of single-digit number facts (numbers) and the fluency in standard algorithms (whole number operations) "give students power over numbers" and are key "basic skills and knowledge that a solid elementary school mathematics foundation requires," writes Dr. Wilson. He continues, “The case for the importance of the standard algorithms for whole number operations cannot be overstated. They are amazingly powerful.... They give the operations structure.” Indeed, standard algorithms should be taught immediately. Inefficient, alternative algorithms should be pushed to the side as enrichment. 

Leslie Valiant (Probably Approximately Correct), Professor of Computer Science and Applied Mathematics at Harvard University, points out that even though there has been tension between learning and reasoning, the "primacy will be given to learning, but reasoning will still be essential." This fits the basic idea in cognitive science that learning knowledge in long-term memory is the solid rock on which higher-level thinking is made possible. Note. Jason Zimba, one of the two major writers of Common Core math standards, recently acknowledged, "The standards also allow for approaches in which the standard algorithm is instructed in grade 1, and in which only a single algorithm is taught for each operation." Zimba's clarification connects well with the traditional teaching of arithmetic, often called old school. Old school math works well when taught well and is solidly supported by cognitive science, that knowledge, both factual and procedural, is the rock on which higher-order thinking is made possible. Just because something is old school does not mean it deserves to be tossed in the waste can, which is what hip reformers have done. Old school, in the teaching of arithmetic that prepares students for algebra, needs to be restored starting in 1st grade! The problem is that the "old school" teachers have all retired. There is hardly anyone left in teaching. 

Barry Garelick (Teaching in the 20th Century) is retired, has a math degree and real world experience applying math as an analyst. He embarked on a second career as a secondary school math teacher, but he has encountered difficulty finding a full-time teaching position where he lives in California. Administrators say he is too old to be hired as a full-time teacher or he is too "old school," or both, so he began as a substitute teacher for the 2013-2014 school year. He had two assignments: a long-term substitute math teacher at a middle school for 6 weeks and later at a high school for the entire 2nd semester.

Garelick explains that students should not be expected to solve far-fetched problems without prior knowledge and experience, only "variants of well-studied problems." He observes that good multiple-choice questions are valid ways to ascertain a student's math content achievement and reasoning. In addition, Garelick points out that the eight Common Core Standards for Mathematical Practice (SMPs) were written by non-math people and have been strongly criticized by several mathematicians and others. The SMPs represent mathematical expertise, but kids are novices. Also, during Common Core training, Garelick was told that "the content standards require students to work in groups, discuss, conjecture, critique each other arguments and that teachers are to be guides on the side." These ideas are unconvincing and do not come from content; they come from the SMPs, which are hardly effective in the teaching of mathematics to novices. In short, the SMPs are the essence of Common Core--not the content. However, in my opinion, the SMPs are unrealistic because they represent mathematical expertise. Kids are novices, so the SMPs are not a good fit. 
  
[Comment. The SMPs are nonsense mandates. Common Core reformers believe kids should become little mathematicians, that is, mathematical experts. Students should invent or discover math, work in groups, do lots of projects, critique mathematical arguments, and focus on novel or real-world problems that are far from the problem types they have studied. The mandates for mathematical expertise are unrealistic because kids are novices, not experts. Moreover, basic mathematics is not a matter of opinion or argument. It is based on true statements and consists of both factual and procedural knowledge. Ian Stewart points out, "Mathematics deduces new facts from old ones."

One SMP mandate is to make arithmetic [or any math] intentionally hard so that students struggle and learn to persevere. How stupid! The logic is flawed. Why make arithmetic harder and more complicated that it really is? Why insist that students learn inefficient, non-standard algorithms that they will never use for operations? Why give students far-fetched problems without the prior knowledge or experience needed to solve them? Is it any wonder that for multiple decades our kids have stumbled over simple arithmetic or algebra and lag far behind their peers in other nations?

