**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 ride a bicycle without a bicycle.*

Likewise, you can't learn to do arithmetic without doing a lot of arithmetic.

Likewise, you can't learn to do arithmetic without doing a lot of arithmetic.

**1. College Trajectory (High Skills)-->4-Year University**

Some charter and many independent schools require all their students to take Algebra-1 in 7th or 8th grade. It is the college trajectory, and it has worked very well. Most public middle schools had an Algebra-1 course for the best students.

**I don't advocate college-for-all.**Some kids need a trajectory that leads to calculus in high school, and Common Core isn't it.

**It is hard to judge which students should follow this trajectory.**

*Kids who take advanced math courses (Algebra-2 with trig, precalculus, and calculus) and core science courses in middle school (physical science) and high school (chemistry and physics) go on to earn a bachelor's degree.*Also, 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.**Average kids can do these things when they are

__taught well__and

__work hard__to achieve.

**2. Community College (Middle Skills)-->2-Year College**

Learning

**m**

**iddle skills**are valuable for many students. They are learned through

**community college programs,**and, sometimes, through limited apprenticeships. To enter community college, a student needs to be good at math, which has been a

**stumbling block**. Students are often sidetracked to the

**remedial math**

**rut**. It is clear that K-12 schools must do a better job teaching kids math, and Common Core is not it. Our high schools should offer a

**vocational alternate to college for all**.

I recommend that students should pass an

**arithmetic and**

**Algebra-1 achievement test**to enter a community college, and, if the student needs more math, then it would be taken as part of a 2-year associate's degree or certificate.

**What is clear is that most future jobs will require some college, but how much college?**

**Note.**Earning

**a bachelor's (4-year) degree at the university "is not the right answer for everyone." Many students would be far better off "by robust technical training [community college] that will lead them to middle-skill jobs," observe**Newman & Winston

**(**

*Re skilling America*

**, 2016).**Tn short, there are good jobs available for those who have the

**right middle skills**via an associate's degree or certificated program.

*These jobs do not require a 4-year bachelor's degree.*

3. High School Diploma or GED (Low Skills)

3. High School Diploma or GED (Low Skills)

****ToBeContinued 7-13-16

**1. This is reform math, not standard arithmetic.**

**2. It's inefficient and wastes instructional time.**

*The arithmetic taught should be useful and efficient. The “many ways” of nonstandard, more complex, alternative algorithms of reform math to calculate simple arithmetic problems are not efficient.*

**In short, reform math has screwed up the learning of standard arithmetic, which is essential.**

**Why would anyone suggest that this is a good way to teach novices subtraction?**Still, it is typical of

**reform math**.

*Making Number Talks Matter*(Humphreys & Parker, 2015) shows five different ways to subtract, including Decomposing the Subtrahend (shown above), but

**not**the standard algorithm. The book focuses on contentious mathematical practices (Common Core) and implies that standard arithmetic (old school) "destroys a child's intellect, and, to some extent, his integrity."

**What nonsense!**The reason kids are poor at basic arithmetic ("fragile skills") is that they do not practice enough, or it hasn't been taught well.

**The idea of drill-for-skill to automate essential fundamentals in long-term memory has fallen out of favor for decades due to reform math and progressive ideology.**

**[**

*Number Talks*cites a so-called

__common mistake in middle school math__: 1/3 + 1/3 = 2/6 = 1/3. How can this be a common mistake in middle school?

*Even 1st graders don't make this kind of error.*

**If middle school kids are making this mistake, then it shows how badly arithmetic had been taught in elementary school. Their reasoning is wrong!**]

The reason that kids have “fragile skills and shallow understanding” is that basic arithmetic skills have been taught poorly, especially via reform math and its methods.

**College professors lament that students can’t do simple arithmetic without reaching for a calculator (e.g., 0.256 x 100), which is directly linked to 25 years of reform math.**

**Kids can't calculate!**Standard arithmetic can be taught poorly, too, but, under the

**reform-math regime**, math achievement has been static because the intent was not stellar achievement; it was equal outcomes, says

**Thomas Sowell**, and socialization via group work.

*Consequently, reform math has not prepared more students for Algebra-1 in middle school.*

**The "one-size-fits-all" C**

**ommon Core state standards embrace reform math and the nonstandard, inefficient algorithms--more of the same.**

**Math Matters (More Than You Think)!**

Kids who take advanced math courses(Algebra-2 with trig, precalculus, and calculus) and

