I would like to address a convention in mathematics that needs to change; BEDMAS. For those who are unfamiliar with this it stands for: Brackets, Exponents, Division, Multiplication, Addition, Subtraction. This is how I was taught order of operations. However, division and multiplication actually are completed from left to right, and the same goes for addition and subtraction. Therefore, we really need students to memorize also BEDMAS, BEMDAS, BEDMSA, depending on the order of the operations.

Also there could be operations inside brackets, so in actuality all that must be memorized is B(BEDMAS)EDMAS, B(BEMDAS)EDMAS, B(BEDMSA)EDMAS, …

I hope we are all confused as I am.

I have had a conversation recently where I was told, “If we don’t focus on order of operations then 3 + 2 * 10 could be 23 or 50, and there can only be one right answer”. Now without a context 3+2*10 is 23. However, students need to know WHY to put brackets, and when to add sometimes before we multiply. As for 2 examples:

If you are taking a cab which has a base charge of $3, and $2/km, then it would cost you $23 to take the cab 10 km.

If you are planning a party for 10 people and want to supply cookies, which costs $2/person, and cake, which costs $3/person, the total cost would be $50.

I understand that we need a procedure to solve questions that have no meaning, but for students who are just learning a new concept there should always be meaning in their learning.

This idea of memorization through acronyms can been in many subjects:

HOMES – Great Lakes

Mrs. Vasquez Eats Many Juniper Seeds Until Nurished – Planets.

I could go on and on. I am not saying knowing the order of things is unimportant but should not be more important than the meaning and application of the knowledge memorized. For my own students, I would rather they understand why the order of the planets are important for the equilibrium of our solar system than just the order of the planets.The sad truth is that the only time students are going to see a question, as 3+2*10, is when they need to answer a skill testing question for a prize. Are we teaching critical thinkers, or lottery winners?

Great post and couldn't agree with you more. I feel that math in education is only being taught under the "knowledge and understanding" category and students are being denied the thinking, application, and more importantly the communication of mathematics. I've blogged about conceptual vs. procedural teaching that you may want to check out http://wp.me/pXAmd-1a.

ReplyDeleteThanks for your reflection!

Wonderful post. In the 1960s, we memorized by acronym (George Evans Old Grandmother Rode A Pig Home Yesterday and A Rat In Tommy's Hat Might Eat Tommy's Ice Cream), but I was lucky to have teachers who also let us think and figure things out for ourselves. My least favorite words heard in math class are, "Here's a trick you can use to..."

ReplyDeleteNice post DM,

ReplyDeleteA couple of points.

1. I believe the Mrs. Vasquez thing is not an acronym, but rather a mnemonic (can someone come up with a mnemonic for mnemonic?).

2. 3+2*10=50 (it is not 23). Since there are no brackets, the convention is that you perform the operations left to right. Your example of the taxi fare puts brackets around the 2*10.

I admit I am splitting hairs a bit. However, mathematics is a precise language and therefore hair-splitting is entirely appropriate.

Having said this, I think it's a bit silly to spend a lot of time memorising BEDMAS or FOIL for that matter. Why not spend some time understanding what is happening?

You wrote: "I would rather they understand why the order of the planets are important for the equilibrium of our solar system"

Really? AFAIK, even the three-body problem is not understood, let alone a ten-body problem...

Order of operations is not really about arithmetic. If you only need math to calculate cab fares and plan dinner parties, then it doesn't really matter how you write the operations. We need order of operations in order to read and understand algebraic equations. For example, you might need to use the amortization formula in order to calculate the monthly payment on a home loan. This is a very complicated equation, and someone who lacked an understanding of order of operations would not be able to use the formula correctly.

ReplyDeleteI do agree that the acronym BEDMAS and its variants should be junked. We also need to get past the idea that "order of operations" implies a fixed ordering of the steps. For example, it does not violate order of operations to do 3+4+5*6 = 7+5*6 = 7+30 = 37, even though a narrow reading of BEDMAS would say that 5*6 must be done before 3+4.

To play devils advocate, Dave, what is the best way for kids to remember that multiplying goes first ( yes Indy, the answer is 23). I believe it's been too long since you two have taught middle school! Convention is that with no brackets, multiplying and dividing are done first, in the order they appear. This is a topic where a deeper understanding doesn't seem to make sense; I mean, why does it matter which operation comes first, it depends on the circumstance. Every problem is different and it's more important that students understand why the order is important and why they get different answers. BEDMAS is a mathematical rule and one that needs to be remembered. We teach in grade 7 Dave, that the DM is evaluated in the order they appear followed by AS in the order they appear. Brackets also come first, so students understand that. So to sum up, students do only need to understand BEDMAS (and actually, some people use PEMDAS).

ReplyDeleteThough you have clarified about BEDMAS but still i am confused about the concept behind this.

ReplyDeleteBEDMAS or PEMDAS or whatever is not so much about the math as it is about communication - communicating ideas and results.

ReplyDeleteWhy do we have grammar?

Why do we have appropriate ways of spelling words?

Communication!

