**First-Grade Arithmetic & Algebra**

**Math builds the brain. It makes you smarter. There is no substitute for knowledge in long-term memory and the practice that gets it there. Mathematical thinking comes from mathematical knowledge in long-term memory, not thin air.**

*E*

*mbrace the power of memorization and practice for mastering content.*

**My First-Grade Arithmetic & Algebra page**provides basic information and advice for parents, teachers, administrators, school districts, and boards of education regarding early math/algebra education.

The webpage explores some of the math content I had taught to

**1st-grade**

**students**in the early 80s and the spring of 2011. I was influenced by the

*Madison Project*(1957) and

*Science-A Process Approach*(1967). I fused basic algebra ideas to standard arithmetic starting with ordinary 1st-grade students. My

**Teach Kids Algebra (TKA)**project had two major objectives:

**(1)**to broaden the math curriculum to include algebra (i.e.,

**abstract experiences**) in the early grades (1-3) and

**(2)**to respond to the reform math movement and Common Core standards, which were not world-class math. Also, reform math did not teach standard arithmetic for mastery and used minimal guidance methods that were inefficient. The result has been

**flat achievement**according to national and international tests.

*While kids in some nations are leaping ahead in math and reading achievement, most U.S. kids are crawling if that.*

*The order of topics on this page is random. There is repetition, probably too much. Excuse errors. I am in the process of removing some of the content.*You can contact me via email (ThinkAlgebra@cox.net).

**Note:**Basic math doesn't change. It endures. "A true math statement will remain true forever." (Quote: Edward Frenkel,

*Love & Math*)

**TKA**provides young children with

**abstract experiences**such as variables, substituting into equations, the truth set of an open sentence, simple arithmetic of signed numbers, etc. Algebra obeys the axioms of arithmetic. Like the Madison Project,

**Teach Kids Algebra**(TKA) regards "

**content and method**as equally important and inseparable." And, the method I used was explicit instruction via worked examples (content). No calculators. No manipulatives. No group work. First-grade students were given a two half-hour sessions a week for a total of 7 instructional hours.

**First-Grade Student in TKA (2011)**

The first task was for 1st-grade students to plot points in Quadrant I given the x-y coordinates, connected the points in the order given to form a geometric figure, then find the

**perimeter**of the figure.

**"To understand mathematics means to be able to do mathematics."**(Polya)

**Finding perimeters is a good example.**

**"You learn only through mastery.**

**"**(Engelmann)

**"The [math] concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."**Math Professor W. Stephen Wilson

The Asian nations work on mechanics of arithmetic first with explanations later. At times, there is memorization without understanding,

**but understanding is embedded with usage**. Even though I ask 1st graders to memorize**5 + 7 = 12**doesn't mean they don't understand it. Indeed, the concept of addition is straightforward on a number line. Also, the idea that**24 is 2tens+4ones**by place value is a 1st-grade understanding of place value as applied to double-digit numbers. Good math teachers also lead with worked examples to support some level of understanding. Knowing how something works is functional or practical understanding. Also, "understanding" is a vague, relative idea that is difficult to define and measure.**In math, knowing facts and efficient procedures (algorithms) in long-term memory builds the foundation for understanding, applyling, and reasoning in math. ***Students cannot apply something they don't know well.**Indeed, the memorization of essential facts and procedures plays an important role in doing math at an acceptable level.***Can students do arithmetic?****G. Polya**(

*How to Solve It*) points out, "Mathematics, you see, is not a spectator sport.

**To understand mathematics means to be able to do mathematics.**And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems." For example, you don't understand addition unless you can do addition such as finding the

**perimeter**of a rectangle, which should be a routine problem in 1st grade. Students should be able to do the addition using standard methods at the level needed. They should know that they can add the numbers in any order, which is an important property of addition. Also, it is vital to automate the mechanics of standard arithmetic such as adding vertically 26 and 35 starting in the 1st grade.

**Polya explains that the teaching of problem-solving should begin with simple, routine problems, slowly adding complexity.**

Math is more

**abstract**than other subjects, so it is harder to learn. It takes more practice to master than other school subjects.

**Edward Frenkel**(

*Love & Math*) writes, "People think they don't understand math, but

__it's all about how you explain it__to them." For decades, K-8 teachers have had difficulty explaining math. Sadly, in many progressive K-5 classrooms, basic arithmetic has been downgraded or not taught for mastery, that is, not practiced enough to stick in long-term memory.

*Instead of learning standard arithmetic,*s

*tudents are taught alternatives (i.e., reform math).*However, there is no substitute for factual and procedural knowledge in long-term memory and the practice-practice-practice that gets it there. Indeed, schools should teach the mastery of necessary skills first, not a bunch of reform math fluff and its alternative, nonstandard algorithms.

**What's the unknown?**

As early as possible, students should

**recognize the fundamental problem patterns**for addition, subtraction, multiplication, and division, which are straightforward. The difference is the numbers--whole numbers or integers, fractions, and decimals.

**Students should extract the numbers from the word problem, determine the pattern (operation), and write a simple equation in one variable that models the situation.**What is the

**unknown**(

**)? For example (1st grade), Jane has nine apples. Bill gives her seven more apples. How many apples does Jane have now?**

*x***What is the pattern?**

**What is the unknown (**The student should recognize the pattern (addition), extract the numbers (9, 7), and write an equation with a variable:

*x*)?**9 + 7 =**.

*x**At first, to add 9 + 7, the student can use a number line until she memorizes the fact.*Answer the question: Jane has 16 apples.

**Pattern recognition**is important when teaching problem-solving.

*Turning words into symbols is a key math skill that requires lots of practice and review.*

**Notes**

**1. Reform math**asks students to calculate 9 + 7 using three steps:

Add 1 to 9 to make 10.

Subtract 1 from 7 to make 6.

Add 10 and 6 to make 16.

"Gee, isn't that cool?" (No, not really.)

"It shows understanding." (At what level? How do you measure understanding?)

**Note:**9 + 7 = 16 should not be a multistep calculation.

*The addition fact should be memorized!*

**2. I count three calculations that clutter the working memory.**If the goal is to stamp 9 + 7 = 16 into long-term memory permanently, then the calculating strategy of reform math is counterproductive because it

**delays the memorization**of 9 + 7 = 16. Children should know 9 + 7 = 16 instantly from memory.They should not be calculating it every time the fact is used.

**Learning is remembering from long-term memory.**Indeed, single-digit number facts are needed in long-term memory to perform the standard algorithms. Also, students should not use a calculator, counting on fingers, or reform math strategies as a crutch. Retrieving facts instantly from long-term memory is the goal. Retrieving must be practiced.

**3.**

*For little kids, the number line visually shows how 9 and 7 are 16 without being mathy (e.g., set theory, etc.).**The number line is essential mathematics.*

**ONE**is the fundamental organizing principle in math, not the reform math ten frame found in 1st and 2nd grade textbooks. I recommend taping a

**0-20 number line**on each student's desk. The reform math strategies of making a drawing or writing an explanation are not necessary, either, because they can increase cognitive load. (Sometimes,

**Richard Feynman**, the brilliant, Nobel prize-winning physicist, would tell his students, "I do not understand it." But, I can do it.)

**Children are novices in math, so their understanding is limited or functional at best, which is okay.**

**The way children and engineers use math is about the same.**It is based on the assumption that all the procedures, techniques, and algorithms learned are valid.

**William Byers**(

*How Mathematician Think*) points out that school math is algorithmic, which is the "how" not the "why" of math. The "why" should be left to those studying mathematics at the university.

**Ian Stewart**explains, "One of the biggest differences between school math and university math is proof.

**At school we learn how to solve equations or find the area of a triangle; at the university, we learn why those methods work and prove that they do.**Thus,

**understanding**for school children is knowing when to apply the right algorithm and perform it quickly to get the right answer. Children also need to understand that an idea in math is built from other ideas.

Reform math overemphasizes understanding, which is very difficult to quantify, at the expense of performance.

**In math, students should learn how to apply a procedure correctly and be able to do the procedure rapidly (competency)**. Thus, factual and procedural knowledge is the backbone of learning school math. In school math, the algorithmic thinking approach (i.e., the performance) does not require much understanding because it is mechanical. However, students need to master a certain number of algorithms so they can be applied automatically to problems.

**In short, you have to be able to use and apply the algorithms you studied.**

*"Applying" is the understanding students should have.*If an algorithm works, then move on; however, the algorithms you use should always be the most efficient ones, which is a reason that mathematicians recommend that students should learn the

**mechanics of the standard algorithms** from the get-go.

*The standard algorithms always work.*

Here is something to think about:

**Dr. H. Wu**(UC-Berkeley) points out,

**"Computational facility on the numerical level (arithmetic) is a prerequisite for facility on the symbolic level (algebra)."**Another thought: Educators in competing nations embrace the power of memorization and practice to help young children master arithmetic and algebra fundamentals. In contrast, memorization and practice have fallen out of favor in many American classrooms.

**National Math Panel 2008 Quote:**

"What is developmentally appropriate is not a simple function of age or grade, but rather is largely contingent on prior opportunities to learn” (Duschl et al., 2007). Claims based on Piaget’s highly influential theory, and related theories of “developmental appropriateness” that children of particular ages cannot learn certain content because they are “too young,” “not in the appropriate stage,” or “not ready” have consistently been shown to be wrong. Nor are claims justified that children cannot learn particular ideas because their brains are insufficiently developed, even if they possess the prerequisite knowledge for learning the ideas. (End of Quote)

**[ Note:**Kids do not grow into math or abstract thinking as Piaget postulated. The reason students at an early age don't know much math has little to do with being developmentally ready or too young. It is that they weren't taught the content. In contrast to Piaget, I embrace the view of

**Jerome Bruner**"who argued that kids are capable of learning nearly any material so long as it is organized, sequenced, and represented in a way they can understand."

****

*It is the reason that I was able to teach Algebra to 1st and 2nd grade students. Algebra is abstract, but it obeys the rules (axioms) of arithmetic. That's the link.*

**]**

Math Panel Quote:

Math Panel Quote:

**"Students learn by building on prior knowledge**, extending as far back as early childhood. Learning and development are incremental processes that occur gradually and continuously over many years.

*Even during the preschool period, children have considerably greater reasoning and problem-solving ability than was suspected until recently.*

**"**

**Math Panel Quote:**

**"For all content areas, practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees up working memory for more complex aspects of problem solving."**

*The Old School ideas of memorization and drill to develop skill are not obsolete. They are firmly supported by the cognitive science of learning.*

*The Math Panel rejected Piaget's claims and reinforced the teaching of standard algorithms, not reform math, but I was disappointed in its grade placement of content, which was not based on international benchmarks.*

**Singapore 1st-Grade Arithmetic, China, Piaget, Work Ethic, Gross Underachievement, etc.**

The content children learn in the 1st grade dramatically impacts their future learning.

The content children learn in the 1st grade dramatically impacts their future learning.

**Zig Engelmann**pointed out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning."

**In Singapore**and other nations, kids memorize the addition facts in 1st grade and learn the mechanics of formal algorithms (+, -). Also, they practice subtraction and multiplication (repeated addition) and write and solve equations in one variable from word problems. Simply, they drill to develop skill. How many 1st grades in America do that?

