Monday, October 24, 2011

Science in Elementary School

Kids are left behind in science.

What has happened to science in elementary school? 
With an emphasis on math and reading, science has been pushed to the side in many elementary school classrooms [1]. There is no time and little equipment. Often, K-8 teachers do not know enough science to teach it well. Furthermore, the math needed to do science is seldom taught or introduced. Science is highly mathematical, yet K-8 science textbooks and teachers seem to limit the math. None of this is new. Science has been neglected for decades. 

The thinking required in science, however, is different from the thinking done in school math. Mathematical statements are shown to be true or false by following a set of rules called number properties (axioms) of operations, equality, etc. [2]. One true statement forms the basis for another true statement and so on. This is the way math knowledge builds: one idea builds on another. In short, the rules (properties of numbers) in math do not change. The definitions in our number system do not change either. A fraction (rational number) will always be represented by the quotient of two integers (a/b, b ≠ 0). Equivalents like 3/4 = .75 = 75% will not change; 7 is always 6 + 1, etc. Equivalency and substitution are important ideas in mathematics. 

In science, however, there is no "true or false" like in school math. Instead, there are observations and inferences. There are no absolutes in science. The rules in science can change based on "partitioning by scale." In short, Newton's laws of motion still work, but not at the atomic (very tiny) scale.

There are facts, such as the number of protons in the hydrogen atom, etc. Kids must know facts (background knowledge). In addition, they also must know how to measure (make observations), how to draw valid inferences (conclusions) based on observations (data), how to minimize confirmation bias and errors in experiments, how to do the required math, how to communicate results with charts and graphs, and how to distinguish between correlation and cause-effect. Above all, students must learn to distinguish between observation and inference. But, this is not the organizing principle of science textbooks. Too often, TV programs, documentaries, news programs, textbooks, and other materials blend the two. Often, students interpret an inference as fact. 

Students should do science projects that have clearly defined independent and dependent variables and control.  Moreover, students should be taught what the late Richard Feynman calls intellectual honesty in science, something that is often lacking. In real science, we bend over backward to prove our conjectures wrong through experiments. We also present data that does not support our conjectures. We do not fudge data. And, we do not extrapolate beyond known data, i.e., make an inference based on an inference. An inference based on another inference is a misguided conclusion. Such extrapolation of data is more common than most people think. A common example would be a computer model of a complex system [stock market, weather, etc.] that attempts to forecast the future [based on the past] and makes unproven claims. This is not science; it is speculation

Feynman states, in a lecture, that physicists guess theory, then they test it. "If [we guess a theory that] disagrees with experiment, then it is wrong." Science is not based on, authority, opinion, consensus, or political agenda. It is based on an experiment. Furthermore, experiments must be repeatable and peer-reviewed. Lisa Randall, a particle physicist, writes, "People too often confuse evolving scientific knowledge with no knowledge at all and mistake a situation in which we are discovering new physical laws with a total absence of reliable rules." 

Dr. Randall clarifies, "Science evolves as old ideas get incorporated into more fundamental theories. The old ideas still apply. The wisdom and methods we acquired in the past survive. Today's methodology began in the seventeenth century." Kids should study Newton's laws of motion because they still apply. Randall says that we can still measure pressure, temperature, and volume because they are real quantities. In short, fundamental scientific knowledge is important and should be stressed in school. 

Regrettably, many of the elementary school science textbooks I have seen are incomplete and often perpetuate misconceptions. They are almost math-less, which misleads students. The real world is explained (modeled) through equations. The textbooks do not teach what science really is. One fundamental idea is that scientists try to prove ideas wrong, not right, by carefully crafted experiments; i.e., science does not prove anything right. Scientists seek out counterexamples and correct itself by getting rid of false ideas. According to Karl Popper, every theory must be falsifiable.

