**Learning is remembering.**It is auto recall of pertinent facts and procedures to solve problems. This is especially true for arithmetic, algebra, and higher math.

**Knowledge Is King.**There is no substitute for knowledge in long-term memory and the practice that gets it there.**Many educators no longer teach for mastery of content but for proficiency levels on state tests.**The state tests are based on state standards via reform math. Proficiency levels often vary from state to state and are unreliable.**How should we prepare kids for the future jobs?**

**Local Control. What's That?**

The

**Every Student Succeeds Act**(ESSA), which replaced No Child Left Behind (NCLB), marks the end of local control. Schools are not independent. They must comply with federal regulations if they accept government money. ESSA is intrinsically linked to standardized testing based on benchmarks that are

**not world class**. Moreover, the reform-minded

**test makers**interpret the Common-Core-influenced state standards as reform math, not as standard arithmetic that should be mastered.

**T**

**hus, the test controls, limits and warps the math curriculum to fit testing, not mastery.**

*There is no local control, not at the state, district, or school level.*

*Almost everything that is taught in the classroom is based on the*A curriculum that is warped to fit a test is a fragmented curriculum that perpetuates lackluster achievement.

**test items**, which is a very bad idea (test prep).*It seems that our*

*public schools will always be deficient in achievement. *

**Algebra-2 (Watered Down)**

Many states such as Arizona have made Algebra-2 a required course for graduation, which is not realistic in my view. Out of necessity, teachers will likely lower the standards for Algebra-2 (i.e., weaken the course) so that most kids can pass and graduate or they will substitute alternatives for Algebra-2 such as statistics and computer science, which are not close in content to Algebra-2. A solid course in Algebra-2 with trig is needed for a robust course in precalculus and, later, calculus. In short, students are not adequately prepared fo higher level math. Algebra-1 is a middle school course for students who are prepared.

**The problem has been that we are not preparing students well enough. For decades, arithmetic has been taught poorly in elementary schools.**

*Weak elementary school math instruction goes up the grades into high school. For example, some precalculus courses at highly rated high schools have weak academic content required for university level calculus courses.*

**Read**

__Algebra-2__**in my Math Notes.**

**Perspective (The Era of Abe Lincoln)**

**Kids in the 1800s learned much more fundamental arithmetic than students today**,

**according to**which "contained exercises on the primary principles, and their applications; together with models of analysis in the shape of

*Ray's New Intellectual Arithmetic*book 1877,*solutions*[worked examples], including fractions, ratios, and percentages."

→ Ray's small 140-page book (4.5 x 7) encompassed the entire 3rd and 4th-grade arithmetic curriculum. Here are two easy 4th-grade questions:

**(1) 3/4 of 24 are 6 more than 2/3 of what number?**

(2) Find the interest on $50 for 6 months, at 6%.

(2) Find the interest on $50 for 6 months, at 6%.

Today's 3rd and 4th-grade students could perform like the children in 1800s if they were appropriately trained in arithmetic starting in the 1st grade, which requires memorization, drills to improve skills, and frequent review. But, they are not. Instead of learning single-digit number facts and standard algorithms from the get-go, students are taught reform math via Common Core state standards, which are below world-class benchmarks, clutter the curriculum, increase cognitive load, and impede student achievement.

**Note:**

**Matthew Lynch**(

*Education Week*) writes that today's students want instant gratification, which has been an excuse for lowering academic standards. Lynch observes, "

**Traditional education is undermined by this growing anti-intellectualism.**Today's students are less inclined to pursue academic achievement if it offers no direct relevance to their daily lives." Regardless of state standards, which are below world-class benchmarks, individual teachers can hold students to their high standards.

**Note:**

**Common Core ( CC) math is not dead in our schools.**

**HELP!**

It is everywhere. It has been rebranded as

**state standards**. The state tests are Common Core reform math tests. Common Core without the testing mandate is dead, but it is not what we have with

**Every Student Succeeds Act**(ESSA), which continues the

**mandatory testing**of No Child Left Behind (NCLB). Also, Common Core is not world-class math starting in 1st grade, according to my analysis in 2011. It is not internationally benchmarked as originally trumpeted.

**How should we prepare kids for the future jobs?**

**Just like we did in past revolutions, but we need to upgrade the math and science.**Students should learn English, math, science, composition, history, literature, economics, geography, finance, computer science the arts, etc. In other words, an excellent liberal arts education.

**Knowledge**has always been the best preparation for the future, but, today, kids aren't learning nearly enough math, science, and other subjects.

Our schools are infused with weak curricula (state standards), ineffective instructional methods (minimal guidance), and inane education policies at the local, state, and federal levels (Every Student Succeeds Act, etc.). Also, unlike high-performing nations, we don't connect education to jobs, economic prosperity or national security like high performing nations. We keep importing talent and setting up shop where the skilled workers and talent are developed--in China, Korea, Singapore, or India! (

*MathNotes by ThinkAlgebra*)

**Note:**

*You should not substitute computer science or statistics courses, which require higher-level math to understand, for Algebra-2. High school statistics is more of a course in using the TI-84 graphing calculator than learning mathematics. In other words, students who passed Algebra-1 do not have sufficient mathematical background to understand computer science or statistics courses. Many states require Algebra-2*or its

*alternatives (so-called equivalents)*

*for high school graduation. Computer science or statistics courses may be alternatives, but they are not equivalents for Algebra-2, which is the required algebra content knowledge for community college and college math placement tests.*

**At the university level, computer science as a STEM major requires university-level calculus, not AP.**

*(Note: AP calculus is not accepted as credit toward a STEM major at many universities such as the University of Texas.)*

**Welcome to My Main Math Page**

**There is no substitute for knowledge in long-term memory and the practice that gets it there.****Memorizing skills and practice are critical to learning and thinking.****Understanding does not produce mastery; practice does!**- Incidentally,
**strict schooling**such as mastery learning, which is practice-practice-practice, does**not dent**a child's curiosity, innovativeness, or creativity as some have claimed. **In math, it is essential that children drill-to-develop-skill!****State proficiency does not mean mastery.**The student needs to know math stuff in long-term memory to apply it.**Learning is remembering**from long term memory.**Knowledge has always been the best preparation for the future no matter the epoch.** Indeed, standard arithmetic and higher level math (algebra, trig, etc.) have endured era after era, that is, their impact has soared over time. Kids don't need less mathematical knowledge (factual and procedural) in long-term memory; they need more math knowledge to prepare for the future. Still, many bad mouth math because they don't like math. Furthermore, teachers are poorly prepared to teach even basic arithmetic. It shows a bias against standard arithmetic and algebra. Many adults boast that they were not good at school math, but you don't hear adults brag that they were lousy at reading or hated reading.**You don't get good at math or reading without memorizing and practicing extensively.**You don't get good at math by not practicing.