**[**

**Note:**

**Lenora Chu**wrote that the Chinese strict school system has surprising upsides, which include "the benefits of memorization, competition as a motivation, and the Chinese cultural belief in hard work over innate talent." Chu put her 3-year-old son in the Chinese public school system in Shanghai. "He developed surprising powers of concentration and became proficient in early math." There is a global race to achieve says Chu.

**]**

****The fact remains that many ordinary U.S. students grossly underachieve compared to their peers in some other nations. We don't expect much from kids academically because educators and parents hold to the refuted Piaget "developmental appropriateness" claim that kids grow into abstract thinking. They don't. Also, children are not fragile as some "experts" claim.

__Let me repeat.__

*The reason students at an early age don't know much math has little to do with being developmentally ready or too young. It is that they weren't taught or exposed to the content.*

Unfortunately, the "work ethic" that built America has softened significantly over the decades. Early on kids are not required to memorize or drill to develop math skills. Moreover, for decades, kids have been praised for no good reason. Often, they get high grades not because of academic scholarship or achievement but because of grade inflation. Competition is considered harmful or harsh for kids. No, competition is good for kids; it motivates them.

The public school system "tends to pass kids whether they learn the material or not," writes

**Larry Winget**(

*Your Kids Are Your Own Fault: A Guide For Raising Responsible, Productive Adults*). Parents need to step up and "make sure that their kids get an education whether their school does its job or not." If your child has a bad attitude, then you, as the parent, need to fix it. Stop blaming the school, the teacher, etc. Stop giving excuses.

**Janine Bempechat**(

*Getting Our Kids Back on Track*) writes that self-esteem is overrated. "We need to worry less about self-esteem and more about competence. We need to expect much more from our children." Professor Bempechat wrote this 17 years ago. Nothing has changed. She pointed out that "children learn nothing from easy assignments" and argued that young children must learn to get through challenges. "We need to raise our expectations and standards for their academic achievement." Homework can be frustrating for children, but over time, it "serves to foster qualities that are critical to learning: persistence, diligence, and ability to delay gratification." Homework, says Bempechat, does not rob children of their childhoods or undermine their love of learning. Parents need to support achievement.

**Note1:**Even though kids are not all equally intelligent, athletic, musical, or creative, most kids starting with the 1st grade can learn the fundamentals of standard arithmetic and algebra at an acceptable level--if the math is taught well and practiced for mastery.

*Kids need to memorize and drill to develop skill.*

**Learning is remembering from long-term memory.**

*For example, if a 1st-grade student*

*doesn't have auto recall of 5 + 7 = 12, then the student hasn't learned it.*Learning is a change in long-term memory.

**Note2:**Not knowing (remembering) the single-digit number facts for instant recall or the standard algorithms may create a cognitive load in working memory that interferes with solving problems and learning.

**Read**You don't want your working memory's

__Cognitive Load.__**limited space**busy with stuff that should have been automated in long-term memory such as the single-digit number facts and standard algorithms.

**Note3: Read **

Note4: Read

__Random Thoughts on First Grade Arithmetic__Note4: Read

__Bad Math Education__**Note5: Read the latest: Children Are Novices.**

**Even though kids are not all equally intelligent, athletic, musical, or creative, most kids, starting in the 1st grade, can learn the fundamentals of standard arithmetic and algebra at an acceptable level--if they are taught and practiced for mastery.**

**The Asian system is built on memorization**, which forces students to store information in long-term memory where it is ready for use to solve problems (Vohra). American educators don't get it, that is, students need to practice for mastery.

**Asian children are taught mechanics of operations first with the explanation later, and it works!**

*We do it backward with understanding first, and it doesn't work. Simply, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory.*(

**Note:**

*Asian students have a lot of intrinsic motivation. They try harder in school and learn more.)*

**There will always be gaps!**In real education, you increase differences (Richard Feynman). We have been "equalizing down" to close gaps in the name of fairness (Thomas Sowell) and feeding all students the same math curriculum starting in the 1st grade ("one size fits all" Common Core and state standards). These are counterproductive approaches. The reformers and professors of education say that teachers should differentiate instruction in a mixed classroom, but it never works well.

**Chester Finn, Jr. and Brandon Wright**write in

*EducationNext*, "Rare is the teacher who can do right by her ablest pupils at the same time she provides slower learners in her classroom the attention that they need." Also, misinformed reformers and others argue that kids need less mastery of traditional arithmetic and its standard algorithms because they can use calculators.

**WRONG!**

**A weak math student with a calculator is still a weak math student!**

**The webpage**explores some of the math content I had taught to**1st-grade**students in the early 80s and the spring of 2011. I was influenced by ideas from*The Madison Project*(1957) and*Science-A Process Approach*(1967). I fused basic algebra ideas to standard arithmetic, starting with ordinary 1st-grade students. It's not that hard to do.****__(My Teach Kids Algebra Program)__**Some Thoughts On Teaching Little Kids Algebra****The number line represents essential mathematics**(e.g., whole numbers, fractions, operations, properties, etc.), but I rarely see it in textbooks. I used number lines in the 1st week of school when I taught a self-contained

**1st-grade**class at a Title-1 urban school in the early 1980s. I taped a

**0-20 number line**to desks the first day of school. Later, a

**-10 to 10 number line**.

In the

**first two weeks**of 1st grade, the number line was used to show magnitude, add one, subtract one, operations such as addition (5 + 2) and subtraction, properties (e.g., 2 + 5 = 5 + 2), add in any order, the location of 1/2, etc.

**The number line is important mathematics.**

**::: Be sure to visit**

__My Contrarian Math Page__for the latest.

**✔︎ Learning the basics of standard arithmetic and algebra is not magic. It's hard work! There are no tricks or shortcuts.**Learning is remembering, which requires a lot of practice for essential factual and procedural knowledge to stick in

**long-term memory**.

**Zig Engelmann**states, "

**Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning.**Concepts are difficult to teach when students have mastered only some of the facts some of the time." In the 1960s, using

**old-fashioned repetitive drills**,

**Engelmann's pre-1st-grade-students**memorized

**number facts and multiples**to do arithmetic, find areas of rectangles, solve equations, calculate fractions, and factor expressions. "Education policymakers have a model of how things should be, but 'should be' is not reality. They believe that it is more important to preserve their flawed understanding of how kids learn than it is to provide effective instruction to kids."

**If learning is remembering from long-term memory, then as Engelmann points out, "You learn only through mastery**

**"**(i.e., practice-practice-practice). He explains, "Education is about teaching, and teaching has been flawed for decades."

*He states that teachers are not teaching for mastery.*Instead, educators often adhere to

**minimal teacher guidance methods**such as discovery, project, problem, inquiry, group, etc., which, as

**Kirschner-Sweller-Clark***concluded, were ineffective compared to

**explicit instruction**via carefully selected

**worked examples**.

**Engelmann**states, "Educational researchers and policymakers do not endorse the most effective programs."

**Beverlee Jobrack**(

*Tyranny of The Textbook*) points out that teachers rarely select the most effective programs. Kids aren't learning essential content for mastery in long-term memory because they don't practice the fundamentals enough. Also, arithmetic basics are not taught well.

**"Certain things are rote, not because you teach them as rote. They are rote because they are rote, such as numbers, properties, math facts, etc."**It is the reason that Engelmann could teach fractions and their operations to kids before they entered the 1st grade!

*It is the reason that I was able to teach 1st-grade students algebra concepts.*

*******Kirschner, Sweller, & Clark**2006: "Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching"

****The way math is taught today in our schools (as

**reform math)**rather than standard or traditional arithmetic is

**substandard**compared to the Asian system.

**The disparity starts in 1st-grade arithmetic.**While Asian students

**excel**at math

*and*problem-solving, American students

**stumble**over simple arithmetic, which is an unacceptable narrative!

**In contrast,**

**Asian children are taught mechanics first with explanation later, and it works!**

*We used to do that!*

*In K-8 high-achieving math students and low-achieving math students are tossed together.*

**Chester Finn, Jr. and Brandon Wright**write in

*EducationNext*,

**"Rare is the teacher who can do right by her ablest pupils at the same time she provides slower learners in her classroom the attention that they need."**

**Reform math has not stressed the mastery of standard arithmetic in long-term memory!!!**

**"In education, you increases differences."**

Richard P. Feynmanwas invited to a conference to discuss "

Richard P. Feynman

**the ethics of equality in education**." He confronted the experts by asking this question. "

**In education, you increase differences.**If someone's good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?" (

*Surely You're Joking, Mr. Feynman!*by Richard P. Feynman,

**Nobel Prize in Physics**)

**"You don't know anything until you have practiced."**(Feynman)

**Sometimes, I use visual aids to demonstrate a key idea in mathematics. The two visual aids I use most are the number line and equal arm balance.**

The

**equal arm balance**is metaphor for the abstract idea of equal in an equation. The idea of equality is that the left side balances the right side of an equation (

**5 = 5**). Algebraic operations are performed on both sides to ensure that the equation remains balanced.

Thus, if I add 2 to the left side, then I must add two to the right side to keep the balance. In symbols: 5

**+ 2**= 5

**+ 2**or 7 = 7. Likewise, if I subtract 3 from the left side, then I must subtract 3 from the right side to maintain the balance. In symbols: 7

**- 3**= 7

**- 3**or 4 = 4.

**In the simple equation x + 4 = 23, algebra allows us to subtract 4 to each side, obtaining x = 19.**That is, to

**undo**add 4, I can

**subtract 4**on both sides. In symbols:

**x + 4 - 4 = 23 - 4**, thus, x = 19. The idea is simple: performing the same operation to both sides of the equation to keep it balanced.

**One of first ideas I teach 1st-grade students is the equal sign.**As students find addition and subtraction facts on the number line and begin to memorize addition facts, they also learn to apply the

**Think Like A Balance**idea to determine if statements are

**true and false**:

**2 + 4 = 7 - 1**

6 = 6 (True)

6 = 6 (True)

2 + 4 = 6 - 1

6 ≠ 5 (False)

2 + 4 = 6 - 1

6 ≠ 5 (False)

Using a

**number line**helps students with

**magnitude**and figuring out number facts. I also introduce

**missing addend**problems based on important concepts such as

**6 + ❑ = 7 - 1**. The early use of

**symbols**to represent a specific number is important arithmetic and algebra.

*If the right side is 6, then the left side must also be 6 to balance the sides. Thus, box is 0.*

**The Asian system is built on memorization, which forces students to store information in long-term memory where it is ready for use to solve problems, explains Arvin Vohra.**

Memorized facts, efficient procedures, important concepts, and key formulas (i.e., basic mathematical content) in long-term memory prevent

**cognitive overload in working memory.**Asian children are taught to memorize, while American children are deficient in memorization skills. In fact, memorization is disparaged by many reformers. American reform methods stress understanding over memorization, and it doesn't work in math. "Memorization is the seat of knowledge," writes

**Barry Garelick**on his blog. "Thinking skills are intertwined with domain knowledge," writes

**Daniel Willingham**.

**Asian children are taught mechanics first with explanation later, and it works! We do it backward, and it doesn't work.**Also, the Asian system drills kids on

**routine problem types**, but American children are asked to solve "so-called" real-world, nonroutine problems with calculators.

