Wednesday, December 1, 2021


Welcome to MathPage1
December 20-21-22-23-24-25-26, 2021

"Algebra is figuring out what you don't know from what you do know." (LT)

Merry Christmas!

This website is for parents and educators who want their children and students to go to college and earn a degree.

Kids don't need screens or devices to learn arithmetic and algebra. Devices are a multi-billion dollar hoax, explains Dr. Nicholas Kardaras. For math, students need effective teaching, a lot of practice-practice-practice, and continual review. Teachers need a blackboard, not a tech screen, and traditional textbooks that students can take home.
Will this 1st grader get to the square roots of negative numbers in high school? 
K-8 students need much more time on task to prepare 
for high school Algebra-2 and precalculus.

Ponder these ideas:

  • Pumping more money into schools for smaller class sizes lacks evidence to justify it. 
  • Both math and reading scores were tanking years before the pandemic. The pandemic harmed achievement, too. 
  • Many gifted and talented programs don't increase math or reading achievement. Say what?
  • Tutoring works when done daily, one-to-one, by a competent tutor, which is rare.
  • Government-paid preschool? Really? Government-paid college debt relief? Really? Why should taxpayers fund preschool or college debt relief? Who has devalued responsibility?
  • The schools should not have been closed! There are many who disagree with me. Guess who got rich with remote stuff? Think, the tech companies? Did kids learn much? 
I have empathy for teachers with everything going on in schools today. Many left teaching. It was different when I started teaching. I also empathize with students who go unchallenged in many of today's classrooms. Today, a 4th-grade student had difficulty with 18 - 2, which is a 1st-grade concept. Kids must be taught better in the early grades, starting with 1st-grade arithmetic. (12-8-21)
Is STEM dead?
Students must be prepared for STEM courses in math and science by high school. Therefore, more students should take Algebra-1 in 8th grade, not fewer. Furthermore, preparatory measures should start in 1st grade with basic arithmetic and parts of algebra, measurement, and geometry. 

Note: When I teach algebra to elementary students in grades 1 to 4, I don't think about social justice, equity, diversity, identity, inclusion, or race. Instead, good calculating skills are needed for solving equations and problem-solving. For equation solving, it's "guess and check" in the 1st and 2nd grades and "undo" (inverses or opposites) beginning in the 3rd and 4th grades. But some astute 2nd graders have figured out the idea of undo. 

Example of 3rd-Grade undo
x + 234 = 693. Solve for x
To isolate x, undo "adding 234" by "subtracting 234" on both sides.
x = 459 to make the equation true. 

Incidentally, I am not into critical race theory. I expect students to learn algebra, both factual and procedural knowledge, such as solving equations. There are no calculators, group work, discussion, or alternative algorithms (Common Core and NCTM). Currently, I meet with 4th-grade students once a week for 60 minutes at a K-8 Title-1 city school. Almost all the students are minorities. I have been doing this off and on since 2011. My Teach Kids Algebra (TKA) program is STEM math for elementary school students. 

Algebra exercises your brain.

More Math, More Science--Not Dumbed Down Stuff
Rebecca Goldin, George Mason University, cited a study that over 1/3 of students "have not significantly improved critical thinking" after four years of college. She writes, "These students had trouble distinguishing fact from opinion, and cause from correlation." In short, they lack quantitative thought, which doesn't grow on trees. According to Goldin, students need more science courses with supporting math courses to develop critical thinking.

Taking College Algebra and Biology 101, or even a "so-called" quant thinking course, probably won't be enough to substantially improve critical thinking. Instead, students should take more math and science courses in college, such as precalculus, calculus, statistics, along with rigorous college-level courses in chemistry and physics, "even if these courses are not required." In short, students should avoid undemanding courses. Instead, they need more math and science courses, "seriously."

Anti-STEM Trend/Equalizing Down/It's Irresponsible!
We can increase math equity by dumbing math down so that almost all students pass, but should we do that? Should we make equity in math mean equal outcomes by radically reducing content? I am afraid that has been happening over the decades. Today, we have woke math. Replacing the "algebra to calculus" pathway in high schools with modern data science is counterproductive. Why? 