****

**The most neglected aspect of learning math is practicing for mastery. Proficiency doesn't mean mastery. A student proficient in math in one state may not be proficient in another state. (Same problem as before.)**Kumon math focuses on learning factual and efficient procedural knowledge in long-term memory. Learning is remembering from long-term memory.

**Kumon practice leads to mastery of fundamental skills.**

**Kumon drills mechanics to develop skill.**

A worked example is given without much explanation. Knowing facts and efficient techniques are vital for learning math well. The student needs to know something in long-term memory to apply it.

**Kumon**may be supplementary, but it is precisely what students need to improve their math skills and mastery. There is no substitute for practice-practice-practice and review of the fundamentals.

**Children don't like practicing, so parents need to take charge.**Often, I am reminded of

**Larry Winget**who wrote

*Your Kids Are Your Own Fault*. Parents need to "raise responsible, productive adults." He writes, "If your kids are a mess, it's time [for parents] to take responsibility for it. We allowed it to happen. While the school system may appear to be failing our kids, it is really parents who are failing by not making sure their kids learn what they need to know to be successful, healthy, and prosperous."

**One of the best Algebra-1 texts was the Dolciani Modern Algebra Book 1 from the 60s and early 70s era.**A newer edition was published in 1981 (not shown). The older Dolciani texts are hard to find.

Today's Algebra-1

**Common Core**editions (such as the Holt McDougal text) are filled with graphics, color, fluff, and extras while the

**Dolciani Algebra-1**text (© 1970, 1973 Edition) is smaller and thinner in physical size and about 270 fewer pages.

**The Dolciani text is packed with practice exercises and word problems.**

The last chapter (Chapter 14) of the Dolciani text is "

**Geometry and Trigonometry**." Kids learn geometry, trig, and resolving vectors to solve problems, including many physics problems. There is also an "Extra For Experts" section in Chapter 14 called "Complex Numbers." In the Dolciani era, kids didn't have calculators. They used trig tables and paper-pencil calculations. Kids do not need a graphing calculator to learn algebra well.

**Note:**An education professor claims that teaching kids math discriminates against children of color. How stupid! It is more rubbish coming from the progressive schools of education. Click:

__.__

**Read More**Note: East Asian students have a lot of

**intrinsic motivation**. They try harder in school and, later, in the workplace. I wish American students had the same level of motivation.

**Most**

**U.S. kids today are more interested in video games, social media, texting, and sports than in academics or school work. The smartphone and social media dominiate their lives. How dull!**

Changes (1-1-18): I split my first-grade page into

**two pages**. FirstGrade continues on FirstGrade-2

Changes (12-28-17): My Stagnation page no longer appears in the Menu. It has been moved to my

__MathNotes__by ThinkAlgebra. (Click

__Stagnation)__

****

**Best Preparation for the Future**

"

**Today is no different from other booming eras**," writes

**Scott Montgomery**, Editor-in-Chief, Education Week--whether it is the radio-television-communication age, atomic age, space age, the computer and Internet age, etc. Math has endured through all these flourishing eras, and the ability to do mathematics well has become increasingly important--not less significant as some claim. Our education system needs international benchmarks of excellence and knowledgeable teachers who can adhere to those standards. In my opinion, we don't have either in K-12 mathematics.

**What is the best preparation for future jobs?**

**The same as in other innovative epochs!**Students should learn math, reading, writing, science, literature, English, languages, philosophy, finance, economics, history, the arts, etc.

**In short, knowledge has always been the best preparation for the future no matter the epoch.**Unfortunately, there is an anti-knowledge and anti-math movement in American education.

**Reform Math zealots**often tell us that children are better off if they develop a deep understanding of math, but we are not told what the "deep" understanding looks like much less how to measure it. In fact, it is challenging to define or quantify "understanding." Are kids better off in reform math with its alternative, nonstandard algorithms? I think not.

**Reform Math is not world class math.**Moreover, achievement has stagnated under reform math beliefs, test-based accountability, and minimal guidance instructional methods.

**Common Core and state standards are taught through a reform math prism.**The math standards and corresponding curricula are not world-class, and the minimal guidance instructional methods (e.g., working in groups, etc.) have low "learning" value. Teaching to the "items on a state test" creates a fragmented curriculum.

**Minimal Guidance = Minimal Learning**

*The Kirschner-Sweller-Clark Equation*

Teaching to "items on the test" is a

**fragmented curriculum.**For years, teachers lacked a coherent K-12 math curriculum and were taught to use ineffective methods of instruction. Algebra-1 is a middle school course for average students who are prepared. It is clear that our elementary schools have not been preparing students for Algebra-1 in middle school. The primary focus of the National Math Panel (2006-2008) was readiness for algebra by middle school. The Common Core reform math people ignored many of the recommendations of the Panel.

In the 50s, when I was in grade school, my understanding of the multiplication algorithm was mechanical. The standard algorithm always worked. I knew nothing about the distributive property until Algebra in 9th grade.

**What I learned in grade-school arithmetic was to apply the correct standard algorithm to the problem.**Writing the steps demonstrated sufficient understanding. We didn't make drawings to "prove" that 3 + 5 = 8; we memorized single-digit number facts. We didn't learn 5 or 6 different ways to multiply; we learned one way in 3rd grade--the standard algorthm--and moved on. In contrast, the reform math people don't like the algorithmic (mechanical) approach even though it worked well with almost all school students when it was taught correctly and

**practiced to mastery.**

**U.S. math programs ran into trouble when schools stopped teaching basic factual and procedural knowledge of arithmetic at the mastery level.**

This knowledge consists of the memorization of single-digit number facts, the mechanics of standard algorithms, the applications of arithmetic through ratio/proportion, etc.

The negative effects of the

**reform math**movement have been shown through national and international tests and the explosion of remedial math at community colleges.

In short, students in some other nations were leaping ahead in achievement while U.S. students were crawling if that. American students were bad at fractions, decimals, percentages, and long division because the education experts argued [wrongly] that memorization and drill were bad for kids.

**Today, students are still bad at basic arithmetic. It is not practiced for mastery.**

**Note:**The Common Core and state standards are not world class. Our students start behind and stay behind.

Also, I believe that ordinary teachers don't know how to teach standard arithmetic or reading. Reading has stagnated, too.

*I do not blame the teachers; I blame the schools of education that trained the teachers and the state departments of education that certified the teachers and set policy.*

**Knowing builds the foundation for higher-level thinking**. The range of cognitive skills (left), starting with a strong base of Knowing, is similar to Bloom's taxonomy. Applying requires Knowing, and Reasoning implies both Knowing and Applying.**Teachers should start at the bottom and focus on Knowing (both factual and procedural knowledge in arithmetic and algebra). This builds the foundation for higher thinking.**Furthermore, knowing something takes substantial practice. You cannot apply something you do not know well in long-term memory.**More is said than done.**It is especially true in education. We say we want students to engage in "higher-level" thinking,

**yet we don't focus on lower-level thinking (i.e., knowing and applying content) that leads to and enables higher-level thinking.**Put simply: our actions do not support our goals.