New Science Framework

The new science Framework from the National Research Council, oddly enough, was written by the Division of Behavioral and Social Sciences and Education and its committee, which is made up of mostly of educators, not real scientists. In fact, the "science content" experts (i.e., the design team) were not allowed in meetings in which the final decisions (consensus) regarding content were made (p. 17). Surely, You're Joking. No! Also, read the K-12 Science Framework.
I was disappointed after reading parts of the new K-12 science framework from the National Research Council (July 19, 2011).  The Framework committee merges science with engineering and technology, skimps over math needed to do science, stresses scientific processes (called "practices" in the document) over content, requires little content knowledge in elementary and middle school, and combines chemistry and physics, leaving important content out. 
In my view, the new Science Framework from the National Research Council is flat. It does not challenge children, and it does not paint a true picture of what science is all about. The fundamental idea, that science does not prove anything right, is missing. The Framework also lacks a historical perspective. Missing are the great scientists and how they changed the focus of science, e.g., Galileo, Dalton, Maxwell, Bohr, Heisenberg, Plank, Dirac, Einstein, Feynman, Higgs, etc. Moreover, the Framework lumps technology and engineering together; however, they are applications or products of science, not science. The framework skimps on chemistry and physics and the math needed to do the science. Is this the best we can do? It is disappointing! 
Note. I wrote an analysis of the Framework last summer (July 2011). It is very long. Here is a snippet: The Framework writers insist that a hypothesis (or theory) is not a guess. It is. This is what scientists do--they guess or make conjectures, then they test to see if the guess can be shown false. If an idea (guess) is not testable, then it is not science. We need to teach the testability principle by experiment to kids learning science. David Deutsch (The Beginning of Infinity, 2011) writes that conjecture (making a guess) is the real source of all our theories. Theories must be testable. He writes, “Knowledge must be first conjectured and then tested.” In science, we do not rely on authority or opinion. We have “a tradition of criticism,” says Deutsch. The bottom line, according to physicist Richard Feynman is, “If it does not agree with experiment, then it is wrong.” In other words, real science self-corrects itself over time. Ideology does not. Science does not prove ideas right; it eliminates wrong ideas. 

Many old ideas (e.g., Newton's laws of motion) are correct but incomplete. The laws work well at one scale, but not at another scale (e.g., atomic). We should not toss out Newton because his "laws" are incomplete. Lisa Randall, a particle physicist, says that many of the old ideas apply and have practical applications at the right scale ("appropriate conditions"). This [the scales] is what we should teach kids. Scales are an organizing principle in science. 

Elementary and middle school kids should learn Newton's laws of motion and the historical contributions of scientists like Galileo. The radical methods pioneered by Galileo in the 17th century are still used today: proof by experimentation (not authority, opinion, or consensus), thought experiments, and the use of technology to extend our senses to make better observations. For Galileo, the technology was the telescope. Technology plays an important role in science, but it is not science. by LT, ThinkAlgebra, July 2011 

If we want students to understand the world, then we should teach them substantially more physics and mathematics early on. Furthermore, we should establish math and science standards that, at the least, match the benchmarks from nations that excel in these academic disciplines. In my view, the new K-12 Science Framework does not do this. The Science Framework is the latest version of science education written by a committee made up of mostly nonscientists. It is off-target because it requires very little knowledge of math needed to do science and very little science content knowledge. The committee's makeup and its frame of mind in composing the framework are troublesome. It is not the best we can do.  It is not even close. And, as ZE"ve Wurman, a critic of Common Core math standards, explains, the conceptual science Framework is "science appreciation" all over again.

Also, read  Most Kids Don't Understand Science by ThinkAlgebra

[1] A report supporting my observations was released at the end of October (Strengthening Science Education in California). The report states the obvious: little science is taught in elementary school. But, neglecting science in grade school is not new. In my experience, not much science has been taught in elementary school for decades. Middle school science has gone downhill, too. There is not enough stress on basic science content, reading science, and learning the math needed to do science. Furthermore, many teachers are ill-prepared to teach science. 10-28-11

[2] There are not that many properties. A few of the basic properties [axioms] of numbers that should be learned in first grade in the first month or two of school are: add zero [identity] property, add one property, commutative property of addition (2 + 3 = 3 + 2), equality [or equivalency] property (2 + 3 = 1 + 4), "add in any order" property (3 + 4 + 7 is 10 + 4 or 14), etc. The idea that teaching arithmetic to 1st graders should use a framework based on number properties, rather than counting, is absent in American programs. Morris Kline writes, "Axioms are suggested by experience and observation. Kline also writes, "Operations on numbers [must] give a result that fits our experience." He states that "axioms are useful when our experience fails us or leaves us in doubt." Indeed, axioms (number properties) come in handy as kids learn arithmetic. For example, 3 + 5 = 10 - 2 is a true statement because of the transitive property of equality. In "little kids" talk, both 3 + 5 and 10 - 2 name the same point on the number line and, therefore, are equal to each other (equivalent). 