**W. Stephen Wilson**, a math professor at Johns Hopkins University, points out that using calculators is

**absolutely unnecessary**for arithmetic and algebra. He writes, "The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."

**While Asian students excel in math and problem-solving, American students stumble over simple arithmetic.**

**Memorizing skills and practice are critical to learning and thinking.**The more "rote"

**East Asian learners**, who memorize and drill-to-improve-skill, have soared far above U.S. students not only in content knowledge and ability to apply and perform mathematics correctly but also

**creative problem-solving**at the Advanced levels (TIMSS). The constructivist view that memorization and paper-pencil standard algorithms are harmful to children

**is bogus**. The "memorizers" in Asian nations are much better at problem-solving than American students raised on "constructivist computational methods" often called strategies. Unfortunately, many schools-of-education no longer teach standard paper-pencil arithmetic, observes

**Bill Quirk.**There is no comparison:

**54% of Singapore**8th-grade students reached the

**Advanced Level**to only

**1**

**. 8th graders (TIMSS 2015). The Advanced Level is loaded with problem-solving.**

**0%**of U.S*Indeed, Asian students are good problem solvers (dominate the top), but American students are not (near the bottom), according to the latest PISA math results for 15-year olds.*

One reason is that our reform math programs and textbooks use the wrong approach.

**Garelick**observes,

**"**Concerning the math books of earlier eras, they started with the

**teaching of the standard algorithm first**. Alternatives to the standards using drawing or other techniques were given afterward to provide further information on how and why the algorithm worked. It is the opposite of how reformers are advising it be done now.

**"**

**In my view, many of the alternative algorithms of reform math are not essential. They clutter the curriculum and waste instructional time better spent on mastery of fundamentals.**

*(Sources of ideas: Quirk, Garelick, Wilson, Willingham, Hu, Vohra, et al.)*

Very young children can learn much more essential arithmetic and algebra early on than teachers have been led to believe or prepared to teach.

*In the *

**first 30 days of schooling***,__1st-grade students__should switch from counting cubes to memorizing math facts (e.g., n + 0, n + 1,**n + 2, a + b = 10, 10 + n**), learning**place value**(12 = 1ten+2ones) and the**standard algorithm**.**Up to 70% of the daily math period should be spent on**

__continual practice and regular review__to cement the basics in long-term memory.**I**

**f you don't remember something, then you****haven't learned it. Learning is remembering.**Not automating math facts so they stick in long-term memory starting in the 1st grade impedes achievement. Unfortunately, U.S. reform math programs belittle "drill to improve skill" as an obsolete method of instruction and poor teaching. Really? In contrast,

**Singapore 1st-grade students**memorize the addition/subtraction facts, practice formal (standard) algorithms, and do multiplication as repeated addition.

**Comment:**The constructivist view that memorization and standard paper-pencil algorithms are harmful

**is bogus**. The "memorizers" in Asian nations are much better at problem-solving than American students raised on "constructionist computational methods" or strategies.

**Asian children are taught mechanics first with explanation later, and it works!**

**We do the opposite, and it doesn't work.**Indeed, many schools of education no longer teach standard paper pencil arithmetic such as long division, etc.

***Note:**The "

**first 30 days of schooling"**is from my 1st-grade self-contained class in the early 1980s. Math was practiced and studied for 50 to 60 minutes daily. Most addition math facts and many subtraction facts were learned by Christmas. The standard algorithm with carry was also learned in the 1st semester.

**Note: How should our 1st graders learn multiplication? In Singapore, 1st-grade students learn multiplication as the addition of identical addends (i.e., repeated addition: 3 x 5 = 5 + 5 + 5 = 15), memorize half the single-digit multiplication facts in 2nd grade and the rest in 3rd grade along with formal (standard) algorithms.**

*The concept of multiplication is simple.*

**Three Urban Legends are difficult to eradicate.**

**Kirschner & Merrienboer**

**(**"The

*Educational Psychologist, 2013)*outline three urban legends in education:**first legend**is one of learners as

*digital natives*who form a generation of students knowing by nature how to learn from new media, and for whom “old” media and methods used in teaching/learning no longer work. The

**second legend**is the widespread belief that learners have

*specific learning styles*and that education should be individualized to the extent that the pedagogy of teaching/learning is matched to the preferred style of the learner. The

**final legend**is that learners ought to be seen as

*self-educators*who should be given maximum control over what they are learning and their learning trajectory. It concludes with a possible reason why these legends have taken hold, are so pervasive, and are so

**difficult to eradicate**." (Quote from the article's Abstract by Krischner & Merrienboer)

The ideas of digital natives, specific learning styles, and self-educators appeal to many progressive educators (and parents), but they are misguided and counterproductive.

**Competency**

"

**If we want students to become competent in arithmetic and algebra, then they need to be more like ballet dancers, gymnasts, swimmers, violinists, chess players, etc.**That is, students need to practice and review so that the fundamentals of math stick in long-term memory (automaticity). Students should drill to improve math skills. Up to 70% of math class time should be spent on practice, review, and study.

**Children are novices and need repetition to learn.**They understand a new idea as it relates to old ideas in long-term memory. Math is hierarchical, sequential, and cumulative. Students will not be prepared for algebra in middle school when basic arithmetic skills are deficient.

**Basic arithmetic starts in 1st grade.**

*Algebra-1 is a middle school course for average students who are prepared.*

**Zig Engelmann**said, "if you want to be smart in the instructional arena, then you have to look at the kid's

**performance**."

*The performance of K-8 students in math and reading has been flat and unacceptable (NAEP, TIMSS).*

**Ian Ayres**(

*Super Crunchers*) writes, "The educational establishment is wedded to its pet theories regardless of what the evidence says.

**For many in education, philosophy trumps results**."

**Ayres**writes, "Engelmann flatly rejects both the child-centered and whole-language approaches. He isn't nearly as famous as Chomsky or Piaget, but he has a secret weapon--data." The evidence that Engelmann's methods work well date back to 1967. Engelmann emphasized "basic skills like vocabulary and arithmetic, not higher-order thinking and problem solving." Engelmann realized early on that the acquisition of basic skills in math and reading provided the framework for higher-order thinking, etc. Children cannot do critical thinking or problem solving if they don't know anything.

**Knowledge was important!**(Engelmann's quote: Ian Ayres,

*Super Crunchers*)

**To learn something is to remember it!**And the key to learning is

**repetition**says

**Maria Brilaki**writing on

*Lifehack*:

**"By actually doing something new over and over again, your brain wires new pathways that help you do this new thing better and faster." Click:**

__Memory & Learning.__

**1. Three Equivalence Relations (Key Properties of Equality)**

**First,**any whole number is equal to itself (7 = 7), which means that

**equality is reflexive**.

**Second,**If a whole number a is equal to a whole number b, then b is equal to a. (If a = b, then b = a), which means that

**equality is symmetric**. (3 + 4 = 7 is the same as 7 = 3 + 4)

**Third,**if two numbers or expressions are equal to a third, then they are equal to each other (If a + b =

**c**and x + y =

**c**, then a + b = x + y, which means that

**equality is transitive**.

**Note.**The key properties of equality include

**all real numbers**, not just whole numbers.

(Reference: R. James Milgram, Stanford.edu)

**Note:**The

**transitive idea of equality**is the underpinning of "

**Think Like A Balance**" in my

**Teach Kids Algebra**program. If the left expression of an

**equation**is equal to

**7**and the right expression of the equation is also equal to

**7**, then 3 + 4 = 8 - 1 is a true statement in mathematics. In short, 3 + 4 =

**7**and 8 - 1 =

**7**, then

**3 + 4 = 8 - 1**because

**equality is transitive (7 = 7)!**In math and solving equations, we need to be make sure the statements are true via the properties of equality and numbers.

Equal Sign (=) Think Like A Balance

**T**he equal sign shows an association or relationship between the expression on the left side and the expression on the right side:

**3 + 4 =**. The equal sign does not mean to complete an operation (i.e., calculate, such as in 3 + 4 = ❑ often found in textbooks) explains

*x*- 1**Dr. H. Wu.**

*For an equation to be a true statement, both the right side and left side must have the same value.*Therefore, ❑ must be an expression equivalent to 7 because the left side simplifies to 7.

**Wu**writes, "Education research in algebra has decided that the students' defective understanding of the equal sign as "an announcement of the result of an arithmetic operation" rather than as "

**expressing a relation**" is a major reason for their failure to achieve algebra."

**2. Subtraction is a Form of Addition**

The second major point in 1st-grade arithmetic is that

**subtraction is defined in terms of addition**: a - b = c is a true statement if and only if [iff] c + b = a. That is, if 8 - 5 = 3 is true iff 3 + 5 = 8 is true. Subtraction should be taught at the same time addition is taught in 1st grade. Addition and subtraction are

**inverses**: a + b - b = a (

**6**+ 4 - 4 =

**6**). If I add b, I can

**undo**it by subtracting b. Everything can be demonstrated on an

**equal-arm balance or a number line**. Note: In algebra subtractions are changed to additions (a - b = a + -b) and divisions are changed to multiplications (a ÷ b = a ● 1/b) Find box: 8 + ❑ = 0. Find box: 8 ● ❑ = 1.

I often remind elementary students to "

**think like a balance" to test whether a math statement is true or false.**They use the same strategy to find the solution to an equation. The concept is "

**equality is transitive**." In the equation

**5 +**

**❑**

**- 3 = 4 + 1**, what specific number is the

**symbol ❑**to make the statement true? There are several concepts in the problem. The idea that

__equality is transitive__is key: If the right side is equal to 5, then the left side must also equal 5 to make a true mathematical statement. The second idea is the

__undo (inverse)__relationship between subtraction and addition. To undo subtract 3, add 3. Thus,

**❑ = 3**. Therefore, 5 +

**3**- 3 = 4 + 1 (True: 5 = 5). There are other concepts involved. Also, for real numbers

**n and m**, only

**one**of the three conditions can be true: either

**n = m, or n > m, or n < m**.

**Practice:**Lastly, to get 1st-grade students

**to think in the basic concepts of mathematics**requires practice-practice-practice. First-grade students can also learn to

**turn words into mathematical symbols**(i.e., the language of mathematics).

**When I am asked how I teach little kids algebra, I can only say "I don't know."**

I never use formal lesson plans. I jot down some notes.

*I am good at explaining things.*I study. One idea leads to another. It is the way my brain works. Don't ask me how. Ideas float in my head unconsciously and sometimes come together in consciousness. I think the starting point is knowing and understanding the mathematics to rearrange and organize it in such a way to present it in the best possible way flowing from one idea to the next in a logical order (sequence), always writing symbols on the board. It is hierarchical learning. I think about sequence and prerequisites a lot and study what others have done.

