Education

Teaching 5- and 6-Year-Olds Serious Basic Math

The second edition of "Breaking Numbers Into Parts": very highly recommended.

|The Volokh Conspiracy |

For several years, my son went to the UCLA Math Circle run by Dr. Olga Radko and Dr. Oleg Gleizer (both mathematics PhDs), whom I've gotten to know well. It's a superb program, which teaches children ages 5 to 17 serious mathematical concepts and mathematical problem-solving, not just drills or algorithms.

Gleizer and Radko now have a second, two-volume, edition of a book based on their work with the math circle, "Breaking Numbers into Parts," aimed at parents and teachers who want to teach 5- and 6-year-olds (here's volume 2). The book isn't written to be read by children, who might not even be able to read well (or to read at all) at that age. But adults can use it very effectively to guide children through concepts such as commutativity, the number line, the properties of odd and even numbers, mathematical operations as a generalizable concept, reflections and symmetries, and more—and, more importantly, to teach children about proofs and about the use of math and logic to solve problems.

If you have young children whom you want to teach about math, definitely get these books.

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  1. “Breaking Numbers into Parts” link is broken.

  2. Sorry, Professor, the link is broken.

    Thanks for pushing this, though – I still have the first edition in an Amazon wish list from when you mentioned it three years ago. It was right before my first was born, so they’re still a bit young.

  3. 2 + 2 = 4

    it ain’t that hard.

    1. 2 + 2 = 4 may not be hard, but why is it true that 1 + 2 = 2 + 1, whereas 1 – 2 does not equal 2 – 1 ? Since EV specifically mentioned commutativity, that’s what popped into my mind.

      1. Silly math geek joke: what’s purple and commutes?

        An Abelian grape.

      2. That is a pretty tough primary school question. I’ll try the first part: If I give you a bowl with 1 grape and a bowl with 2 grapes, it doesn’t matter which bowl I give you first – you’ll end up with the same number of grapes to eat. Each grape counts as one grape, no matter when you got it.

    2. And “2 + 2 = 4″ is only true if you stipulate the base number used;
      And what does Godel’s Theorem say about the force of the term, ” = “?

      1. there are 10 kinds of people; those who know that we use base 10 unless specified otherwise, and those who do not.

      2. There are only three base systems it doesn’t work for: 2, 3, and 4. If you have 4 in the equation you obviously aren’t working in any of those.

        All Goedel says about that is that it isn’t absolutely certain if you base certainty on provability from axioms.

    3. Only for sufficiently small values of 2.

      1. For sufficiently small values of two, 2 + 2 = 3. 🙂

      2. 2 only comes in one size

        1. Yes, that’s what makes “Only for sufficiently small values of 2” a little joke. (Or am I missing your joke?)

  4. I’ve seen Vol. 1 and it’s actually pretty good. The idea — “adults can use it very effectively to guide children through concepts” — is also good… and the book is helping to revive the somewhat lost art of productive parent/child interaction. Others are also trying to rekindle the behavior (see, inter alia, https://tinyurl.com/y4reznrw).

  5. And to think I’ve been suggesting schools should stop trying to push math so early, and delay it until 4th or 5th grade. I’ll need to look.

  6. When I saw “Serious Basic Math” in the title I thought the post would be about teaching axiomatic set theory to first graders, fortunately for them it is not quite that serious or basic.

    1. Based on my experience with my own kids, and hearing from other parents, I do think early math curricula are too heavy on the theory, and too light on the arithmetic. basically, they expect kids to be able to extrapolate from abstract concepts, and solve word (sorry “story”) problems without giving them the basic computational tools to do so.

      Memorization is completely taboo, despite the fact memorization at that age is much easier than abstract thought.

      So they refuse to teach what I consider the “mechanical” methods of computation. In my ideal curriculum, you would memorize the single digit times tables, as well as all the single digit addition permutations (some subtraction too, if you are feeling frisky). Then you learn to line up your numbers, and you can solve any addition, subtraction or multiplication equation.

      Instead they teach a combination of tricks and counting on your fingers (and call them “algorithms”) as the way to solve problems. To me, that is completely ass-backwards. The tricks are useful for doing arithmetic in your head, but only after you know the “mechanical” method, because the mechanical method works every time, and the tricks do not.

