HN Books @HNBooksMonth

Doug Lemov's book "Teach Like a Champion" is great: https://www.amazon.com/gp/product/1118901851/ref=dbs_a_def_r...

I'd also recommend "Teaching as Leadership": https://www.amazon.com/Teaching-As-Leadership-Effective-Achi...

This is an excellent question because it allows the student to demonstrate an understanding that (a) digits have different 'meaning' depending on the context - the 5 'means 500' and the 8 'means 80'; and (b) there is a difference between replying with a true statement and answering correctly.

Bobby has made a true statement that is related to the question, but hasn't completely answered the question. The correct is that the value of 5 in 582 is 500, and an even better answer would be "the value of 5 in 582 is 500 because the 5 is in the hundreds place".

This distinction is something that many teachers overlook. There is an excellent handbook for teachers called "Teach Like A Champion" [1] which emphasises that "Right is Right" - here is an excerpt:

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Right Is Right is about the difference between partially right and all-the-way right—between pretty good and 100 percent. The job of the teacher is to set a high standard for correctness: 100 percent. The likelihood is strong that students will stop striving when they hear the word right (or yes or some other proxy), so there's a real risk to naming as right that which is not truly and completely right. When you sign off and tell a student she is right, she must not be betrayed into thinking she can do something that she Cannot.

Many teachers respond to almost-correct answers their students give in class by rounding up. That is they'll affirm the student's answer and repeat it, adding some detail of their own to make it fully correct even though the student didn't provide (and may not recognize) the differentiating factor. Imagine a student who's asked at the beginning of Romeo and Juliet how the Capulets and Montagues get along. “They don't like each other,” the student might say, in an answer that most teachers would, I hope, want some elaboration on before they called it fully correct. “Right,” the teacher might reply. “They don't like each other, and they have been feuding for generations.” But of course the student hadn't included the additional detail. That's the “rounding up.” Sometimes the teacher will even give the student credit for the rounding up as if the student said what he did not and what she merely wished he'd said, as in, “Right, what Kiley said was that they don't like each other and have been feuding. Good work, Kiley.” Either way, the teacher has set a low standard for correctness and explicitly told the class that they can be right even when they are not. Just as important, she has crowded out students' own thinking, doing cognitive work that students could do themselves (e.g., “So, is this a recent thing? A temporary thing? Who can build on Kiley's answer?”).

When answers are almost correct, it's important to tell students that they're almost there, that you like what they've done so far, that they're closing in on the right answer, that they've done some good work or made a great start. You can repeat a student's answer back to him so he can listen for what's missing and further correct—for example, “You said the Capulets and the Montagues didn't get along.” Or you can wait or prod or encourage or cajole in other ways to tell students what still needs doing, ask who can help get the class all the way there until you get students all the way to a version of right that's rigorous enough to be college prep: “Kiley, you said the Capulets and the Montagues didn't get along. Does that really capture their relationship? Does that sound like what they'd say about each other?”

In holding out for right, you set the expectation that the questions you ask and their answers truly matter. You show that you believe your students are capable of getting answers as right as students anywhere else. You show the difference between the facile and the scholarly. This faith in the quality of a right answer sends a powerful message to your students that will guide them long after they have left your classroom.

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[1] http://www.amazon.com/gp/product/1118901851

⬐ haberman

Something about this rubs me the wrong way. It reminds me of the first homework assignment I did for CS in college. My solution was correct, but was marked down for using an O(n^2) algorithm instead of O(n). Nothing in the original assignment said anything about a requirement to use the most efficient algorithm.

If the question is "how do the Capulets and Montagues get along," the answer "they don't like each other" is not partially right, it is right. It is not detailed or sophisticated, but the question didn't ask for detail or sophistication.

If we're going to call an answer "partially right" because it misses some nuance that could be explored in greater depth, than we could play that game all day long with anything that has ever been written. For example let's take your answer:

> The correct is that the value of 5 in 582 is 500, and an even better answer would be "the value of 5 in 582 is 500 because the 5 is in the hundreds place.

