33 thoughts on “A Problem With Equality”

  1. Every child can be a math wizard. Anything is possible if you lower your standards far enough.

    American kids may not be up to snuff with unimportant things like the equal sign but I’ll bet they’re far better than those foreign kids at putting a condom on a banana and in self esteem. There are only so many hours in a school day and you have to have your priorities straight. [/sarc]

  3. “Dewey’s project is complete.”

    But this article is about the equals sign. Didn’t Dewey invent the decimal point?

  4. I’d be confused too if I were growing up now.

    Post modern postulates:

    Man = Woman

    Animal = Human

    Patriotism = Taxation

    These are arguably more important for the future citizen to understand than the mathematics of dead white males.

  5. “But, but…wasn’t it Einstein that said that it was all relative? I mean, you know, in, like, what dimension are you equalizing? I read on the internet that in the fifth physical dimension 1 + 1 = 3 because they use like this totally radical base 8 numbering system which is supposed to be way more harmonious with the cosmic forces and dimensions.”

    Test Subject Ken, a product of the 1980s public education system

    Seriously, though, Ken was a product of innovations like Honors, TAG, and AP classes, although many of the teachers had a hard time keeping up with us TAG kids. Since the military didn’t want me, I ended up putting my mad math skills to use in the financial sector, and it’s why I get things like structured debt products and interest rate swaps dumped on me – because no one else, younger or older, wants to deal with them.

    The problem with equality is clearly not particularly attributable to the Millenials. They’re not the ones in D.C. ‘equalizing’ the expenditures with the available revenues. That rests on the shoulders of different generations.

  7. This could possibly stem from over-exposure to “computer science” earlier and earlier in school where students are taught that D = D+1 is logical and acceptable. Perhaps we need a national move to change the syntax to D <= D+1 so as to avoid the cross domain confusion.

  11. MfK,

    [[[[Didn’t Dewey invent the decimal point?]]]

    It was Melvil Dewey invented the Dewey Decimal System, not John Dewey….

  12. I’m reminded of an experiment done by the computer science department of a British university.

    Somehow they managed to convince the facility to require all first year students to take a programming class (I’m guessing it was in the 80s).

    At the start of the semester they handed out a questionnaire. There was 10 questions, all of the form: What does A = B do? Of course the symbols were different for every question but the assignment statement was the same for all of them. The answers were multiple choice, and the same for each question:

    a) moves the value in B to A
    b) copies the value in B to A
    c) moves the value in A to B
    d) copies the value in A to B
    e) something else

    The symbols were somethings like: fruit, clothing, animals, as well as letters both Latin and Greek.

    Note that anyone with a basic understanding of the mathematical equal sign but no concept of programming would probably have some difficulty with this problem, wanting to say “false” for a lot of the answers, but seeing as that wasn’t offered as an answer no-one did.

    At the end of semester they gave out the same questionnaire and collected the results. Unsurprisingly, the students who did well in the class showed the most improvement. Most the students who did below average in the class also showed improvement.

    But among the population of students was a bunch who did terribly both at the start of semester and at the end of the semester.. these people were easy to predict: they were the ones who answered inconsistently. They answered b) for Apple = Orange but they answered c) for Car = Truck or something else for A = B.

    For these people the idea that symbols can be *meaningless* seems to be too hard of a concept to grasp and they’ll never be programmers, or likely, mathematicians.

  13. …they’ll never be programmers, or likely, mathematicians.

    I wouldn’t be too sure about that. I’m certain I ran into more than a few during my software development career.

  14. mpthompson, yeah, I wanted to say “good programmers” but I thought it was a bit of a misnomer. Perhaps “economically interesting” is better but then what do you call the mathematicians 🙂

  15. Wife and I support the Separation of School and State organization. Not sure the current system can be rehabilitated given the amount of money and power at stake.

  16. We homeschool our kids and use a math curriculum that’s strong on the equals sign thing. I’ve been really diggin’ how much it focuses on balancing equations (even at an elementary school level) instead of treating equals as “this is where the answer goes”. The curriculum is called “Making Math Meaningful”.

  17. Interesting, I was taught that 1 + 1 = 10.

    Yep, there’s 10 kinds of people in this world: those who understand binary and those who do not.

  18. …they’ll never be programmers, or likely, mathematicians.

    They’ll grow up to be journalists!

  19. “It was Melvil Dewey invented the Dewey Decimal System, not John Dewey….”

    It’s just no fun when you have to explain it. In fact, it’s a little scary *that* you have to explain it…

  20. As an aside, mathematical equivalence and orderings are two of the most powerful tools of the modern mathematician (and these carry over into other fields that use any serious mathematics, such as the hard sciences). If you can’t grok equality and the greater than/less than signs, then you aren’t going to have a chance with virtually all of modern mathematics.