Zig Engelmann rebuts the "dreaded" and "distasteful" SMPs especially for little kids: "Committees keep writing standards that are not based on empirical evidence of what children are able to learn about math and the specific technical details of instruction that cause the learning." Zig also points out, "The approaches that require less instructional time are superior to approaches that required more time." Thus, the popular inquiry, discovery, project, problem solving or other similar "minimal teacher guidance" approaches, along with the inefficient, alternative algorithms often taught as part of Common Core, misuse instructional time and often lead to "too little" learning. Efficiency in learning is vital, but not in the minds of reformers who stand by Common Core.  

Ian Stewart (Letters to a Young Mathematician) writes, "Mathematics deduces new facts from old ones.... New mathematical ideas build on older ones... Everything must fit together logically.... Mathematics happens to require a lot of basic knowledge and technique."] End of Comment 

::::: 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?" [Comment. Like many parents, mathematicians, too, are deeply troubled with the content and methods being taught in school math. Inefficient, alternative methods for calculating (e.g., lattice, partial products, etc.) are commonplace in reform math, including Common Core, which doesn't stress standard algorithms. Reform math ideas, often championed by professors in schools of education, greatly influence the curriculum, but the ideas just don't work well and squander valuable instructional time. 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 required for algebra, but reform math slows progress and doesn't get them there. Under a reform math regime, kids never seem to master simple arithmetic. In 1st grade, students should begin to learn the five building blocks of arithmetic as applied to addition and subtraction.] See W. Stephen Wilson below. 

For at least 50 years, reformers, professors in schools of education, and others have belittled the traditional teaching of arithmetic, calling it obsolete and old school (throw it in the trash can), and branding it as poor teaching. I disagree with those who claim that traditional teaching of arithmetic is poor teaching or obsolete because the fundamentals of arithmetic and algebra, which need to be automated through memorization and  practice, don't change. 

You don’t need a calculator to do math well, but you do need to memorize key facts and practice procedures a lot. "Algebra boils down to solving for x and working with formulas," writes mathematician Steven Strogatz (The Joy of x). You don’t need to be a genius or have a Ph.D. to do math. With a place-value system, ordinary children all over the world learn to do math. The catch is that you need to know some math in long-term memory, mostly arithmetic, algebra, and parts of measurement and geometry, to perform math and solve problems. In short, you need factual and procedural knowledge in long-term memory. And, the best way to get key math knowledge, such as single-digit number facts and standard algorithms, both of which are essential, into long-term memory is through memorization, practice, and repetition--not group work, discovery activities, discussions, or projects. In short, you have to memorize, study, and practice a lot of math calculations and word problem types. On the other hand, you do not need to know why "everything in math works the way it does," but it helps. "Even things one learns by rote represent the substrate, the raw material, of understanding," writes Barry Garelick (Teaching Math in the 21st Century). The more math you learn, the smarter you become. Also, it is fanciful to believe that students can do critical thinking to solve math problems without knowing the prerequisite math content.

The idea that you can do math without knowing math originates from trendy reform math programs such as Pearson's Investigations, which is still used in many schools. The reforms emphasize critical thinking at the expense of learning and mastering arithmetic and algebra [knowledge], which, in turn, are needed for problem solving. The "conventional wisdom" today is that kids don't need knowledge in long-term memory to be successful in the future, which is the reason that memorization and practice have been slowly disappearing from many classrooms over the years. The conventional wisdom, of course, is dead wrong!  The traditional teaching of arithmetic (old school) works and reform math ideas haven't worked well for multiple decades.  

::::: W. Stephen Wilson, Professor of Mathematics, Johns Hopkins University writes that the five building blocks of knowledge and skills in elementary school mathematics (K-5) are: 
(1) numbers, 
(2) place value, 
(3) whole number operations, 
(4) fractions and decimals, and 
(5) problem solving. 
Early elementary school basics or building blocks do not change and should be taught to all students. For example, the memorization of single-digit number facts (numbers) and the fluency in standard algorithms (whole number operations) "give students power over numbers" and are key "basic skills and knowledge that a solid elementary school mathematics foundation requires." Professor W. Stephen Wilson explains, "Mathematics is built level by level. Multi-digit addition and multiplication are built up from single digit operations using the place value system and the basic properties of numbers such as distributivity. The general operations reduce to the single-digit number facts." The whole-number standard operations (procedural knowledge) depend on instant recall of single-digit number facts (factual knowledge) from long-term memory. Teach basic arithmetic first (to automaticity), and focus on one efficient method to calculate each operation (the standard algorithms).