Ok let's play this game. "Partially correct," I say. "First of all, the fact that a 'hundreds place' exists at all is thanks to the relatively recent invention of Arabic numerals around 500 A.D. and its innovation of a numeral for zero, which previous number systems such as Roman numerals lacked. Secondly, the reason the third digit represents hundreds is because we use base 10 numbers by convention, which likely traces back to our hands and their 10 fingers. Other bases are used in different contexts, most notably hexidecimal in software engineering, in which case the '5' would be in the 256's place, representing 1280. Thirdly, the convention of writing digits is most-to-least significant order appears arbitrary -- it is hard to argue that it is linked to the order of text directionality, since the same order is used in both left-to-right (eg. Latin) and right-to-left (eg. Arabic) writing systems."

Solid B+ answer though. Don't worry, I have faith in your ability to produce a truly correct answer someday.

⬐ avmich

This sounds rather fishy to me, and I don't know where to start, so I probably won't. Just one quote which particularly feels wrong:

> the teacher has set a low standard for correctness and explicitly told the class that they can be right even when they are not.

That "explicitly" word strongly suggests to me the author doesn't know what he's talking about.

⬐ jacobolus

In other words, the student (“Bobby”) understands the subject, understands the gist of the question, and makes an answer which demonstrates that understanding, but the question is ambiguously/confusingly worded, such that “it’s in the hundreds place” doesn’t fully satisfy the teacher’s pedantic expectation of a “right” answer. Thus the teacher needs to tell the student he is wrong and ask leading questions until the student guesses his way into “correctness” and hopefully eventually memorizes the specific pattern desired by the teacher, like a trained dog going through a list of tricks to get to the treat (“no I said lie down, and you are merely sitting”).

Spending time on the difference between the phrases “a 5 in the hundreds place” vs “5 hundred”, when both the student and the teacher understand them to mean the same thing is a waste of people’s focus. Instead the student could be doing something much more interesting, such as examining what the digits would mean in a base twelve system, discussing how a number is transformed when multiplying/dividing by ten, learning what happens to the digits in higher places when working with modular arithmetic, learning how to use an abacus to keep track of the place values, or developing an algorithm to transform back and forth between explicit counters like pebbles in a dish and their written decimal representations, etc. etc.

On the other hand, the question asked of droopybuns’s daughter is sort of interesting from a philosophy/sociology/pedagogy point of view. There’s not really much math content in it, but getting students to think about the ways the social context shapes expectations about right answers and effective ways to navigate a society full of bureaucrats with sticks up their asses is definitely worthy of discussion.

⬐ droopybuns

Best response yet :)

Finaly spoke with the teacher:

Bobby is wrong.

The "value" of the 5 is 500. They are learning number placement right now: 582 = 500 + 80 + 2

So the thing she is supposed to say is that the 5 is in the hundreds place, so it should be equal to 500.

I loathe that they assign the word "value" when the concept is contextual. The value of 5 is axiomatic, as is the value 582. When I explained this, the teacher's only response was to argue that this standard is almost verbatim outlined by the school district. <s> It must be right then! </s>

This teaching risks proving that 5=500.

Markmcc has an excellent counterpoint to my frustration, and I plan to use it to explain the objective of the question to my daughter.

⬐ jacobolus

A better wording for the question the teacher was really trying to ask might be:

“What quantity does the symbol ‘5’ in the decimal number ‘582’ represent?”

Then the answer is unambiguously ‘five hundreds’.

⬐ droopybuns

Much better. Thank you for expressing the crux of the confusion & proposing a better solution.

If you asked an experienced effective high school teacher "how do you do it?" What could they tell you? It took a lot of work to learn how to teach: experimentation, analysis, etc...you simply have to learn how to teach!

Take a look at Teach Like a Champion 2.0 by Lemov. It answers this precise question! (http://www.amazon.com/Teach-Like-Champion-2-0-Techniques/dp/...).