Mathematician Morris Kline (Mathematics for the Nonmathematician) states that operations (let's say, fractions) are designed to "fit experience." Arithmetic facts and operations are learned mostly by rote, but students should also be aware of the axioms or properties (e.g., commutative property of addition and multiplication) that govern operations to determine whether or not the mathematics is correct. For example, I can explain why 1/2 of 1/3 is 1/6 on the number line, but this type of understanding does not come into play when students are multiplying fractions (e.g., 2/3 x 3/4). In short, when applying the multiplication of fractions algorithm, students do not think in terms of marking off 3/4 of one whole on a number line, then dividing each fourth into thirds, which gives 12ths (but from 0 to 3/4, there are nine equal parts or ninths. Converting 2/3 to 9ths = 6/9). Counting over 6 tick marks, you end up at 6/12 or 1/2, etc. Sounds confusing? It is to many kids. 

Furthermore, making a number line model for fractions with larger numerators or denominators becomes a total mess. The multiplication of fractions algorithm can be inferred from a number line demonstration. Students should be taught to depend on efficient methods (algorithms, step-by-step procedures, operations on numbers, or recipes) that produce correct answers fast

The multiplication of fractions algorithm can be "formulated" by the number line idea and other clues by the 3rd or 4th grade. For example, ½ of a number (e.g., 1/2 of 10) produces a smaller number, not a larger number (½ of 10 is 5; it means ½ x 10 = 5). We know this by experience and develop a multiplication of fractions algorithm so that the answer is always correct. (See Example 2 below)

In division, 5 oranges divided into halves is 10 (halves). In arithmetic, this is 5 ÷ ½ = 10. To divide by ½ gives the same result as multiplying by 2/1 (the reciprocal of the divisor. This is invert and multiply). The algorithm for the division of fractions is formulated to fit experience. In short, 5 ÷ 1/2 = 5 x 2/1. Thus, to divide by any number, multiply the number by the reciprocal of the divisor and then apply the multiplication of fractions algorithm. In short, students change division to multiplication. 

In Example (1), adding the fractions should produce a larger fraction. Thus, the idea of adding the numerators and adding the denominators does not work because it does not fit our experience. Adding fractions can be represented by adding lengths on the number line. The answer is greater than one, not less than 1. The algorithm does not work. 

In Example (2), multiplying the numerators and multiplying the denominators works 
(fits our experience). It is the algorithm that kids are taught to use when multiplying fractions. A fractional part of any number (fractional part must be less than 1) produces a smaller number. (But, 3/2 x 7/5 will produce a larger number because 3/2 is 1 + 1/2 and 7/5 is 1 + 2/5. This is consistent with our experience when both factors are greater than 1.)

Algorithms (operations on numbers) must fit experience, be efficient, and produce the correct answer. 
10-24-11, 10-28-11, 11-1-11, 12-223-11

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Photo Credit: Hannah by LT
©2011 LT/ThinkAlgebra

Friday, July 15, 2011

Problem Solving

Reform math programs launch into "problem-solving" backward by de-emphasizing the grade-by-grade mastery of basic arithmetic knowledge, both facts, and procedures, which are the very essentials needed for problem-solving. Polya's problem-solving strategies work only if the student has sufficient prerequisite knowledge needed to solve a specific problem. Thus, problem-solving is always deeply rooted in background knowledge in long-term memory. The primacy of background knowledge cannot be over-emphasized. Elementary school teachers should focus on making sure students master the fundamentals of arithmetic, both factual and procedural knowledge, starting in grade 1. A curriculum that is focused on problem-solving strategies and light on content knowledge does not cut it.  ThinkAlgebra [Draft I]
Let’s start here . . .
“Mathematics has the dubious honor of being the least popular subject in the curriculum . . . Future teachers pass through the elementary schools learning to detest mathematics . . . They return to the elementary school to teach a new generation to detest it.” You would think that this was written in 2011, but it wasn't. G. Polya was so concerned about math education that he wrote it in the 2nd edition preface (1956) of his famous book, How to Solve It. Actually, Polya quoted it from a study reported in Time magazine. Not much has changed in the past half-century. I think some teachers are doing a great job teaching math. We just do not have enough of them.    