On the simple examples given above (cab fare and cookies) it seems obvious that the equations listed could only be solved one way. But they only make sense solved that particular way if we write out the whole story problem. Mathematicians are by nature very lazy and do not want to write out every little detail. so to communicate the desired meaning of the mathematical expression, we need standard ways to interpret it. Hence BEDMAS/PEMDAS.

In more complicated expressions that are routine in science and engineering, the order of operations is essential to make sure everyone understands the authors meaning.

So the basic concept that needs to be taught is that communication is important and one of the ways we accomplish this in math is the order of ops.

JaR

Engineer/Math-Science teacher

Colorado

I heartily agree with the overuse of acronyms and silly memory tricks in mathematics. I strongly urge teachers to stop using FOIL for multiplication of two binomials, when the distributive property is certainly well within the grasp of any algebra student who has reached the point of needing to multiply polynomials. In fact, this "trick" becomes useless when one of the polynomials in the product is a trinomial or any other higher order polynomial. Students will recall a "trick" long enough to get an answer on paper, but they will remember the process longer if they understand the concept.

ReplyDeleteWe must teach conceptually so that students learn conceptually and can communicate mathematics verbally, graphically, analytically.

L.Bryan

Math Teacher

Tennessee

Keep teaching BEDMAS middle school teachers! I use SAMDEB to teach algebra in high school. If students don't know BEDMAS they'll miss the beautiful symmetry of it all in high school.

ReplyDeleteFor starters, Indy Lagu, you are wrong, the answer is 23 and it CAN only be 23. The order of operations is not left to right, and in this case adding those brackets makes NO difference. You are simply proving the point of needing BEDMAS because you clearly don't understand order of operations AT ALL.

ReplyDeleteSecondly, acronyms and mnemonics are useful tools for initially learning challenging concepts or for memorization of multiple things. The brain on average can only remember 7 +- 2 things at a time so grouping concepts into one chunk allows for the brain to process and work better. I will agree since division and multiplication are preformed together it can be confusing, but that is all in the teachers ability to explain math so students understand.

Lastly, Dave your party example describes a COMPLETELY different equation and it would look like this (2+3)*10 = 50. For a math teacher that is simply disappointing.

End Rant.

Hello friends,

ReplyDeleteA fellow teacher sent me the link to this post after a lesson in my classroom. I had no choice but to teach order of operations. My grade 6 students were involved in a group problem solving activity. One group could not understand why their answer was not the same as the group beside them who did the same thing but they did it on the calculator that corrected for order of operations. It lead to a lesson on the order of operations and how important it is to do things in order that makes sense. So I say...(and I speak only as a math lover and teacher, but I am not an expert.)..... 1)Teach the acronym if you want and don't if you don't. 2)Teach math in real contexts. The two CAN coexist. Talk about making too much out of nothing. I love math because it makes sense. The arguments above seem silly to me:)

Wow. It's sad how few people here understand how math equations are read and understood. If a mnemonic helps someone to understand a concept, then it's a good tool. Math is math anywhere on the planet and there will only be one correct answer in any of the above equations. Brackets have two purposes: 1- they help to make a formula easier to read (is this case they are not required) and 2 - they indicate that the operations inside are meant to be done against the natural order of operations.

ReplyDeleteYou didn't even solve the second example about the party properly. The correct equation for what you are solving is actually:

ReplyDelete(10*2) + (3*10).

So no....23 and 50 are not correct for the original expression AT ALL.

If 2 + 2 always equals 4 we wouldn't be where we are today. Mathematical tecniques taught today weren't taught 50 yrs ago & won't be 50 yrs from now. Math goes down better when it's fun. So relax and teach others with kindness. Thank you

ReplyDeleteThe guy that wrote this is a teacher? I sure hope not. If he is, hopefully he's been fired, or only teaches kindergarten. I'm not sure how one makes it through high school, let alone earns a teaching degree, without having a clear understanding of the importance of order of operations. Has the man never learned algebra? If this guy truely is a teacher, this may help explain why American students (and general public, for that matter) lag behind the rest of the deveopled world.

ReplyDeleteAnd yes, I know: Spelled truly wrong, extra comma, etc., etc..

ReplyDeleteMy English teacher was as bad at English as this guys is at math..

Ok, not gonna lie.... Spell check has ruined any ability I once had to spell propery...

I was reading up the history of the order of operations the other day, and I would have to say both David Martin and Anonymous are correct. Computing is largely responsible for the rigid rules of BEDMAS, or whatever you want to call it. Communication is the key here and some level of consistency is needed - therefore the rules.

ReplyDeleteThe use of 2 + 3 * 10 = 50 dollars can be correct depending on the interpreter. However, this is when using brackets for the party example (2*10) + (3*10) is a better way to communicate as to avoid confusion. There is still only ONE answer, but for the sake of consistent interpretation, I would rather use the (2 + 3) * 10 people = 50 dollars or 2 * 10 + 3 * 10 = 50.

Rules are rigid at times to help maintain some form of consistency in communication. However, teachers still need to be ever-observing their students to try to see if they understand what it means. I have attended one of Martin's seminars on math assessment for teachers and his use of oral exams has inspired me to want to use that as a tool for assessing student understanding.

disclaimer: I am a science guy, not a math guy. In science, we really, really try hard to communicate how we got to point B from point A.