Tuesday, January 25, 2011

Reforming Math, 4

(Continuing from Reforming Math 1, Reforming Math 2, and Reforming Math 3.)

My latest e-mail to the Head of School at Natural Friends:

Here's a list of basic math concepts, with sample questions that all graduating NF kids should be able to do, WITHOUT CALCULATORS. To the best of my knowledge, none of the following is currently being taught at NF. If you have kids who can do all this, ask whether they study math outside of NF, for instance with Kumon, tutoring, or help from their parents.

I.) Standard algorithms.

     1.) standard long division algorithm
         try this.) 2226 ÷ 3.

     2.) standard multiplication algorithm
         try this.) 23 x 3560.

     3.) standard subtraction algorithm.
         try this.) 3246 - 1634

     4.) standard algorithm for taking an average
         try this.) what's the average of {10, 100, 36, 84}?

II.) Fractions.

     1.) fractions are a form of division.
         try this.) how can you write 7 ÷ 4 as a fraction?

     2.) general concepts: x/x = 1, x/1 = x, etc.

         try this.) How would you write 4 as a fraction?

         try this.) Complete this sentence: "multiplying by 1/2 is the same as dividing by _."

         try this.) Complete this sentence: "any number divided by itself is _."

         try this.) Complete this sentence: "any number divided by 1 is _."

         try this.) How would you write "one-half of 37" in mathematical notation?

     3.) multiplicative inverse (reciprocal) of a fraction.
         try this.) 5/4 x _ = 1

     4.) how to convert a fraction to a decimal
         try this.) convert 4/7 to a decimal.

     5.) how to divide one fraction by another
         try this.) 4/7 ÷ 2/3

     6.) how to cross-cancel when multiplying fractions
         try this.) 4/5 x 2/3 x 5/4
         try this.) What is one-half of two-thirds of three-fourths of four-fifths of 5?

     7.) multiplying a whole number by a fraction.
         try this.) 4 x 1/2 = _

III.) Decimals.

     6.) how to multiply two decimals
         try this.) .25 x .75

     7.) how to divide two decimals
         try this.) .75 ÷ .25

IV.) Shortcuts with 10.

     8.) how to multiply by multiples of 10
         try this.) 234567 x 100.

     9.) how to divide by multiples of 10
         try this.) 234567 ÷ 100.

Readers -- what math are your kids missing?


  1. From PsychMom:

    Should be able to do all that by the end of Grade 6?
    I know for a fact that my 4th Grader can't do any of that.

  2. Suburban Chicken FarmerJanuary 25, 2011 at 11:19 AM

    Here in southern California, my fifth grader is doing all of that.

  3. MY older kid is in seventh grade, and is definitely not as proficient with the standard algorithms for division and multiplication as she should be.

    In our curriculum they really stress lattice multiplication, and some cumbersome way of division, I can't remember what it is called.

  4. PsychMom, a lot of this stuff would happen in 5th or 6th grade. The kids should be solid on it to do well in 7th grade pre-algebra.

    For 4th grade, I would mostly hope for proficiency with +, -, x, ÷ whole numbers.

    KD, one of the many ridiculous facets of constructivist math is its infatuation with the lattice method. There's a reason the standard method was invented -- it's quick and efficient.

    Suburban Chicken Farmer, do you know what math curriculum your school uses?

  5. PsychMom adds...

    Where's all the smoke coming from then when I ask her to add a two two-digit numbers?

    Proficient is not a word I would use to describe my daughter's mental math skills. Though I do hear squeals about loving math.

    Geez...I don't know what's going on...

  6. PsychMom -- I wouldn't expect a 4th grader to be able to add 2 2-digit numbers in her head, necessarily. That's why we've got paper and pencil.

    Does the school expect her to do "mental math" only?

    What's the curriculum?

    If she loves math, I guess that's a good sign.

  7. Suburban Chicken FarmerJanuary 25, 2011 at 2:11 PM

    Math Connects Macmillan/McGraw-Hill.
    We do run into strategies which my husband and I weren't explicitly taught in grade school or don't particularly care for... like some of line plots or front-end v back end estimating without a decent rationale. My math prof friend says she has a heck of a time with her ed. college students being unable to functionally estimate anything. Anything. Yikes, I hope we all know, estimating is a very important, maybe the most important ability.
    Just this morning my son was dividing fractions and zipped right through by cross canceling/cross reducing.
    I like algorithms. I believe they should often be proved and justified so students know why it is what it is.

  8. ***
    I like algorithms. I believe they should often be proved and justified so students know why it is what it is.

    I agree absolutely.

    You're using an old-fashioned, not "constructivist" math program. I'm sure it's not perfect, but it probably does a reasonable job at basic skills.

