I made my favorite pumpkin muffins this morning, for the first time in quite a while, but I made them differently today.

The recipe calls for one cup of pumpkin from a can. A can of pumpkin contains about a cup and a half, and what are you gonna do with an extra half cup of pumpkin? So I always put in the whole can.

Today however, I was using pumpkin from the freezer. Last fall, I turned a couple of pie pumpkins into pumpkin puree and froze it for exactly this purpose. The bag of pumpkin puree I was working with contained two cups. So I made two batches, using the prescribed one cup per batch.

They are good, but they are not quite as good as the ones that have a cup and a half of pumpkin.

If you spend any time baking, you will surely run across claims that baking is different from cooking. Baking requires more precision and following of directions than other types of cooking, you’ll be told.

Similarly if you spend any time learning math, you will surely run across claims that learning math is different from other intellectual activities. Learning math requires precision and following of steps, you’ll be told.

These are lies. My deliciously moist pumpkin muffins prove that this is so.

This is what I love so dearly about Eugenia Cheng’s book *How to Bake Pi.. *She writes that in both math and baking, decisions have consequences. Sometimes these consequences are undesirable, such as bread that doesn’t rise or arithmetic that is inconsistent.

In math as in baking, you need to follow instructions carefully in order to achieve the known result. If I want muffins that are exactly like the ones in the original recipe, I need to use one cup of pumpkin. But here’s the secret we don’t let you in on: If I use the whole can of pumpkin, I will still get pumpkin muffins. They will be different ones, but they will still be pumpkin muffins.

The difference between baking and math is that you nearly always see the natural consequences of the decisions you make, while we structure most people’s experiences with math in ways that hide those consequences.

If you leave out the sugar, your muffins will not be delicious. They may have structural problems as well. You notice these consequences and you tend to try to figure out what went wrong.

If you claim that , the only consequence is that someone else tells you that you are wrong—whether teacher, tutor, or back of the book. In Cheng’s book, she describes treating math exactly like baking. She pushes her students to consider the natural consequences of their claims. If , then when , . If , then there are going to be lots of troubles later on.

Go read her book. Then make some pumpkin muffins. And please don’t listen to Chris Kimball; the man is a total killjoy.

If x = –1, then 0 = 2 is the problem – yeah?

Heh. Good catch

Aaron. In a world where 0=2, pretty much anything’s possible though, right?Actually, the world where 0 = 2 (modulo 2, characteristic 2) is pretty cool, but it is a decidedly different “flavor.”

My personal favorite for natural consequences for (x + a)^2 = x^2 + a^2 is to give students a right triangle with (for example) side lengths 6 and x and hypotenuse length x + 2 and ask them to solve for x. Then I sit back and enjoy. This is especially fun because I’ve already conditioned them to question whether I’m giving them an impossible problem, so they’re really happy to catch me out. Ho ho ho.

I don’t understand your comment, Julierwright. 6-8-10 seems like a standard right-triangle solution answer to me. How are the students “catching you out”?

Guess I should have spelled it out a little more. If they make the standard mistake, they set up the equation 6^2 + x^2 = x^2 + 2^2. Then when they try to solve it and get 36=4, they think it’s an impossible problem. So I either share somebody else’s 6/8/10 answer and have them see it works and ask them to investigate what went wrong, or ask them to describe what they did and break it down in detail.

I am so happy you wrote about this book. I have been in love with “How to bake pi ” for some time and gave it to all my fellow math teachers for Christmas. I love the way she connects math and baking and love the way she puts math into ways that others can access it. I also feel like she and I would be best friends if I knew her. :-). Thanks for your great blog and noticing always!