*We say one thing, then do another.*

**Unfortunately, the progressive reformers continue to pitch evidence-lacking ideas and ineffective programs, which impede children's achievement.**

**Immanuel Kant**

**wrote that thought (e.g., critical thinking, problem-solving, analysis, etc.) without content is empty.**To learn something means remembering it from long-term memory such as the single-digit number facts and standard algorithms in arithmetic. Learning requires effort, memorization, drill to develop skill (practice-practice-practice), and review. The main focus has been on reform-math alternatives, not standard arithmetic or standard algorithms.

**In the self-contained classroom, the focus has been on giving everyone the same. Also, much of the attention is given to low-achieving students with very little attention paid to our high-ability students.**

**Thomas Sowell**(

*Dismantling America*) called it a "fallacy of fairness." He points out that "equalizing downward by lowering those at the top" is a "crazy idea taught in schools of education." Our best students should receive accelerated instruction, but this is rare in K-8. Sowell explains, "Fairness as equal treatment does not produce fairness as equal outcomes."

**Our best students should be taught a totally different curriculum in math and other disciplines.**

Moreover, teachers are often required to teach to

**"items on the test"**often using inferior methods of instruction (minimal guidance methods of instruction); consequently,

**many students never master arithmetic**. In effect, the math curriculum is

**fragmented and below world-class benchmarks**.

*The disparity begins in the 1st grade.*

**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**.

If a student counts on his fingers to solve 5 + 7 each time he needs it, then it stays in the working memory. The single-digit addition fact does not move to the long-term memory without much practice and review.

**In other words, the student hasn't learned 5 + 7 = 12 because he can't remember it instantly. Learning is remembering from long-term memory.**To reduce

**cognitive load**(

**John Sweller**) when solving problems, students should automate single-digit number facts and standard algorithms as early as possible.

**Carl Bereiter**states, "The limits of what children can do or learn are essentially unknown." Indeed,

**we should not limit students to so-called grade-level assumptions, but isn't that exactly what we often do?**

Student progress should be based on mastery, not age or grade level.

Student progress should be based on mastery, not age or grade level.

**Digital Devices & Tech**

Bill Gatessaid in an interview, "

Bill Gates

**I still believe that sitting down and reading a book is the best way to really learn something.**And I worry that we’re losing that." He is referring to a physical book, not reading a book on a screen.

**Also, in my contrarian opinion, kids**

**don't need to use digital devices in lessons. Likewise, they don't need to use calculators to learn arithmetic or graphing calculators to learn algebra.**

*Sadly, the new College Board*

**SAT**allows graphing calculators for half the math questions.**Zig Engelmann**points out, "

**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." Again,

**we should not limit students to so-called grade-level content.**

**If learning is remembering from long-term memory, then 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."

**Engelmann**states that we talk about issues in education but never about teaching, itself. He points out that teachers seem fixated on grade-level beliefs and bad learning theories. Teachers rarely teach for mastery and are told that technology is the silver bullet. Educators often prize

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

**Kirschner-Sweller-Clark***point out, are ineffective compared to

**explicit instruction**via carefully selected

**worked examples**.

**Engelmann**states, "Educational researchers and policymakers do not endorse the most effective programs," and teachers rarely choose them (

**Beverlee Jobrack**,

*Tyranny of the Textbook*)

*.*Kids aren't learning essential content for mastery in long-term memory because they don't practice the fundamentals enough. Arithmetic basics are not taught well in the primary grades.

**Jobrack**writes, "Recognize that learning is work and that engagement comes through

**student achievement**....

**Education should continually be upgraded, not continually reinvented and reformed.**Experience does matter."

**Moreover, the curriculum is flawed.**"Common Core [i.e., state standards] are based on the idea that the way to improve learning is to challenge students and make it more difficult or rigorous so that students can learn more difficult concepts. It doesn't work that way." Why would teachers make content more difficult than it is so that students struggle? It's one of those inane "mathematical practices." If children get a good foundation in basics first, then more difficult content can be learned.

**Engelmann**explains, "A steady diet of 'struggle' slows up mastery." Furthermore, all those alternative, nonstandard ways to do the operations clutter the curriculum and increase cognitive load to slow the mastery of standard algorithms.

**In summary, n**

**ot knowing the times table for instant recall, for example, creates a cognitive load that interferes with learning and solving problems (John Sweller).***The same applies to number and equality properties, and the standard algorithms.*

**Also, "Certain things are rote, says Engelmann, 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"

**✔︎ Asian nations focus on mechanics first (the "how").**

The mechanics are practiced for

**mastery**in long-term memory. The Asian system works! In contrast, the American system of

**reform math**has been ineffective and backward, because it stresses "understanding" and "alternative algorithms," not the mechanics of standard arithmetic. For example, the

**standard algorithms**, which are core arithmetic, are delayed or pushed aside in reform math.

**✔︎ Competency**in math requires memorization and practice-practice-practice. Unfortunately, memorization and "drill to improve skill" are considered obsolete and poor teaching by progressive reformers, but they are wrong.

**Memorizing skills and practice are critical to learning and thinking.**As the late

**Richard Feynman**used to say,

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

**✔︎ International tests**show that Asian math students

**excel at factual and procedural knowledge and problem-solving**while American students stumble over simple arithmetic. For example, 54% of Singapore 8th-grade students scored the

**Advanced Level**to only 10% of U.S. 8th graders (TIMSS 2015). The Advanced Level of TIMSS is loaded with problem-solving, and it is a valid gauge of a nation's math program.

**Asian students have a lot of intrinsic motivation. They try harder and learn more in school; consequently, they score the best on international math tests.**

**✔︎ Ian Stewart**(

*Letters to a Young Mathematician,*2006) 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 university, we learn

**why**those methods work and prove that they do.

**"**Children are not little mathematicians; they are

**novices**, not experts, and there is a huge difference.

**✔︎ 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.**

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

(Richard Feynman, Nobel Prize, Physics)

**✔︎ Teachers should help bright kids excel, not to let them fend for themselves.**

✔︎ 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 the math is taught well and practiced well.

✔︎ 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 the math is taught well and practiced well.