After harsh criticism, Jason Zimba, one of the two main writers of the Common Core math standards, acknowledges flexibility in interpreting the standards. He points out, "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." David G. Bonagura Jr. writes in the Wall Street Journal, "Contrary to today's education theories, memorization is critical in the classroom and life." He also writes in the National Review Online, "All students must learn to perform the basic mathematical operations of addition, subtraction, multiplication, and division in order to function well in society. Knowing why these operations work as they do is a great benefit, but it is not essential. In mathematics, concepts are often grasped long after students have mastered content--not before." Indeed, understanding grows gradually and changes over time, says the late Robert B. Davis (The Madison Project). 

Memorization plays a major role in learning basics.

Comment. Many claim that Common Core is about critical thinking, but the truth is, students can't do much higher-level thinking without a solid bedrock of lower-level thinking. Basic mathematical knowledge, both factual and procedural, which are lower-level thinking, need to be in long-term memory to enable higher-level thinking. Bonagura reminds us that higher-level thinking skills such as critical thinking or analysis are "impossible without first acquiring rock-solid knowledge of the foundational elements upon which the pyramid of cognition rests. Memorization is the most effective means to build that foundation." Indeed, "Memorization deserves to be reinstated to its foundational role in learning," not only in mathematics, but also in other disciplines such as science, English grammar, spelling, literature, and history, etc.

Regrettably, memorization and practice needed to automate fundamentals of arithmetic in long-term memory have fallen out of favor in many modern classrooms. Indeed, Common Core does not stress standard algorithms and substitutes inefficient, alternative algorithms for whole number operations, which is a faulty approach. 

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. 

(Note. Instead of teaching the standard algorithms straightforward, which requires students to memorize single-digit number facts for instant recall, Common Core often introduces inefficient or alternative algorithms, including the partial product and partial quotient algorithms. Consequently, students never get to or master the standard algorithms. The long-division standard algorithm, for example, is a lot more than just doing division. Students must have instant recall of single-digit multiplication facts, be able to subtract correctly, and be fluent in using the standard algorithm for multiplication. In addition, fractions are quotients, and long division changes them to decimal form. Why is 1/4 = .25? For example, 3/7 means 3 ÷ 7, which, through long division, shows an interesting repeating pattern: .428571428571.... "The long-division standard algorithm easily shows that rational numbers give rise to repeating decimals. It, by its very nature, also teaches estimation and begins to prepare students to understand convergence, a basic step toward calculus," writes Dr. W. Stephen Wilson.)

Reform Arithmetic vs Traditional Arithmetic
(Investigations, Grade 5, a popular reform math program)

Solution 2 is the standard algorithm, which should be the first and primary procedure for doing multiplication. It requires instant recall of single-digit multiplication facts and place value knowledge. In contrast, Solution 1 is an example of an inefficient, cumbersome procedure from reform math. Similar inefficient, alternative algorithms are promoted by Common Core. Common Core does not stress standard algorithms.

Note. Incidentally, the 5th grade multiplication problem from a popular reform math program is a typical 3rd grade practice problem in the traditional teaching of arithmetic. One of the best ways to practice the retrieval of single-digit number facts from long term memory is by stressing and practicing the standard algorithm, which is based on auto recall of single-digit number facts. 

In many of today's classrooms, literacy and numeracy are not that important. The trend against knowledge has been an epic error. Ze've Wurman says that teachers coming out of schools of education have an "intellectual emptiness" and are taught that "working well in groups is more important than knowing arithmetic, or that students being excited about learning trumps knowing history or science." But, engagement is engagement and not the same as learning, which makes changes in long-term memory. Learning requires memorization, practice, and repetition. Working in groups has nothing to do with learning math. The focus in school should be on literacy and numeracy, but, too often, it is not. (Quote Source: Ze've Wurman's "Forward" to Barry Garelick's new book Teaching Math in the 21st Century.) 

