On Math Education

When I was a kid, my life was mathematics.  When I was little, my mom would hand me sheets of addition and subtraction problems that did not involve carrying or borrowing, and I would fill them out for her, and she would give me more.  That’s my earliest memory.  Second earliest is the procession for President Kennedy after his death.  Third earliest might be my Dad changing my brother’s diaper.  My mother once said, “You must have remembered it because of uniqueness.”

Anyway, when I was a kid, before I went into first grade, my mom taught me multiplication and division.  My Dad taught me some heuristic rules around percentages.

So when I came into first grade at age six, I was shocked that no one else could do math.  I was good enough with math and reading that the special education teacher took note of me, and helped give me my unusual education 1967-1969.  Though I was in a normal classroom, I was a “group of one.”  I had my own special reading and math books.  The teachers pushed a variety of enrichment materials to me, but it was like subpar homeschooling.  I was on my own, and no one would correct my work.  For a little kid, I was pretty motivated, but it would have helped a lot to have more adult interaction.

SRA helped on the reading side, and there were later SRA attempts at math though I don’t think that did much.

In third grade, they gave me a programmed instruction curriculum in math, in addition to the ordinary class.  At some points, I acted as a tutor to other students.  The programmed instruction was modeled after the “new math” fad.  I could get it, but at the time, I realized that I was so different from my peers, that I knew that if I could get it, that did not mean that others my age could get it.

Then in fourth grade, they mainstreamed me.  I spent time playing around with how to do square and cube roots by hand.  Tedious, but not that hard to do.

In fifth grade, my Father brought me an algebra book that used programmed instruction.  I puzzled over it, and didn’t get it until I talked to an older friend about it who told that “x’ is a number that we do not know, but are trying to calculate.

There’s more to this story, but I will drop it, lest I bore you…

What I experienced as a child affected me.  I could see the abstraction of math while young, and it amazed me.  But now look through my eyes as I find out that I am unusual.  There is a normal track for math, and a normal way to teach it.  As I tried to tutor my classmates, I realized math was not intuitive for almost everyone else as it was for me.

I became a good math tutor.  Parents would hire me , and ask me what my rates were.  I don’t know how I thought of this, but I said, “Five dollars per sitting.  A sitting could be five minutes or two hours. If I lose their attention span the sitting ends.”  That motivated the parents to motivate the kid.  After one short sitting, future sittings got longer.

As an adult, I married a math teacher.  She admits that I am the better with math, and that I often come up with creative ways to teach concepts that she could not.  She is still quite good with teaching math such that all of our biological and adopted children made progress in their percentile scores in math and other topics as they grew.

I think that I know math, and how to teach math.  I have done it while young, and older with my older children.  I have never taken a course in education, and thus my views of pedagogy have never been sabotaged by what is taught in most colleges regarding teaching children.

Some may think this assessment too harsh, but remember, we had the “New Math” in the Sixties.  It was a disaster.  For me, a math prodigy, thinking about math through the lens of set theory, it was challenging and interesting.  To most students, it was deadening.

So now we have the evil “Common Core Math Standards.” [CCMS] When I was a kid we joked about Communist plots to destroy America.  Well, I think the Communists are pretty weak in general — they don’t understand the nature of man.  But here we are trying to make kids try to make adult judgments regarding math.  That’s just plain stupid, because it doesn’t get the way children develop.

The Holy Grail of Critical Thinking

I don’t think critical thinking can be taught.  If you are smart enough, you will think critically. If not, no.

I say this as one where my wife and I have homeschooled our eight children for 18 years, and as my children get older they disagree with us to varying degrees.  We taught them well.  Sadly, some disagree with our premises.  But, they are all smart and the seven that have gone through standardized testing have all shown significant progress, moving 25% or more in the percentile rankings from elementary school to high school.  My wife teaches very well, and I support her. Please also note that five of the children were adopted, and the same effects happened with them.

But the CCMS flips things on its head.  Children need to learn facts.  They can absorb facts because they are easy for the young  to absorb.  Drill on math facts is a very good thing because it eliminates a hurdle to learning more in math.  Once you know the basics, the mind is capable of absorbing more abstract reasoning.

It is the opposite of what the experts say.  Math should focus on the concrete with young children, and as they get wiser, on to things that are more abstract.  They should not begin with abstraction, and try to move to the concrete.