**Also, c**

**hildren learn by repetition. The math skills and factual knowledge should come before teaching**

**applications.**

*Schools should teach the mastery of basic skills first, not a bunch of reform math fluff and its alternative algorithms.***"**The

**great power of mathematics**is that it allows us to translate things that we see in the everyday world into mathematics. Then using all the stuff we learn in our math classes, we can analyze, answer things, and make new insights into our world and our lives." (

**Edward Burger**, Professor of Mathematics at Williams College)

*Students should write equations that depict problem situations by turning words into*

**math symbols**.**Problem Situation 1**

Jane has some pencils (

**). Bill gives her 7 more. Now she has 13 pencils. How many pencils did she have to start?**

*x***-->**

*Turning words into symbols:*

**.**

*x*+ 7 = 13**Problem Situation 2**

I am thinking of a number (

*). If I double it and add 3, I get 19.*

**n**

**-->**Turning words into symbols:

*n*+*n*+ 3 = 19**Problem Situation 3**

**A.**Use the equation

**y = x + x - 3**to build an

**x-y**table for x = 2, 3, 4, 5, 6, 7.

**B.**Graph the number pairs in Quadrant-I to sketch a picture of the equation.

**C. If-then S**

**tatements**

If x = 25, find y.

If y = 27, find x.

If y = 97, find x.

**Note:**1st-grade students solved the equations using guess and check, number facts they had memorized, rules (properties) of numbers and equality, inverses, and the algebraic rule for substitution. Also, you should introduce more complicated equations: Find the number

**n**that solves the equation

**n + n - 3 = n + 5**. Students would

**guess and check**.

The symbol

**n**is a specific number. (FYI: n = 8, 13 = 13) In school math, it is often called a variable, but it is actually a symbol for a specific number.

Cognitive Scientist

**Daniel Willingham**(*American Educator*| Summer 2008) wrote:**"Recognize that no content is inherently developmentally inappropriate.**If we accept that students’ failure to understand is not a matter of content, but either of presentation or a lack of background knowledge, then the natural extension is that no content should be off limits for school-age children."**The main reason children have difficulty with a math topic is that they have not learned the prerequisites.**

➡ 1st Grade Teach Kids Algebra (TKA), Spring 2011

Fusing Algebra to Standard Arithmetic.➡ 1st Grade Teach Kids Algebra (TKA), Spring 2011

Fusing Algebra to Standard Arithmetic.

- True False & Equality (=) "Think Like A Balance"
- Equations in One Variable, Guess & Check
- Equations in Two Variables (Input-Output Model)
- Function Rules & Building x-y Tables
- Plotting Points in Q-I & Finding Perimeters
- Graphing Linear Equations in Quadrant I
- Given y, Find x (Reverse, Undo) & Steepness of a Line (Slope)

**Note:**Each of the two 1st-grade classes had 14 half-hour lessons. Each of the two 2nd-grade classes had seven hour-long lessons. I appreciated the cooperation of the four classroom teachers at the urban Title-1 school in Tucson. Almost everything I introduced to students involved basic**arithmetic**such as the auto-recall of single-digit addition facts, number and equality properties, and so on.

**Comment:**Kids can learn much more complex math ideas at an earlier age, I thought.__Algebra can activate cognitive growth__, I thought. Ordinary students in ordinary classrooms can learn algebra, I thought. Think, algebra in the 1st grade, I thought, which is the reason my website is named**ThinkAlgebra.org**.*In my "early algebra" enrichment lessons (Teach Kids Algebra), 1st-grade students solved equations using guess and check, the rules of arithmetic and equality, variables, the rule for substituting, inverses, and single-digit number facts (memorized).**Find the number***n**that solves the equation**n + n - 3 = n + 5**.*1st-grade students also learned functions (input-output model), function rules, built x-y tables of linear functions such as y = x + x - 2, and plotted pictures of linear functions (Graphing in Q-I).***Counting**

Counting is a fundamental skill but by the 1st grade the memorization of basic facts; place value; rules or properties of numbers, operations, and equality (=); and standard algorithms are much more powerful. Counting on and using fingers help kids in preschool and K to learn numbers. For example, 12 + 3 means to start at 12 and count three more (12 --> 13-14-15).

**Zig Engelmann**had preschool and K students memorize multiples to do multiplication such as areas of rectangles. Kids learned to count by 2s, 3s, 4s, ..., 9s from memory. Thus, they could figure out 4 x 5 by counting four 5s from memory: 5 10 15 20. In short, students learned the multiplication table.

**Vector [Number Line] Approach (Direction and Magnitude)**

At the start of 1st grade, an excellent approach to counting is using a

**number line 0 to 20**. Start at zero then move right to 12 and hop three more to 15. (12 + 3 = 15). The vectors are 12 units right, then 3 more units right, and the net effect is 15 units. The number line can be used for simple subtraction (14 - 5 = 9). The vectors are 14 units right then 5 units left, and the net effect is 9 units. Also, it expands for fractions, decimals, and negative numbers. Learning magnitudes is an essential skill. As a counting method, however, the number line is limited.

**Combining Like Terms**

In the standard (vertical) addition algorithm, students learn to add ones to ones and tens to tens, etc., which is similar to combining like terms in algebra.

**Also, it is essential that students memorize single-digit number facts to use the standard algorithm, which is a fast, efficient way to count.**Most addition facts can be learned by Thanksgiving. Flash cards and using the standard algorithm have been effective ways to practice and review addition facts.

Standard algorithms should be the primary method of calculation from the get go. Standard algorithms should be taught and practiced for mastery

**first**. Consequently, students need to memorize single-digit number facts and practice-practice-practice. Repetitio mater studiorum!

*****

**Kids are novices and learn by repetition.**

**Higher-level thinking skills, without substantial knowledge in long-term memory supporting them, are at best superficial.**

The world-class

The world-class

**1997 California math standards**stated that students should know the standard algorithms for whole numbers (addition, subtraction, multiplication, and long division) no later than the end of 3rd grade.

*****Repetitio mater studiorum is Latin for "Repetition is the mother of learning."

Identify 30% of the arithmetic content that will have 70% of the impact on achievement, then spend 70% of class time on the practice and review of the 30% for mastery. The grade levels are a guide. In the

**2nd and 3rd grades**, more complex addition and subtraction (2387 + 4877 and 3067 - 1698), column addition (25 + 87 + 94 + 62), and money ($3.56 + $8.74) should be practiced.**Note: In Singapore, 1st-grade students learn multiplication as the addition of identical addends (i.e., repeated addition), memorize half the multiplication facts in 2nd grade and the rest in 3rd grade along with formal (standard) algorithms.****(1) The standard algorithm is efficient and the best model for place value. It simplifies regrouping (See the 12?)**

**and**

**requires the automation of single-digit addition facts.**

*It is a very fast way to count when numbers are larger. It should be the priority.*

**(2) Doing arithmetic using the standard algorithm in the first semester of 1st grade should be the clear end result.**

*There is much to learn in the*

__first semester__when teachers focus on the end result (standard algorithm). What content do students need to learn to reach the result? There are some helpful ideas on this page. (Perhaps, I should write a 1st-grade primer for parents and educators or a workbook for students.)**(3)**

**To learn something is to remember it!**And the key to learning is

**repetition and regular review!**

**(4) Addition is not understood if you can't do it.**

__Students struggle in arithmetic because they do not know the single-digit number facts from long-term memory. The standard algorithms are easy to learn, but single-digit math facts must be practiced and regularly reviewed. (The standard algorithm is a good way to practice and review single-digit number facts along with flash cards.)__

******Comments:****Children need to drill to improve skill and competency.**

**Unfortunately, competency in standard arithmetic has not been the primary objective of reform math.**

*Incidentally, p*

*ractice does not make perfect; it makes*__improvement__, which requires__focused effort__.**Your job as a parent is to "raise responsible, productive adults."**

__Children do not like practicing, so parents need to call the shots.__

**Memory learning in arithmetic or math is always associated with**

**prior knowledge**, i.e.,

**prerequisites**, as one idea builds on another.

__You understand a new idea as it relates to old ideas in long-term memory__

__.__A typical lesson (shown below) from

*Science-A Process Approach (*illustrates a

**1967**)**sequence of prerequisites (Gagne) and the lesson's behavioral objectives (Mager).**

Note:

**P**

**erformance-based learning objectives**(

**Mager**) are characterized by

**action verbs**such as

**order, demonstrate, name, identify, construct, etc.**

**Mager's idea of performance-based means that the objectives are specific, performed by the Learner, and easily measured.**In short, "the objectives are not a description of the course materials or something that the instructor does."

**For example, what exactly must the Learner be able to do (perform) in arithmetic?**Examples are shown below from a science lesson (SAPA: Using Numbers 8, 2nd Grade, 1967), which I used in

**1st-grade arithmetic**in the early 1980s.

**Comment:**Students used a

**number line**(-10 to 10) to demonstrate how to determine sums of integers. Also, for Using Numbers 7, the procedure demonstrated by my 1st-grade students was the standard algorithm without carrying and later with carrying, etc. To perform the addition

**standard algorithm**well, students needed to memorize the

__single-digit addition facts__and a

__place value system__.

*My 1st-grade students worked very hard to learn basic arithmetic.*

Part C is 2nd Grade from Science-A Process Approach (1967). The sequence shows the prerequisites and how the they link together. The learning (behavioral) objectives use specific action verbs. In math, the correct sequence is crucial. Incidentally, in the 1982-1983 school year, I shifted Using Numbers 5-6-7-8 down to my self-contained 1st-grade class. Also, I did Communicating 11 and Measuring 14 in 1st grade. In the spring of 2011, I did the equivalent of Using Numbers 8, Communicating 11, and much more with two 1st-grade Title 1 classes at an urban school.

**To learn something is to remember it!**And the key to learning is

**repetition**says

**Maria Brilaki**writing on Lifehack:

**"By actually doing something new over and over again, your brain wires new pathways that help you do this new thing better and faster."**

"

**Practice through attentive repetition develops permanence in memory.**"

*(Quote: Committee of 15 Report, 1895)*In today's progressive classrooms, students do not practice enough to automate the factual and procedural fundamentals of standard arithmetic in long-term memory.

__Unfortunately, memorization and practice (i.e., repetition) of fundamentals (i.e.,standard arithmetic) have fallen out of favor in many modern classrooms.__It is one reason that our students stumble over simple arithmetic. Contrary to the American instruction,

**Singapore 1st-grade students**memorize the addition facts and related subtraction facts and learn formal algorithms for addition and subtraction. They are also tracked. Dr. H. Wu (UC-Berkeley) points out, "Computational facility on the numerical level is a prerequisite for facility on the symbolic level." In short, students who know arithmetic number facts and procedures to automaticity (numerical level) do well in algebra (symbolic level).

**The importance of place value is often overlooked.**

The standard algorithm embodies the place value system and should be taught first, not the alternative strategies of reform math which are extras. For example, students do not need to make drawings of facts and ideas they know. These are extras (i.e., reform math) and are not needed. When I taught 1st grade in the early 1980s, the standard algorithm was introduced in the 1st marking period. Students memorized single-digit addition facts.

**"Standard arithmetic algorithms can only be understood in the context of the place value system," writes mathematician W. Stephen Wilson. "The place value system is mathematics."**A beginner concept in 1st grade is adding ones to ones and tens to tens. Another key concept is that 12 ones means 1ten+2ones or t + 2, etc.