Professor Wilson says that we have lost the pro-arithmetic war. “How can I teach serious college level mathematics to students who are ill-prepared?" Ill-prepared means that the students don't know enough arithmetic or algebra to do college level mathematics. This is part of a larger trend in education, an assault on knowledge and liberal education, which extends from K-12 into the colleges and universities. The university should be more than a vocational school, says Fareed Zakaria (In Defense of a Liberal Education).
----   

::::: Assault on a Liberal Education: Math, Science, and the Humanities
Many of today's "digital" students and reformers do not grasp or savor the intrinsic merit of math, science, and the humanities such as literature, history, art and music. Students no longer read books to advance their knowledge. The reformers say the era of books is gone. Will Fitzhugh (The Concord Review, 2005) writes, “Some educators have decided that we are in a “post-literate” era, where books, writing, and all those things we associate with the literate era of the past, are no longer so important." He continues, "Many educators also seem to have acquired the idea that students need almost no knowledge, because they can always “look it up,” presumably online." Regrettably, reformers have waged war on knowledge, even though domain knowledge in long-term memory is the rock on which higher-level thinking is made possible. There has been a shift from a knowledge-based education to a skills-based, practical education, along with a unintelligent bias of "out with the old" and "in with the new." 

"Out with the old," even if it worked, is twisted thinking and progressive ideology. If an idea, approach, or strategy is new, or uses the latest technology, or is technologically hip, then it must be innovative and good for digital kids. Yet, in education, almost all new ideas, reforms, or innovations have not grown to be productive programs. They are trendy fads that come and go. Similar to many progressive reform ideas that failed in the past, evidence isn't needed, just good intentions. Reformers assert that education needs to enter the 21st century--not redo the 20th century or the 19th century--no matter what. Old stuff, they say, has little value in today's world. Reformers chuck out old technology (blackboards, white boards) and stuff they don't like, such as long division or cursive writing. They cut back on the humanities and belittle arithmetic and algebra, etc. Liberal arts colleges are now less popular. The reformers, I think, are wrong! How dull this world would be without liberal education (science, math, and the humanities such as literature, languages, history, art, music, etc.). 

In contrast, scientists did not toss out Newton's Laws of Motion & Universal Gravitation because they are old. They work at the right scale and are key parts of classical physics. Likewise, educators should defend, maintain, and build on what has worked in the past, not discard it. Indeed, long-established teaching of traditional arithmetic has worked well when taught well and practiced well (i.e., for mastery). In traditional teaching of arithmetic, for example, the automation of factual and efficient procedural knowledge needed for problem solving is a good fit with the findings in cognitive science. Yet, the reformers call the traditional teaching of arithmetic obsolete and old school and brand it as poor teaching. 

::::: Fundamentals in math don't change. 
The math content in school math, such as, but not limited to arithmetic, algebra, and geometry, is old. Nonetheless, the traditional teaching of math is certainly not obsolete or outdated, as many reformers claim, because it forms a necessary foundation for higher math. The newer discoveries in math are for mathematicians (Ph.D. level) and, therefore, are far, far above the K-12 school or regular collegiate math curriculum levels, so there is little chance that the latest math content will reshape or influence the math fundamentals taught in K-12 school math. The idea that math knowledge is changing so rapidly that it is impossible to keep up with is nonsense at the K-12 school math level because the basics do not change. The fundamentals may be old, but they won't change because math content is hierarchical, cumulative and logical, that is, one idea builds on another, etc. The generation of the natural numbers start with two assumptions that 0 + 1 = 1 and 1 + 1 = 2 or n +1 = n'.