**Why students should master arithmetic and algebra!**

**Knowing math and having financial savvy can greatly improve your quality of life. Math and science open the door to economic opportunity. Click for additional information!**

**Unfortunately, most of us don't know how percentages work.**

**If we did, then we would never take out a seven-year car loan (or any car loan), pay the minimum on credit card purchases, use "payday" or "title" loans, etc. Most don't understand the science of money such as the**

**miracle of compounding, which is exponential growth, not linear change.**

**Note:**Compared to students in some other nations such as the East Asian nations, our kids don't know much math or science. It's not that they lack intelligence; it's that they are not taught. Teaching students something is hard work. The bottom line is that

**knowledge in long-term memory is critical to thought**, explains

**Daniel Willingham**.

*Starting in the 1st grade with standard arithmetic, knowing math is important.*

**How am I supposed to learn math without a calculator?**

**John Saxon**wrote

**Saxon Algebra-1**in 1981 as a reaction against the

**constructivist-**

**reform-math philosophy and calculator use.**The Saxon Algebra-1 text is different from the later NCTM

**reform math Algebra-1 texts**. The reform texts of today require graphing calculators.

**No calculators are used in Saxon Algebra-1**

**.**

*****The Saxon focus is on learning the fundamentals through repetition and review, not technology.

**The "practice sets" in the Saxon books concentrate on remembering via repetition, which is the essence of learning.**They were designed to help students become good problem solvers through

**cognitive support**via worked examples and practice. "

**[**

**Benjamin]**

**Bloom**tells us that it is necessary to '

**overlearn**' to achieve automaticity," writes

**Frank Wang**, President of Saxon Publications, Norman, Oklahoma. Basics need more than mastery; they need a step up to automaticity. Also,

**Wang**points out that "

**Books that try to teach applications and concepts at the same time often fail.**

**The teaching of concepts must come first.**" (Wang refers to defective U.S. math textbooks.) But teaching concepts is not nearly enough.

**Students must learn mechanics**, too. In short, students must be able to perform and apply mathematics efficiently. Concepts are easy to learn; however,

**students cannot perform math and solve problems without knowing mechanics**. Briefly, kids cannot apply facts and skills they don't know well. Likewise, teachers cannot teach factual and procedural knowledge they do not know well.

*****

*In the Third Edition (2003) of Saxon Algebra 1, Lesson 114 uses a scientific calculator and a graphing calculator. Many teachers skip the lesson.*

**Furthermore, we can build engagement through student achievement**, not via gadgets such as computers, laptops, tablets, calculators. Students don't need manipulatives or calculators to learn basic arithmetic; they need a number line. Also, any child can improve in arithmetic and algebra with proper instruction and plenty of practice.

**Calculator use covers up weak math skills.**

Mathematician

**W. Stephen Wilson**points out that calculators are "

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

*The reality is that the more "rote learners" of the East Asian nations have excelled in factual and procedural knowledge*I have found over the past 35 years that most parents have been led to believe that their children are doing well in school (grade inflation); however, the performance data from the national and international tests (NAEP and TIMSS respectively) show otherwise.

**and creative problem-solving**(TIMSS, PISA) leaving most American students in the dust.**

**What about P-21 Skills?**

Daniel Willinghamwrites on the

Daniel Willingham

*Britannica Blog*, "Thinking skills are intertwined with domain knowledge. Critical thinking in one domain does not apply to another.

**Knowledge is critical to thought**."

**Willingham**, a cognitive scientist, explains, "

**21st-century skills**(such as working well in groups, managing projects, or developing leadership) will not be developed simply by putting people in groups or asking them to be leaders." He says that we don't know how to teach P-21 skills like we teach algebra and reading. "It's not that the ideas are bad, but they clearly are

__not workable__in the way that seems obvious: we want students to be able to do

**X**in the world, so stick more

**X**in the classrooms. If it were that easy, it would have worked by now, because it has been tried many times before." What kids need is

**content knowledge**in math, both factual and procedural.

**Indeed, knowledge in long-term memory is critical to thought and learning.**

**What kids don't need**is a blend of group work, project learning, discovery learning and other constructivist minimal guidance methods. They don't need "fairness policies" where kids of diverse knowledge and achievement in math are lumped together in the same math class.

**Are we instructing our elementary students so that they will be ready for Algebra-1 in middle school and Algebra-2 and higher math in high school?**

*The simple answer is No.*If educators continue to use

**inferior methods of instruction**, state standards that are below world class, and a reform math curriculum, then students will not advance to Algebra-1 in middle school. The aim of the

**National Math Advisory Panel's**recommendations (2008) was to get more kids ready for Algebra-1 in middle school, not fewer. After ten years, we have not done this.

The reality is that the more "rote learners" of the East Asian nations have excelled in factual and procedural knowledge

**and creative problem-solving**(TIMSS, PISA) leaving most American students in the dust. (Worth Repeating!)

Commenting on Common Core and state standards based on Common Core,

**Ze'ev Wurman**points out,**"One-size fits all represents****a giant step backward**. Children in Singapore and South Korea, for example, master introductory algebra in eighth grade or earlier.**In today's economy, more advanced mathematics is a prerequisite not only for college admission but also for most vocational and technical-training programs."**We need a curriculum that**"**emphasizes pre-algebra concepts in the early grades, as early as the 3rd grade, with the goal of getting as many eighth-graders as possible into Algebra I. Indeed, "No child rises to low expectations." The recommendation of the**National Mathematics Advisory Panel**(2006-2008) was to get more kids into Algebra-1 no later than the 8th grade.**Note:**I introduced algebra concepts (Teach Kids Algebra) in 1st grade in 2011. Source of Quotes: Ze'ev Wurman and Bill Evers, CITY article, "Out of the Equation" 2012; Context: The 1997 California standards were world class but were replaced by Common Core, "a giant step backward.")**Forgetting is easy.**Learning something--so it sticks in long-term memory--requires substantial repetition, practice to improve, regular review, and effort.

**If you don't remember something, you haven't learned it. Learning is remembering from long-term memory.**

**Kids need to learn straightforward standard arithmetic, well-established algorithms, and routine applications to prepare for algebra**, not alternative arithmetic as in constructivist reform math. Reform math includes minimal guidance methods such as discovery learning, many different nonstandard algorithms that clutter the curriculum, real-world nonroutine problems that often require calculators, etc.

I used the

**0-20 number line**in the 1st grade to teach magnitude, one more than (add one), math facts, and the fundamental ideas of addition, subtraction, and multiplication. Later, it was used to teach fractions and their operations.**The number line is prime mathematical content.***But, I seldom see number lines in K-5 math textbooks.***Barak Rosenshine ("Principles of Instruction") explains, "**

**Students need cognitive support to help them learn to solve problems.**The teacher modeling and thinking aloud while demonstrating how to solve a problem are examples of effective cognitive support.

**Worked examples**(such as a math problem for which the teacher not only has provided the solution but has clearly laid out each step) are another form of modeling that has been developed by researchers.