The reason is that "data science–-computer science, statistics, and artificial intelligence--is built on the foundations of algebra, calculus, and logical thinking," according to the letter signed by 1,470 STEM professionals and STEM educators (and rising) who vehemently oppose California's proposed woke math framework. 

The open letter reads, "Reducing access to advanced mathematics and elevating trendy but shallow courses over foundational skills would cause lasting damage to STEM education in the country and exacerbate inequality by diminishing access to the skills needed for social mobility."

"While these mathematical fields [i.e., algebra and calculus] are centuries old and sometimes more, they are arguably even more critical for today’s grand challenges than in the Sputnik era." Thus, any student who wants to pursue a STEM field should have the opportunity to take algebra and calculus in high school. But students must be prepared properly. Indeed, dumbing down the curriculum is illogical, off-target, and stupid. Richard Rusczyk (the Art of Problem Solving) explains that AP Calculus is for average students who are prepared. in fact, many colleges and universities do not accept AP-Calc toward STEM credit. Instead, STEM majors must take the university's calculus courses. 

Sowell: It's a Crazy Idea! 

Years ago, Thomas Sowell pointed out that "equalizing downward, by lowering those at the top" is a "fallacy of fairness." Equalizing by dumbing down the curriculum is a "crazy idea taught in schools of education." (2010)

Let's educate children and get rid of the junk that dumbs down the curriculum in the name of equity and diversity. Students must be prepared to take more difficult courses in math and science. Not all students will get to calculus in high school, of course, but more students should get to precalculus, a course that stresses advanced algebra and trig. Many independent schools require precalculus for graduation. Why? 

The reason is that 74% of students who take a solid precalculus course in high school will more likely earn a bachelor's degree in four years. Thus, solid math courses in high school matter. Grades matter. Homework matters, too. "Homework might seem like a waste of time, but it teaches you content, time-management, and discipline--all of which you’ll need in college."

Things must radically change in K-7 to prepare students for algebra by 8th grade. For example, in my opinion, 7th-grade students should have access to a solid prealgebra course. (Textbooks like Glencoe/McGraw-Hill Pre-Algebra 2003 are difficult to find. Each lesson has over 60 practice problems.) Also, there is plenty of Algebra-1 in solid Pre-Algebra courses. 

Going Beyond the Curriculum
STEM Math for Grades 1-5: Teach Kids Algebra (TKA)/Feedback
TKA started in 2011 as a reaction against Common Core reform math and minimal guidance methods. I fused basic algebra ideas with standard arithmetic. I taught algebra ideas by explaining worked examples (models) with practice and feedback in class. The importance of traditional arithmetic was stressed, starting with automating single-digit math facts that supported the standard algorithms. It begins in 1st grade. 

Note: Most kids were behind in math before the pandemic (NAEP); however, the pandemic effect put students behind even more. Unfortunately, in my opinion, it was a lost year for many students. For example, I was able to teach much more content to 4th graders a few years ago than 4th graders today. Elementary school classroom teachers struggle to help kids catch up, but I fear many arithmetic gaps will remain.

Note: Giving immediate feedback (below) is important when young children learn algebra fused to traditional arithmetic. TKA is STEM mathChildren need guidance and encouragement. They must also learn persistence

Immediate Feedback

It's Ideology! And, it's wrong!

Today, much of education is not about teaching kids to read, write, do arithmetic or about achievement, merit, and excellence. Say What? Instead, schools have become a bastion of equity, diversity, identity, and inclusion. Everything boils down to race. The liberal ideology is strongly supported by teacher unions, schools of education, federal and state education departments, many school boards, and others, including the media. I believe education has gone down the rabbit hole, an abyss. Commonly, there seems to be an inverse relationship between excellent academics and equity. 

For decades, the "equity" reformers have dumbed down the math curriculum (the academics). The latest act declares math as racist, a blatant attempt to dumb down math even more. Equity has evolved to mean the same outcomes, which is impossible unless content and standards are lowered substantially. But, students are not the same in academic ability. Some students are better at math than others. Some students have better teachers. Also, some students are not smart enough while others don't study enough to become smart enough. Equal outcomes or equal growth is a preposterous idea. 