This cycle has been entrenched in education for at least five decades because schools of education are not selective or academically demanding. I do not blame teachers; I blame those in charge of selecting, training, and certifying teachers. If we want better teachers in elementary and middle school math, starting in 1st grade, then we need to educate and train them better and weed out teachers who dislike mathematics, who are mathphobic, or who demote the importance of mathematical [content] knowledge. Poyla believes a teacher's knowledge of mathematics and attitude toward mathematics rub off on students. He writes, "Yet it should not be forgotten that a teacher of mathematics should know some mathematics and that a teacher wishing to impart the right attitude of mind toward [math] problems to his students should have acquired that attitude himself." The book was first published in the U.S. in 1945.

Polya poses a problem.
The length of the perimeter of a right triangle is 60 inches and the length of the altitude perpendicular to the hypotenuse is 12 inches. Find the sides?

A student cannot solve this problem without substantial knowledge of high school mathematics (algebra and geometry). But, isn't prerequisite knowledge necessary for any math problem, at any level, even for routine problems? Knowledge first! In mathematics, students should start with basic arithmetic and routine problems to build a storehouse of knowledge and experience in long-term memory before moving to more complex problems that take more insight. The idea that students can do problem-solving without fundamentals in place is illogical and backwards, yet this is what many teachers think. Elementary students should focus on mastering basic arithmetic and routine word problems, grade by grade. This requires solid practice.

I pose a chemistry problem.
Calculate the grams of hydrogen required to produce 82.000 grams of ammonia from nitrogen and hydrogen gasses.

Would you attempt to solve this routine chemistry problem without knowing the fundamentals of high school chemistry? Of course not! Solving problems requires domain-specific knowledge. Moreover, learning to solve routine problems, whether they be in chemistry or elementary school arithmetic, presupposes both knowledge and practice.   

I pose a Latin problem.
Ego vos hortor ut amicitiam ombibus rebus humanis anteponatis. Sentio equidem, excepta sapientia, nihil melius homini a deis immortablibus datum esse. 

Would you attempt to translate Latin without knowing the fundamentals of Latin? Of course not. Translating Latin requires domain-specific knowledge. 

Knowing builds the foundation for higher-level thinking.
You cannot apply something you do not know well.

Knowing builds the foundation for higher-level thinking . . .
The range of cognitive skills (right), starting with a strong base of Knowing, is similar to Bloom's taxonomy. Applying requires Knowing, and Reasoning implies both Knowing and Applying. Teachers should start at the bottom and focus on Knowing (both factual and procedural knowledge in arithmetic and algebra). This builds the foundation for higher thinking, such as Applying and problem-solving. In TIMSS, Applying is solving routine problems. (This is problem-solving.) Furthermore, knowing something takes substantial practice. You cannot apply something you do not know well [in long-term memory]. 
We tend to believe what we think, but our assumptions are often wrong. 
Sweller, Clark, and Kirschner [2] write that the results of research in problem-solving in mathematics are “both counterintuitive and contrary to many widely held views. For example, many educators assume that general problem-solving strategies are not only learnable and teachable but are a critical adjunct to mathematical knowledge.” The assumption is wrong and unproven. It is a gross misinterpretation of Polya by many math educators, special interest groups (e.g., P21), and others. 
Practice Fundamentals
Knowledge Is Key
According to Polya, when a student attempts to solve an unfamiliar problem, the student should think of a related problem and then, by analogy, try to solve the original problem. He writes, “We may consider ourselves lucky when, trying to solve a problem, we succeed in discovering a simpler analogous problem.” But, thinking of a simpler analogous mathematical problem, solving it, and applying it (“extrapolating” it to solve the original problem) require specific content knowledge. Moreover, a simpler analogous problem can have both similarities and dissimilarities. As one reads Polya’s book, it becomes clear that students need extensive mathematical knowledge and “determination” to do problem-solving. In short, problem-solving in mathematics requires sufficient mathematical knowledge in long-term memory, practice, and “determination.” There are no shortcuts. 

Sweller, Clark, and Kirschner point out, “There is no body of research based on randomized, controlled experiment indicating that such teaching [generalized problem-solving strategies] leads to better problem-solving.”  They observe, “Recent reform curricula both ignore the absence of supporting data and completely misunderstand the role of problem-solving in cognition.” 
Note. Reform math champions and elevates the idea of “general problem-solving skills,” group work, spiraling of content, and calculator use. At the same time, these “problem-based” programs minimize or demote essential content knowledge. This is no accident; it is by design. In my view, this upside-down relationship is a crucial flaw in reform math programs. Students must practice content to learn it and to apply it. 