**Addition is not understood if you can't do it.**

**Note:**In the early 1980s, my 1st-grade students found that using a number line and counting cubes were limited when combining larger numbers. In the 3rd week of school, I proposed a problem similar to 32 + 65. The 6-year-olds worked in groups because there were not enough centimeter cubes to go around. Each group counted out 32 cubes, then 65 cubes, combined them, and counted the total. It was a chore. Needless to say, there were different answers. I said, "I can add the numbers just by looking at them: 97." It was like magic to them. The students were astounded that I could add 32 and 65 that fast without counting cubes and wanted to know how to do it. (They initially thought that I cheated.) It sure beats the one by one counting method, which is inefficient and of limited use.

**First-Grade Early Algebra**

Teach Kids Algebra Project (TKA)

Teach Kids Algebra Project (TKA)

I created

**Teach Kids Algebra**(TKA) to support math education in Title-1 schools. Some fundamental ideas of algebra (e.g., functions, steepness of a line, etc.) "in some intellectually honest form" (Jerome Bruner) can be taught to ordinary 1st-grade students who know some basic 1st-grade arithmetic, such as place value and the instant recall of addition facts from long-term memory. I thought that algebra could be reachable by very young children through

**standard core arithmetic**because the foundation of algebra rested on the shoulders of arithmetic. It could start in 1st grade, I thought. The "early algebra" project called Teach Kids Algebra (TKA) began for 1st and 2nd graders in the spring of 2011. I figured out efficient, straightforward ways to present abstract algebra in a logical sequence to very young students.

**Notes:**

(1) Since 2011, the TKA project has continued off and on. For example, in the 2016-2017 school year, I teach TKA early algebra to 2nd- and 4th-grade students of color at a city Title-1 school. I was asked to continue TKA in the 2017-2018 school year.

(2) Also, by "

**core arithmetic**," I mean "standard arithmetic with standard algorithms." I do

**not**mean reform math with its multitude of alternative strategies (algorithms) found in Common Core or state standards.

(3) Also, learning anything new requires effort, persistence, and practice. "Don't give up. You can figure it out!" We need to get children out of their comfort zone.

The initial TKA school project involved two 1st-grade classrooms, two 2nd-grade classrooms at the same Title-1 school, and a 3rd-grade class. The curriculum for 1st and 2nd graders, which consisted of 7 hours of instruction spread over 2 months, was the same for both grades.

e.g., x • x - 3 • x = 10, and much more. Incidentally, there are two roots (zeros): x = 5 or x = -2. The 3rd graders looked for positive whole number roots. I wonder how many high-school seniors can solve x^2 - 3x = 10? Third graders also worked with equations that involved multiplication, reciprocals, and fractions (12 • 4 • ❏ = 12) and other types.

I knew what I wanted to teach 1st-grade students (i.e., real algebra content), but I had to figure out how to teach it to little kids. TKA was a blunt reaction against reform math and its methods and philosophy. It contradicted the status quo of constructivism, Piaget's stages, minimal guidance methods such as discovery, Common Core, reform math and its instructional methods and philosophy, etc.

TKA highlighted symbolic mathematical language (i.e., abstraction).

**Photo:**TKA also included a 3rd-grade class, which I met twice a week from mid-January through May 2011. Often the 60-minute math period stretched to 90 minutes. While 1st-grade students were working with linear equations such as y = x + x - 2 to make tables and graphs, this 3rd-grade class was figuring out simple quadratic equations using guess and check,e.g., x • x - 3 • x = 10, and much more. Incidentally, there are two roots (zeros): x = 5 or x = -2. The 3rd graders looked for positive whole number roots. I wonder how many high-school seniors can solve x^2 - 3x = 10? Third graders also worked with equations that involved multiplication, reciprocals, and fractions (12 • 4 • ❏ = 12) and other types.

**TKA: Against the Status Quo and Ed Establishment.**I knew what I wanted to teach 1st-grade students (i.e., real algebra content), but I had to figure out how to teach it to little kids. TKA was a blunt reaction against reform math and its methods and philosophy. It contradicted the status quo of constructivism, Piaget's stages, minimal guidance methods such as discovery, Common Core, reform math and its instructional methods and philosophy, etc.

**TKA did not involve group work, calculators, manipulatives, or minimal guidance methods.**Instead, it focused on symbolic mathematical language (abstraction), clear explanations with worked examples (explicit instruction) and practice sheets with feedback. TKA was designed to get children out of their comfort zone and stretch their working memory. Learning core arithmetic and algebra well requires effort, persistence, and practice.**Kids can learn much more essential arithmetic and algebra early on than teachers have been led to believe or are prepared to teach.**Unfortunately, reform math is all about alternatives to basic arithmetic and not mastering core arithmetic that kids need to know early on to get to Algebra-1 in middle school. Also, Common Core and state standards are typically interpreted through the lens of reform math and its alternative strategies (algorithms) and philosophy.

**Abstraction, Abstraction, Abstraction**TKA highlighted symbolic mathematical language (i.e., abstraction).

**Numbers are abstract. Operations are abstract. Properties are abstract, and so on.**The power of mathematics is that it is abstract and should be taught that way. Indeed, 3 + 6 is 9 no matter what is counted; i.e., 3 + 6 = 9 is applicable to hundreds of problem situations. Incidentally, the classroom teachers were astounded at the "advanced" content students had learned in a short period.**Competency (Effort Fosters Excellence)**

Thus, we put away the centimeter cubes (manipulatives) to focus on learning the single-digit addition facts and the standard algorithm in the 1st marking period. By Christmas, students had automated most of the addition facts and some of the subtraction facts, but students never stopped practicing and reviewing them to gain more competency. Also, they learned a simple "carry" idea in the standard algorithm, the equality concept (=), "

**add in any order**" property of addition, and much more. Other properties (mini identities) such as

*x + y = y + x or x + 0 = x*were important, too.

**Up to 70% of each daily math class was practice and review.**

**The focus was**

**on competency, which requires considerable effort!**Practice makes you more competent. It is important to drill-to-improve-skills.

**Effort**is a key ingredient. "Effort fosters excellence" (competency) and "can inspire you to keep learning," writes

**Daisy Yuhas**(March 2017 issue:

*Scientific American Mind*).

**The problem situations students solve should increase in difficulty.**

Start with routine problem situations that use numbers such as

**measuring**(length in cm, mass in grams, and liquid volume in milliliters), finding perimeters of polygons (geometry), problems that involve money, etc.

**My 1st-grade students (early 1980s) did a lot of measuring at the beginning of the school year.**

**Practice makes improvement!**

*If we truly want students to improve in the*

__core arithmetic that prepares them for Algebra__, then they need to practice-practice-practice, which is how novices learn.**Note:**By core arithmetic, I do not mean Common Core or Common Core influenced state standards.

**Measuring & Counting**

"In daily life, we don't just count objects, we also measure quantities such as length, perimeter, area, weight, and time," writes

**Luke Heaton**(

*A Brief History of Mathematical Thought*, 2017). By the first grade, instead of counting one by one using centimeter cubes (manipulatives), counting reaches the next stage, which is

**adding abstract numbers: 2 + 3 = 5.**The equation is

**abstract**and can apply to thousands of real situations. Here is one: Jane has 2 apples. Dan gave her 3 more apples How many apples does Jane have now? Students abstract the numbers and use

**symbolic mathematical language**(

**2 + 3 = ❏**). Then, solve for the unknown:

**❏ = 5**. Lastly, students "undo" the abstraction to get back to the real situation:

**Jane has 5 apples.**Counting faster to do arithmetic involves learning number facts and the standard algorithms.

*In short, the memorization of single-digit number facts is critical for the "doing of mathematics" in 1st grade. The standard algorithm (place value system) counts numbers larger than single digit facts rapidly but uses single-digit facts and the place value system to do it.*

**The doing of mathematics and the understanding of mathematics are rooted in symbolic mathematical language (i.e., abstraction).**

**Thus, one simple, abstract math fact such as 2 + 3 = 5 can apply to thousands of real situations (problems).**Abstraction is the power behind the problem-solving and the understanding of mathematics. Students need to learn, use, and understand symbolic mathematical language. Moreover, the

**behavior**of numbers is governed by a

**system of rules**, such as the number properties and the equality properties. One simple property of addition is

**n + 0 = n**. Another is

**a + b = b + a**, and so on.

**Morris Kline**(

*Mathematics for the Nonmathematician) explains,*"Mathematics proceeds by deductive reasoning from explicitly stated

**axioms**," such as the Commutative Axiom of Addition shown above. He points out

*, "*When a child learns that

**5 + 5 = 10**, he acquires in one swoop a fact which applies to hundreds of situations. Part of the secret of the power of mathematics is that it deals with abstractions." Students can disregard what has been counted, explains

**Luke Heaton,**because numbers and operations are abstractions, so "it doesn't matter whether you are counting apples, pears, or people." Indeed, the abstraction 2 + 3 is still 5 no matter what is being counted.

**Kline**states, "Whole numbers, fractions, and the various operations with whole numbers and fractions are abstractions. ... One must distinguish the purely mathematical operation of adding 3 and 10 [3 + 10 = 13] and the physical objects with which these number may be associated."

**Heaton**writes that "even the greatest mathematicians needed to be taught the rules." All students should be taught the rules and the symbolic language of mathematics starting in 1st grade. Little kids can think about numbers abstractly because they can relate the numbers to concrete objects, but relating numbers to concrete objects is not necessary.

**To play chess well, you need to know and apply the rules of the game. To do math well, you need to know and apply the rules of math.**

Understanding single-digit number addition such as 3 + 6 is easy and obvious. It is a matter of using a number line and combining.

**It is not difficult for**

**1st-grade students**to hop along the number line to visualize the adding (combining) of two single-digit numbers. A simple number line from

**0 - 20**should be taped on each 1st-grade desk on the first day of school and used to solve simple equations such as

**3 + 5 = ❏**and 2 + 4 + 7 = ❏. [Subtraction: 8 - 5 = ❑; 8 - 3 = ❑]. Kids should start to learn facts such as x + 0 = x (identity) x + y = y + x (commutative),

**add 1 to get the next number**: x + 1 = [next number]; n + 2, x + y = 10, etc. (

**Note:**If students learn the "make 10" facts early, then they can figure out

**3 + 6 + 7 = ❏**by adding 3 + 7 first to make 10 and adding 6 to 10 to make 16. The "

**add in any order" property of addition**is a powerful idea.)

**The Standard Algorithm Is Primary**

1st Grade: Standard Algorithm

**The standard algorithm is efficient and the best model for place value. It simplifies regrouping**.

*The idea behind the algorithm is to add ones to ones and tens to tens, etc., because this is how the number system works.*The grid shows that when ones are added to ones, there are

**12ones**, which is automatically broken into

**1ten 2ones**. (See the

**12.**) Record the 2ones in the one's column and "carry" the 1ten to the ten's column (

**carry mark**). Then add tens to tens for a total of

**3tens**. The answer is 32, which means 3tens 2ones.

**The standard algorithm is economical and helps automate fundamentals.**It is what we want first graders to do. 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 and should be presented at the beginning of the school year when students start memorizing single-digit addition facts.

