Mr Honner

My Tests are So Hard

Everywhere I’ve taught there have been teachers who brag about how hard their tests are. It’s always a central part of their identity as a teacher, of how they see themselves, and how they want to be seen. They proudly consider themselves more rigorous than their colleagues.

But nothing could be easier than making a test hard. You can just put more questions on it than can be reasonably handled in the allotted time. Or put problems on that haven’t been emphasized in class or practiced enough. Or problems that test edge cases and not core ideas. Or problems from the next unit. Or problems you simply haven’t prepared all students to handle.

I’ve seen teachers do all these things. It’s not rigorous. It’s lazy. You know what’s truly difficult? Writing a test that is fair, representative of core ideas, and appropriately challenging.

By MrHonner, 2 yearsJanuary 14, 2024 ago

Teaching

Taught Helplessness

I’m currently reading “The Design of Everyday Things” by Don Norman and it’s interesting to think about what the theory of product design has to say about instructional design.

For example, the author discusses how “learned helplessness” can result from poor design. A product whose functionality isn’t discoverable, and that doesn’t provide good feedback, will be frustrating to use, so users will likely give up after trying and failing a few times.

Just as I was making the connection to teaching math in my mind, the author himself brought up math instruction as a common example of “taught helplessness”: When math is presented as unintuitive, and poor or misguided feedback is given, students are likely to just give up. The problem is amplified by the linear way in math is usually taught. In many classrooms, if you don’t understand what happened yesterday, you will probably struggle to understand what is happening today.

Originally posted on Mastodon.

By MrHonner, 2 yearsJanuary 7, 2024 ago

2024 and Differences of Squares — Solution

The new year 2024 is a difference of squares, $2024 = 45^2 - 1^2$ , which got me thinking about a fun little number theory problem:

Is there a largest number that can not be expressed as the difference of squares? If so, find it. If not, prove no such number exists. Good luck, and happy new year!

As promised, here’s my solution.

There are infinitely many numbers that can not be expressed as a difference of squares. In fact, we can completely characterize numbers that can be expressed as a difference of squares and those that can’t. It all starts with factoring.

Differences of squares have a useful structure that can be exposed by factoring:

$a^2 - b^2 = (a + b) \times (a - b)$

We can leverage this structure to answer our question.

Suppose n can be factored as $n = s \times t$ . If n can be expressed as a difference of squares, then we can also write

$n = a^2 - b^2= (a + b) \times (a - b)$

Now set $(a + b) = s$ and $(a - b) = t$ . This gives us the system of equations

$a + b = s$

$a - b = t$

We can solve this system by adding and subtracting the equations. Adding gives us $a = \frac{s+t}{2}$ , and subtracting gives us $b = \frac{s-t}{2}$ . This shows us how to express n as a difference of squares: Just factor n into $s \times t$ , compute $a = \frac{s+t}{2}$ and $b = \frac{s-t}{2}$ , and then $n = (a + b) \times (a - b) = a^2 - b^2$ .

There’s only one thing we have to worry about: a and b must be integers. But as long as s and t have the same parity — that is, s and t are both even or both odd — then the sum and difference of s and t will both be even, and so $\frac{s \pm t}{2}$ will be an integer.

This means that n can be expressed as a difference of squares if and only if we can write $n = s \times t$ where s and t are both odd or both even. This is usually possible, often in multiple ways. But there’s one situation when it isn’t: When n is divisible by 2 exactly once. When this is true, then however you factor n into $s \times t$ , the lone factor of 2 will end up as a part of either s or t, making one of them even and the other odd. Thus, in this case, it’s impossible to factor n so that the two factors have the same parity, and so it’s impossible to express n as a difference of squares.

This gives us a complete answer to our question: A number n is not expressible as a difference of squares if and only if it is divisible by 2 exactly once! In other words, every odd number times 2 is not expressible as a difference of squares, and every other integer is.

As an example, given $n = 105 = 3 \times 5 \times 7$ , we can factor $105 = 7 \times 15$ which gives $a=\frac{15+7}{2}= 11$ and $a=\frac{15-7}{2}=4$ , and sure enough, $105 = 11^2 - 4^2$ . Notice, we could also write $105 = 5 \times 21$ , which gives $a = \frac{21+5}{2}=13$ and $b = \frac{21-5}{2}=8$ , so $105 = 13^2 - 8^2$ .

On the other hand, it isn’t possible to do this at all for 6. There are only two factorizations, $6 = 6 \times 1$ and $6 = 2 \times 3$ , and in both cases the factors have different parity, so the a and b we need won’t be integers.

