# A Tablespoon of Salt

I enjoy cooking – I find it really relaxing. There’s something nice about getting to switch to autopilot and follow a recipe after a long day of work.

A few years back I found a soup recipe (I can’t remember which one, unfortunately) that looked really, really good. I invited some friends over for dinner on the promise that I’d trade them food and shelter for good conversation. Nowadays, I think I have a pretty good handle on the basic steps in making soup, but at the time I was still getting a handle on cooking and found myself essentially following recipes more or less verbatim.

One of the first steps in this particular recipe was to cook a bunch of aromatics over low heat, along with some salt to draw out the water and speed things up. The recipe recommended adding in a teaspoon of kosher salt. I misread the directions and instead added a tablespoon of table salt.

Needless to say, although everything smelled amazing, when it came time to serve it up we found that it was essentially inedible. I had to apologize for keeping people waiting, but we managed to salvage the evening by heading downtown to a restaurant where the food was made by trained professionals.

So what went wrong here?

There were two immediate errors. First, I had mixed up tablespoons and teaspoons. That made my measurements off by a factor of three. I also used table salt instead of kosher salt, which has smaller crystals that can pack more tightly, effectively multiplying the oversalting amount by something like an extra 25 – 40%. So I had essentially added in around four times the amount of salt necessary. Oops.

But let’s take a step back. Why didn’t I notice that I was about to make this mistake? Well, I was new to cooking, and I hadn’t made enough recipes to realize that a full tablespoon of table salt is a lot of salt. That’s three times the daily recommended salt intake for a person. (That point is particularly relevant, since I was making food for three people!) In a sense, I hadn’t built up an intuition that would cause me to pause for a minute, say that something didn’t feel right about adding that much salt, and then double-check the recipe.

It also didn’t help that I had never really stopped to think about the difference between kosher salt and table salt. I had just thought that salt was salt was salt and that was that. It had never occurred to me that different types of salt had different properties and different use cases, or that recipes would specify different types for a good reason. (Or, rather, that they often are different and often are there for good reason. In some cases, salt is salt is salt. Just not this case.)

It also didn’t help that I hadn’t yet built up enough cooking experience to realize just how powerful salt is. There are some ingredients you can accidentally triple without breaking things. If I used too much oil in this recipe, I probably wouldn’t have noticed because the last step was to immersion blend everything and chances are it would have (mostly) emulsified in. It just would have been super rich. Salt is not one of those ingredients, but I hadn’t learned that yet.

And finally, it really didn’t help that this was a recipe I’d never tried out before. At the time I was making this recipe, I hadn’t yet developed enough experience to be able to stop and taste something partway through a recipe and figure out whether it needed some sort of adjustment. In this case even, when I was thinking “wow, those aromatics by themselves are really unpleasantly salty!” I didn’t have the context to know “and that’s not what they’re supposed to taste like!”

The reason I’m mentioning this story is that it began with me making what, in some sense, is a pretty minor error: using the wrong amount of salt. But really, this error indicated that I still had a lot to learn about cooking, namely:

• A tablespoon of salt is a lot of salt – probably way more than what you’d ever need in soup for three people.
• Kosher salt is not the same as table salt, and the two aren’t one-for-one interchangable.
• Oversalting something can totally wreck it, and you have to be very careful about how much salt you add to something.
• You should not invite people over and serve them an untested recipe!

Over the years both taking and teaching classes, I’ve found that there are certain mistakes you can make that seem minor, but which totally break whatever it is that you’re working on. Often times, the reason that those mistakes seem minor is because there’s a lack of a deeper understanding of some key concept or concepts, and really the true error is something more fundamental. I call these errors “tablespoon of salt errors.” A tablespoon of salt error is one where

• the error appears superficial to the person who made it;
• the error is sufficiently serious that it’s difficult, if not impossible, to recover from it without starting over; and
• the error reflects a fundamental process error or a lack of understanding of some key concept.