      1. Saxon math continues ues tonhave the best results, see Oklahoma study.

      2. My mother taught me to memorize the addition and multiplication tables up to 9+9 and 9×9, before I started kindergarten. It took the schools until 7th grade to catch up, and that was only because I was moved up a year.

        1. Kintucky?

          In West MI, we were through the 12s times table in early third grade (89-90) (well, I was, but everyone was expected to be up to the 9s times table at the time).

  7. Thanks for this. Sent it over to AOC’s office. Figured she could use to learn some basics about math. Such as add $10 billion in tax revenue, refund $3 billion, and you get a net gain of $7 billion. And zero tax revenue means zero revenue to refund which means zero net gain.

    1. Or this one: raise taxes by one trillion dollars. Spend 170 billion trillion dollars on your New Green Deal, while simultaneously setting civilization back two centuries. Things get worse.

  8. I haven’t read the book, so maybe this is covered. But I remember a first day speech given by an excellent math teacher in 10th grade geometry. She asked the class to give a definition for mathematics. She said “Biology is the science of living things, physics is the science of mass, energy, and motion, Chemistry is the science of the elements and how they combine and react with other elements. So, what is mathematics?” Many guessed that math was the science of numbers.

    She said “No. English Literature is not just the study of letters of the alphabet. A lot of math uses numbers, but there are whole branches of math, like geometry, where numbers are often unnecessary. We use numbers as an easy way to introduce you to patterns – 2 + 2 = 4, and it will ALWAYS equal 4. The important thing is to recognize the patterns. Math is basic to every other science because it is the science of PATTERNS. Recognizing patterns, understanding patterns, manipulating patterns and drawing logical conclusions from patterns. Math is logic, the ability to think rigorously, and you’ll use it every day of your life, to tell time, to make decisions on what to buy, what job to take, how to budget your time, how best to spend or invest your money. Math can be hard, but nothing worthwhile is easy. So, let’s get to work learning some math.”

    Once I understood what math was really about, it became much easier, and worth the effort.

    1. Mathematics is what the universe was built on.

  9. It was Ph.Ds who created the ‘New Math’ of the 1960s, which was a disaster. They asked Ph.D mathematicians what students should learn, and they suggested a curriculum for future Ph.D mathematicians. So, we got set theory, dropped in from the sky and divorced from the rest of the curriculum. As one professional mathematician said long after – even he didn’t have any use for set theory.

    Presumably, this program is better than that one was. In any case, I wouldn’t boast about it being developed by Ph.Ds. As we said in grad school – Piled High and Deeper.

    1. It was MDs (medical doctors) who participated in and helped develop the eugenics platform of the early Twentieth Century. This is why I no longer give any deference to those who hold a medical degree. When I need consultation about a health issue or surgery I just consult the people via the internet. We should probably just do away with “experts” all together and just start electing doctors and teachers.

      I’m with you man … down with experts and their fancy initials!!

      1. Schr?dinger, is that you?

      2. PhD mathematicians dont know to teach. They know how to do advanced math.

    2. Tom Lehrer had the final word on the New Math.

      https://www.youtube.com/watch?v=UIKGV2cTgqA

    3. the “new math” wasn’t taught in schools until the mid-70’s though, fortunately for me

      1. Stupid “new math.”

        What did all those 60s and 70s students get out of it anyway.

        I mean besides:

        Developing the Internet
        Creating the basis for all the personal computing power we have at hand today
        Designing satellite communications
        Inventing today’s medical technologies
        Gaining a better understanding of the Universe

        1. Seeing as how ARPANET was created in 1969 and was the difficult part of “developing” the internet, I rather doubt any 60s or 70s students did much to help, as the year the first student who had learned only New Math came of majority would be just a bit before 1970.

          As for personal computing, the biggest contributor to that was the miniaturization and power reduction of integrated circuits, especially CMOS which was invented in 1963. The path from there has been remarkably straightforward: pretty much everything is CMOS except in non-consumer applications, just really tiny. Advances in photoresist in the 70s and 80s were very good, but they came from chemistry and had very little to do with math. The basis, certainly, was done by 1963, when we already had MSI and CMOS.