While technology can advance rapidly, the fundamentals of arithmetic and algebra stay the same. For example, multiplication is still multiplication, but simplified and much improved (i.e., standard algorithm) over the ancient Babylonian method. Unlike the ancient Babylonian method of multiplication using a table of squares (about 2000 BC), which is complicated, today's standard method of multiplication uses memorized single-digit multiplication facts, which makes it fast, efficient, and easy to learn. Students should learn and practice for mastery the most efficient procedural methods for operations, which, typically, are called the standard algorithms. (Unfortunately, Common Core reform math adds unnecessary complications that slow the acquisition of key content in long-term memory. Students learn inefficient, alternative algorithms. Common Core reform math does not simplify or improve mathematics. My message to reformers is simple. Don't make simple arithmetic harder than it is.) In short, no matter what the reformers claim or believe, the fundamentals of arithmetic and algebra are essential and need to be taught efficiently and mastered by most students to prepare for the future.   

FYI: One formula for Babylonian multiplication is 
ab = [(a + b)^2 - (a - b)^2]/4. 

----------

There have been many attempts to upend knowledge in schools and replace it with [out of context] skills. A basic tenet of cognitive science is that broad knowledge is needed for problems solving and critical thinking to meet the needs of the future. The knowledge that kids learn in school has not changed very much over time. Even though kids are bombarded with information, broad knowledge can only be obtained through the thoughtful study of the liberal arts.  Thus, the progressive, anti-knowledge movement, which is prevalent in schools today, is irrational. Knowledge in long-term memory fires up creativity, innovation, and problem solving. This is basic cognitive science, that higher level thinking (problem solving or critical thinking) is the product of lower level thinking (knowledge).

Lisa Hansel (The Core Knowledge Blog) wrote, "I hear this all the time that information is growing at a shocking rate, and that today's knowledge will be out of date before students graduate. Obviously, students don't need knowledge, they need to learn how to find knowledge." The solution is Google, reformers say. Well, actually, no! Kids need to study the liberal arts in a well-rounded curriculum to gain organized knowledge in long-term memory. Googling information is not the same as acquiring knowledge in long-term memory, which takes practice.

In short, very little of the information born from the latest, cutting-edge research affects elementary and secondary education or even college, say Bruyckere, Kirschner, & Hulshof (Urban Myths about Learning and Education). "The fact is that much or most of what has passed for knowledge in previous generations is still valid and useful."  In my opinion, it is irresponsible to disparage knowledge or to say that kids don't need knowledge. Knowledge is the rock on which higher level thinking is made possible.


Please excuse typos and other errors in this post. Rough Draft
Model Credit: Remi

©2015 LT/ThinkAlgebra
Comments: ThinkAlgebra@cox.net

DeepUnderstanding?

This post is a collection of random thoughts, both new and old, in no particular order. 
Please excuse typos and errors. 

For at least 50 years, progressive reformers have attempted to downgrade knowledge by attacking the teaching of tried-and-true traditional arithmetic, calling it obsolete and old school and branding it as poor teaching. The progressives have denigrated memorization and practice for mastery, which drive essential mathematical knowledge, both factual and procedural, into long-term memory for instant use in problem-solving.  

It is what we have known down through the ages: Math knowledge in long-term memory fires up problem-solving, creativity, and innovation. In short, without knowledge ( i.e., lower level thinking), there is no higher-level thinking. Also, "It is pretty hard to understand mathematics without doing some mathematics," writes Professor Jordan Eilenberg (How Not To Be Wrong: The Power of Mathematical Thinking).

Note. The standard algorithm empowers students and requires automation of number facts. 
It should be the primary method of calculation from the start of 1st grade, yet very few teachers teach traditional arithmetic, which has been shunned as "old school" and branded as "poor teaching." It's not true! The fact is that high-level thinking requires a solid foundation in low-level thinking, which is called knowledge. Kids can't do arithmetic unless they know some arithmetic--factual and procedural knowledge--in long term memory. Indeed, memorization and practice-practice-practice are essential for building a storehouse of knowledge in long-term memory needed for problem-solving. Memorization and practice are not obsolete, as some claim. They are essential and are firmly supported by cognitive science.  It is important to acknowledge that memorization and practice-practice-practice build fundamental factual and procedural knowledge in long-term memory--knowledge that engages and enables problem-solving in mathematics, says Daniel Willingham, a cognitive scientist. 