Ze'ev Wurmanin a recent blog (7-16-11), writes, "The overwhelming majority of children can reasonably easily learn what we teach in our K-12 schools, given competent teachers and effective teaching methods." But this is not what happens in K-12. Most of our students remain mediocre at best (TIMSS) and only about 30% are proficient in math (NAEP). Furthermore, tens of thousands of incoming students flood remedial math courses at community colleges. Wurman says, "[The] cause must be in how we teach our students in school and outside it." We do not come close to teaching content that nearly all Singapore students learn in grades 1-9. By 9th grade, virtually all Singapore students (99.9%) have covered all of Algebra 1 and Geometry, according to Wurman. (Note. Ze'ev Wurman was one of the writers of the California math standards adopted in 1997. While most states continue to use the NCTM reform math framework, California dropped it in 1997 because its test scores plummeted. The 1997 California math standards were among the best in the United States. Furthermore, they were benchmarked to top-performing nations. Kids in top-performing nations do algebra in middle school. The California 1997 standards put Algebra 1 in 8th grade and Geometry in 9th grade. However, recently, I am sad to say, California replaced its excellent standards with mediocre Common Core math standards. Wurman has been an outspoken critic of Common Core math standards.)
Without explicit guidance
The belief in reform math is that, if students can learn problem-solving strategies and “discover” solutions to problems “without explicit guidance,” knowledge, or instruction, then learning math content (e.g., basic arithmetic) is not that urgent or important. This methodology is not “the most effective or efficient way to learn mathematics” and partly explains why most students are not proficient in mathematics (NAEP). What happens in reform curricula is that students do not learn enough content to support cognitive problem-solving. There are many math programs that emphasize a problem-based approach. This sells textbooks but does not produce “excelling” math students. In fact, reform math programs have produced a flood of remedial math students. For example, in 2009, nearly 90% of incoming students at Pima Community College (Tucson) were required to take remedial mathematics. 
Inverse [Upside Down] Relationship
An emphasis on problem-based curricula often displaces mastery of key math content. This inverse or upside-down relationship is found in most reform math programs. Not only is this inverse idea a basic philosophy in NCTM math standards, but it also carries over to Common Core by delaying fluency. [3] Math educators call it spiraling

In reform math, grade-level mastery of arithmetic is not the goal. If a student does not learn addition in 1st grade, it is repeated in 2nd, 3rd, and 4th grade (it spirals). Indeed, in the new Common Core math standards (left), students are not expected to be fluent in addition and subtraction until 4th grade. This is a nonsense approach. 
To be effective, a curriculum that is strong is problem-solving must also be strong in computational skills and fundamentals. Knowing is the foundation for higher-level thinking. So-called “generalized problem-solving skills” are not a substitute for mastery of fundamental content.  
In arithmetic and algebra, problem-solving is deeply rooted in the background or content knowledge (both factual and procedural) and not in general [problem-solving] strategies as embraced and advocated by many math educators, textbook writers, ed school professors, and special interest groups.
Problem solving in math is domain-specific, but math educators act as if it is not. 
Skill in problem-solving in mathematics requires substantial domain-specific schema [background knowledge], not “domain-general.” Students cannot apply the mathematics they do not know well. Students should be taught to solve routine problems to build an arsenal of background knowledge for more complex problems.

Worked Examples
Sweller, Clark, and Kirschner write, “But domain-specific mathematical problem-solving skills can be taught. How? One simple answer is by emphasizing worked examples of problem-solution strategies.” In other words, students can learn problem-solving skills by studying worked examples that exemplify them. This requires diligent practice. In mathematics, problem-solving skills are deeply rooted in content [knowledge]. Moreover, these skills take time to develop.  Sweller, Clark, and Kirschner write, “There are no separate, general problem-solving strategies that can be learned.” 

