**Welcome to First Grade, Page 1**

**The Seeds of Innumeracy Are Planted in Lower Elementary School**

When we think of innumeracy, we think of adults, but innumeracy starts in the 1st grade with reform math. Schooling in the early years should focus almost entirely on

**numeracy and literacy,**which are highly useful skills. Children who don't memorize the multiplication table or learn the standard algorithms by the 3rd grade are delaying useful math skills. Learning half of this and a little of that is a poor start to developing numeracy.

**Acquiring skill in standard arithmetic is a function of mastering single-digit math facts beginning in the 1st grade.**Math facts in long-term memory are the prerequisites to standard algorithms. In short, mastering the number facts is fundamental arithmetic. The facts are essential and must be continually practiced and reviewed until they stick in the memory. Students should drill to develop skill. Unfortunately, the mastery of essentials is not the primary goal of reform math, which has dominated elementary school classrooms for decades. The latest version of reform math is Common Core or state standards and their tests. Reform math clutters the curriculum with so-called "standards of mathematical practice" and many "alternatives algorithms." It confuses novices. In first grade, learning only a few of the easy addition facts is insufficient and not close to mastery. This is where innumeracy starts. Learning is remembering from long-term memory.

*Much is taught, but little is learned.*

**Note: Poor numeracy**is one reason that U.S. 4th-grade students cannot find the interest on $50 for 6 months, at 6%.

**(**Problem from

*Ray's New Intellectual Arithmetic, 3rd & 4th grade,*1877. In the 19th century, Ray's textbooks were popular in American schools.

**Indeed, in the old schools, literacy and numeracy were learned.)**

This

(FYI: I taught SAPA

In my early algebra program, I fused basic algebra ideas to standard arithmetic starting with two 1st-grade classes at a Title-1 urban school. I have returned to teaching basic algebra ideas to a 1st-grade class as a guest teacher (2018). A list of topics from my 1st-grade

**webpage**explores some of the math content I taught to typical**1st-grade****students**in the early 80s and the spring of 2011. I was influenced by the*Madison Project*(**Robert B. Davis**, 1957),*Science-A Process Approach or SAPA*(**Robert****et al. and AAAS 1967),****Gagne****Zig Engelmann,****Richard Feynman,**and others**.**In 1st-grade**SAPA**,__taught were arithmetic or math related (Numbers, Graphs, Measurement, and Geometry). The math in SAPA was far ahead of the typical curriculum of the time.__*four of the six science processes**In short, SAPA students learned the math that was needed to do the science.*There is nothing like it today. In fact, mathematics has been missing from elementary school science programs*for decades and decades*.**In SAPA, you teach a lot of basic arithmetic ahead of the school's curriculum, including parts of algebra, measurement, and geometry.**If you can't measure it, it's not science. If you can't falsify it, it's not science.**It's speculation!**Junk science, "so-called" research studies, and opinion float around disguised as science. Statistical correlations are just that, correlations, not cause and effect. Anecdotal evidence is not evidence; it is opinion. Something that is "statistically significant" does not mean it is useful in the classroom or even important. In short,**"Science doesn't prove anything true--all it does is get rid of false views"**and hopefully gets a tad closer to the truth, explains**Nigel Warburton**(*A Little History of Philosophy*). Likewise,**Charles Wheelan**writes, "Statistics cannot prove anything with certainty." Some scientists and researchers are biased, fudge the data, extrapolate beyond the data, use bad samples, and pass correlation off as cause and effect. Scientific integrity was a big concern of Nobel physicist**Richard Feynman**who wrote, "We really ought to look into [education] theories that don't work, and science that isn't science."**Be skeptical of claims made by scientists, studies, researchers, pundits, thinktanks, or so-called experts.**(FYI: I taught SAPA

**K-6**in the very late 60s when I was head of the science department at The Wyndcroft School. My salary was $3,800, not nearly enough to keep me there. In December of 1969, I accepted a position to teach 7th-grade biology (and later mathemathics) in a small school district in Wilmington, DE. My salary increased over 200%. I had an emergency (temporary) teaching certificate and took courses at the University of Delaware in the early 70s to earn a regular teaching certificate. I never majored in educaton.)In my early algebra program, I fused basic algebra ideas to standard arithmetic starting with two 1st-grade classes at a Title-1 urban school. I have returned to teaching basic algebra ideas to a 1st-grade class as a guest teacher (2018). A list of topics from my 1st-grade

*early algebra program*of 2011, known as**Teach Kids Algebra**or TKA, can be found on**.**__First Grade, Page 2__*With a few modifications, I used most of the same curriculum (lessons) I developed seven years ago.***9 + 7 = 16:**The number line is important mathematics! Indeed, the single-digit number facts can be shown on a number line. Using a number line leads to the memorization of single-digit facts. Unfortunately, it is seldom used in today's reform math programs. Students in 1st grade should memorize the addition facts for auto recall and learn the

**standard algorithm**no later than Christmas for larger numbers.

*The standard algorithm is an efficient*

**place value system**

*for combining ones to ones, tens to tens, etc. It uses single-digit numbers facts from long-term memory.*The paper-pencil standard algorithms are the best tools for beginners to do basic arithmetic.