**Worked examples allow students to focus on the specific steps to solve problems and thus reduce the cognitive load on their working memory.**Modeling and worked examples have been used successfully in mathematics, science, writing, and reading comprehension.

**"**

**Some educators waste a lot of instructional time teaching the four Cs--critical thinking, communication, collaboration, and creativity (P-21 skills)**, often at the expense of teaching kids content in math, science, history, and literature and of the importance of CDE--concentration, discipline, and effort.

**David Goodman**(

*A Moron with a Computer Is Still a Moron*, 9-5-11) writes, “Education begins with discipline and an extended concentration span.” In other words, effort is required to pay attention.

**The problem with P-21 and its four Cs is that the skills are taken out of context and not deeply rooted in content kids must know (e.g., mathematics, science, history, literature, etc.).**Students cannot think critically about anything unless they have substantial background knowledge of the subject. Students are not thinking critically; they are pretending (i.e.,superficial thinking). Moreover, critical thinking (problem-solving) is different for different domains. For example, in mathematics, problem-solving involves axioms, logic, deduction, proof, algorithms, and counterexamples. In science, problem-solving involves observations, measurements, inductive inference (conjecture), theories that predict, and experimentation to support scientific ideas.

You cannot teach creativity. It comes from studying something for years. Kids are not taught arithmetic to become more creative. They are novices.

Some teachers seem obsessed with “engaging the student” to the point at which engagement, itself, becomes the central thrust of the lesson rather than the lesson's content; however, engagement or "skill in using tech" is not the same as math or science learning and achievement.

**Kids Can Learn Harder Stuff--if they work hard & if their teachers work hard!**

**Build Mastery Through Practice!**

Some educators say that little kids cannot learn algebra or memorize multiplication facts because it is too hard. I often hear, “Kids are not developmentally ready for that.” Or, “That’s too advanced.” Or, “Our kids cannot compete with Singapore kids.” Or, "Rote learning is bad learning." It’s bunk! It is low expectations! And, it is contrary to experience and cognitive science. Repetition, multiple exposures, effort, and extensive practice are important in learning. Students need strong teacher guidance, explicit instruction, a world-class curriculum, and lots of practice to learn arithmetic and algebra well.

*In short, our students need to master a world-class curriculum starting in grade 1 to stay competitive. It takes effort, work, and the development of smart thinking. Students cannot apply arithmetic or algebra that they do not know well. Thus, acquiring background*

*knowledge*

*quickly is critically important.*

*Student achievement should be the highest priority in our classrooms.*

*What has gone wrong?*

*It boils down to*

*curriculum*

*, which is the content in the textbook or the math program.*

**Beverlee Jobrack (**

*Tyranny of the Textbook*), observes that "standards, achievement testing, technology, and professional development [and other reforms] have had little effect on improving student achievement" (p. xx).**(From**

__Math Notes)__It also boils down to

**instructional methods**. Often, teacher use the least

**effective methods**of instruction such as minimal guidance and group work. The popular idea that hands-on or discovery learning is the best way for children to learn math

**lacks evidence**. Children learn very little from the minimal guidance methods.

**Have such reforms as "Laptops for All," technology hype, and engagement zeal gotten in the way of learning?**

**Larry Cuban****(Stanford) says that reformers often recycle ideas that didn't work in the past. He writes, "Policies can at best be only hunches about what will work in schools, and even the best guesses, grounded in all available evidence, are no guarantee of success."**

**Indeed, e**

**ducators should be skeptical of reforms and "don't swallow the hype."**For example, NCTM-based reform math didn't work in the past, but it is back via Common Core and state standards. Another example is personalized learning. Individualized Prescribed Instruction (IPI) failed in the 70s.

*Technology has not erased the woes in math education and has not lived up to its hype.*We have had calculators and computers in classrooms since the 80s or sooner. And, while there have been incremental improvements, in general, our math scores on national (NAEP) and international tests (TIMSS, PISA) have been flat and disappointing.

**Calculators are "absolutely unnecessary."**

**W. Stephen Wilson**, a math and education professor at Johns Hopkins University, points out, “I have not yet encountered a mathematics concept that required technology to either teach it or assess it.

**The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them.**There might be teachers who can figure out a way to enhance learning with the use of technology, but it’s absolutely unnecessary."

**It is regrettable that the NCTM reform math standards (1989) advocated the use of calculators starting in Kindergarten.**The idea was that kids could use calculators as a substitute for learning the basics of standard arithmetic in long-term memory. Many progressive educators decided to dump or marginalize standard algorithms. Indeed, the writers of EngageNY Common Core (Eureka Math) stated that if elementary students had difficulty with arithmetic, then they should use calculators. Many reform math programs such as

*Investigations*require calculators. Today, calculators are used on the

**SAT, AP, and GED**exams and even on some state tests. The rise of so-called "real world problems" with messy numbers has been an excuse for calculator use in K-8.

**The use of calculators covers up weak math skills.**

**Students**raised on calculators and graphing calculators since middle school and

**earlier**are shocked when their Calculus 101 professor at the university doesn't allow calculating devices on exams. Moreover, many universities do not accept high school AP Calculus for credit toward a STEM major. Students need to take the university's calculus course.

**How do schools boost graduation rates?**

They lower standards, inflate grades, misname math courses with scant content as college prep, and implement credit recovery schemes. If these students enroll at a community college, they will need extensive remediation in math and other basics. Gee, isn't that what has been happening for nearly two decades?

**Many students who enroll in college are not prepared in math or reading.**

::: Model: McKayla 4th Grade

**6-26-17 Notice:**ThinkAlgebra is undergoing major changes. Some of the content in CommonCore, FirstGrade, Snaps, and the Main (Index) page has been deleted. Expect more content to disappear. Also deleted were these pages: TKA and CognitiveScience.

*Click*

**Memory & Learning**

"

**To learn something is to remember it.**" If you can't instantly recall 6 x 7 = 42, then you haven't learned it.

At odds with the champions of discovery learning and other minimal guidance instructional methods,

**active teacher guidance during instruction**, not group work, should be the primary approach in our classrooms; however, the curriculum taught explicitly via worked examples needs to be world-class and properly sequenced. Identify**30%**of traditional mathematics that will have a**70% impact on achievement**, then allocate 70% of class time on the practice and review of the 30% for mastery emphasizing that**learning is remembering**from long-term memory.*The 30% includes the memorization of single-digit number facts and learning the standard algorithms from the get go.***Standard Algorithms Should Be A High Priority!**

*Despite that, they are not emphasized in reform math classrooms.*

**Starting in the 1st grade, the standard algorithms and the supporting single-digit math facts should be a high priority through 3rd-grade core arithmetic.**The standard algorithms should be taught first along with the place value system, but they are not in many classrooms. Also, the standard algorithms for multiplication and long division should be learned no later than 3rd grade.