Who are the "people" who say math is racist or "showing your work" and "getting the right answer" is white supremacy? They are divisive fools and know little of math or how children learn arithmetic.

Reflections of an Education Contrarian: A Divergent View

I am an education contrarian. My views often diverge from mainstream education "group think" and orthodoxy. For example, I teach 4th graders real algebra. Also, by 1st or 2nd grade, students work with expressions and variables (symbols that represent a number), evaluate expressions, solve addition and subtraction equations (functions), using guess and check, and later inverses (3rd grade). Given a linear equation (y = mx + b) and x-values, the student builds a table of values (x - y) and extracts the (x,y) number pairs to plot points in Q-I, even in 1st grade.

1st-Grade Algebra: Teach Kids Algebra - TKA

In TKA, the focus is on variables, order of operations, and functions.

Functions: Equation-Table-Graph (e.g., y = x + x + 2, y = x + x - 4, etc.) 

Functions in the 1st Grade (STEM Math)
Equation, Table Building, & Point Plotting (Graph)

Algebra is figuring out what you don't know from what you do know. (LT) That's why we study algebra! It's why I teach 1st graders functions (equations-tables-graphs) fused to the arithmetic they are learning. Ian Stewart points out, "Mathematics happens to require rather a lot of basic knowledge and technique." But, first, students have to know something in long-term memory: facts and proceduresLearning arithmetic or algebra well involves memorization, repetition, review, and practice-practice-practice. In short, hard work and persistence!

Note: I am teaching two 4th-grade classes at the Title-1 urban school. Almost all students are non-white. I am a volunteer, guest algebra teacher and give hour-long lessons once a week. 12-1-21

Teach Above the Curriculum!

I am increasingly concerned about academic achievement in schools, especially mathematics and science. Kids can learn more in-depth, key math content earlier and faster. I challenge teachers to stretch beyond state standards and progressive reform math. Teachers of 2nd graders, for example, should teach above the 2nd-grade curriculum and dive into the middle of the 3rd-grade curriculum. Students in 2nd grade should memorize half of the multiplication facts and calculate with the standard algorithms. 

One of the most debilitating trends in many progressive schools that teach reform math is their excessive emphasis on calculators as early as K. This nonsense originates from the National Council of Teachers of Mathematics (NCTM). It is because calculators "enable students to do arithmetic without thinking about the numbers involved in a calculation." For this reason, it makes sense to use them, let's say, in a high school science class (chemistry and physics). But for elementary students, even middle school students, the main goal of math education is to get students to think about numbers and learn arithmetic--both factual and procedural knowledge.

In K-7, toss out the calculators, most manipulatives, minimal guidance methods, and group work. Instead, explain worked examples (models) and give plenty of practice. Students should practice 30 problems a day--at least 2/3 should be "review" of key topics.

We need to train teachers better. They should be required to take a precalculus course in college, along with college-level chemistry and physics, even future 1st-grade teachers. Many educators disagree. Still, K-8 teachers must know math and science to teach math and science.


(1) Students can learn more mathematics than the current curriculum.

(2) Students can learn more math earlier than previously thought (e.g., the beginnings of multiplication and algebra in 1st grade). 

(3) Students can learn more math faster.

Note: Teach Kids Algebra (TKA), my algebra program for grades 1 to 5, started in mid-January 2011 with two 1st-grade classes (7 Lessons), two 2nd-grade classes (7 Lessons), and one 3rd-grade class (twice-a-week Lessons). The algebraic thinking lessons introduced true/false statements (=), variables, and the algebraic rule for substitution. Then, it quickly expanded to functions: equation-table-graph. TKA is STEM mathematics for very young children. Over the years, most students learned the algebra content with once-a-week, hour-long lessons. TKA in 4th-grade classes started late this school year (November 2021). TKA for 2nd grade will begin early next year. 