First-grade students should figure out (via number line) 
and memorize n + 2 by the 2nd or 3rd week of school. 
Teach place value as applied to operations: 13 = 1ten+3ones 
or t + 3. Grids are often helpful for beginners.

 1. Jason Zimba, one of the two major writers of Common Core math standards, recently stated, "The standards also allow for approaches in which the standard algorithm is instructed in grade 1, and in which only a single algorithm is taught for each operation." I am a proponent of teaching the standard algorithm from the get go in 1st grade. Students are novices; they do not need a perfect understanding of regrouping or place value to use the grid. Understanding grows slowly. At first, only a functional understanding is needed. The standard algorithm is the focused goal for calculating and should be presented at the beginning of the school year.  Traditional arithmetic when taught well works well.  

Bloom's Taxonomy in TIMSS

2. Tom Loveless writes, "Deeper Learning is the current term for an old idea.  The notion is that schools spend too much time focused on the acquisition of knowledge, especially knowing facts.  In the past century, several alternatives have arisen to dethrone the prominent role of knowledge in schools: project-based learning, inquiry and discovery learning, higher-level thinking, critical thinking, outcome-based education, and 21st Century Skills.  Now it is deeper learning." Deeper learning has been yet another attempt to turn Bloom's learning taxonomy upside down and upend knowledge. According to the fundamentals of cognitive science, it is the lower level thinking (i.e., knowledge) that enables and fires up higher-level thinking, such as problem-solving, creativity, and innovation. Why are some educators continually trying to marginalize the prime importance and impact of factual and procedural knowledge in learning arithmetic well?

3. Kirchner-Sweller-Clark (Why Minimal Guidance During Instruction Does Not Work...) write, "Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process.” Students' desks are put together for group work and inquiry activities, which often use manipulatives. This configuration and minimal guidance methodology are not conducive to learning. The teacher is not the academic leader in the classroom. The teacher's role has been diminished to that of a facilitator.  High-level thinking requires a solid background of low-level thinking, which is called knowledge.

4. Michael E.Martinez (Future Bright) writes, "Research supports a view of intelligence as both lower-order and higher-order. The mind's ability to engage in higher-level operations must in some way rest on a foundation of lower-level functions." This instructs teachers to focus on lower-order thinking first, that is, both factual and efficient procedural knowledge in arithmetic and algebra. Always remember, kids cannot operate on knowledge they don’t have. 

✏️ Traditional arithmetic is tried-and true-and works well when taught well. But teachers coming out of schools of education have been taught to teach reform math by progressive-minded professors and to shun traditional [old school] arithmetic. Unfortunately, teachers are taught to discourage stacking numbers (vertical addition). They have been told that the standard algorithm is not an acceptable strategy, even though it is the "simplest, most efficient method for solving addition problems." Instead of traditional arithmetic, children are taught reform math strategies that use graphical methods (drawings) or counting and this (somehow) "magically" shows "deeper understanding," but it doesn't. The approach is often tedious, mostly unnecessary, and ineffective.

The reality is, kids can't do arithmetic unless they know some arithmetic (factual and procedural knowledge) in long term memory. And if they can't do arithmetic well, then they won't do algebra well, etc. Indeed, popular, minimal guidance instructional methods, such as project-based, problem-based, inquiry, discovery, constructivist, etc. are mostly ineffective and a waste of valuable instructional time. The popular methods are a mismatch to what most kids need, especially poor kids from single-parent households whose opportunities for success (i.e., catching up to the middle class) as adults are limited. Disadvantaged children, who come from one-parent households and are miseducated in schools will not have a shot at success. Poor children will become poor adults and dependent on government assistance programs. They will need more than ability and hard work.
Kids, today, need to learn more math than previous generations because high-tech jobs demand it. 