Think about it for a moment: would you rather hire a guy who understood the basics of your business, or hire a guy who had a theory about your business, but did not understand the basics?  You would hire the former if you were smart.

Understanding the basics is important, and sadly, we have gotten away from it in the last 60+ years in math.  We did much better in the past, and we paid teachers less in real terms back then.  The colleges that teach teachers should be dismantled, and teacher accreditation should be eliminated, because there is no clear value created by accreditation.

We need education to be more like home schools, where teachers train students for many years in elementary grades K-8, where tutoring plays a large role. Understanding the student, and consistent mentoring makes a far brighter student.  Eliminating credentialing would being in brighter, more motivated teachers that ignore the idiocy to the teaching colleges.

Now maybe there is a home and private-schooling cabal, pushing CCMS in an  effort to destroy the public schools, or at minimum, assure that those who go to public schools will be peons to those who don’t go there.  I really doubt that, because those who don’t send their kids to public schools are more upright than those that don’t.  They are putting forth extra effort for their kids, versus people who don’t care.

Practical Advice

I live in Howard County, one of the richest counties in America.  Unique to Maryland, the counties are the school districts.  We homeschool in a county that is one of the best around.  We are evangelical Christians, but most homeschoolers here are secular. Why?

The school districts adopt a variety of dumb ideas like CCMS that hinder reading and math.  If I could vote to eliminate the public schools I would do so not out of ideology, but raw incompetence.

Most parents that we interact with are either hiring a tutor in math, or doing it themselves.  If that is the case, why not homeschool? It’s a lot easier than it looks, and it doesn’t take a lot of effort to outperform the public schools.

The main reason is the two-earner household.  Most could tighten their belts, and have one parent stay at home to teach the kids.  The second trouble is child control/discipline — my comment is if you are firm with them as parents were prior to 1950, you should have no problem, but consult your local statutes to figure what you have to hide from.

With CCMS many families will have to do one of the following:

  1. Tutor their children in math
  2. Pay someone to tutor their children in math
  3. Homeschool
  4. Start a war against those that set the public educational standards


Children are not capable of absorbing abstraction.  Every real parent knows that.  Fight the educated idiots who are trying to ruin math education with their misbegotten theories that do not understand math or kids.


  • tspare says:

    Good post once again. I wish more school boards can read this. My kids is at one of the top elementary school in Ontario, Canada and their math education is horrific. The curriculum seem to believe that if you show different instances of a math construct, the kids will figure out the rule. For example, doing enough 2+2+2 or 3+3+3+3, you well see the pattern and you will figure out the rules for doing multiplication. Its only after my parents helped with math drills such as memorizing multiplication table; Doing repeated long series of addition, subtraction, multiplication and division that my kids got over their math hurdle. Now they find math class very easy. Maybe my kids are not the brightest but this do not appear to work for majority of the kids in their class. I think the thing is that if you are not good at the basic operations, your mind is not free to learn the higher level constructs and is always spending lots of energy getting stuck the basic steps. The key thing that saved us was that we believed if a grade 1-3 student says that he/she is afraid of math and are not good at it, its not the kids problem but the teacher or the course material (In talking to the teacher they said they had to teach what is given by the school board so its really the course material). So we actively went about fixing it without blaming the kids.

  • cold.as.ice says:

    We also homeschooled. In general kids enjoy learning. Ours did.

    A great view of professional education comes from a teacher. http://econfaculty.gmu.edu/wew/articles/12/SchoolsOfEducation

    We were able to select our own text books. To be free of the PC junk that can be over 50% of a math text book. See http://www.siam.org/news/news.php?id=240 and then ask your public library for a copy of the book.

  • Greg says:

    Public schools exist for public unions first, student needs are a distant second.

    And the majority of parents measure school “success” based on how much money the government throws at the education budget (even though more than half of said budget goes to administrative overhead).

    Teachers quickly learn that the way to make real money is to become a bureaucrat — actually teaching students is for suckers / idealists, and pays a LOT less.

    U.S. Education isn’t about education anymore. Its about feeding the public employee unions.