There’s an interesting resemblance here to Euclid’s Formula for generating Pythagorean triples. There’s also an interesting follow-up question about how many different ways a number can be expressed as a difference of squares. And since the numbers that answer our question are those that are divisible by 2 exactly once, I wonder what properties numbers that are divisible by 3 exactly once have.

Thanks to everyone who contributed on the Mastodon thread! There are some cool ideas there as well, so be sure to check it out.

Challenge Numbers

2024 and Differences of Squares

The new year is one of my favorite kinds of numbers: a difference of squares!

$2024 = 46 \times 44 = (45 + 1) \times (45 - 1) = 45^2 - 1^2$

This observation got me thinking about what kinds of numbers can be written as the difference of squares. For example, $3 = 2^2 - 1^2$ , $5 = 3^2 - 2^2$ , and $16 = 4^2 - 0^2$ , but it is impossible to write 6 as the difference of squares of integers.

So here’s a little mathematical puzzle to start the new year: Is there a largest number that can not be expressed as the difference of squares? If so, find it. If not, prove no such number exists. Good luck, and happy new year! I’ll give my answer in a follow-up post.

Related Posts

2024 and Differences of Squares — Solution
The 2017-2018 Conjecture
Happy 2016!

Resources Writing

Books I Read in 2023

Here’s a selection of books that made an impact on me in 2023.

Superforecasting tells the story of psychologist Philip Tetlock’s project to systematically evaluate the predictions of experts. What exactly does an advisor mean when they tell the President that a military operation has a “good chance” of being successful? It wasn’t so long ago that no one thought to even ask such a question, and as Tetlock shows, the consequences couldn’t be more real. Tetlock’s work led to the development of the Good Judgement project, a forecasting competition designed to identify the characteristics of “superforecasters”, individuals with a quantifiable talent for predicting how world events will unfold. It’s a great book, and one I was partly inspired to read because of my involvement in a student forecasting tournament based on the Good Judgement project (where our students took first and second place overall!)

I finally read The Lady Tasting Tea, by David Salsburg, which had been on my list for a while. It’s an excellent history of the development of statistics, told through fascinating characters and interesting anecdotes. It immediately joined my student lending library, and it also took second place on the list of statistics book I’ve learned the most from (to be fair, it’s first on the list of books I’ve learned from by reading it rather than writing it). It’s full of fun little details, like this quote I shared with my linear algebra class the very day I read it.

Manil Suri’s The Big Bang of Numbers was a highlight of my year, both because it’s a unique book that tells a whimsical origin story of mathematics full of insight, humor, and even spirituality, and also because I had the privilege to interview Manil Suri at the Jaipur Literature Festival this past year (video here).

Made to Stick, by Chip and Dan Heath, is the book that I’ve been thinking most about as a teacher this year. Why do silly ideas go viral while important ideas struggle to take hold? It’s a question every teacher should consider, and Made to Stick offers lots to think about. Months later I’m still pondering the power of generative metaphors, story as simulation, and why people should care about duo piano. Thanks to my (former) student Satvik for the recommendation!

Atomic Habits by James Clear was another book that had been on my list, and it was a worthwhile read even if confirmed many previously held beliefs. A core idea of the book is very mathematical, namely that the impact of habits compound like interest in a bank account, generating exponential (hopefully, positive) personal growth.

I feel a bit funny saying that John Fleischman’s Phineas Gage was probably the best book I read this year, seeing as its target audience is middle school students. But this infamous story of the 19th-century man who blasted a railroad spike through his brain and lived to tell about it is so vivid, engaging, and expertly contextualized in the history of science that it keeps coming up in thoughts and conversations.

In fiction, I enjoyed Susanna Clarke’s Jonathan Strange and Mr. Norrell, a fantastical history of magic in Britain told through the story of the apprenticeship-turned-rivalry of the title characters. Amor Towles’s A Gentleman in Moscow was one of the loveliest books I’ve read in a long time. And I enjoyed every moment of Elmore Leonard’s Get Shorty. Leonard also authored the stickiest advice I read in 2023: don’t write the parts that the reader skips.

In addition to getting back to my year-end lists (sorry, 2022) I’ve been trying to be more active on Bookwyrm, the ActivityPub-based social network dedicated to reading and reviewing. You can find me there at phonner.

And thanks, as always, to the (shamefully underfunded) Brooklyn Public Library for making it so easy to read this year and every year.

My Tests are So Hard

Taught Helplessness

2024 and Differences of Squares — Solution

2024 and Differences of Squares

Books I Read in 2023

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