**Note:**I taught a 1st-grade self-contained class in the early 80s. My students memorized the addition facts and used the standard algorithms for larger numbers by Christmas. They also learned metric measurement by measuring mass, liquid volume, and length (g, mL, cm, m), negative numbers (number line), fractions, perimeters, money, etc. Multiplication was introduced as repeated addition. Also, I taught my

**Teach Kids Algebra**program (

**TKA**) to two first-grade classes in 2011. Currently, as a guest teacher, I am teaching my

**TKA program**to multiple grade levels at a Title-1, urban K-8 school, including a

**1st-grade class.**

"The

Thinking skills such as "critical and creative thinking, and problem-solving" are all the rage in today's classrooms but "are not productive educational aims" as

There are no generalized thinking skills independent of domain content knowledge much to the chagrin of teachers, parents, administrators, school districts, special interests, organizations, pundits, and

**domain specificity of skills**is one of the most important scientific finds of our era for teachers and parents to know about," but they don't.Thinking skills such as "critical and creative thinking, and problem-solving" are all the rage in today's classrooms but "are not productive educational aims" as

**E. D. Hirsch, Jr.**(*Why Knowledge Matters*) points out. "Thinking skills are rarely independent of specific expertise." For example, domain knowledge of math, lots of it, enables and supports thinking skills (i.e., problem-solving) in mathematics.

**Thinking skills are domain specific.**There are no generalized thinking skills independent of domain content knowledge much to the chagrin of teachers, parents, administrators, school districts, special interests, organizations, pundits, and

**reformers**who push "thought without content."**Anders Ericsson and Robert Pool**(*Secrets from the New Science of Expertise*) write, "There is no such thing as developing a general skill." In other words, students cannot solve a trig problem unless they have lots of trig knowledge in**long-term memory**, perseverance and experience solving trig problems.*Likewise, a*Regrettably, most U.S. 1st-grade students are not expected to memorize the addition facts, practice the standard algorithms, or solve perimeter problems. They don't use number lines. Also, the

**1st-grade student**cannot solve**perimeter problems**without knowing and applying basic arithmetic.**mechanics**of standard algorithms should be taught and

**mastered first**with the explanation later.

*The standard algorithms are efficient and mechanical. (All*

*algorithms*

*are mechanical; i.e., they are programmable.)*Moreover, to perform standard algorithms fluently requires the automatic recall of single-digit number facts.

Typical 1st-grade students in my

**Teach Kids Algebra**program (TKA) plotted points, drew the line segments to form a rectangle, numbered it, and calculated the perimeter. (Spring 2011)

In TKA, no calculators, no manipulates, no group work, no Common Core, etc. Students sat at individual desks.

**You don't improve achievement by lowering standards or measuring inputs rather than outputs.**You don't improve performance by cluttering the curriculum with "standards for mathematical practice (expertise)" and many "alternative, nonstandard algorithms" often at the expense of learning essential standard fundamentals. Kids are not experts or little mathematicians; they are

**novices.**

Good education means you increase differences, points out

__. Like musical ability, artistic ability, or athletic ability,__

**Richard Feynman****academic ability**varies widely.

*No ability develops without good teaching and practice-practice-practice-with-feedback.*Some kids will do math better than others. Closing achievement gaps with equal resources (

**inputs**) hasn't worked, not even if you lower standards and "

**equalize downward**

*by lowering those at the top*," writes

__(__

**Thomas Sowell***Dismantling America, etc.*). Equalizing downward is a "

**fallacy of fairness**" and bad policy. Our best students need to shine, but, too often, they do not.

In math, for example, the focus should be on

**performance**--the doing of math to solve problems. But this doesn't happen without knowledge and skill-practice-with-feedback.

**Anders Ericsson and Robert Pool**(

*Peak*) write, "The bottom line is what you are able to do, not what you know, although it is understood that you need to know certain things in order to do what you are able to do."