On the other hand, memorization and practice are not obsolete; they are essential for putting necessary [key] knowledge into long-term memory.  Students acquire number sense and become better at mental math because they have memorized number facts. In short, memorized facts enable students to become fluent in standard algorithms and better at problem solving. 

Traditional Arithmetic
Get to the standard algorithm as quickly as possible.
In the early 80s, I gave a problem to my first-grade class in the first-month school. What is 32 + 37?
Background: Kids were already in counting exercises in the first full week of school and used cubes to measure, to count, and to work equations I made up. For example, I would write an equation on the board, such as 3 + 8 - 5 = box. The 3 means to put 3 cubes in the counting circle; the +8 means to add 8 more cubes in the same counting circle, and - 5 means to take out (remove) 5 cubes from the same counting circle. I circulated the room to make sure students were following the operations correctly. The box, which is a variable like x, means how many cubes are in the counting circle? After the three operations, the students would count the centimeter cubes in the circle, which is 6, so box = 6. The equations became more complex:  12 - 7 + 10 - 3 = x.

In a couple weeks, I gave them 31 + 27 = x. (Note. Kids started to memorize n + 2 addition facts. They also learned that adding 1 generated the next whole number: n + 1 = n'.) Kids were put in groups because there were not enough centimeter cubes to go around. (Note. By this time, students were used to breaking down numbers by place value. Thus, 31 was 3tens+1ones or 3t + 1 (We write ordinary numbers, such as 25, in positional notation.)  The counting method describes above took a lot of time, and, often, the answer was not right. I asked students to do it again to see if they came up with the same answer. Some groups did, some did not. No matter, counting to add larger numbers was an arduous process for 6-year-olds. Counting is not an efficient method for adding and subtracting.

I wrote the following on the blackboard.

I said that I can add these number in about 2 or 3 seconds and without counting cubes. They were skeptical, yet curious. "The answer is 58," I said. 

How did you do that? Students wanted to know! NOW! Remember the addition facts we started to memorize? I demonstrated how to do it by adding ones to ones and tens to tens, using memorized single digit math facts. It was a revelation to the students. There is a simple way to add, rather than counting cubes or making drawings, etc. We put away the cubes. And, so the standard algorithm was born. Place value is very important: the idea that 58 means 5tens 8ones or 50 + 8.  

The standard algorithm empowers kids!

A grid is helpful for some students.

I showed the algorithm first and explained why it works once students were good at using it through practice. For practice, a grid is helpful for novices to separate tens and ones.  Give practice problems in horizontal form for kids to rewrite in vertical form. Procedure: add the ones, then add the tens. 

At first, explaining why something works is not as important as making sure students are competent at using the algorithm. To use the algorithm well, students need to automate addition facts. Students should master the algorithm without carry before proceeding to the carry concept. 

Note. Children do not need perfect understanding to move forward. Understanding grows slowly. For example, Isaac Newton did not know why his calculus worked; it just did.  (The conceptual understanding for calculus would not be developed for another 200 years.) However, Newton knew how to apply his calculation methods to solve physics problems. His calculus answers always matched the experimental values. In short, the calculus always worked. 

An important addition application for 1st graders is to find the perimeters of polygons. Also, 1st graders should do a lot of word problems. They need to learn when the add, or subtract, or multiply (repeated addition) from word problems just like Singapore 1st graders. 

Also, 31 + 27 = x is an algebraic equation and, therefore, follows the rule of equality, which states that the left side must be equal to the right side for the equation to be true: 58 = 58. Furthermore, to solve the equation, the student needs to do arithmetic. A numerical equation that shows the solution would be 31 + 27 = 58.

Word Problem: Jill has 31 pencils, and Jack gave her 27 more pencils.
How many pencils does Jill have now? Write an equation that shows the solution.

The equation is the model.

To me, writing an equation that shows the solution to the arithmetic off to the side is showing your work. The operation required to solve the word problem is addition, which the equation clearly shows. It is called mathematical language, and it speaks for itself. The number and operation symbols convey the concepts. 

Note. All these things require practice-practice-practice so they stick in the long-term memory.


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