    And if future voters (aka students) are terrible at math? Well that makes stealing from them even easier

  • etaoinshrdlu2 says:

    David, I appreciate seing your thoughts on education. Congratulations on writing about life in addition to the financial topics. I’m not sure who’ll see them, but you might nudge some people into checking out homeschooling, whether Christian or secular. As a former homeschooling dad I strongly agree with everything you said. When a public school teacher is good and can find a minute for a gifted student it does help. I had one or two like that I think. But how often either the teacher is not good or just has no time or energy to add something. At home, adding special things can truly be the norm. But the new homeschoolers or interested “candidates” need info and support. Sadly, I just do not see much hope for reforming the public schools anywhere. The problems are too big and the obstacles are too entrenched and conflicted.

  • Bill Edgar says:

    I taught high school mathematics for 30 years in a large suburban Philadelphia school district. Eighty per cent of our students went on to college. I taught Algebra I to the average students, BC Calculus to future Ivy students, and courses with names like “Refresher Math” to students who needed courses with names like “Refresher Math.” My own five children went to a different inner ring suburban school district with over fifty languages and no racial majority, where fewer than half went on to a four year college. Here are a few observations on math education, in no particular order, based on my experience with both school systems.
    1. Few actual math teachers embraced the New Math fifty years ago or embrace today’s “critical thinking” or “discovery” fads. In each district there is something of a war between the teachers and their supervisors. Most high school math teachers like math, are reasonably competent at it, after a few years teaching know what their students can and can’t handle, and are not natural innovators. The new ideas come from math education professors at the university level.
    2. By grade eleven, the smart students have very little idea just how difficult it is for average and below students to grasp math concepts. They don’t sit with them in class, they certainly don’t talk with them about math. (At the same time, the average kids in the same comprehensive high school have no idea just how smart and how hard working the smart kids are.) The only ones who have a real grasp of the distance between the average and the superior and the below average students are the teachers. (And, thanks to how the normal distribution functions, the distance in a BC Advanced Placement Calculus class from top to bottom is greater than the distance between the average BC student and the average high school student.)
    3. College professors come from the smart group of students. They didn’t know the math ability of the average, let alone the below average, students when they were in high school. Most math education professors taught high school for a year or two a long time ago, if they taught that much. They therefore simply don’t know what average kids are like, how they learn, or why they learn, and they don’t know that they don’t know. Most of their “knowledge” comes from narrowly focused, usually not very well designed, research that is much less rich and complex than the practical knowledge of actual teachers about students and how they learn. It’s easy for such people to dream up clever curricula, such as the New Math deriving everything from set theory, or the present fads that emphasize “discovery” and “critical thinking” and not know that they are writing for the top ten or twenty percent at best.
    4. When actual high school math teachers protest that this stuff won’t work, they get dismissed on the grounds that they are “afraid of change,” don’t want to work, and are basically inferior material anyway. (If they weren’t so inferior they would not be doing the low status work of teaching in a public high school.)
    5. There are other, even worse problems with math education: most elementary school teachers fear math and are not very good at it. So they tend to skimp on math time. And middle school is the worst part of American education, since it was designed by the psychologists in the 1970s to emphasize the emotional and social over the academic, so the math teachers there don’t dare demand a whole lot from their students.
    6. Unfortunately, the solutions to the math teaching troubles in our country are not going to be found in home schooling, because most home schooling parents are not mathematically strong themselves. In general, when I got a student in one of my high school classes who had been home-schooled, or who had been in a private school, he was stronger than his new peers in verbal skills and weaker or much weaker mathematically. David and his wife are obvious outliers in their ability to teach their own children mathematics.

  • cig says:

    If teaching was so easy that everybody could successfully home school, why is it that people who do it for a living don’t manage? Even with somewhat broken institutions, it sounds far fetched. Your spouse may be a talented teacher, but is everybody’s?

    Division of labour is generally a great thing, people good at teaching teaching more than their own kids and people good at brain surgery cutting more than they own brains is a good idea, rather than everybody doing DIY-everything and being, on average, more mediocre at it than those who specialise, however imperfect they are.

    Also a problem with homeschooling is that it deprives the kids from social interaction (warts and all) that’s preparation for adult life, regardless of the teaching side of things (that’s perhaps less of a problem in large families).

    • Homeschooled students students on average get more socialization and better socialization than those in public schools. They tend to be less peer conscious, and better at interacting with people of all ages, not just their peers.

  • cold.as.ice says:

    10 things my kids missed by being homeschooled.