**Core Knowledge**(

*What Your First Grader Needs to Know*):

**"**In school, any successful program for teaching math to young children follows these three cardinal rules:

**(1) practice, (2) practice, and (3) practice**. In school, first graders should practice math daily, to ensure that they can effortlessly and automatically perform the basic operations upon which all problem solving and other sophisticated math applications depend. Some well-meaning people fear that practice in mathematics--for example, memorizing the addition and subtraction facts up to 12, or doing timed worksheets with 25 problems--leads to joyless, soul-killing drudgery. Nothing could be farther from the truth. The destroyer of joy in learning mathematics is not practice but anxiety--the anxiety that comes from feeling that one is mathematically stupid or lacks any 'special talent' for math.

**"**

*(Do not confuse Core Knowledge with Common Core.)*

**Kids should use symbols starting in 1st grade!**

Dr. H. Wu,Department of Mathematics, the University of California at Berkeley, wrote,

Dr. H. Wu,

**"**There is an effort to put 'algebraic thinking' into all grades. If I understand this term correctly, it means looking for patterns [e.g., 1, 6, ___, ___, ___.], working with manipulatives [e.g., algebra tiles, etc.], and using technology [e.g., graphing calculators, etc.]. The intention is laudable, but this kind of algebraic thinking is

**not enough**to promote the learning of algebra from a

**mathematical perspective**; it must go further in the direction of

**making use of symbols**and computing with them whenever it is natural to do so. I suggest, for example, that the next time you teach the primary grades,

**instead of writing 15 + _____ = 22, try instead, find a number**

*x*so that 15 +*x*= 22.**"**In short, students should use symbols and compute with them starting in

**1st grade**. (Incidentally, the pattern could be 1, 6, 11, 16, 21; or it could be 1, 6, 6, 6, 6 from the repeating decimal 1/6 = 0.1666; or there could be another pattern or no pattern at all.

**We over-stress patterns, manipulatives, and technology.**)

In my "early algebra" program, concepts such as variable, true/false, equality, algebraic rule for substituting, functions, finding function rules, building x-y tables, and plotting x-y graphs were introduced in

**1st grade**. Indeed, algebra topics are accessible to very young children through standard arithmetic becausd the properties (axioms) are the same. Much of the "algebra" I had presented to very young children involved doing arithmetic. First-grade students

*who are learning standard arithmetic can solve simple equations like*

**n + n + 5 = 30 - 5**using*intelligent guess and calculation, that is, guess-and-check arithmetic, memorized number facts, the rule of substitution, and the idea of equality (=).*The left side of the equation must equal the right side (25) to make the equation true.

*It*

*only*

*happens when n = 10.*

**Note:**The H. Wu quote above was from a presentation to the powerful National Council of Teachers of Mathematics (NCTM). But

**I have yet to find an elementary school textbook develop the**

__mathematical perspective of algebraic thinking__that

**Dr. Wu**referred to (e.g., find a number n so that 6 + n = 15, or find the number n so that 4 + n = 12 + 7) in 1st grade or any elementary school math textbook. Also, I have not found the types of equations shown below in textbooks. [Box (

**❑**) is a variable like

*.]*

**x****Using**

**symbols**

**is important. Here are a few**

**1st-grade open sentences**

**from my**

**TKA early algebra program**.

First-grade examples.

*Find box to make the open sentences true. Apply the rule for substitution.*

1. 3 + 12 = ❑ + 3

2. 5 + (❑ + ❑) = 25

3. ❑ + ❑ +

**❑ - 7 = 5**

4. 16 -

4. 16 -

**❑ = 16 +**

**❑**

*And, much more.*

**Note:**

*A math sentence is either*

**true**(3 + 2 = 5),**false**(2 + 3 = 4), or**open**(2 +**❑**= 14)

*. To find box (❑), which means to make the open sentence true, kids use guess and check,*

__, memorized number facts, true-false and equality ideas. (Box is a variable like x.)____rule for substitution__In 2nd grade, I Introduced equations that involved multiplication such as the open sentence

**❑ • ❑ + ❑ = 42**as

**part of my "early algebra" TKA lessons.**

**Note:**The x^2 + x = 42 quadratic equation (x^2 + x - 42 = 0) has two solutions: x = 6 or x = -7. I don't expect 2nd-grade students to guess-check -7 or use a factoring method for a solving

**quadratic equation**; however, they can find

**box = 6**by guess-check.

**(Note:**Some of the ideas were from the 1957

*Madison Project*, but I rewrote them for the 1st and 2nd grades. It was assumed that half the multiplication facts were memorized in 2nd grade.

**)**To find box (

**❑**), which means to make the open sentence (equation) true, kids should use guess and check,

__algebraic__

__, true/false and equality ideas, and__

__rule for substitution____memorized number facts__. (Note: box is a variable like

*x and*

*represents an unknown number.*)

**"**

**Critical thinking does not exist as an independent skill."**

**Students need to drill (spaced repetition) to boost skill and performance in arithmetic.**According to

**reform math**advocates, children should do what mathematicians do (processes), such as described in the so-called "Standards for Mathematical Practice" (SMP) of Common Core state standards, but children are novices, not little mathematicians.

__Learning school math and "thinking" like a mathematician who has had years of specialized study are entirely different.__

**The purpose of school math should not be to change students into little mathematicians but to make them competent in standard arithmetic and algebra**to prepare for high school math (Algebra-2, Precalculus), chemistry, physics, finance, statistics, etc. and higher level university math (e.g., calculus, etc.) needed for their chosen fields of study. Mathematicians live in a world of abstraction and symbols and seldom use numbers.

**School kids are novices and need to learn numbers, arithmetic, algebra, and practical applications well.**

Students don't study arithmetic to argue from evidence or to become more creative or inventive. Math is not a matter of opinion. Kids study arithmetic to learn

__factual and standard procedural knowledge__needed to solve a range of problems. Also, arithmetic in long-term memory is the

__building block__for higher-level math such as algebra, trig, and calculus.

**Note:**

**Basic school arithmetic**is more than sums. It includes multiplication, long division, fractions-decimals, ratio-proportions, percentages, and parts of measurement, geometry, and algebra, incuding negative numbers and equation solving. In short, students need to learn the fundamental

**ideas, skills, and uses**of mathematics.

**W. Stephen Wilson**, a mathematician, wrote, "What is striking about reform math is that the

__standard algorithms__are either de-emphasized to students or withheld from them entirely. ... [which] is in contrast with a more traditional focus on understanding and mastery of the most efficient mathematical algorithms." He also indicated, "The staunchest supporters of reform math are math teachers and faculty at schools of education."

**Professor Wilson**points out that the reform math approach has produced kids who have

__weak arithmetic skills.__

**Students will not be prepared for algebra in middle school when basic arithmetic skills are deficient.**Indeed, contrary to reform math, students must memorize single-digit number facts, become fluent in using the standard algorithms from the get go and learn much more such as parts of algebra, measurement, and geometry. To sum up, novices do not need to learn more complicated, alternative (nonstandard) algorithms. They are extras that clutter the curriculum. They increase cognitive load and push aside the mastery of standard arithmetic, i.e., factual and standard procedural knowledge.

"

**Mathematics builds new ideas on old ones**. ... It is hierarchical and logical. Everything fits together," writes

**Ian Stewart**(

*Letters to a Young Mathematician*, 2006). Like most mathematicians, Stewart thinks beginners should

__learn arithmetic basics first__. Contrary to Common Core (reform math), you don't ask novices to attend to precision, use appropriate tools strategically, construct viable arguments or argue from evidence, show many different ways to prove an answer, and a bunch of other extras. In short children should not be asked to prove their answer correct by making a drawing, using several different algorithms, or writing a paragraph.

**Arithmetic is not a matter of opinion.**

*Parents will be happy to know that 3 + 5 is still 8 and should be memorized in 1st grade.*

*Furthermore, the standard algorithms for addition and subtraction should also be practiced and learned in 1st grade.*To learn something is to remember it.

**Many topics are taught but not learned.**Learning number facts, for example, requires continual practice and review.

**Stewart**observes, "Math happens to require rather a lot of basic knowledge and technique." Thus, automating factual knowledge and standard procedural knowledge in long-term memory should be the focus in

**1st grade**and above. Students should memorize single-digit number facts and practice standard algorithms for each operation.

In

**S**

**ingapore**, 1st-grade students do that, but not U.S. students. Singapore 1st-grade students memorize the math facts, both addition and subtraction, and drill to improve skill. They also learn formal (standard) algorithms, multiplication as repeated addition, write an equation for a word problem, and much more.

**Note:**Singapore students coming into 1st grade with

__deficient numerical skills__are placed in a pull-out class for math with another teacher with the aim of catching them up. It's called

**tracking**.

A

**1st-grade**question would be:

**What is 3 and 9, less 5, times 3?**

Incidentally, 1st-grade students should use

__memorized facts from long-term memory__, not fingers or counters. Many questions are multistep. Singapore kids memorize and drill to improve skill, but these do not dent their creativity, inventiveness, curiosity, or problem-solving abilities as some claim wrongly. It certainly didn't hurt

**Einstein**and other brilliant scientists of his era. Also, on international tests,

**Singapore students**are the best in math content knowledge

**and problem-solving**.

*They leave U.S. students in the dust.*

(Note: The commas indicate the order in which the operations should be done:

**3 and 9**are 12; 12

**less 5**is 7;

**3 times 7**is 7 + 7 + 7 = 21.)

Standard Arithmetic Fundamentals Do Not Change!

W. Stephen Wilson, Professor of Mathematics, Johns Hopkins University, writes that the five

Source of Wilson Quotes: "Elementary School Mathematics Priorities" by

*But, reform math tries to circumvent the fundamentals of standard arithmetic with alternative, nonstandard procedures!*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: numbers, place value, whole number operations, fractions/decimals, and problem-solving.**Early elementary school math basics or building blocks do not change and should be taught to all students in a straightforward manner, says Wilson.**For example, the memorization of single-digit number facts (numbers) and the fluency in standard algorithms (whole number operations and place value) "give students power over numbers" and are the critical "basic skills and knowledge that a solid elementary school mathematics foundation requires."**Wilson explains, "The general operations [standard algorithms] reduce to the single-digit number facts." It's fantastic!**Teach standard arithmetic first (for automaticity), and focus on**one efficient method to calculate each operation**(i.e., the standard algorithms).Source of Wilson Quotes: "Elementary School Mathematics Priorities" by

**Dr.****W. Stephen Wilson****(Notes:****1.****Jason Zimba**, one of the two main writers of the Common Core math standards, acknowledges, "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."*Unfortunately, it is not the way Common Core math is most often interpreted.***2. David G. Bonagura Jr.**writes in the*Wall Street Journal*, "Contrary to today's education theories,**memorization**is critical in the classroom and life."**)****[Note.**Indeed, at least

**140 years ago**, American

**2nd-grade students**had learned the following:

**(1)***How many are 2 and 5, less 3, multiplied by 9, divided by 6?*

**(2)**What will 63 marbles cost, if 14 marbles cost 2 cents?**]**

*Source: Ray's New Primary A*

*rithmetic (1st/2nd grade) 1877*

**A 1st-grade question would go like this: How many are 5 and 3, less 4, multiplied by 3**

**.**(The student should approach this problem as follows: First add 5 and 3, which is 8, then subtract 4 from 8 to get 4, and then multiply the 4 by 3 to get

**12**.) In

**Singapore**, 1st-grade kids learn addition, subtraction, and multiplication (as repeated addition). They memorize single-number facts, work with "formal algotithms" (i.e., standard algorithms), write equations in one variable from word problems and solve them.