*Kids don't stop learning and improving because they have reached a limit; they stop learning and improving because they stopped practicing skills, write Ericsson and Pool.*

**Any child can get better at arithmetic by practicing certain skills with feedback. **

*Knowledge is important, but it should not be the end. Skills are needed, too. You need factual and procedural knowledge to develop the math skills.*

**In short, you drill to build skill.**The goal should be mastery, not state test-based proficiency. Furthermore, the math skills are specific. Starting in 1st grade,

**some**of the skills include memorizing the single-digit number facts, applying the place value system, and learning the standard algorithm. These

**beginner skills**are needed for doing arithmetic in 1st grade. So, why are our 1st-grade children not learning them?

**Sadly, first-grade students can't do basic arithmetic and parts of algebra, measurement, and geometry unless they memorize the addition single-digit facts and learn the standard algorithm. To develop excellent math skills requires drill with feedback at school and at home.

*There are no shortcuts! Learning is hard work.*

**"Knowing" is Preparing for the Future**

**Knowledge**has always been the best preparation for the future no matter the epoch, but, today, kids aren't learning nearly enough math, science, and other subjects that make up a

**liberal education**.

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

*The Knowledge Requirement*) writes, "The conclusion of

**cognitive research**concerning skills is this:

**Broad knowledge of many domains**is the only foundation for wide-ranging problem-solving and critical-thinking skills." Hirsch points out, "Teaching strategies [such as those used in math and reading] instead of knowledge has only yielded an enormous waste of school time." Strategies or techniques for reading, for example, are main-idea finding, close reading, and others, so they are practiced to death. Also, the so-called "mathematical practices" and alternative "nonstandard algorithms" are the strategies in math. These strategies, procedures, or approaches do not work well in the actual teaching of reading and math. Read

**Future**.In math,

**knowing facts and procedures**(i.e., standard algorithms) in long-term memory builds the foundation for understanding, applying, and reasoning in math.

**Students cannot use something they don't know well in long-term memory.**Indeed, the memorization of essential facts and efficient procedures plays a pivotal role in performing math at an acceptable level. The learning objective should be the

**mastery**of content in long-term memory, not proficiency on state tests. Learning is remembering from long-term memory.

****

**In a postmodern world, educators have substituted critical thinking for knowledge.**We are told that learning facts is not that important. Individual interpretation and opinion are much more important. Really?

**Immanuel Kant**(1724 - 1804) once wrote that "thought [critical thinking] without content knowledge [facts] is empty."

**One problem has been that opinion is often disguised as fact in the media, social media, and in some textbooks.**One opinion seems as good as another. Belief and anecdotal "evidence" aren't evidence of anything. Still, they are often accepted or promoted as proof.

Click:

__Read what I wrote about 1st grade in 2013.__

If it were up to me, I would replace smartboards with old-fashioned blackboards, reform math with textbooks that stressed standard arithmetic and mastery, and minimal guidance instruction with explicit teaching and worked examples.

**I think we should look to the past to move forward.****Old School**memorization, drills, repetition, and practice-practice-practice forced essential factual and efficient procedures into long-term memory for use in problem-solving. Also, I would dump standardized testing at most levels. Indeed, schools and teachers should fit the curriculum to the students who walk through the door and make their own tests to track student progress. Furthermore, OECD researchers have observed, "Attending**orderly classes**[Old School] in which students can focus and teachers provide well-paced instruction is beneficial for all students, but particularly so for the most vulnerable students." And, instead of putting money into tech-tech-tech, we should be putting money into better teaching and textbooks that focus on subject mastery.**Note:**We used to teach the standard algorithms for multiplication and long-division in the 3rd grade. Also, the memorizing single-digit number facts and practicing the mechanics of the standard algorithms for addition started no later than the first marking period of the 1st grade.

**Aside:****Fareed Zakaria***(In Defense of a Liberal Education)*wrote, "Chinese students spend 25 to 30% longer a year in school than their American counterparts. They're two years ahead in math because they've taken at least two more years of math!"**R. James Milgram**, a researcher and outspoken mathematician at Stanford, says that American students are at least two years behind their East Asian peers in math by the 4th grade. Also, he writes, "[There are] certain**key topics**that you have to carefully teach all the way to real mastery in these early grades. These key subjects include fractions and above all ratios, rates, percentages, and proportions."*But first, kids need to master whole-number arithmetic.*Unfortunately, mastery has not been the primary goal of reform math.**Milgram states that Common Core and state rebrands "will produce inferior math learning." State standards are not benchmarked to world-class standards.****The Best Preparation for the Future**