  9. I live outside of the US and at school we are just told to use our calculators when dividing by decimals! However, I have just realised that dividing decimals uses more or less the same principles as dividing whole numbers, so I'm curious as to why they avoided teaching us.

  10. KD -- The lattice thing drives me insane. I wish someone would explain the rationale for teaching multiplication that way.

    Someone told me that at the junior high level Everyday Math is a thing of the past and standard algorithms are back. Can you confirm? My guess is that those are the years when the district actually tries to get the kids up to speed with these basics. I'm not sure I'd think that's such a bad idea, as long as the elementary program were run in a way that didn't breed math anxiety or bad habits (and I'm not sure it is). My math motto in elementary school would be "First do no harm."

  11. PsychMom adds:

    I don't know what the curriculum is, FedUpMom. I'm afraid to ask for fear of frightening someone by asking a question.

    And with the two-two digit numbers....she can't do it on paper reliably either...the idea of "carrying one over" is verboten..

  12. Chris -- Everyday Math only goes through the 6th grade, so by definition it's over by junior high.

    If you expect junior high math to line up in any reasonable way with elementary math, forget it. The junior high curriculum will probably assume the kids know the standard algorithms.

    I've said it before, and I'll say it again -- just get the Singapore Math workbooks and start working through them with your kids. If your kids are bright, which I'm sure they are, it doesn't even have to take much time.

  13. Our school uses a watered-down constructivist curriculum in the elementary years (up to grade 6), which means there are a lot of frustrating, pointless homework questions, but in the end the standard algorithms are taught. (There is no mention of the lattice method in my daughters' textbook series, for instance, just a complicated explanation of why the standard algorithm works.)

    That said, my girls--currently in grade 6--don't know all the stuff on your list. They could answer most of the questions, but were especially weak on the fractions part. I think that might be because last year, their disorganized teacher got behind in math and just skipped the fractions unit. It never seemed to occur to her that it was perhaps a more important unit than . . . oh, I don't know, advanced probability (5th grade version). So the girls haven't had any lessons on fractions since grade 4, and even then I suspect the teacher may have skipped some of the material she was supposed to cover. (One of my beefs with French Immersion, is it makes everything more difficult and time-consuming to teach. I've always thought, math should be taught in English, at least.)

    Another thing is, with respect to some of the questions on your list, for instance how to divide two decimals, my daughters can do it only because my husband and I have taught them. What often happens is, the girls are assigned a homework question which asks them to do something--e.g., divide two decimals--that they haven't yet been taught. The textbook expects them to use a calculator, but the teacher often tells them not to. So we just teach them how to do it. This happens over and over again, which is one of the many reasons why I loathe their textbook!

    Anyway, thanks FedUpMom, for continuing to raise questions about "reformist" math. I look forward to hearing how the Head of School at "Natural Friends" responds to your latest email.

  14. Chris, there a several math classes to choose from at the junior high level, so I'm not sure how the algorithm issue is approached at each level.

    My daughter is in taking Pre-Algebra. I think the assumption is that you are proficient in some method of computation(but they don't spend very much time on it)....I've seen my daughter occasionally use the lattice multiplication method, but I think it is her choice to use what she wants. The class also assumes that you are comfortable with using fractions.

    It is definitely different from Everyday Math. I'm not sure if this makes things any clearer.

  15. Looking forward to seeing how your Head of School responds! From what I can tell, the Quaker schools are particularly enamored with constructivist / Reform Math.

  16. Regarding Quaker schools, there's one such school that's fairly well known: Sidwell Friends in Washington DC--the one where President Obama's daughters attend. Last I heard, they were using Investigations, supplemented by Everyday Math, a more horrendous combination I can't even begin to imagine.

    See: http://www.ednews.org/articles/obama-sidwell-friends-and-the-achievement-gap.html

  17. Right, Quaker schools want to like constructivist math because it rings a lot of "progressive" bells. It sounds right to left-wing types.

    However, I can say that an academically serious Quaker school will find ways around it. Friends Omphalos, for instance, officially uses Everyday Math, but they must supplement like crazy. They teach standard algorithms, and the kids are prepared for an intense junior-high math experience.

  18. My goodness, there must be a lot of Quaker schools where you live! Are they actually "Quaker" in any meaningful way? (I don't think we have Quaker schools at all here in Ontario.)

    But back to the issue of math: FedupMom, do you know which level(s) of the Singapore math workbooks cover the type of fraction questions you list above. I'm thinking of buying a workbook or two, to help my daughters catch up in this area.

  19. Great question, northTO. Fractions are vital. Singapore Math 5A has a lot about them.

    Singapore Math

  20. Thanks for all the comments -- I've got ideas for a bunch of posts!

  21. I teach "developmental math" in a community college -- my entering students can't do most of these. (And, yes, many of them are right out of high school.)