21st-century skills (P21), the latest foolish fad
The idea in 21-century skills is that these skills can be learned outside of domain-specific content knowledge. I think not. 
[1] G. Polya's How To Solve It was first published in the United States in 1945. Polya was concerned about math education, the training of teachers, and how teachers influence the attitudes of students.  
[2] Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics by Sweller, Clark, and Kirschner, in Doceamus, November 2010. 
[3] The Common Core Math Standards (2010) continue the NCTM spiraling approach. Delaying fluency makes no sense. 

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Updates: 7/15/11, 7/16/11, 7-17-11, 8-4-11

Sunday, June 19, 2011

Some thoughts on teaching little kids algebra

Teach Kids Algebra Project (TKA: 1st, 2nd, 3rd Grades)
I shall never forget these kids! 
The idea that little kids cannot learn algebra ideas is nonsense.
TKA is a response to reform math and Common Core.

Special Insert 10-30-17
"Equations are the lifeblood of mathematics, science, and technology," points out mathematician Ian Stewart (In Pursuit of the Unknown, 2012). It is the reason that students need to learn to write equations, rearrange them, deal with them and become skilled in solving them, all paper-pencil, not only for math class but also for chemistry and physics classes, etc. Also, knowing trig is important in physics courses. Right triangle trig used to be taught in good 7th-grade pre-algebra courses, but it is no longer the case. (Also, the 1970 Dolciani Algebra-1 textbook included a chapter on Geometry & Trigonometry. The trig problems were physics problems that are found in algebra-based physics courses. Other problems involved finding the vertical and horizontal components of a vector and the resultant of vectors. Students learned to solve trig equations that solved physics problems.) 

The use of graphing calculators and the shift away from solving equations using traditional algebra have produced weak algebra students over the years. It is one reason that early algebra is important in our schools.
End Insert 

TKA 3rd-Grade Students (2011)

Algebra is arithmetic plus variables. A variable is a symbol, such as x, y, or ☐, that represents an unknown number. For little kids, it is important to stress that a variable is a number
Students start with numerical equations (true/false) and the idea of equality, go to equations in one variable, proceed to equations in two variables (functions: input-output model), etc. Little kids solve equations using guess and check, memorized math facts, number and equality properties, procedures, and logic. This prepares them for algebraic methods for solving equations (e.g., the addition property of equations, etc.) by 3rd or 4th grade. Algebra ideas are not difficult if students know math facts (automaticity) and pay attention in class.  

● Variables, Equations, tables, graphs

Little kids, including typical 1st-grade students, can 
1. solve a range of linear equations using guess and check, number facts, and properties; 
2. write equations in one unknown to model a word problem (The idea is to translate words into symbols);  
3. build tables that show numerical relationships; and 
4. plot points and graphs of linear equations.
The idea of substitution is an important preparation for Algebra I. 
● Student Comments
1. "When you give us work it's fun because we don't know the answers." 
MV, 3rd Grade TKA Student
2. "I look to you to keep my brain working. I was able to understand what you were teaching. You have helped me with math I used to struggle with. The math that you taught me is amazing. You have taught me never give up, keep on trying. Thank you for all you have done for me."  MR, 3 Grade TKA Student

