Witch 109: Bad Language

This is a continuation of a previous WitCH (and PoSWW) on the Logic and Proof chapter of VicMaths, Nelson’s Specialist Mathematics Year 12 text. The previous WitCH comprised the first part of Section 3.1, titled Conjectures, together with some associated exercises. The remainder of 3.1 covered conjectures proper, including examples and counterexamples and the like:

Yes …

On to 3.2, titled The Language of Proof. Below are the, um, highlights from this section. We’ve restrained ourselves and not included associated exercises.

12 Replies to “Witch 109: Bad Language”

  1. I only looked at the first box. They are so confused, it is criminal.

    Simplification applied to sentences in the logic is a fairly advanced part of the subject. If they aren’t dealing with it in terms of logic, then they should treat arithmetically equivalent sentences as the same (i.e. use an equivalence relation), like for instance rationals.

    Would they say a = 1/2 does not imply a = 2/4? Because maybe it implies a = 3/6?

    I honestly don’t know based on reading that, because they seem to flip flop from one line to the next. And if I’m confused by it, I bet teachers and students will be too. Atrocious.

  2. Oh no, what is the logical equivalence garbage? Same truth value? So now there are only three statements up to logical equivalence?

    This is what obsession with truth value and lack of understanding will get you.

    Why can’t they just use a standard book or at least consult a logician.

  3. A number is divisible by 4 if and only if it is divisible by 2 and by 2.

    What we can see here is the difference between a proof and mathematical reasoning.

  4. After the revolution, the authors of this chapter should face charges of crimes against humanity.

  5. There are bits (OK, very small) that I actually find OK in this section.

    They are lost in a quagmire of rubbish, so maybe the authors got lucky more than properly understanding this.

    Mixing the Boolean and classical logic is dangerous, and the examples here illustrate why!

    “If and only if” is fine as a logical linking phrase. Using the equivalence operator instead from Boolean logic instead of the classical logic symbol is the beginning of a slide downwards into mess.

    The heading “Verifying De Morgan’s Laws” would be funny if they weren’t trying to be serious…

    1. No, I don’t think the authors were lucky. Unlike 3.1, there is some solid structure underlying 3.2, which makes it hard to get the thing completely wrong. But, it was a good faith try. And “Verifying De Morgan’s Laws” is funny. One can’t not laugh.

  6. @Grant: For the manufacture of weapons of maths destruction?

    @Glen: No, this is what obsession with writing a curriculum full of ill-conceived thought bubbles will get you. Using a standard book or at least consulting a logician is useless because the material this thought bubble has imposed on the curriculum requires at least a term to teach properly and teachers have about 2 weeks.

    @Marty: Yes, its fools proof.

    @Marty: One cant not laugh. Lucky for me there are other textbooks that help me understand the double negative!

  7. With the death of geometry, students no longer see much in the way of proofs in Years 7 to 11. Then in Year 12 we jump to a formal treatment with abstract notation. Doomed to fail even with good teaching.

    The jump from using maths as a problem solving language to constructing logical systems, (theorem +proof)^\infty, causes much heartache with university undergraduates; real analysis or linear algebra in second year? Better to turn them off maths in Year 12!

  8. Wow, is this from an actual textbook and not some draft up for a first review?

    I am not a native English speaker but I do not think that the authors interpret the concept of necessary condition the same way I do. How does the claim in the 13th line work? “a+4=9 is a necessary condition” because a=5 can also lead to other conditions that are equally valid?? Eh??

    I love worked example 9. Great example that you can use in class to discuss what you can screw up writing proofs. Typo in the question btw as an ‘is’ is lacking.

    They do not prove that if n^2 is even then n is even. Only stating that squaring an odd number gives you an odd number is a great example of not proving anything. For such a starting example it is actually quite advanced to prove this part of the claim. Starting with “Suppose n^2 is of the form 2m” leading to “zo n is even” needs a proper argument using (prime) divisors.

    The 3rd step is hilarious.
    – Suppose n=2m as n is even.
    – then n^2=4m^2 is even because this gives n=2m and that is divisible by 2…
    of course it is, you defined it that way. Oh my…
    So they even screw up the easy part of the proof (n^2=4m^2=2*(2m^2).

    Personally, I would also want a proper conclusion as a final step to finish each proof when teaching proof writing.

    I am still trying to figure out example 10. What is the point of introducing a bunch of tokens when statements A and B are not defined in the process? And how (and why?) do you “randomly” remove tokens 2, 4 and 5?

    It reads like a practical joke, not a textbook.

    1. Thanks, Deejay. No, it’s not a draft although, as I suggested in comments on the follow-up post, it was probably a very rushed job.

      Regarding your comments:

      *) a + 4 = 9 is both necessary and sufficient for a = 5.

      *) Yes, example 9 is a disaster. They *almost* get the first bit right, but the punchline is so badly expressed that it is wrong. And the second, easy part is a mess.

      *) Yes, Example 10 is like a bad LSD trip.

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