4th Grade TKA/Title-1/Urban School
Almost all students are minorities.
April 2012

Math is used extensively in science, but science isn't taught that way in our elementary and middle schools. There was one exception: The math needed to do the science was integrated into Science--A Process Approach (1967, SAPA K-6). So, for example, 4 of the 6 processes taught in 1st grade were math or math-related: 

  • 1. Using Numbers, 
  • 2. Measuring, 
  • 3. Communicating (graphs), and 
  • 4. Using Time/Space Relationships (geometry). 

Furthermore, in SAPA, 1st-grade students learned negative numbers. Today, there is nothing like it. However, some SAPA math topics are part of TKA, especially integers, linear functions, and graphing. Much of the math taught in SAPA was above the current curriculum. 

Algebra is about figuring out what you don't know from what you do know. That's why we study algebra! 

What has happened to math education in the US? 
We chase after test scores instead of credible academic achievement and continue a sameness ideology via Common Core or state standards. Teaching to the test is a flimsy curriculum and a lousy way to teach mathematics to novices. Also, we expect kids to do critical thinking without sufficient background knowledge in long-term memory, even though, according to cognitive science, factual knowledge in long-term memory must precede higher-thinking skills. (Willingham)

Minimal teacher-guided instruction--popular in reform math--is ineffective and inefficient and contributes to substandard achievement, according to researchers Kirschner, Sweller, & Clark. Kids cannot solve math problems on a near-empty math tank (the long-term memory). For example, "thinking" will not help a student solve a trig problem without specific, prerequisite trig knowledge in long-term memory and practical experience working with similar trig problems.

Drills & Lessons

In 1895 the "Committee of 15," sanctioned by the National Education Association (NEA), made recommendations for elementary schools (grades 1-8). Arithmetic should be taught beginning in 1st grade with 60 minutes of oral drill daily plus five textbook lessons a week to prepare students for geometry and physics in 7th grade and algebra in 8th grade. These recommendations make good sense even today, but, unfortunately, the wisdom of old souls has been replaced by a progressive agenda of reforms and ideology that don't work (e.g., Common Core taught as reform math). 

I read comments in other blogs that are opinions or off the mark. I often exclaimed, "What is this person talking about?" When I write about education, I write from things I have done in the classroom. I was not born when the Committee of 15 laid down its recommendations, but if you want kids to learn stuff, then the Committee of 15 was correct, daily drills with five weekly lessons. I often implemented daily practice with daily lessons when I was a classroom teacher.

Don Lee at claims, "Diversity, Equity, & Inclusion (DEI) are the top concerns for many School District leaders when purchasing from K-12 education companies and organizations." Really? In short, DEI is about race, not educational products that focus on reading, writing, and arithmetic, which, I think, should be the top concerns but often aren't. For example, national and international tests show that most kids are not learning the fundamentals of arithmetic. Yet, instead of upgrading the math curriculum for all students starting with 1st-grade arithmetic, educators and policymakers have been sidelined with DEI

The progressives have proposed that no entrance tests or exams are needed to enter the University of California system. "Any exam would be prone to bias, say officials. That is, it wouldn't show equal outcomes by race, ethnicity, and family income." (Source of the quote: Joanne Jacobs' blog12-7-21.) 

In short, equity is redefined as equal outcomes. Without equal outcomes, there is no equity. Really? Instead, you have a dumbing down of curriculum, especially math and science, and a ton of grade inflation.  

The progressives want to make grading fairer by not reporting Ds and Fs. Really? "Not reporting Ds and Fs is the equivalent of lying about a student's progress," observes Debora Rinehart, a math and science teacher at St. Theresa School, a Catholic school in Oakland., who opposes the idea. (Source of the Quote: Joanne Jacobs' blog)

The test of the morality of a society is what it does for its children.

Dietrich Bonhoeffer, Theologian

Rabbi Hillel wrote that the major principle of the Torah is also practical guidance: "What is hateful to you, do not do to your neighbor." Perform acts of kindness to practice this commandment, and help others fulfill their potential, which is the prime reason I went into teaching. 

LT Retired

©2021 ThinkAlgebra/LT