**[**

**Note:**First-grade students in my

**Teach Kids Algebra**program (2011) wrote equations based on a relationship in a table of values (x-y) and more. (e.g., To find

*y*, start with

*x*and

**add 3.**The "add 3" equation is

**y = x + 3**.) The math was abstract and content centered. The

**principal idea**was to show that the graph of an algebraic linear function really is a line in the coordinate plane and vice versa. Figuring out function rules, writing equations in two variables, building a x-y table of values, and point plotting (x, y) are essential to learning some of the basics of algebra. In the 1st and 2nd grades, students can grasp a functional understanding of some of the required algebra ideas and skills. It's a start.

**]**

**1st-grade students can link abstract numbers to (1)**

__Note:__**letters**(variables), (2)

**relationships**(functions, x-y), and (3)

**pictures**(graphs). Start with the meaning of numbers as place value (12 is 10 + 2 or t + 2), single-digit facts (from memory), and standard algorithms that are based on place value. Introduce variables early: n + 3 = 7.

**Mathematician W. Stephen Wilson points out, "**

**The place value system**You cannot teach mathematics without the place value system, standard algorithms, and the other building blocks."

__is__mathematics!**For example, In the early 80s, I introduced 1st-grade students to two-digit numbers by**place value

**and the**standard algorithm (tens-ones) in the first month of school when students had begun memorizing some of the addition facts. (At first, teach the standard algorithm without carry. Once students know more facts by heart, then introduce "carry" concept.) Students should think this way:

**23 is 2tens+3ones**, 35 is 3tens+5ones, and 58 is 5tens+8ones. In short, they need to know that

**23 means 2tens+3ones**

**or 2t +3, and so on.**

**Place value**Remember, kids are novices; they will not have perfect understadning, which grows slowly over the years. In the 1982-1983 school year, I made practice sheets. It was extra work, but my 1st-grade students learned arithmetic well.

__is__mathematics.**In the past, the standard algorithm was always the key model for**

**place value**. First-grade students should systematically memorize addition facts starting in the second full week of school. Ask students to figure out the simple number facts, such as "n + 2" and "make ten" patterns on a number line. Use vocal exercises and

**flash cards**to "

**drill-to-improve-skill**" on a daily basis to cement the number facts to long-term memory. Students should practice number facts at school and home. Single-digit number facts--automated in long-term memory--are needed for the

**standard addition algorithm**.

**Addition is not understood if you can't do it.**Thus, students need to be able to perform arithmetic well.

__One of the best ways to__. The concept of addition is easily shown on a number line.

**practice the recall of number facts**is to perform the standard algorithm**"Extracting a math problem from a word problem requires a high level of critical thinking," writes Professor Wilson.**It is called mathematizing. Students should extract numbers and write an equation to

**symbolize and model**the word problem. Start with easy problems. (Jayne has 5 candy bars. Jim gives Jayne 2 more candy bars. How many candy bars does Jayne have now?). The extracted math problem (an equation) is

**5 + 2 = ❏**, which students should write on paper. "Five plus two equals box."

**Box is a variable. A variable is a number.**The

**mathematizing process**requires lots of practice and immediate feedback from the teacher.

*Students need to practice writing equations even though the answer is obvious.*(Note: In this case, the students solve the equation using

**memorized number facts**: n + 2 facts. The students could also write

**5 + 2 = 7**. First-grade students in Singapore and elsewhere do this.)

*Answer: Jayne has 7 candy bars.*

Later, word problems can use larger numbers. (Jane has 23 pencils. Joe gives her 35 more pencils. How many pencils does Jane have now?) Soon, students should learn the "carry" idea. First-grade students should also work on subtraction situations, such as

Later, word problems can use larger numbers. (Jane has 23 pencils. Joe gives her 35 more pencils. How many pencils does Jane have now?) Soon, students should learn the "carry" idea. First-grade students should also work on subtraction situations, such as

**missing addend**equations.**Problem Situation: If I add 7 to the number n, I get 12. (n + 7 = 12)**

**The Core Knowledge Blog**

According to

**E. D. Hirsch Jr.**(

*Why Knowledge Matters*, 2016), “American teachers (along with their students) are being blamed for intellectual failings that permeate the system within which they must work.” The real problem, says Hirsch, lies in “

**failed educational theories**,” in particular:

1. Early education should be appropriate to the child’s age and nature, as part of a natural developmental process.

2. Early education should be individualized as far as possible—to follow the learning styles and interests of each developing student.

3. The unifying aim of education is to develop critical thinking and other general skills.

**These ideas, while attractive, are misguided**, says Hirsch.

(Source:

*The Core Knowledge Blog*)

**Note:**These theories [stages of natural development (Piaget), learning styles, thinking that is independent of content knowledge, and others] are not supported by the

**cognitive science of learning**. Simply, they don't work well in the regular classroom. Still, they stick to education like glue and significantly influence schooling.

Note: We do not live in Lake Wobegon where all the children are above average.

*Not every child can be a doctor, play at Carnegie Hall, be an Olympic gymnast, or go to college. Indeed, dreams often do not match the child's abilities.*"Abilities not only vary; they vary a lot," says

**Charles Murray**. Moreover,

*"*Low intellectual ability is the reason why some students don't perform at grade level," writes

**Charles Murray**(

*Real Education*, 2008).

**Note: Practice makes improvement possible, but it does not create talent.**The ability has to be there, to begin with, explains

**Ian Stewart**(

*Letters to a Young Mathematician*, 2006). Indeed, ability matters a lot.

**Lastly, an ability will**

**not blossom**

**without expert instruction**

**and repeated practice of fundamentals (i.e., drill to improve skill).**

**Hirsch**

**writes,**"

**Critical thinking does not exist as an independent skill**. The

*domain specificity of skills*is one of the most important scientific finds of our era for teachers and parents to know about, but it is not widely known in the school world.” Hirsch points out, "

**The basis of skills is specific domain knowledge**." The stress on

**all-purpose critical thinking skills**, which do not exist,

**has been a significant waste of instructional time. Critical thinking must be linked to or sprout from a specific domain of knowledge.**

__Many lessons found in 2nd-grade__

**Note.***enVision Math*textbook, such as

**addition with a simple carry and subtraction with a simple borrow**, should have been learned in 1st grade.

**The standard algorithms are based on place value.**

Proficiency with the standard algorithms (left), however, requires the auto recall of single-digit number facts stored in long-term memory. Unfortunately, tried-and-true

**standard arithmetic**(e.g., memorization of single-digit number facts, standard algorithms, drill-to-boost-skill, etc.) has been pushed out, or delayed, or downplayed by

**constructivist reform math**people who advocate alternative, more complicated procedures to do simple arithmetic and minimal guidance in group work. Both are bad ideas for beginners.

**The topics students can learn, even at age 5 and 6, let's say algebra ideas linked to arithmetic, is not a function of age (Piaget's natural stages) but a function of background knowledge. Students are ready to learn a topic if they have the right prerequisite knowledge. (See my**

__CommonCore__page!)Moreover, first-grade students come to school with a built-in

__logarithmic number line__(unequal units),

**but in arithmetic, the number scale is linear (equal units)**and needs to be taught that way. The distance between 2 and 3 is ONE unit or just plain 1. To get the next number (n'),

**add 1**to n: n + 1 = n'. (FYI: n' is read n prime) Thus, 2

**+ 1**= 3; 87

**+ 1**= 88, etc.

**Linear number lines**should be used from day-one in teaching arithmetic to first graders, but I don’t see number lines in math textbooks.

*Counting starts at 1, but number lines and rulers start at 0.*

My direct experiences with

**1st-grade students**--in the late 1960s (

*Science--A Process Approach, SAPA*), the early 1980s (self-contained), and in the spring of 2011

**(**

**Teach Kids Algebra or TKA program**

**)**--have demonstrated, at least to me, that very young children can learn complex content at some level of understanding, although incomplete, not only in mathematics but also in physical science. Early algebra lessons challenged elementary school students in grades 1 to 5, developed mathematical ability and thinking, found hidden talent, and gave advanced math opportunities to underrepresented students.

Furthermore, I firmly believe that

__excellent math students__—like good gymnasts and pianists—are the

**product of excellent instruction and practice**. And, similar to very young musicians, very young arithmetic students need to

**drill to improve skill and**

**performance**so that they can

**achieve, advance, and excel.**

**Indeed, good math students are the "**

**product of training**

**, not of some inborn genetic programming,"**

**writes**

**Anders Ericsson**

**(**

*Peak*, 2016)

**.**

The

**math training**should start in 1st grade or earlier. (Indeed, many Asian parents teach their kids numeracy skills long before they enter school and push their kids into early piano or violin lessons because they think the music lessons will help their kids learn mathematics better.) In Asian schools, kids memorize the essentials of arithmetic because “facts in long-term memory must precede higher-level thinking skills,” says

**Daniel T. Willingham**, a cognitive scientist. But, it is memorization with some level of understanding, which is imperfect and can vary widely, even day to day.

**Simply,**

**learning is not linear; i**

**t's jerky!**Moreover, good teachers realize that conceptual understanding is far from perfect and

**develops slowly over time**. Educators should worry more about a child's

**performance**than their understanding, which is difficult to measure.

**Cognitive Scientist**

**Daniel Willingham**(

*American Educator |*Summer 2008) wrote: "Recognize that

**no content is inherently developmentally inappropriate**. If we accept that students’ failure to understand is not a matter of content, but either of

**presentation or a lack of background knowledge**, then the natural extension is that no content should be off limits for school-age children." The main reason children have difficulty with a math topic is that they have not learned (automated) the

**prerequisites**in long-term memory. Very young children can learn substantially more arithmetic, algebra, and geometry than most educators and parents think. The myopic view of educators, curriculum people, textbook writers, policymakers, and parents has led to

**low expectations**for a wide range of students, including average students.

**In arithmetic, kids need better content (curriculum) that is taught well (instruction), and they need to drill to improve skill.**

*"*Claims based on Piaget’s highly influential theory, and related theories of “developmental appropriateness” that children of particular ages cannot learn certain content because they are “too young,” “not at the appropriate stage,” or “not ready” have consistently been shown to be wrong. Nor are claims justified that children cannot learn particular ideas because their brains are insufficiently developed, even if they possess the prerequisite knowledge for learning the ideas." (Source:

**National Mathematics Advisory Panel**Report 2008)

**Kids are novices, not pint-sized mathematicians.**

*Despite that, we ask novice students to explain (prove) their answers by making diagrams or writing paragraphs as if they were experts, which they are not.*In contrast, showing steps should be more than enough.

