How School made you Worse at Math

This video was made for the #SoME1 Summer of Math Exposition by Grant Sanderson of 3Blue1Brown. Thank you for giving me the motivation to finally make something.

Timestamps:
0:00 - Introduction to PEMDAS
0:38 - Previous Explanations

1:37 - Ambiguities in PEMDAS
2:16 - PEMDAS isn't Universal
3:21 - PEMDAS is Unintuitive
3:56 - Multiplication by Juxtaposition
5:12 - Equal Precedence isn't Intuitive
7:17 - Solutions
8:27 - Just Teach the Order of Operations

3Blue1Brown:

Why aren't you making math videos?:
The Summer of Math Exposition:

The How and Why of Mathematics:
PEMDAS is wrong:
The Problem with PEMDAS: Why Calculators Disagree:

minutephysics:
The Order of Operations is Wrong:

The Website Ambiguous PEMDAS by Oliver Knill was an important source for me in the ressearch for this video: Check out his website for much more info on the troubles with PEMDAS:

#SummerofMathExposition
#3Blue1Brown
#Math
#PEMDAS
#OrderofOperations

[interrogatix]

Okay wow, I was not expecting this video to get this much traction. I was just kinda happy that a few people on the 3Blue1Brown discord server liked it and then just kinda left it at that, thinking this was a fun little project to flex my video making skills and not much more. But we're now at almost 3000 views (Edit: scratch that, I'm absolutely floored right now) and still growing and I'm not really sure how to think about this. I wanna say something corny like "thank you so much, it means alot to me, etc." but that feels weird while this is still somewhat surreal to me. I wasn't planning on making anything more for the channel in the near future so don't expect anymore videos soon, but I will definitely make more at some point.

So in order to not dupe myself, instead I'm just gonna ask, what did you think of the video? Particularly if there were any specific parts which especially grabbed your attention or that you didn't think worked so well. These two points, especially the latter, are in my experience the best kinds of criticism you can get.

So yeah... thank you... I guess?


Also on an only slightly unrelated tangent. I think Henry (Minutephysics) has a problem with communicating his intent because he's made the exact same mistake again in his most recent video on The Butterfly Effect:

He once again tries to explain how something can lead to misinterpretation, explains the problem and then people retort with "But that's not how I interpret it".

I previously thought he must've misunderstood how PEMDAS works which left his argument open to be rebutted but now I'm convinced he's just really bad at clarifying what he's even trying to argue. Commentors think he's arguing that The Butterfly Effect is bad based on his own faulty interpretation, completely missing that the fact that the butterfly effect can so easily lead to misinterpretation in the first place is the real problem with it. EXACTLY the same thing that happened with the PEMDAS video.

Tangent over. Also I now hate the word "misinterpretation", it's annoying to type.

[TheDiego]

I didn't realize how lucky I was to have my teacher just explain the rules to me outright instead of bothering with an acronym

[alnaskabeer]

We had a C programming lab session for us this semester, and one of the questions was to evaluate a expression. It was not like a function or something to do with variables, it was just a math expression. All we had to do was copy the thing and equate it to a variable, and print it. And the answer we got (the actually mathematically correct answer which the computer and the calculator gave) was rejected by the teacher because it didn't follow the BODMAS rule... It wasn't anything ambiguous at all, she genuinely thought that division had higher precedence compared to multiplication. We ended up deliberately adding brackets to get the "correct" answer. This is so sad.

[johnbarnhill]

I always overuse parenthesis just to make sure there’s no possible ambiguity

[jacemandt]

The basic problem here is that it's not an *order* of operations, it's a *hierarchy* of operations. Syntax, both in language and in math, creates structural trees out of linear input. Multiplication doesn't precede addition, rather, they're more closely "glued together" in the structural tree.

I wrote an article about this for the Mathematics Teacher journal that was published in 2017, but I think it's unavailable if you're not a member of the NCTM. It doesn't address the problem of multiplication by juxtaposition, but it does argue that PEMDAS isn't a useful approach, and suggests that students be taught to seek out terms and factors first, and then the terms and factors within those, and so on. This duplicates the hierarchy, rather than suggesting that it should be a linear order.

[bens]

A big problem for learning is the oversimplification of subjects. I think "PEMDAS" is a great example of how making things "easier" can really mess you up when it gets more complicated.

[brutusmagnuson]

This is why I use the heck out of parenthesis. I’m also a programmer, so using brackets becomes useful, or everything turns into a mess.

[SteinGauslaaStrindhaug]

My take as a programmer is it's parenthesis first and if it's still ambiguous, you haven't used enough parenthesis. Relying on order of operations which differ slightly in almost all programming languages (because a language syntax will end up having one even if you don't think about it because _something_ has to be evaluated first); is madness!