Knowledge has always been the best preparation for the future no matter the epoch, including the booming Atomic, Space, Communication, Computer-Internet, and AI eras. Furthermore, the importance of math has endured and soared over the years. Kids need more mathematical knowledge than before, both factual and procedural, to prepare for the future, not less. We need to teach basic math for mastery, not test-based proficiency. Click & Read:

Knowledge has always been the best preparation for the future no matter the epoch, including the booming Atomic, Space, Communication, Computer-Internet, and AI eras. Furthermore, the importance of math has endured and soared over the years. Kids need more mathematical knowledge than before, both factual and procedural, to prepare for the future, not less. We need to teach basic math for mastery, not test-based proficiency. Click & Read:

__Future__**<-- Note:**

**Starting in the 1st grade, students should practice the basics for mastery in long-term memory, not for test-based proficiency. Mastery is needed, not proficiency levels that can vary from state to state or year to year.**

**We should prepare kids for the future just like we did in past revolutions, but we need to beef up the math and science.**Today, kids need more math than in previous generations or eras. Students should learn English, math, science, composition, history, literature, finance, economics, geography, languages, the arts, etc. In other words, a well-rounded liberal education. Unfortunately, the elementary curriculum has been warped to accommodate state testing, says

**Fareed Zakaria**

*(In Defense of a Liberal Education).*

Also, we have this idea that science is relevant only to scientists and math is relevant only to mathematicians--a "huge loss to society as a whole," says Zakaria. In my view, demoting math and science knowledge in our K-12 and college classrooms is not the way to prepare students for the future. Common Core and state standards are not world-class math. (Aside: People forget that "Mark Zuckerberg studied ancient Greek intensively in high school.")

**FirstGrade 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.**

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

**Note:**January 1, 2018. I split 1st grade into two pages.

*This page continues on*

__FirstGrade-2.__**

**My FirstGrade Arithmetic & Algebra page**provides 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 typical

**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 typical urban 1st-grade students.

My 2011

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

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

**abstract experiences**) in grades 1-3 and

**(2)**to respond to the reform math movement and Common Core standards, which were not world-class math.

**Note:**

**Reform math,**which dominates most classrooms, did not teach standard arithmetic for mastery and used minimal guidance methods that were inefficient. The consequence 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.*

If you seat kids in small groups or at tables, then "[they] would rather pay attention to one another than to the blackboard," writes

**David Geary**(

*Scientific American Mind*2011). Learning standard arithmetic takes effort, and it involves the memorization of single-digit number facts and the practicing of standard algorithms starting in 1st grade. The importance of place value is often overlooked.

**The standard algorithm embodies place-value.**Also, if learning is remembering in long-term memory, then we are not teaching children to learn.

**In 2015,**

**Siegfried**

**Engelmann**wrote that K-8 math "students should be

**grouped homogeneously**, placed in the instructional programs

**according to their skill level**, and taught

**at a rate**that assures they will

**perform at about 100%**by the end of the lesson." The

**methods**used to teach the

**content**must be efficient and work well.

**We don't do anything like this in our schools;**consequently, our students

*grossly underperform*.

**I think Engelmann is one of the few people in education who makes sense.**

**Dr. Robert Davis**wrote, "The effectiveness of a program must be judged not by what was taught but rather by what is learned."

*Davis implies a*

**performance test.**

***Indeed, in the American system of reform math, much is taught, but little is learned. (Our kids do poorly on national and international tests, especially at the advanced levels. Starting in the 1st grade, American students are not learning the math that their peers in some other nations learn.) Click on the*

**MathNotes***page.*

**Focus on Performance!**

Instead of focusing attention on understanding (as in reform math), which is difficult to measure and prone to different interpretations, we should be much more worried about

**lackluster performance**in arithmetic and algebra fundamentals as measured by both national and international tests. We can measure and evaluate

**performing**in math, but we cannot do that with an ambiguous verb "to understand."

*We need to stress performing math well beginning in the 1st grade.*

**Click/Read:**

**Focus on Performance.**

**Teach Kids Algebra (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.

*Students did practice sheets on their own. I walked around the room giving individual help.*First-grade students had two half-hour sessions a week for a total of 7 instructional hours.

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

One of the tasks (upper left) 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. It was part of the last lesson, which was a

**Culminating Activity**for 1st grade.

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

*Finding permeters is an example.*

2. "You learn only through mastery." (Zig Engelmann)

3. "The [school math] concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them."

**Quote3:**

**Math Professor W. Stephen Wilson** refers to the frequent use of scientific calculators, graphing calculators, smartphone calculators, online calculators, and other technology often found in K-12 classrooms.