Kids who take advanced math courses

**core science courses**in middle school (physical science) and high school (chemistry and physics) go on to earn a bachelor's degree. (Note. These courses are not integrated science or conceptual courses.) Solid 8th grade physical science and high school chemistry and physics courses require a

**good math background**. Indeed, the math courses taken in middle and high school affect the probability of obtaining a college degree and the student's future earnings. Math matters, but not all math courses are equal. (Note. Not all courses labeled as college prep are college prep. Many pre-algebra courses are labeled as algebra.) Students need to take

**advanced**math courses.

**This is contrary to Common Core's doubtful rhetoric (guesses) about college and career readiness and the Next Generation science standards.**A college degree (associate's, bachelor's, or master's) is your best bet for the future job market, especially in the managerial and professional fields, not a high school diploma.

According to the

__, "Those with at least some college education have captured 11.5 million of the 11.6 million jobs created during the recovery. While jobs are back, they are not the same jobs lost during the recession.__

**Georgetown University report****The Great Recession decimated low-skill blue-collar and clerical jobs, whereas the recovery [January 2010 to January 2016] added primarily high-skill managerial and professional jobs."**

If your degree is in an

**overloaded field**(e.g., some STEM fields) or the wrong field, then you will be lucky to find a job. Many students follow their dream and end up living at home with a worthless degree, large student-loan debt, and no real job prospects. Kids need to be practical and major in a field that will put food on the table.

**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.**Average kids can do these things when they are

__taught well__and

__work hard__to achieve.

**Note.**Parts of this page have been removed. If you would like a pdf of the original, please email ThinkAlgebra@cox.net. I will continue to shorten content. 6-1-16

Starting in the

Also, teaching many different thinking strategies, along with confusing, complex, nonstandard calculation procedures to do simple arithmetic, such as those found in

-----

**1st grade**, I believe it is essential to establish a solid foundation of**standard arithmetic**in long-term memory, and then start applying the arithmetic students have learned through**routine problems**with increasing levels of difficulty. A**hierarchically sequenced****curriculum**requires a**sharp focus on achievement**, the rapid mastery of**factual and efficient procedural knowledge,**instructional methods based the**science of learning**, and strong teacher guidance, which means**the teacher should be the academic leader, not a facilitator as taught in ed school. "****Minimal guidance in mathematics leads to minimal learning**," write**Sweller, Clark, and Kirschner**. Indeed, minimal learning has been the narrative for decades.Also, teaching many different thinking strategies, along with confusing, complex, nonstandard calculation procedures to do simple arithmetic, such as those found in

**reform math**via Common Core or state standards, instead of**straightforward**knowledge and skills in**standard****arithmetic**, has been a colossal waste of valuable instructional time.**Kids are novices and need background knowledge. They need to drill-for-skill, so essential factual and procedural knowledge sticks in long-term memory.**

In grades 1 to 4, the automation of standard arithmetic and the learning of algebra fused to arithmetic often boil down to hard work and persistence.In grades 1 to 4, the automation of standard arithmetic and the learning of algebra fused to arithmetic often boil down to hard work and persistence.

-----

**The latest K-12 fad is****personalized learning**, which would require the complete computerization of schools. In a few decades, maybe half the teachers would be cut. The downsized teaching staff would "remake schooling" as an "adjunct faculty" and would supervise or facilitate the personalized learning, suggests**Mark Naison**, a college professor at Fordham University.**I think it is a truly bad idea from special interests**(aka corporate-based reform), and I doubt it would happen on a wide scale, but suppose I am wrong? Suppose schools districts across the US adopted personalized learning, thinking it would be the future in education?**-----**

Note.Topics are presented randomly. Please excuse typos, errors, andNote.

**redundancies.**

**Notes.**

(1) We need to get disruptive kids out of the classroom, stop playing good behavior games, stop testing for so-called "soft skills" that are hard to define and measure, just as "understanding" and "creativity" are almost impossible to measure.

(2) K-5 kids don't need group work and discovery or collaborative activities. They need academic achievement in basic reading, math, and writing.

(3) College readiness and career readiness are

(4) Kids need a liberal education, which includes math and science, along with the humanities and the arts. In addition to math and science instruction, make sure K-5 students have music, art, PE, library, recess, English grammar, spelling, history, and geography.

(5) To get to algebra-one in middle school, elementary school students should automate standard arithmetic first, not reform math. By the end of 3rd grade, students should be proficient in all four whole number operations (aka standard algorithms, place value, and single-digit math facts, and the rules of arithmetic).

(6) Students are not the same, so the one-size-fits-all ideology in Common Core or state standards is nonsense.