● Effort, persistence, and practice (EPP)
The idea that kids need innate ability to learn arithmetic and algebra is bunk. Kids need the skilled teaching of content and lots of practice to master arithmetic and algebra. Furthermore, they must work hard, i.e.,  effort, persistence, and practice count. Indeed, according to Daniel Willingham, a cognitive scientist, the “vast majority of K-12 students” can learn arithmetic and algebra; however, this should not imply that learning math is easy because learning math does not come naturally. Willingham points out, “It takes time, effort, and mastering increasingly complex skills and content.”  Math is hierarchical and should be taught so that one idea builds on another. It is logic. The logic begins with the transitive property of equality (Think Like A Balance) and true and false statements. Reasoning in math (new true statements are linked to other true statements) is different from reasoning in science (inferences are based on observations). In short, problem-solving is always domain-specific and requires domain knowledge.
● Equivalency (=)
Equivalency, which is a fundamental concept in mathematics, is seldom taught well. For example, 3 + 5, which is 8 {true}, is equivalent to 10 - 2, which is 8 {true}.
Therefore, 3 + 5 = 10 - 2 is a true statement. 
The logic behind the true statement is the transitive property of equality: two things equal to the same thing {8} are equal to each other. In Teach Kids Algebra, 1st, 2nd, and 3rd-grade students apply the "equivalency idea" (Think Like A Balance) and guess and check to find an unknown: 3 + 5 = x - 2. It is not possible to determine whether this statement is true or false without substituting a number. The box is a variable like x, and it can represent any number. But, to make a true statement, x is 10, not 8.
● Algebra, the Higher Arithmetic
The algebra lessons I give to 125 1st, 2nd, and 3rd-grade students require them to use number facts, number procedures, and number properties or laws (arithmetic). In short, algebra is built on arithmetic. As Morris Kline states, “Algebra is the higher arithmetic.” If we want kids to learn algebra, then they must be good at arithmetic. In primary school, kids who have auto recall of number facts and experience with standard algorithms do better and understand more. 
● Mini-Lessons: Lecture And Feedback
Manipulatives and calculators are not used. Students do not color things, paste things, cut things out, or work in groups. I lecture (explain how things work with examples), write stuff on the board as I explain things (visual-auditory), and ask students questions as I go (interaction). Then, I hand out problems for students to try on their own (guided practice) and roam around the room talking to students, giving feedback, and providing individual help. The classroom teacher helps a lot too. All this is accomplished in a 30 minute period, which is often too short. Explicit instruction works best
FYI: I met one 3rd grade class twice a week for at least one hour each time. Typically, the session would last 15 to 20 minutes more than an hour We always ran out of time. (Ms. S., the classroom teacher, was generous with time.) The students work hard for an hour straight, usually longer, but they do not seem to mind. Time passes quickly. I was not only teaching fundamental algebra ideas, but I was also teaching persistence and effort. One 3rd grade student wrote to me, "I had a hard time with algebra, and because of you I got better with algebra. I enjoyed learning because you made my brain work." 
At first, the math is challenging, but I keep reassuring and encouraging students, often one-to-one, giving them support: "Try a different number. Do not give up! You can do this. Let me show you how. Try this. You can learn this--it's great stuff." And, they did. The success of the program hinges on talking to individual students and giving them important feedback and encouragement. Kids need adult support. 

● The Equation As Model: Translating words into symbols
When I present a word problem, let's say in 1st or 2nd grade, I model it with an equation. For example, Jill has some pencils. Bill gives her 5 more pencils. Now Jill has 12 pencils. How many pencils did Jill have before Bill gave her pencils? Jill has "some pencils" is represented by a variable I call x. Step-by-step, I piece together an equation: x + 5 = 12 on the board by asking questions: Do we add or subtract 5? What do we do with 12? The equation is the model. Kids do not need bar models to understand simple problems. They learn to write equations. They learn to translate words into symbols. It is important that students identify the unknown in a word problem. This starts in 1st grade. Identifying the unknown and solving problems from an algebraic perspective makes sense.  

For older kids (3rd grade), the equations become more complicated. Write an equation, then solve it. I am thinking of a number. Six less than triple a number is 15. What is the number? 
Equation: x + x + x - 6 = 15
Solution: x = 7  by inspection (7 + 7 + 7 - 6 = 15; 15 = 15). 

Note. Being able to translate words into symbols (equations) and being able to solve the equations is what algebra is all about.

Note. It is okay for students to struggle because math is not always about getting the right answer. It is about developing young minds and improving their effort, persistence, and reasoning in solving problems. Furthermore, math builds the brain. It makes kids smarter. In short, a cognitive struggle is a good thing.
● Myth
Willingham also observes that “our society has accepted the fact that math is not for most us;” however, he says that this “notion is a myth.” The idea “I am not good at math” is rooted in our culture. We need a radical change in our attitudes toward learning and schooling. 