          By coincidence, satellites for communication were being created in 1963. It should have been fairly obvious that New Math had little to do with satellite communications, considering that New Math was given prominence due to hand-wringing over Sputnik.

          “Inventing today’s medical technologies” is incredibly vague, considering that many or even most of them didn’t require math to invent, and in a literal sense “today’s” medical technologies were probably invented by people in their 40s and 50s; the youngest person to go all the way through New Math would be about 62 now.

          On your last point, even a modicum of better understanding would make your statement true. It’s toothless.

          1. I do find it amusing that if you hadn’t qualified your second point with “the basis for” it would actually be somewhat correct! Mass-market PCs were kicked off in 1977 and Wozniak was subjected to New Math. Bill Gates was the biggest player in making PCs cheap and widely available and he was also schooled in New Math.

            1. Your anecdotes are near-genius intellects – at least. Somehow, I doubt either benefited from any reforms during their times in school. They flew through topics and were years ahead of their classmates by the fifth grade.

              1. Absolutely. The folks who designed PCs were a varied bunch but also all very intelligent. Wozniak and Jobs made the Apple II and they very much fit the genius-in-the-garage stereotype but the TRS-80 was made by a more standard corporate design team; Peddler, the guy behind the PET, was somewhere in the middle. All of these were released in 1977.

                I doubt that New Math helped much with anything here; they weren’t really even using much math. Everything was just about how chips played together.

        2. Seeing as how ARPANET was created in 1969 and was the difficult part of “developing” the internet, I rather doubt any 60s or 70s students did much to help, as the year the first student who had learned only New Math came of majority would be just a bit before 1970.

          As for personal computing, the biggest contributor to that was the miniaturization and power reduction of integrated circuits, especially CMOS which was invented in 1963. The path from there has been remarkably straightforward: pretty much everything is CMOS except in non-consumer applications, just really tiny. Advances in photoresist in the 70s and 80s were very good, but they came from chemistry and had very little to do with math. The basis, certainly, was done by 1963, when we already had MSI and CMOS.

          By coincidence, satellites for communication were being created in 1963. It should have been fairly obvious that New Math had little to do with satellite communications, considering that New Math was given prominence due to hand-wringing over Sputnik.

          “Inventing today’s medical technologies” is incredibly vague, considering that many or even most of them didn’t require math to invent, and in a literal sense “today’s” medical technologies were probably invented by people in their 40s and 50s; the youngest person to go all the way through New Math would be about 62 now.

          On your last point, even a modicum of better understanding would make your statement true. It’s toothless.

          1. Seeing as how ARPANET was created in 1969 and was the difficult part of “developing” the internet

            The italicized statement is conclusory yet lacks supporting argument or data.

            1. Then it’s pretty obvious you don’t know much about the history. ARPANET implemented TCP/IP, which was developed as part of DARPA. Everything in the internet is based on TCP/IP or wrappers for TCP/IP. The difference between now and then is that the hardware is better and it’s not just military access.

      2. It got thrown at me in the 7th grade, in 1962. Some had trouble with it. The weirdest part to me was understanding geometries in terms of lines, rays, and half-lines. I don’t think I’ve stumbled over a half-line in over 50 years now. But most of the rest of the concepts were useful, even if distracting at the time. Perhaps I was immune, because I think I had already learned my multiplication table thru 25, and the powers of 2 up to 65,536

        1. Ok I’ll bite, what the heck is a half line? A line that terminates at a point on one end? Isn’t that a ray?

      3. We saw it in my Boston 8th grade class in 1967. It was just a taste, but yes, we did get a lesson in sets. And then went back to dealing with decimals and fractions and whatever else came before algebra.

  10. Sally Brown (of Peanuts fame) on The New Math

    http://assets.amuniversal.com/…..1dd8b71c47

  11. It seems to me that there’s a much more important problem than young children lacking basic arithmetic, and that is teenagers being allowed to graduate high school without being sufficiently numerate to calculate the interest on a home or car loan. If you can’t do that, you’re not going to see through the kind of deceptive spending bills that states frequently put to voters.

  12. I don’t think that teaching 5- and 6-year olds is a good idea. Imagine that your child comes and said: “Can you Do My Assignment For Me” – Yes, of course! and you view site for help. Seriously, difficult math is not a good idea for kids.

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