**Teach Kids Algebra (TKA)**

The students in the two 1st-grade classes were in

*Everyday Mathematics (*. Even though EM's is weak on standard arithmetic, I liked parts, such as the

**EM**)**table building**(in-out model) and "Find the Rule" exercises in which students would figure out the rule and complete the in-out table. The in-out idea fit my presentation of functions for 1st-grade students, but I introduced more complex relationships between x and y than EM. Also, TKA students built tables using equations such as

**y = x + x - 3**. They plotted x-y points in Q-I to make a picture (graph) of the function. I sent some of the lessons to

**Don Cohen**(The Math Man) who was greatly interested in my work with very young students. He wrote several small books such as

*Calculus by and for Young People*. He passed in 2015.

**Note:**EM (1st Grade) uses the

**number line**for skip counting and adding and subtracting whole numbers. Fraction strips were used to compare fractions. There are other parts I liked.

(Incidentally, the 1st-grade TKA students were ordinary, urban Title-1 students in mixed classrooms.)

**We don't ask students to memorize something without some level of practical (i.e., functional) understanding. Understanding requires**

**knowledge in long-term memory.**

Students in the Asian nations learn the mechanics of arithmetic first with explanations later, and it works. At first, there is only functional understanding. Even though I asked 1st graders to memorize

**5 + 7 = 12**, it doesn't mean they don't understand it at the level needed to move on. Indeed, the concepts of addition and magnitude are straightforward on a

**number line**. The idea that

**12**

**by place value is a 1st-grade level grasp of place value as applied to double-digit numbers. Also, good math teachers lead with worked examples to support understanding.**

**is**1ten+2ones**Knowing**

**how something works is a functional or practical understanding**

**.**Also, children never attain a perfect understanding of anything. There may be different levels of understanding (e.g., a 4th-grade understanding of place value is different from a 1st-grade understanding of place value),

*but the levels are hard to quantify*. I don't think anyone has a deep understanding of anything.

**The reality is that any u**

**Without knowing, there is no understanding or thinking. In other words, understanding is a fledgling idea at best and develops slowly over time with much study and experience.**

**nderstanding requires knowledge.**The more knowledge, the more understanding is implied, that is, understanding is relative to knowledge.**Being able to perform and apply math to solve problems implies some level of practical understanding.**

**In math, knowing facts and efficient procedures (i.e., standard algorithms) in long-term memory builds the foundation for understanding, applying, and reasoning in math. **

*Students cannot apply something they don't know well in long-term memory.*

*Indeed, the memorization of essential facts and efficient procedures plays an important role in performing math at an acceptable level.*

**Can students perform arithmetic?*(If you can't do vertical (column) addition, then you don't understand addition.)*

**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 and know that they can add the numbers in any order, an important property of addition. Also, it is vital to automate the mechanics of standard arithmetic such as adding 26 and 35 vertically 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 many nonstandard alternatives (i.e., reform math).*However, there is no substitute for factual and efficient procedural knowledge in long-term memory and the practice-practice-practice that gets it there.

*Indeed, schools should teach the mastery of necessary facts and skills first, not a bunch of reform math fluff and its alternative, nonstandard algorithms (often called strategies).*

**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 calculations:**

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." (Really? 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 should be fixed in long-term memory to perform the standard algorithms. Also, students should not use a calculator, count on fingers, or use reform math strategies as crutches.

**Retrieving facts instantly from long-term memory is the objective.**

*Retrieving must be practiced and start at the beginning of 1st grade.*

**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 plain

**0-20 number line**on each student's desk on the first day of 1st grade. The reform math strategies of making a drawing or writing an explanation are not necessary because they often increase cognitive load.

**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 Mathematicians Think*) points out that

**s**

**chool 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 perform the procedure rapidly (competency)**. Thus, factual and efficient procedural knowledge is the backbone of learning school math. In school math, the algorithmic thinking approach (i.e., the performance) does not require deep understanding because it is mechanical at first. 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**

**apply and perform the algorithms**

**you studied.**

*In short, "applying knowlege" to solve math problems is the understanding students should have.*The algorithms you use should always be the most efficient ones, which is a reason that mathematicians recommend that students 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"

**Minimal Guidance = Minimal Learning (The Ki**

**rschner-Sweller-Clark Equation)**

****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 from 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**

**(**

*Educational Psychologist, 2013)*outline three urban legends in education:**"**The

**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**.

**"**(Long 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).

**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.

*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.*

Changes made on 1-1-18, 2-3-18, 3-4-18

Changes made on 1-1-18, 2-3-18, 3-4-18

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