*The learning requires memorization, drill-to-improve-skill, and regular review.*For decades, reform math has embraced alternative strategies, calculators, and minimal guidance methods--not standard algorithms and straightforward, explicit teaching. The standard algorithms, along with fractions, place value, and other key math topics are the

**core arithmetic**needed to prepare for algebra.

**The result of the reform math approach has been a massive number of students placed in remedial math at community colleges**, even up to 88% of incoming students (Data: Pima Community College 2014).

**Note1:**Common Core and state standards based on Common Core are often interpreted through the lens of reform math.

**Note2:**

**Regarding reform math**, no one uses the area strategy to multiply or partial quotient strategy to do long division, yet these reform math procedures are taught in our elementary schools along with a hodgepodge of other cumbersome, alternative processes or procedures to do simple arithmetic.

Traditional

**4th-Grade**Core Arithmetic 1877 (19 Century America - Arithmetic)**[Find the interest of $80 for 7 months, at 6 per cent.]***Ray's New Intellectural Arithmetic***1877**(3rd & 4th Grade) Note: Dr. Ray died in 1855, but his arithmetic textbooks, which were used extensively in the 19th Century, have lived on.

**Red Flag: U.S. Bombs PISA**

**U.S. math test scores for 15-year-olds (10th Grade) bombed to the bottom half of the 72 participating nations and regions in the 2015 PISA international test given every three years.**

**Larry Cuban**(Stanford) points out in his blog (12-3-16),

**“**The research supporting “personalized” or “blended learning” (and the many definitions of each) is, at best

**thin**

**.**Then again, few innovators, past or present, seldom invoked research support for their initiatives.

**”**The same goes for multimedia (Clark & Feldon). Evidence does not seem to matter much to educators. It seems clear that the emphasis is on using tech to engage students rather than on learning content knowledge in long-term memory needed for critical thinking (i.e., problem-solving).

**Stagnant national and international test scores in math and other subjects demonstrate it! Notes:**Thoughts without content are empty. Engagement is not the same as learning. Using tech does not mean better achievement.

****

*Please excuse typos and other errors on this page.*

**East Asian countries dominate the 2015 TIMSS Math results. U.S. students not only lag behind, but they are also not in the same ballpark.**(The more "rote" East Asian learners, who memorize and drill-to-improve-skill, soared far above U.S. students not only in

**content**

**knowledge**and ability to perform mathematics correctly but also

**problem-solving**at the Advanced levels.)

The

**2015 TIMSS**mathematics results show that our

**4th graders**dropped since 2011 (scale scores: 541 to

**539**) and were barely

**treading water**if that, but our

**8th graders have improved**somewhat since 2011 to 2015 (

**509 to**

**518**). (

**Note:**But, the 8th-grade students dropped on the

**2015**

**NAEP Math, a national test,**from 285 in 2013 to 282 in 2015; the 4th grade-students scored 240, the same as in 2007. And, only 25% of 12th-grade students were proficient or above in NAEP math.

**Note: Trends in Math and Science Study or TIMSS is an important international test given every four years. The TIMSS scale center point is 500. Some of the 2015 results were announced at the end of November 2016. The NAEP is a national test given every two years. The National Assessment of Educational Progress is often called The Nation's Report Card).**

**We have spent an enormous amount of money to tread water (flat achievement)!**

**4th-Grade TIMSS 2015**

Since 2011,

**Finland’s**4th-grade students dropped (545 to

**535**) and fell behind the U.S. (

**539**). The Finnish 8th-grade scores were not released.

**At the 4th-grade level, U.S. students were not in the same ballpark as the East Asian nations:**Singapore 618, Hong Kong SAR 615, S. Korea 608, Chinese Taipei 597, and Japan 593. The next were Northern Ireland 570 and the

**Russian Federation 564**. The gap between Singapore and the U.S. is 79 scale points.

**8th-Grade TIMSS 2015**

**U.S. 8th graders showed improvement since 2011**(509 to

**518**), but they were still far behind the high performing East Asian countries. Our 8th-grade students are over 100 points below Singapore. The Scale Scores were: Singapore

**621**, Korea 606, Chinese Taipei 599, Hong Kong SAR 594, Japan 586. The next were the Russian Federation 538 and Kazakhstan 528. The gap from Japan to the Russian Federation was 48 points. The gap between Japan and the U.S. is 68 scale points.

The

**argument**(aka excuse) has been that US kids had never scored well on international tests.

**Is it a sign that U.S. math learning is declining? No!**In fact, our students have made incremental improvements over the decades. The

**stagnant test scores**do indicate that our students haven't gotten much better while students in other nations have soared in math achievement. America is not economically competitive.

**The reality is that children in East Asia have been best at math for over a couple of decades.**

There are

**reasons**why U.S. students lag behind their Asian peers. The Common-Core-influenced math curriculum is

**below world class**! Many teachers implement

**inefficient minimal guidance instructional methods**(group work). The trend in U.S. education has been to eschew memorization and practice, which are needed for the automation of fundamentals of

**standard arithmetic**in long-term memory.

**Furthermore, most of our K-8 teachers have been**

**trained poorly**in both math and science in schools of education, yet, elementary school teachers are asked to teach all subjects. Teachers are unaware of the

**cognitive science of learning**.

*In other words, we have not done what is needed to improve math achievement significantly. We have been running in place.*

**In short, we teach math badly, starting in 1st grade.**Kids have not advanced like their Asian peers. We seem more interested in small class size,

**technology in the classroom**, and test-based reforms than in developing high-quality teachers. In contrast to American classrooms, which are filled with tech (tablets, laptops, computers, etc.), little tech is used in Asian classrooms. Also, different from Asian nations, our schools of education are not selective. Sadly, the teaching profession has been riddled with problems.

*11-30-16, 12-1-16, 12-2-16. 12-3-16*

Notes:

**More TIMSS data will be released in late January 2017. I am interested in the percentage of students who reached the Advanced Benchmarks, which, I think, is the best way to judge the effectiveness of math programs from various countries. In a U.S. national math test (NAEP) given every two years, the 4th-grade students went from 235 in 2011 to 240 in 2015, which is down from 242 in 2013, while the 8th-grade students went from 279 to 282, which is down from 285 in 2013.**

**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.

**Who calculates using area strategies, partial quotient strategies, or a hodgepodge of other cumbersome, more complex strategies (procedures) to do simple arithmetic? [No One!]**

Students need to learn standard algorithms from the get-go! But, most do not! Instead, they are taught reform math that stresses strategies over content.

Students need to learn standard algorithms from the get-go! But, most do not! Instead, they are taught reform math that stresses strategies over content.