(7) Equalizing downward is an illusion of fairness and a nutty idea taught in schools of education.

(8) Teachers should implement the

(1) We need to get disruptive kids out of the classroom, stop playing good behavior games, stop testing for so-called "soft skills" that are hard to define and measure, just as "understanding" and "creativity" are almost impossible to measure.

(2) K-5 kids don't need group work and discovery or collaborative activities. They need academic achievement in basic reading, math, and writing.

(3) College readiness and career readiness are

*not*equivalents.(4) Kids need a liberal education, which includes math and science, along with the humanities and the arts. In addition to math and science instruction, make sure K-5 students have music, art, PE, library, recess, English grammar, spelling, history, and geography.

(5) To get to algebra-one in middle school, elementary school students should automate standard arithmetic first, not reform math. By the end of 3rd grade, students should be proficient in all four whole number operations (aka standard algorithms, place value, and single-digit math facts, and the rules of arithmetic).

(6) Students are not the same, so the one-size-fits-all ideology in Common Core or state standards is nonsense.

(7) Equalizing downward is an illusion of fairness and a nutty idea taught in schools of education.

(8) Teachers should implement the

__cognitive science of learning__, not ideology.

College Ready = Career Ready (Really?)

*Achieve --> Common Core***College ready and career ready are not equivalents, yet the one-size-fits-all ideology behind the idea of "sameness" thrives in American classrooms.**What does all this have to do with Common Core? In 2008,**Achieve**laid the foundation for what is called Common Core, including the one-size-fits-all ideology and the advanced mathematics keystone.**Put simply,****Common Core formulates college readiness as being equivalent to career readiness, which is a stupid idea.**The one-size-fits-all ideology does not fit reality.*The*__same curriculum__is taught to**all students**without regard to individual achievement, ability, and goal; i.e.**,**those who plan to attend a 4-year university, a 2-year community college, a vocational school, or those who plan to enter the workforce without postsecondary education.**Common Core explicitly states that it is not for STEM students.***Apparently, its so-called "advanced math" cornerstone does not extend much beyond elementary algebra.***The Common Core ideology makes no sense. It caters to the narrow,**skewed vision**of special interests and elites that advocated "advanced math for all students."***But, what is advanced math in Common Core? Presumedly, it boils down to the mantra***Algebra for All**,*but, in my view, first-year Algebra isn't advanced math.*(**Confusion**: Do the Common Core reformers rebrand statistics and probability as advanced math? These topics are found in pre-Common Core Algebra 2 textbooks, along with trig and other topics.)**Advanced math is a STEM sequence: Algebra 2, trig, precalculus, and calculus.***In the real world, s**ome students don't need advanced mathematics (e.g., Algebra 2), and it should not be a requirement for a high school diploma; however, it should be a requirement for STEM studies at the community college and university levels.**More information is found in Math Notes:***Click Here****.****(Aside.***David Hume***wrote, "A wise man proportions his belief to the**evidence.")****It is 2016! The same old problems persist. Kids are crawling instead of leaping in math. Achievement has been stalled.**

**Good grief, t**

**he N**

**AEP math test scores in 2015 for 4th and 8th-grade students are lower than in 2013.**

**There is a war on excellence.**Indeed, reform math via Common Core or its rebranded configuration (aka state standards) promotes mediocrity, not excellence, in learning. Reform math methods ignore the cognitive science of learning and suggest that learning standard math well is not that important. Reform math focuses on nonstandard, complicated ways to do simple arithmetic, pushing the standard algorithms to the back burner.

**Put simply, reform math obstructs the fast learning of core standard arithmetic.**

**Reform math via Common Core or Common Core rebranded as state standards**, which is the case in almost all states, confuses beginners and overloads their working memory.

__Students are novices, not little mathematicians.__

*Put simply, reform math makes simple arithmetic excessively complicated.*

*Many students are held back academically because they get the same curriculum, which is the mantra of Common Core reform math progressives.*Also, in reform math via Common Core, the

**standard algorithm**is merely one of the many ways--often

__not__the preferred way, the reformists say--to calculate.

**I**

**ndeed, reform math via Common Core or state standards often supersedes simple, old-school arithmetic with complicated procedures (models, strategies).**

For decades,

**fractions and long division**have been sidelined and taught poorly in elementary schools. The widespread practice has hindered a child's numerical development and ability to do algebra and higher math. It is a primary reason that U.S. students are not prepared for algebra by 8th grade. In short, most US students are not taught enough math content. Students in other nations routinely do algebra in middle school, but our students don't.