For example, when I was an undergrad, I was taking a class on advanced object-oriented programming techniques in C++. One of the assignments asked us to simulate objects moving around in a shipping network. I had a helper function that checked whether the weight of a packet was zero and then returned one of two different values based on this. The code essentially looked like this:

```WeightClass weightClassFor(unsigned weight) {
return weight = 0? ZeroWeightClass : NonzeroWeightClass;
}```

There’s an big error here. Specifically, I’m using a single-equals sign (“assign weight to be zero, then evaluate to zero”) rather than a double-equals sign (“compare weight against zero.”) This code always returns the NonzeroWeightClass, which isn’t correct.

I spent many hours working on this project, submitted it, and earned a solid C for it. The error was localized specifically to the above function, and changing that one equals sign would have fixed the issue and passed all the functionality tests.

In the only time I can ever remember doing so as an undergrad, I went to the TAs to beg and plead for some partial credit back, since after all, it was a one-character error! It’s pretty clear what I meant to do, after all.

But the thing is, this wasn’t a one-character error. Somehow, I had written some code that didn’t work, and I hadn’t sufficiently tested things before I submitted it. In reality, the root causes of the error were more accurately things like these:

• I wasn’t compiling with warnings on the maximum setting. A good compiler probably could have caught this and said something like “so… you’re sure you want to have a comparison statement that always does the same thing?”
• I hadn’t tested things thoroughly enough. This code clearly doesn’t work, but I had written a whopping zero unit tests for my code and therefore didn’t have a way of noticing.

This is a classic tablespoon of salt error. It seems like a minor error (I missed an equals sign). It broke everything pretty seriously (a quarter of the tests failed, despite the rest of the code being essentially right). And it indicated a serious process error on my part (where were the unit tests? why didn’t you compile with warnings as errors?)

I’m now completely convinced that the TAs were justified in taking off a bunch of points, since there was something fundamentally broken with how I approached the assignment. Plus, I now know not to make that error again!

Now that I’m teaching classes rather than taking them, I’m noticing that tablespoon-of-salt-errors are extremely common, especially in discrete math and proofwriting, where very, very simple mistakes can totally derail a proof.

For example, consider the following proof about multiples of three. The relevant definition we give to students is that an integer $n$ is a multiple of three if $n = 3k$ for some integer $k$.

Theorem: If $n$ is an integer that is not a multiple of three, then $n^2$ is not a multiple of three.

Proof: Consider an arbitrary natural number $n$ where $n$ is not a multiple of three. This means that $n \ne 3k$ for any integer $k$. Consequently, $n^2 \ne 3(3k^2)$ for any integer $k$. Therefore, if we let $m = 3k$, we see that $n^2 \ne 3m$, and therefore $n^2$ is not a multiple of three. \$\qed\$

There’s a nice rhythm to the proof, and if I were to read this, I’d think that the (hypothetical) student who wrote it made a good-faith effort to solve the problem. I’d also think that they made a pretty big logic error: just because you can’t write $n^2 = 3(3k^2)$ for any integer $k$ doesn’t necessarily mean that you can’t write $n^2 = 3m$ for some integer $m$ that isn’t of the form $3k^2$. Using that same logic, I could prove the (incorrect) claim that if $n$ is not a multiple of nine, then $n^2$ is not a multiple of nine either:

“Theorem:” If $n$ is an integer that is not a multiple of nine, then $n^2$ is not a multiple of nine.

Proof: Consider an arbitrary natural number $n$ where $n$ is not a multiple of nine. This means that $n \ne 9k$ for any integer $k$. Consequently, $n^2 \ne 9(9k^2)$ for any integer $k$. Therefore, if we let $m = 9k$, we see that $n^2 \ne 9m$, and therefore $n^2$ is not a multiple of nine. \$\qed\$

The theorem here is not true (pick $n = 3$, for example), and the proof follows pretty much the same line of reasoning as the one from earlier, so the logic can’t possibly be correct.

Superficially, this error isn’t all that deep. “Oops, I just made a mistake in that last step.” But really, this goes a lot deeper and hits at the challenge of working with arbitrarily-chosen values. If you want to rule out all choices of $m$, you have to really show that no choice of $m$ is going to work, not just find a family of infinitely many numbers that don’t work. There’s also another issue with setting $m = 3k^2$ without actually saying what $k$ is in the first place (is it a placeholder? you generally can’t do that to placeholders.)

This is a tablespoon of salt error. The error seems superficial (either “you can’t use inequalities that way” or “not all integers $m$ can be written as $3k^2$ for some integer $k$.”) The error completely breaks the proof (the same reasoning can be used to prove patently false statements without much modification). And the error indicates a fundamental misunderstanding of some key concept (arbitrarily-chosen values, placeholder variables, etc.)

A question that I keep grappling with as an instructor, and which I’m not sure I’ll ever really figure out the best answer to, is how to assign grades to student work with a tablespoon of salt error in it. I remember my own frustration as a student getting slammed for mistakes that I thought were pretty minor. I also remember how valuable those experiences were, though, since they really forced me to admit that there were things I didn’t yet know and they helped me figure out where I needed to study up.

Figuring out how to handle this case in a way that pushes students to do better without making them feel overwhelmed by what they don’t yet know is a challenge. It’s exacerbated in the tablespoon of salt case because the grading feels crazily disconnected from reality.

There’s a case currently up at the US Supreme Court (District of Columbia v. Wesby) where someone hosted a party in a vacant house that they had no business being in. The police came, questioned everyone there, and arrested everyone on suspicion of trespassing. In DC law, trespassing requires a mens rea on the part of the perpetrator – they have to know that they’re not supposed to be there. The question before the Court is about probable cause. The police report seeing a group of people in a house whose situation was so rough that the partygoers should have known that they were breaking it. The partygoers reported that they thought they were invited to a house they were allowed to be in. The disconnect between these narratives is currently being decided by nine very intelligent legal minds and will have a huge impact on law enforcement.

From the perspective of an instructor, a student who made a tablespoon of salt error has a serious gap in their understanding that needs to be corrected. From the perspective of the student, they made a minor error and the graders harshly took off way too many points for it. It’s a clash of narratives, which is never fun.

I’m going to keep tuning my approaches to handling cases like these. I figured I’d post this concept in the meantime in case the metaphor proves useful in other contexts.