  21. What does “a=b” do?

    I am a dinosaur who learned Pascal and still uses Pascal, and that operation compares the contents with “a” with the contents of “b.” If they are the same, the expression returns Boolean “true”, otherwise Boolean “false.”

    In C/C++/Java, they tell me, it copies the contents of “b” to overwrite the contents of “a”, and then it returns as a function return-value the contents of “b.”

    As to the distinction between “moves” and “copies”, you have kind of lost me there. In file systems, there is a distinction between “move” and “copy” in that if you do a “move”, the symbol “b” becomes invalid whereas in “program source code”, you never invalidate the symbol “b” as referencing a memory location unless it goes out of scope.

    What am I missing?

  22. What does “a=b” do?

    It actually does depend on context, which language, etc. Even if you accept that it evaluates to a boolean, you still have to define what constitutes equal. Equal could be either having the same content or having the same address or possibly something else (programmers being much too creative for most people in that respect.)

    It does make it difficult to communicate if you don’t first agree on the meaning of your symbols.

    Who doesn’t love redefining 5 to the value of 4 in Fortran and watching others suffer as a result… or my friend that redefined my keyboard before a test so I had no access to essential letters to finish my test. With friends like these…

  24. Back when I was a junior instructor at my alma mater my boss used to remind me: never underestimate the stupidity of your students.

  25. In programming, the equal sign can have different meanings depending on the language.

    In languages like C, C++ and Java, the equal sign is an assignment operator. It’s used to assign a value to a variable, such as a = b; meaning the variable a is assigned the value of variable b. If you’re doing a comparison in if() statement, you use two equal signs, such as if (a==b)

    Other languages like the Basic dialects use the single equal sign in different ways. It’s still an assignment operator such as a = b or a = a + 5 (a statement that makes no sense in algebra but is prefectly legal in programming languages.) For things like comparison in If statements, you still use the single equal sign, e.g. If a = b then…

  27. I’m not convinced that the education system is so screwed up. Yes, it has its weaknesses and inefficiencies, and the hierarchy of the values taught doesn’t match my own hierarchy of values, but they’ve made a lot of progress in education since the 70s. Specifically, I recently took a calculus refresher course in my doctoral program. After I got over the fact that my classmates weren’t even *born* when I took my various calculus classes in engineering school in the 70s, I was amazed at how well the instructor taught the course. I repeatedly sat in awe, saying, “is THAT what they were trying to teach me back then?” and “how did I miss the mathematical meaning of integration when so many teachers covered it back then?” Of course my brain was older this time around, and I was arguably more motivated, but I looked at my kid’s calculus III curriculum and was amazed at how far it’s come in 30 years.
    Of course, now they have things like instructors who speak english and actual textbooks, but this is icing on the cake of an evidently-improved curriculum.

    that being said, I’m sure there’s some sort very un-PC puns I could make involving the article’s emphasis on equalities and my new understanding of integration…

  28. heh, Lynne, my strongest memory from high school mathematics is getting 100% on a 30 question exam on polynomial differentiation. To this day I can’t understand how anyone could not have gotten 100%. It’s not like there was any trick questions.

  29. Lynne, I am not surprised that higher level courses have, in fact, improved since the 70’s. But, I truly do believe that the conceptual underpinnings (like the meaning of the equals sign) are being taught poorly or not at all at the elementary school level. This leaves only the brightest students actually attaining a good grasp of these fundamentals.

  30. The worst thing is that I think this was done mainly because they thought these concepts were too difficult to grasp and they wanted the students to learn to “do the math” so they taught them to “do the math” without teaching them WHY it works and WHAT IT MEANS. The result is that the kids make good progress in the early years “turning the crank” on methodologies they have been taught, but then have no foundation to move into higher mathematics. It invariably catches up to them in algebra when they no longer just have numbers to crunch and must fall back onto the rules of mathematics which they were never taught.

  31. I failed a math test once… which was not possible when I was a kid. So I checked my answers and went to the teacher and rechecked my answers. They were all correct but different from everyone else in the class of about 30 students…

    They had ALL copied the answer from the back of the book which had all wrong answers. It never occurred to me to look in the back of the book for answers.

    They kept their A’s.

  32. I know this is a few days after the rest of the comments, but I am a product of the “modern” public school system and I remember the day (not the date, but the experience) of realizing that “1 + 2 = ___” was the same as “1 + 2 = 5, true or false?” I’d always seen the equals sign as sometimes “equal” and sometimes “the answer goes here” depending on context. That was a good day.

  33. Being an old fart when I see 1+2=5, I see an abomination. Not a boolean expression at all. However, as a programmer I have to change it to (1+2=5) to be comfortable with it (actually, still an abomination because it’s all constants. I’d have to just replace it with false but still include the parens.) Code that doesn’t include the parenthesis around a conditional just looks too ugly for me to stand. I am genetically unable to leave it that way. In C it’s a new story altogether (because now it’s an assignment and the compiler should reject it.)

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