**Standard arithmetic has been marginalized by Common Core's version of reform math.**

**The early mastery of standard arithmetic--not reform math--is required to prepare for algebra. Certainly, the "mantra that one-size-fits-all model [cannot] possibly do justice to the diversity of academic subjects," says**

**Gerald Graff [1].**Also, another problem is the dissimilarity of knowledge, achievement, and academic abilities or skills of students. Therefore, the students who walk through the school door often vary widely in academic ability, so a one-size-fits-all formula is a poor fit.

**The one-size-fits-all Common Core doesn't fit low ability kids (two much content) or high ability kids (too little content) in math.**Even

**average kids**don't master enough standard arithmetic when it is taught through the lens of reform math. In short, there should be different levels of

**instructional objectives**to (better) match students, but that's not the "one size" Common Core way.

Even though reasoning abilities have increased on the

**Weschler Intelligence Scale for Children or WISC**since 1950, students have

*not improved much*in several key WISC subtests:

**Information**(basic knowledge),

**Arithmetic**, and

**Vocabulary,**says

**Mark Bauerlein**

**[2]**

**.**He writes,

**"The lesser subtest outcomes [Information, Arithmetic, & Vocabulary] explain why academics have stalled for U.S.**

*For example, in 2015, "both the 4th- and 8th-grade students score lower in NAEP mathematics than in 2013."*

**John Dewey still lives on ...**

**Progressives believe that utopia is possible in education: everything is relative.**"For them, science is just another opinion...so their

**core issues are fairness and equality, not excellence**," observe

**Berezow & Campbell**(

*Science Left Behind*)

**[3]**.

*It is the same problem we have with progressive education today:*Berezow & Campbell explain, "But rather than keep what worked and improved what did not, Dewey set out to reshape education from the ground up...

**skewed priorities**.**It was not set up to improve learning**; it was actually designed for social acclimation reasons on the latest pop psychology." John Dewey lives today. Is it any wonder that the education reforms are bizarre and don't work well?

**Learning is stalled.**

**Footnotes**

[1] The quote from

**Gerald Graff**is from his chapter on the new anti-intellectualism found in

*The State of the American Mind*, edited by Bauerlein & Bellow, 2015.

**[2]**The quote from

**Mark Bauerlein**is from his chapter on the new anti-intellectualism found in

*The State of the American Mind*, edited by Bauerlein & Bellow, 2015.

**Bauerlein makes an excellent point that helps explain the reason that academics have stalled in the US.**

**[3]**

**Berezow & Campbell**writes, "This is the crux of science [and math] education as an issue in American life. It is not a matter of promoting excellence; it is a matter of pursuing political priorities. To progressives, the focus is not on providing a quality science [or math] education for students... There is a war on excellence."

**The progressives dominate education policies and ignore the cognitive science of learning because it doesn't fit their framework or agenda**.

**Excellence is not necessary.**

------------------------------------------

**Do not confuse low-income kids with low-ability kids.**

*I have found many low-income kids who learn math faster and better than their peers; however, their academic growth in math is often stymied by a range of factors, such as the following:*

*(1) the slow pace of instruction;*

*(2) being asked to "teach" other kids (group work);*

*(3) topical redundancy (spiraling curriculum);*

*(4) the idea that every child gets the same;*

*(5) "equalizing downward by lowering those at the top" in the name of fairness;*

*(6) teaching reform math; and*

*(7) teaching to the test.*

**[Aside. The**

**inference**

**that Common Core reform math and its standardized testing will jump-start stalled achievement so that all students will become college- and career-ready without remediation is untested,**

**unproven**,

**and far-fetched**.

**The inference is pure speculation--a guess--and not supported by valid evidence.**

*We already know that progressive reform math (NCTM) failed in the past and that standardized testing, with implied consequences, and teaching-to-the-test do little to improve actual achievement in math.*

**Standard arithmetic**has been marginalized by Common Core's version of reform math. Contrary to Common Core, the early mastery of standard arithmetic--not reform math--is required to prepare for algebra. American educators don't get this, but Singaporean teachers do.

**]**

## Kids Must Memorize Times Tables & Master Fractions!

Fraction Magnitudes - 2nd Grade, LT

**Fractions and long division are key building blocks in a young child's numerical development. For decades, our math programs have marginalized their importance, but research has shown that this was an epic mistake. Kids must memorize times tables to do long division, fractions, and algebra.**

*A violist won't get to Carnegie Hall without memorization and years of practice, and kids won't master arithmetic or algebra without memorization and years of practice. Indeed, you won't get good at anything without memorization and practice, lots of it, whether it be violin, mathematics, gymnastics, Latin, piano, Physics, and so on.*

**Our kids**

**may not be the next Mozart, Newton, or Murdock, but, "through effort, [they] can develop passable skills in music, math, and writing"**(Breznitz & Hemingway,

*Maximum Brainpower*, p. 192).

