Order of Operations Debate

8/11/2012

I have never introduced students to order of operations. I have always received students after their initial introduction. I always feel like receiving students after their rote memorisation of BEDMAS (or PEDMAS if you call 'em parentheses, or are very apologetic about your kin), means that when I try to let them explore they just default to what they memorised, but they cannot recall that after the lesson. It makes me feel stuck between a rock and a hard place, but I still wonder what the initial introduction can look like.

Initiate a Debate

One introduction that I have just thought of this morning, I'd like to try. It's heavy on symbols, which I am afraid of, but I think it is basic enough that it might spawn some conversation and debate. I just want to share it and see what you all think. I like it because I feel it introduces students to some manner of proof, and following logical principles, and I think it would be very interesting to see students' misconceptions about the order of operations. Here goes:

Premise 1
Ask the class what is 5+12? What is 12+5?
Can we agree that 5+12=12+5?

Premise 2
What is 3x4? Can we agree that 3x4 is another way of writing the number 12?

Conclusion
I am going to write something on the board. Tell me if you Totally agree, somewhat agree, somewhat disagree, or totally disagree...

5+3x4=3x4+5

Have students get in groups and say, "give me a knock down reason why you think you are right."

The students that I imagine

The agree-er- This student believes it, and would focus on the fact that 3x4 is another way of writing 12. This student would be good to play devil's advocate with.

The choose your own adventure - This student says, "If you decide to multiply first it can work, if you work left to right it doesn't." This is pretty easy to counter, ask them to try both ways on just one side of the equation. You get that 5+3x4≠5+3x4, "That seems ridiculous doesn't it? So which one should we choose?"

The reader - We read a book left to right, so we read math sentences left to right as well. This student gets that we need to have a common way to solve all math problems, but does not quite understand the separate operations. (This is the student that I need help with!)

Uncharted Waters

This is where I have never been and wonder how worthwhile this exchange would be. Would students logically piece it together? Would most students be an agree-er, a choose your own adventure, or a reader? How would I encourage each of these groups to prove themselves and come to a consensus? What questions would you ask your class in this discussion or is this just a bad introduction?

What I Hope Students Can Learn?

I have been thinking that I would much rather introduce Order of Operations, as terms rather than BEDMAS. This would prepare them for algebra, and would provide a more conceptual understanding of the process. I do not know what the progression would look like, but I assume that beginning with multiplication and addition would be the natural place to start. Where would you go next? Subtraction? Maybe you are thinking, "Timon you are CRAZY, this is HOW you should do order operations." Please tell me about it; I want to know.

6 Comments

Matt Vaudrey link

8/11/2012 02:44:22 pm

I like it. I would have a few equations (with multiple terms) in my back pocket before beginning the debate. Taking student responses after each round, introduce hairier and hairier problems, with discussion and response each time. By the time you're done, you've built conceptual reasoning, order of Ops ability, and critical thinking skills, all before lunch.

Timon Piccini link

8/12/2012 03:57:29 am

Ya, I guess the tough part for me is that it is a notation. I could have the very same logic which runs like this...

Premise 1
3 x 12 = 12 x 3

Premise 2
2+1 = 3

Conclusion
2+1x12=12x2+1

I guess the discussion goes, "Well which one is right?" And then discuss why a universally accepted method for calculating these is necessary.

8/12/2012 04:02:13 am

I like this http://mathforum.org/kb/message.jspa?messageID=1132320 .

Kelly Holman link

8/16/2012 05:05:51 am

Another way to do it would be to have a multi-step word problem, and ask students to write it as one number sentence. Now, if they follow somebody else's number sentence, will they come up with the same answer?

Maybe I'm crazy, but I've thought about just skipping order of ops and teaching a little algebra first. Then when we substitute numbers in, what order do we do things. The order of ops appears arbitrary, but it's actually inherent in algebra. In 3 + 2a, 2a is a term which must be resolved before you can add it to the 3. I also point out that 2 is physically closer to a than it is to 3, which serves as a visual reminder. I would save the other rules for when we actually encounter them in algebra.

It's always bugged me that books teach order of ops as pre-algebra, at a time when it has no context for students. Maybe the newer books are different.

8/18/2012 05:01:37 am

Alright, Mr. Picc. I tried this with my 8th Graders yesterday. Here's how it went down.

First, I went through the two premises... premisi... the two statements and conclusion, and the class gave me a look that said "Yeah... we did this last year." Which is true; they did. We went on to try some more challenging problems, check with your neighbor, yadda yadda. It's the first week of school and want them engaged more than I want them challenged.

With my Algebra Concepts class (8th grade students taking Pre-Algebra again under a different name), I gave them
5+3x4=
3x4+5=
and said, "Do both of these and tell me what you get."

The discussion that followed was perfect: Some got 60 and some got 17 and both explained very clearly how they got it. After the 17-ers explained, my 60-ers nodded and "ohhhhh"-ed and explained very clearly how their mistakes happened.

Then I spun it again:
2+4+3/3=
3/3+2+4=
This time, they saw the risk and divided first, explaining it to their partner like they've known it for years. We tackled harder and harder problems, then played Order of Ops Bingo.

All in all, your idea blossomed into a great lesson.
Again.
Keep it up.
~Mr. V

8/18/2012 10:18:14 am

Great to hear Matt!

I love that the discussion worked. I think it will be a great review of the order of operations. I'm going to do the same with integers. I look forward to it!