    Before I start, let me tell you about
    something I missed growing up. Not
    far from where I lived was an ordinary railroad crossing. Occasionally
    a train would hold me up, but nothing
    worth mentioning. One day a pick-up
    truck had a personal and unpleasant
    encounter with the train. Growing up
    I [fortunately] missed a personal encounter with a train. The missing out was a good thing and by intent. In
    like fashion our kids have missed out.

    Let’s not even explore what they
    missed academically such as Science
    where Darwin is a god not to be questioned or History with an agenda of either showing how bad was the
    Christian foundation of our Nation or
    else that it wasn’t a Christian foundation after all. With homeschoolers outscoring their public school peers by 81 points on the SAT, it’s hard to see what we missed there. So, let’s look at socialization and see what our kids missed out on.

    (1) Does your child know the etiquette of a drug buy? Do you count
    the money first or check the quality of
    the product? How do you keep from
    getting shot if the deal goes bad?
    Some social skills are good to not

    (2) How to rob a 7-11. Who watches
    for the police? How do you know
    which ones are easy targets or not?

    (3) All the vices they would have
    been encouraged to try, so they would
    know if they wanted to participate or
    not. Sounds crazy, but in a valueless
    environment, how do you know right
    from wrong, especially if it feels
    good?“A University of Pennsylvania
    report says more than 300,000
    American kids are involved in the sex
    trade. This occurs in middle and high
    school, apparently with kids trading
    lunch money for sex.”

    (4) How to roll a joint. Did your child
    pick up the key social skill of rolling and smoking marijuana cigarettes? How about snorting cocaine or shooting crack?

    (5) For sure our kids missed out on
    Planned Parenthood based “education”
    on why abortion is not only a choice but also a right. Well if it isn’t a right, they can be taught that it should be a right.
    Planned Parenthood clinics are responsible for the abortion of over 3 million children over the past 25 years, grossing $69 million in abortion income for 2000 alone. Planned Parenthood is also the source of much of the Public School system’s sex education.

    (6) Professional educators backed by the National Education Association (NEA). To see what this can bring to a child’s education, let’s look at NEA’s own words. “I think it’s a pretty clear signal that the organization recognizes there are some pretty serious needs for gay and
    lesbian children in school — and
    employees.” –Ms. Penny Kotterman,
    chair of the NEA’s Task Force on Sexual Orientation, announcing that the National Education Association (NEA) approved a measure to “encourage” schools to incorporate “HomoPromo” materials in classroom discussions of homosexuality.
    NEA’s focus extends beyond alter-
    nate lifestyles, for example to integrity. In Montgomery County, MD, some of the NEA’s finest “professional” educators took it upon themselves to improve the performance of their students by taking questions from their school system’s Comprehensive Test of Basic Skills and giving them to their students as “class assignments.” When some students thought questions on the annual basic
    skills test looked too familiar, they reported it to administrators. It cost the state $600,000 to reprint the test using new questions. Well at least some of the students had more integrity than their teachers.

    (7) Protection from radical Christian
    concepts. It would be OK for them to
    learn New Age, Islam, Atheism – any-
    thing but Christianity. Your kids missed out on the wisdom of the school system protecting them from dangerous views. Not only would your kids be “protected” from prayer and the Bible, but also “dangerous” groups such as the Gideons.

    (8) Active recruitment by “social”
    groups such as the Blood and the Crypt. Don’t know what these are? They are youth [criminal] gangs. Think of them as Mafia apprenticeship programs. What kind of apprenticeship did your kids
    have? Do your kids know how to do a
    drive-by shooting?

    (9) The chance to get beat up and have
    someone steal their lunch money. The
    closest thing you’ll find in homeschooling is HRS harassing a homeschooling family for “neglecting” their children’s education.

    (10) Zero tolerance for weapons. For
    example: An 11-year-old at Oldsmar
    Elementary School (Florida) was
    arrested, handcuffed and hauled away
    from his school after he was caught
    drawing a picture of a soldier with a gun. “There were some drawings that were confiscated by the teacher,” said
    principal David Schmitt. “The children
    were in no danger at all. It involved no
    real weapons.”

    A few other examples of this idiocy:
    Four kindergarten students playing cops-and-robbers were given three-day suspensions; an 8-year-old boy was suspended for three days after pointing a chicken finger at a teacher and saying, “Pow, pow, pow”; two second-graders playing cops-and-robbers were charged
    with making terrorist threats.