[georgeparkins]

Mathematicians, like writers, all too often need to understand that they're responsible for communicating their ideas clearly. If a sentence is unclear you don't blame the language, or blame the reader, you rewrite it.

[minimumlikelihood]

Being clear, unambiguous is the most important. I never hesitate to add a redundant parenthesis, either in papers or in programming. Also, in a lot of cases, units force the order of operations.

[christoskettenis]

The problem lies in the fact that the slash doesn't make clear what is being divided and what's not. As an engineering and high school tutor I despise the slash and/or obelus symbol when used in grades older than third. Use ONLY fractions for division, which will also assist with simplifications before multiplication because it makes the numbers multiplied smaller. If you must use the "kid" symbols, then make sure that everything divided is in brackets indicating the denomator and anything else exists on the numerator.

[RoseInTheWeeds]

When I first learned PEMDAS in grde school literally added my own brackets PE(MD)(AS) in my notes cause it seemed the most obvious way for me to remember their equal weight. Nice to see others agree!

[BrosBrothersLP]

i love the fact that here in germany in the last years of school, and at university we basically just didnt bother. Every division gets written as a fraction and that way no ambiguity ever occurs

[martinepstein]

I think the true but pedagogically wrong answer is: if you're solving an actual problem and not just doing an arithmetic exercise then ambiguous notation doesn't matter so much. You'll know whether a/bc means (a/b)c or a/(bc) by knowing where the expression came from.

[zachtemoro]

You don’t understand how happy I feel after the gaslighting of telling me I’ve been doing this problem wrong. It’s not that hard to understand left —> right and that M & D and A & S are equal, however I had people tell me I’m wrong and that I “didn’t follow PEMDAS”. I’m not a math major, but math has always been my strong suit, so to have so many people incorrectly correct my math was extremely confusing

[nataliaborys]

As someone who's not a native english speaker and was not taught math in the english-speaking countries...
I never knew there was an acronym for it lol. My teacher just said "First brackets, then multiplication and divison, then addition and subtraction." No silly acronyms, literally just the rule, and many example exercises to solve until it's kinda like muscle memory.

[Polar_Onyx]

throwback to when my elementary teacher told me you can't subtract more than what you started with. I mean why not instill a deep sense of academic distrust in students when they first start doing school, nothing could possibly go wrong.

[imacds]

In my elementary school we had this program called AM (Accelerated Math) where computers would generate custom math tests for students to progress at their own pace. I got really far ahead in 3rd grade and encountered order of operations, which normally isn't taught until 4th grade. My 3rd grade teacher misremembered PEMDAS and explained it to me as a strict order, so multiplication before division and addition before subtraction. It was really annoying getting scores of 28/30, and I didn't realize what I was doing wrong until it came up in the main curriculum a year later.

[PyroMancerk]

Teach them correctly from the start... Wow who would have thought!
It reminds me of back when I was in school and I had an interest in math so I was learning some more advance things at home than our teacher was covering. When I pointed out that you can have a larger number subtracted from a smaller number in math class and it would be negative the teacher just told me that we weren't covering that now as it would be too confusing to people.

Over the years though I've come to think the teacher was completely wrong. Having your students learn "truths" that are false does far more harm. With any sort of learning environment the goal is to build on the lessons that came before. But when you basically throw out the lessons of what came before, or do it in a different way that ends up confusing people. It's like with algebra where some of students have trouble grasping that there are now letters in math when really it's something they have been doing already. In grade school you get a problem like 3 + __ = 7 and you are taught how to solve it. Now that same student gets to Algebra class and sees 3 + x = 7 and they are like WTF there are letters in math, this is complicated.

The constantly changing rules not only adds complexity to those who may already be struggling but it also serves to highlight to all of the students you can't trust your teachers. They are just there to push what's written in the book which itself is potentially questionable. I've had classes where teachers told us not to look up answers in back of the book because they were not always correct, they claimed it was done to catch cheaters but I suspect it was done because of the order of operations being done differently. But at the end of the day it creates a sense of distrust between layman and supposed experts in their fields when the average layman's experience comes from school which had many cases of contradictions and down right falsehoods.

[joda]

I once taught a first grader about how to use negative numbers in about 5 to 10 minutes,
and he proceeded to calculate problems using them correctly. All it took was for me to say that:
Positive numbers are basically like arrows pointing from 0 to the right, and negative numbers just point to the left.
Adding any 2 numbers means to put the base of one of them to the tip of the other, and see where the new tip points.
The best thing about this is that it even works for euclidian vectors and is very intuitive for a child to learn.