If US educators continue to teach elementary school arithmetic as

**reform math**, which stresses multiple strategies more than standard algorithms, such as in Common Core and state standards, then math achievement will remain

**flat**. "Despite huge increases in federal involvement in education, student performance in the United

States has remained stuck at

**average levels**since the late 1960s and early 1970s, observes

**Vicki E. Alger**(

*Failure,*2016). Some of the most recent reforms promoted or funded by the federal government or government agencies--such as the massive U.S. Department of Education with a $70 billion budget--include Common Core, NCLB (now Every Student Succeeds Act), Race to the Top, mandated testing, reform math, use of technology, inclusion policies, and others.

**There are always strings attached to federal dollars.**

**Standard algorithms**depend on the instant recall of

**single-digit number facts**. K-5 students need to

**automate the fundamentals of arithmetic**(ideas, skills, and uses) through consistent practice and use. But, educators continue to

**waste valuable instructional time**on "strategies" that will never be used because they are cumbersome, inefficient, and nonessential. From the get-go, starting in the first marking period of 1st grade, students need to learn single-digit number facts, standard algorithms, and essential content knowledge to move forward like their peers in top-performing nations. We need to

**upgrade curriculum to world class**and apply instructional methods that are explicit, productive, and competent (i.e.,

**strong teacher guidance**).

*Common Core and state standards are not world class.*

Also, we have been on the wrong road for decades. We jump around from blended learning to personalized learning, project-based learning, group learning, and so on. But, students don't get much better! Recently, I read that students should read from books and screens. After all, we are told, it is the digital age and reading instruction should change. "It ["The Changing Face of Literacy"] finds that, while experts quibble over what it means to be digitally literate, they agree on one thing: even the youngest children should be learning literacy with a mix of print and digital texts."

**Really?**

**Well, maybe the so-called experts are wrong!**

**Boaler and Zoido**write, "Research shows that an emphasis on memorization, rote procedures, and speed impairs learning and achievement."

**What a ridiculous statement!**(It is typical of reform math ideologues like Boaler.)

**Research has not shown this.**

*In fact, memorization of single-digit number facts, becoming skilled in using standard algorithms, and learning arithmetic content (ideas, skills, and uses) in long-term memory boost achievement and enable problem-solving. To get better at arithmetic requires practice-practice-practice, which has fallen out of favor, along with memorization, in modern classrooms. Memorization is critical in learning and life.*

11-24-16

**Rick Hess**,

*Education Next*, writes that the current education reformers are “passionate, Great Society liberals who believe in closing achievement gaps and pursuing equity via charter schooling, teacher evaluation, the Common Core, and test-based accountability." They say, "[No] reasonable person can disagree” with this. Really? The belief is brazen and presumptuous because many believe that the Common Core and test-based accountability, for example, have been counterproductive, along with reform math.

The

**evidence**that the math reforms will close achievement gaps is

**scant**. It's utopian thinking promoted by progressive ideologues. But, we do not live in

**Lake Wobegon**where all children are above average in intelligence. There will always be gaps (inequalities) because

**good education increases differences**, says Nobel-prize winning Physicist

**Richard Feynman**. The well-meaning pursuit of inclusion and social justice in the classroom, which

**Thomas Sowell**calls "

**equalization crusades**," is not real equity and is often divisive.

**In the real world, abilities and attitudes vary widely.**Some kids are better at math than others. Some kids study more than others. Some kids become elite Olympic gymnasts, but most never come close. Unfortunately, too many kids struggle with basic arithmetic because of the way it has been taught and presented, i.e, as reform math. Average children can learn arithmetic and algebra. Even if resources and opportunities were the same,

**Sowell**explains,

**"Fairness as equal treatment does not produce fairness as equal outcomes."**Musical ability, artistic ability, writing ability, athletic ability, academic ability, and so on vary widely in a population. Abilities need to be trained.

__Understanding does not produce mastery;__

**practice does**!Many parents are not teaching their kids how to behave, succeed, and achieve in school or life.

*Teachers can't do it all.*If we want students to do higher-order thinking, then they need to start with lower-order content-rich knowledge (lower level thinking). There is no substitute for

**knowledge**in long-term memory and the practice that gets it there.

11-23-16

**We are not all equally creative.**Some believe that creativity comes out of thin air. It doesn't! Knowing stuff is essential to creativity. Knowledge matters!

**"You need to look back at the old things," the traditional stuff, to move forward. Einstein didn't toss Newton in the waste can; he built on Newton's incredible insights.**Indeed, we should

**imitate**the best from the past and "stand on the shoulders of giants." Tradition is not tossed out in highly creative cultures.

*Indeed, something new is not necessarily better.*In fact, most innovations in education flop because they were contrary to the cognitive science of learning and depended too much on

**anecdotal data**. Progressive reformers in education may have been well educated, but they often lacked wisdom and were too quick to toss out traditional arithmetic and instructional methods that had worked in the past. Their

**disruptive tactics**have been counterproductive in math education.

**Newstok**draws an extreme metaphor for today's "googling," i.e., finding information on the Internet with a search engine. He writes, "If you knew no words in a language, having a dictionary wouldn't help you in the least since every definition would simply list more words you don't know.

**Likewise, without an inventory of knowledge, it's frustratingly difficult for you to accumulate, much less create, more knowledge."**

*Knowing stuff counts.*

**Crowding**

It seems that educators are pretty good at addition.

**Over the past 50 years, a lot of stuff, some of it nonessential, has surged into the curriculum and classroom.**It includes the

**extras**in arithmetic and algebra (e.g., reform math, Common Core, so-called mathematical practices, state standards that are below world class, more complicated and multiple ways to do simple arithmetic). Intrusive, also, are government mandates and programs, disruptive evidence-lacking innovations, popular classroom practices that don't work, minimal guidance during instruction, etc.). But, I should also count the

**gadgets**in the classrooms, including computers, tablets, smartphones, graphing calculators, smart boards, Internet, online software and learning programs, etc. Furthermore, education is overflowing with

**failed theories and wrong ideas and reforms**, yet they still stick like glue. Test-based accountability has complicated curriculum and instruction and led to teaching to the test.

**In schooling, we are fixated on gadgets and test-based reform, but "tech and test" are not the solution; they, I believe, are part of the problem.**

*Gadgets have not transformed education and don't improve achievement.*

**"**

**Critical thinking does not exist as an independent skill."**(E. D. Hirsch, Jr.)

The Common Core math standards, now called state standards, were not benchmarked to the standards from top-performing nations. In fact, Common Core benchmarks were lower than the

**1997 California math content standards**, which were world class and emphasized the fast learning of

**standard arithmetic**, including the standard algorithms to operate on numbers, to prepare more students for Algebra-1 by middle school.