● Cognitive Horsepower 
Attention is essential for learning. Attention is controlled by something called executive function. Gary Stix (How to Build a Better LearnerScientific American, August 2011) says that executive function encompasses important cognitive attributes such as the ability to "be attentive, hold what you have just seen or heard in the mental scratch pad of working memory, and delay gratification. These skills (being attentive, holding stuff in working memory, and delaying gratification) often predict a child’s success in school. 
Children who have difficulty concentrating also have difficulty learning mathematics. Teaching little kids algebra is a blast, but learning algebra ideas requires mental fluency with arithmetic facts and procedures and sufficient concentration to hold stuff in working memory.  
In Teach Kids Algebra lessons, for example, students deal with a lot of new stuff all at once in a short time. This s t r e t c h e s working memory and pushes students to concentrate. Knowledge of key math facts and procedures in long-term memory (automaticity) helps a lot because of working memory space, although somewhat "plastic," has limitations. In problem-solving, too often students figure out simple facts that should have been memorized (e.g., 3 + 8). This "figuring" wastes time and working memory space and distracts from solving the problem. Facts should be memorized and stored in long-term memory for a child to excel in mathematics. The focus in working memory should be on solving the word problem, not on figuring out simple facts.   
A child's working memory structure, including the mental scratchpad, is in place by age 6. It has less capacity than an adult's working memory, but it improves somewhat as children grow. Working memory, however, has limitations. It can hold only so much stuff at a given time before becoming overloaded. In contrast, long-term memory does not have this limitation. In mathematical problem solving, it is important that key math facts and procedures are in long-term memory (background knowledge) so that working memory can hold all the essential information from the problem to devise a plan for solving. Also, negative thoughts about math (e.g., math anxiety) can often crowd working memory, which is another concern. (Information from From Stix, Willingham, Beilock)
According to Sian Beilock (Choke), "Working memory is your cognitive horsepower. It involves the ability to hold information in mind (and protect that information from disappearing) while doing something else at the same time." Beilock says that "working memory is one of the major building blocks of IQ." Working memory can be developed, so it is important to practice, stretch, and exercise working memory with challenges to improve cognitive muscle. Beilock points out that "practice shapes your brain." To learn math well, for example, requires both practice and challenges. Also, a student will not learn much math if he is easily distracted or has difficulty with attention in class.
Daniel T. Willingham, a cognitive scientist, states that students learn what they are thinking about, which takes sharp attention. The ability to control attention and hold information are skills that can be trained. Bronson & Merryman (Nurture Shock), suggest that "being able to concentrate [cognitive control] is a skill that might be just as valuable as math ability, or reading ability or even raw intelligence." 
Of concern is that a child's attention span or ability to concentrate has been declining over the past 20 years. The Net-Generation tends to bounce from task to task. Click, Click, Click! They often have difficulty focusing on one task and doing it well. They do not reflect or think through things. Nicholas Carr (The Shallows) says the Net causes brain changes. The biggest change is that students have difficulty concentrating. He observes, "Tests of memorization, vocabulary, general knowledge and even basic arithmetic have shown little or no improvement."  The Net does not make you smarter; school makes you smarter. The Net, says Carr, should not be a "replacement for memory." Remembering is a fundamental cognitive skill that is needed for problem-solving. Math facts and procedures must be retrievable from long-term memory so they can be used in working memory to solve problems.
Aimee Cunningham (Kids' Self-Control Is Crucial for Their Future Success, Scientific American, July 25, 2011) points out that a child’s “self-control is crucial for their success.” She writes [Long Quote], “Self-control—the ability to regulate our attention, emotions, and behaviors—emerges in childhood and grows throughout life, but the skill varies widely among individuals. Past studies have reported that self-control is partially inherited and partially learned and that those with less self-control are more likely to be unemployed, engage in unhealthy behaviors such as overeating, and live a shorter life.” 
● Common Brain Myths
Teachers and parents should be aware of common brain myths. Gary Stix (Scientific American, August 2011) lists five myths from Mind, Brain, and Education Science (Takuhama-Espinosa, 2010). Here are two of the myths. 
1. “Left-brain” and “right-brain” people differ. No. “Brain-imaging studies show no evidence of the right hemisphere as the locus of creativity. And the brain recruits both left and right sides for both reading and math.” 
2. Each child has a particular learning style. No. There is little evidence to support this claim. “For this and other myths, public perceptions appear to have outstripped the science.” Many parents tell me their child is a visual learner or a kinesthetic learner (not an auditory learner). These popular perceptions held by many parents and educators are not backed by evidence. 
In education, "there is an enormous supply of totally untested, untried, and not very scientific methods.”   
(To Be Revised)

LT, Guest Teacher
LT, Founder of  ThinkAlgebra
Updates: 6-19-11, 6-20-11, 6-21-11, 6-30-11, 7-1-11, 7-2-11, 7-9-11, 7-20-11, 7-23-11, 7-24-11, 7-25-11, 7-26-11, 7-31-11, 8-1-11, 9-21-11, Minor grammar corrections made on 4-29-17
Photos by 3rd grade classroom teacher: CSmith