**Gordon Rugg (**

*Blind Spot*) explains

*,*

**"Just because you know something doesn't mean you can put it into words, teach others, or tell others how you're thinking about a problem."**

*Experts often can't explain how they do things or why something they know works.*

It is not difficult for 1st-grade students to hop along the number line to visualize the adding (combining) of two numbers. Once a number fact, such as

**3 + 4 = 7**, is determined, then students need to memorize it. Indeed, 1st-grade student should start**memorizing-and-using single-digit number facts**beginning the second week of school. The standard algorithms and abstract exercises should be used to practice and cement math facts in long-term memory. Also, students should use the**number line**taped to their desks--not counters (manipulatives)--to figure out facts. Number lines start at zero. Moreover, students need to see that there are equal units or distances between consecutive numbers.To practice number facts, I often asked students to measure polygonal curves and, later, perimeters of polygons using centimeter rulers. The idea of perimeter would expand to 3 and 4 line segments (triangle, square, rectangle) and use the idea of

Number + Number = Sum

**chain addition**.**FYI:**A polygonal curve consists of line segments.**At first**, students worked with number combinations they were memorizing, such as n + 2.The 5**+ 2**= 7 math fact is retrieved from long-term memory to find the sum.Number + Number = Sum

**Note:**The idea of adding lengths (distances) to find a sum comes from using the number line.**Mathematical reasoning, logic, and problem-solving come from mathematical knowledge and skill automated in long-term memory, not thin air.**

You can't learn to do arithmetic without doing a lot of arithmetic.

**A student does not need to be gifted or a genius to learn Algebra-1 in middle school, AP Calculus in high school, or grasp some algebra fundamentals in 1st grade. Ordinary kids can do these things when they are**

__taught well__and__practice purposefully__**to improve, achieve and excel.**

**"Almost any child can learn math if it is taught [and practiced] in the right way," writes Anders Ericsson**

*(Peak,*2016

*).*Students need to practice math with

**feedback**to develop and improve skills. While gaining mathematical knowledge is essential, the focus of practice should be on the

**doing of math**(i.e., the performance) and how to improve it. According to

**Ericsson**, there is the tendency to focus too much on knowledge

**at the expense of math skills**. Students should practice the most

**efficient procedures**to do math well.

**Ericsson**says that practice is not always fun, but good students know that improvement in performance (e.g., math skills) requires focused practice with feedback. Good math students are the "product of training, not of some inborn genetic programming," writes Anders Ericsson

**(**

*Peak*, 2016)

**.**

**Comment. Unlike others who write or blog about education, I practice what I preach.**I go into elementary school classrooms and give algebra lessons as a guest teacher.

**I create my own curriculum by fusing algebra content to standard arithmetic. The more arithmetic the student knows in long-term memory, the better.**I look at arithmetic and think algebra.

**Note.**

*My direct experiences with*

**1st-grade students**--in the late 1960s (Science--A Process Approach), the early 1980s (self-contained), and in the spring of 2011 (**Teach Kids Algebra or TKA program**)--have demonstrated that very young children can learn complex content at some level of understanding, not only in mathematics but also in physical science.**Early algebra lessons challenge elementary school students in grades 1 to 5, develop mathematical ability and thinking, find hidden talent, and give**advanced-math opportunities to underrepresented students.

Good math students are the "product of training, not of some inborn genetic programming."

writes Anders Ericsson (

*Peak*

**, 2016).**Ericsson continues, "The right sort of practice carried out over sufficient period of time leads to improvement. Nothing else."

*It's not that kids can't learn algebra ideas in 1st grade; it is that the ideas were never taught or practiced to develop and improve skill.*

**My Teach Kids Algebra (TKA) program, which fastens algebra to standard arithmetic, has shown that the right kind of training and practice boosts a student's ability to learn and do math.**In short, very young children can learn math when it is taught in the right way.

"Practice through attentive repetition develops permanence in

**memory**."

*The concept is automation in long-term memory via drill-for-skill (repetition).*Note. Quote on practice is from the

**Committee of 15 Report, 1895**.

**The equality idea in equations is easily demonstrated using an equal-arm balance and can be taught in 1st grade.**The pans are equal: 4 = 4. If

*I take one cube out of one pan, then I need to take one cube out of the*

*other pan to keep the balance (=).*The idea of equality is essential for balancing equations.

*It is an upgrade, not a reform or reinvention. It is basic reasoning or logic used in mathematics.*The equal sign represents a relationship between expressions on the left and expressions on the right.

**I fused algebra to standard arithmetic.**At first, students used the algebra-rule-for-substituting, memorized number facts, and guess-and-check to figure out equations such as

**n + n + n - 6 = n + 8**. The equal sign is a relationship, i.e., the left side must equal the right side in value.

Also, students built x-y tables, figured out function rules, plotted equations in Q-I, wrote equations for word problems, etc. Calculating skills are always important. Note: Third-grade students should practice straightforward standard algorithms for whole number operations, including multiplication and long division.

The

**undo concept**should start in

**1st grade**: 6 +

__7 - 7__= n.

**n**is 6 because 7 - 7 is 0 and 6 +

__0__= 6.

**Simply, knowledge in long-term memory is the basis for all mathematical thinking and reasoning skills (aka problem-solving)**.

**Daniel Willingham**points out, "

**Factual knowledge must precede [thinking] skill**

**.**"

*Also,*

*I want kids to***think on paper**, so I ask them to take notes in class__starting in 3rd grade__.**]**

*End Excerpt*

**Calculating 7 + 5 in reform math.**

__Strategies Method:__Add 3 to 7 to make 10, then subtract 3 from 5 to get 2, then add the 2 and the 10 to get 12.**The three calculations clog the child’s working memory.**

**The bottom line is that students should memorize and use single-digit number facts, so they stick in long-term memory.**Knowing

**7 + 5 = 12**

**automatically**is a different mental process.

**If 7 + 5 = 12, then 5 + 7 = 12. It is logic.**(Also, if 7 + 5 = 12, then 12 - 5 = 7 or 12 - 7 = 5) Also

*,*

**from****memory**

*is better and more reliable than calculating using strategies. With single-digit number facts in long-term memory, along with place value, the student is aptly equipped to practice and learn the*

**standard algorithms**

*for fluency.*

Algebra ideas are accessible to very young children, including 1st-grade students, through standard arithmetic. Students should learn some standard arithmetic in long-term memory, such as the instant recall of addition facts, to learn some algebra. In my algebra approach (Teach Kids Algebra or TKA), 1st-grade students learn to link

**abstract numbers**to (1)**letters**(symbols: x, y, etc.), (2)**relationships**(functions, equations, etc.), and (3)**pictures**(coordinate graphs). They also learn (1)**rules**(often called number properties) to describe the behavior of abstract numbers (n + 0 = n, etc.), (2) standard algorithms and place value to calculate, (3) the relationship between addition and subtraction, (4) undo (n + 6 - 6 = n), etc.1st & 2nd Grade/Writing Equations

Directions: Write an equation, then solve for

1. If I add 4 to the number, I get 9.

2. If I subtract 6 from the number I get 5.

3. If I double the number and add 2, I get 8.

4. If I triple the number, I get 6.

*From LT's Teach Kids Algebra (TKA) project: May 2, 2011***I am thinking of a number n.**Directions: Write an equation, then solve for

**n**using ideas shown in class.1. If I add 4 to the number, I get 9.

*Response: n + 4 = 9, n = 5*2. If I subtract 6 from the number I get 5.

*Response: n - 6 = 5, n = 11*3. If I double the number and add 2, I get 8.

*Response: n + n + 2 = 8, n = 3*4. If I triple the number, I get 6.

*Response: n + n + n = 6, n = 2***Note.**First-grade students who are prepared find writing equations easy. Soon, using an**input-output**function model (x, y), students work with equations in two variables in the form of y = x + b, then y = mx + b. They build x-y tables and graph simple functions in Q-I.

**Algebra grows out of arithmetic.**

*Kids must be good at*

**standard arithmetic**to learn algebra well.**When arithmetic is taught well, some basic ideas of algebra can be explored in the 1st grade.**

*I know, I did it at an urban, Title-1 school with about 40 first graders in 2011--almost all minority students. Giving very young students abstract ideas in math prepares them for the future. (*

I am thinking of a number(x + 7 = 12)

**x**is a number.)I am thinking of a number

**x**. When I add 7 to the number, I get 12. What is the number?The more I know, the more I can learn, the better I can think,

*which is my*

*summary of the*

**cognitive science of learning**.*For*

*students to learn*standard arithmetic or algebra well and__get it right__, they must be**persistent****and work hard.****In short, grasping****the algebra as it is fused to arithmetic and getting the algebra right are the main goals, which require a persistent mindset and hard work.****Indeed, learning requires**

__practice__, attention, focus, and effort.*Also, the*

*****knowledge and skills taught traditionally in standard arithmetic are often pushed aside or delayed by the "many different" ways, strategies, or models to calculate (aka*

**reform math**), along with the prevalent**minimal-guidance-methods of instruction**(e.g., engaging in group work with discovery, implementing constructivist and hands-on activities, making drawings or visuals, and using calculators and technology).**Okay, how do we get from 4 to 5?**The number line shows how (and why). We

**add 1**, which, on the number line, means you move

**one unit**to the right of 4 to get to 5:

**4 + 1 = 5, that is, to get 5, add 1 to 4.**This is the

**successor rule for integers**:

**a + 1 = a'**, in which

**a**is an integer {4} and

**a'**(a-prime) is its successor {5}. The idea that you get the next integer by adding one to the previous integer is very important, but, unfortunately, it is seldom taught this way in 1st grade. The successor rule works for all integers.

**For example, -6 + 1 = -5 or -1 + 1 = 0.**

*You don't need to call it the "successor" rule. For whole numbers, you can call it the "add 1" rule. Furthermore, what seems obvious to adults, is not always clear to 5 and 6-year-olds.*

**Remember, kids don't think like adults, which is a basic premise of cognitive science, so presentation and explanation are important in teaching young children arithmetic as are "drill for skill," pattern recognition, place value, etc. (Note.**

*The*However, the number line, including negative numbers, is found in a couple lessons from the K-6

**number line**is an excellent teaching tool, but I seldom see it in American or Singaporean first-grade or primary school textbooks.*Science--A Process Approach (SAPA)*, "Using Numbers," Part B (1st grade), 1967. Also, in the SAPA 1st grade lessons, the addition of one- or two-digit numbers with a sum less than 100 was taught, including additions such as 57 + 28 that involve a carrying technique. The importance of math as part of science starts early in SAPA as 4 of the 6 processes taught in 1st grade are math or math related.)

*Please excuse errors. I added some content at the top of the page, but the page is still too long, so more content will be eliminated soon: 10-20-16, 12-11-17*

Changes made on 5-9-17, 5-16-17, 6-7-17, 8-1-17, 9-14-17, 10-23-17, 11-10-17, 11-21-17, 11-28-17, 12-4-17, 12-9-17, 12-11-17

Changes made on 5-9-17, 5-16-17, 6-7-17, 8-1-17, 9-14-17, 10-23-17, 11-10-17, 11-21-17, 11-28-17, 12-4-17, 12-9-17, 12-11-17

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