California rejected reform math based on standards from the National Council of Teachers of Mathematics (NCTM) because students could not do simple arithmetic. Contrary to the NCTM standards, the 1997 California math content standards were well written and stressed the fast learning of factual and procedural knowledge in long-term memory.

**[**

**1997**

**CALIFORNIA 3rd-grade (World Class) MATH Standards**:

**2.4**Solve simple problems involving multiplication of multidigit numbers by one-digit numbers (3,671 x 3 = __).

**2.5**Solve division problems in which a multidigit number is evenly divided by a one-digit number (135 ÷ 5 = __).

**]**

It is unfortunate that California ignored the

**National Math**

**Panel**and scrapped the straightforward, 1997 world-class content standards for inferior Common Core math standards in 2010. (Note. The explicit teaching of standard arithmetic to prepare students for Algebra-1 in middle school had been the principal recommendation of the National Mathematics Advisory

**Panel**of 2008.)

**Nearly 140 years ago, American 2nd-grade students learned multiplication and division in the public schools.**

*The young students memorized single-digit number facts and drilled to improve skill and performance using both abstract questions and word problems.*

**I observed that most of the abstract questions and word problems involved two or more operations (multi-step)**.*The arithmetic content taught today is meager compared to Ray's 1st/2nd-grade 94-page textbook (1877).*

**1. 2nd-Grade Abstract Questions**

*How many are 6 and 5, less 4, multiplied by 7?*

How many are 2 and 5, less 3, multiplied by 9, divided by 6?

How many are 6 x 8 - 4?

How many are 2 and 5, less 3, multiplied by 9, divided by 6?

How many are 6 x 8 - 4?

**2. 2nd-Grade Word Problems**

*:: James bought 3 lemons, at 2 cents each, and paid for them with oranges, at 3 cents each: how many oranges did it take?*

:: What will 63 marbles cost, if 14 marbles cost 2 cents?

:: I bought 2 yards of cloth, at 4 dollars a yard, and 3 yards, at 2 dollars a yard: how much did all cost?

:: What will 63 marbles cost, if 14 marbles cost 2 cents?

:: I bought 2 yards of cloth, at 4 dollars a yard, and 3 yards, at 2 dollars a yard: how much did all cost?

******Source:**

*Ray's New Primary Arithmetic*1877. The text was 4.5 x 7 inches, 94 pages long, and covered all of the 1st and 2nd-grade arithmetic. What a novel idea:

**two grade levels in one tiny book**--no color, graphics, etc., just pure arithmetic! In stark contrast, Pearson's

*enVision Math*2nd-grade paperback (2011) is huge, approximately 11 by 16 inches, and 644 pages long.

**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)**

****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 my 1st graders did not 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 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.**

**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.

**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.**

Common Core reform math or its rebranded configuration (i.e., the state standards) promotes mediocrity in learning, not excellence.

**The standards are not world class starting in 1st grade.**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.

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

**standard algorithm**is merely one of the many ways to calculate and often

__not__the preferred way.

**For decades, fractions and long division have been sidelined and taught poorly in elementary schools.**The lack of adequate instruction 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. Students in other nations routinely do algebra in middle school, but not our students.

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

**The early mastery of standard arithmetic is required to prepare for algebra, but it isn't the main objective of reform math. 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 in the same classroom. 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 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.**"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." Deweyism 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.*

*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*).

**Changes in the math curricula**often come from reformers, textbook writers, researchers, progressive educators, or ed school professors, most of whom have never worked with young children.

**Many of the changes in teaching basic math are evidence-lacking reforms, debunked learning theories, inefficient minimal guidance methods, the latest untested fads, and so on.**Collectively, I label these as

**constructivist**

**reform math**.

*Also, many math programs claim to be "researched-based," but they were never tested extensively in the classroom.*

**In contrast,**I go into regular elementary school classrooms as a guest teacher to teach little kids algebra. The lessons are once a week for an hour.

**I fuse algebra to arithmetic.**To learn algebra, students must know some

**standard arithmetic**in long-term memory.

*For example,*Starting in the first grade, major ideas in algebra are accessible to very young children through standard arithmetic.

**by the end of 1st-grade**, students should have memorized single-digit number facts, performed the standard algorithms that model the place value system, known equivalency ideas (=), worked with properties of numbers and operations, and evaluated numerical and algebraic expressions.**I make up the algebra lessons and try them out**. I do not write a formal lesson plan just some notes and worked examples. A lesson in a 4th-grade class one year will be different in another year. In short,

**I often make adjustments and changes even in the middle of a lesson as I connect new content to content the students already know.**

**Notes:**Some of the ideas I use in my

**Teach Kids Algebra**lessons can be found below and on the ReformMath and FirstGrade pages (See Menu). I do not use calculators, manipulatives, or group work.

**Instead, I explain worked examples, give guided practice, then ask students to complete a practice sheet. I circulate around the room to give students individual help.**

*The inspiration for introducing very young children to algebra came from The Madison Project (1957) and Science--A Process Approach (1967).*

**Likewise,**pilot programs of studies from textbook and software publishing companies even if they seem to be successful do not always scale up. The studies are rarely replicated because it is hard to find an acceptable

**control group**. Testing the effectiveness of math programs is costly. Unfortunately, many math programs that do not work well in the classroom can be found on the "What Works Clearinghouse" list. Furthermore, it is complicated and difficult to put research into practice in the classroom.

**The reasons are simple, however. The teachers are different and the kids coming into the classroom are different.**Also, something that is "statistically significant" does not imply that it is useable in the classroom.

**Note:**Typically, students who struggle in Algebra-1 have deficient arithmetic skills especially fractions-decimals-percentages, ratio/proportions, etc. 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. Kids can do these things when they are

**taught well**. The prerequisites should be

**practiced purposefully**to improve, achieve, and excel.

**Also, high-achieving math students need acceleration that advances them, not enrichment.**

**H. Wu**, UC-Berkeley, talks about the

**problems of implementation**of Common Core state standards. Later, he acknowledged that Common Core will likely fail because K-8 teachers do not know enough math content to teach it well. No amount of professional development will change this, he said.

**Note:**

**Wu**stressed the use of

**number lines**when learning fractions. Fractions are numbers on the number line.

**Tom Loveless**summarizes Wu's view that the number line is much more than a tool.

**The number line is mathematical content.**But, when I look at content in so-called Common Core aligned math programs, I seldom see number lines in K-5.

**I see so-called "strategies" instead of mathematical content.**

**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, Alyssa, Remi, McKayla, Kailey, Gabby, Alex, & kids in my algebra classes***Email LarryT at**__ThinkAlgebra@cox.net__.**Last update: 6-26-17, 9-8-17, 9-10-17, 9-14-17, 9-23-17,**

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**© 2004-2018 LT, ThinkAlgebra**