For decades, fractions and long division have been sidelined and taught poorly in elementary schools. The practice has hindered a child's numerical development and ability to do algebra. It is a primary reason that U.S. students are not prepared for algebra by 8th grade. Students in other nations routinely do algebra in middle school.

## Kids can learn algebra

*Algebra grows out of arithmetic.*

First grader in my algebra class

**The following has been my**

*core premise*for decades: "We begin with the hypothesis that any subject [e.g., arithmetic, algebra, calculus] can be taught effectively in some intellectually honest form to any child at any stage of development." - Jerome Bruner*(The Process of Education, 1960)*

*The implication is that children, even in 1st graders, can learn fundamentals of algebra. As a guest teacher, I teach little kids algebra*(

**)**

__Teach Kids Algebra__*And, yes, 1st graders can learn some fundamentals of algebra, such as*

*(1) numerical relationships (functions),*

*(2) equality (an equation is like a balance),*

*(3) true/false math statements (left=right),*

*(4) the rule for substitution,*

*(5) function rules (x,y),*

*(6) table building,*

*(7) equation writing and solving, and*

*(8) graphing in Q-I; e.g., y = x + x + 2.*

In Teach Kids Algebra,

**algebra is fused to arithmetic**, which makes algebra accessible to very young children. Algebra grows out of arithmetic, so

__good arithmetic skills__are essential and reinforced in TKA. Algebra is a tool for reasoning and requires clear thinking and arithmetic knowledge.

The structure and method of mathematics are that

__one idea builds on another and that everything fits together logically__.

__, which should start in 1st grade, is necessary for reasoning and problem solving. Indeed,__

**Automaticity of fundamentals**__strong math skills through practice__are key to the process.

__Chains of reasoning__connect ideas to each other as complexity builds over time. Understanding, at first, is functional and grows slowly.

**When taught well, math teaches kids to think; it makes them smarter.**

## Automate fundamentals Through Practice

*To move forward, kids need to master [automate] fundamentals through practice. *

US kids need to practice to automaticity.

**Math takes lots of practice and a certain amount of memorization. There are no short cuts. Kids get good at arithmetic or algebra only through practice. Fundamentals must be in long-term memory for instant use in problem-solving.**

**There is no substitute for automaticity of factual and procedural background knowledge in arithmetic.**

*Learn skills all the way to automaticity!*National and international tests show our students lag behind. US kids are weak in both factual and procedural knowledge in mathematics.

**Common Core and state standards based on Common Core are below the Asian level.**

In Common Core, the concentration on strategies to do arithmetic reduces the importance of systematic learning and automation of number facts and efficient math procedures, both of which are critical for

**(1)**

**higher-level thinking**(

__Willingham__: long-term memory learning)

**,**

**(2)**

**creativity**(

__Lemov__: practice to automaticity)

**,**and

**(3)**

**Problem-solving**(

__Polya__: prior knowledge).

__Willingham, Lemov, and Polya say the same thing__

**Note**.**:**

__the automaticity of__

__knowledge (factual, procedural, and conceptual) in long-term memory__is needed for higher-order thinking in mathematics.

**Daniel Willingham**, a cognitive scientist, explains, "If you know that

**9 x 7 = 63**, you need not use valuable

__mental space__[working memory] to do that calculation as part of a more complex problem.

**Knowledge of math facts [automaticity] is known to be an important component of competence in algebra and beyond.**" It is important to tax working memory [exercise it].

Furthermore, Willingham says,

**"Students must have both content knowledge and practice using it."**Indeed,

**thinking well**requires knowing facts and procedures stored in long-term memory. It is important for students to make "

**connections across pieces of information**," writes

**Art Markman**(

*Smart Thinking*).

## Contact ThinkAlgebra

3rd graders in Teach Kids Algebra.

This website is undergoing many changes in Please excuse typos, errors, redundancies, etc.

*LarryT, Founder:**ThinkAlgebra & Teach Kids Algebra**Model Credits:**Hannah, Remi, & kids in my algebra classes***Email LarryT at**__ThinkAlgebra@cox.net__.**Last update: 1-5-16, 3-11-16, 6-1-16****© 2004-2016 LT, ThinkAlgebra**