Please Excuse My Dear Aunt Sally: Passion in Math Class

From Google Images

“Please Excuse My Dear Aunt Sally” or PEMDAS is a new concept being introduced to the fifth grade students this year.  It’s been a few years (okay, more than a few years) since I have last worked with this concept, but I remember the mnemonic as standing for “Parentheses, Exponents, Multiplication, Division, Addition, Subtraction”, aka “Order of Operations.”  Not only have we learned this, but we have also learned about brackets and braces as well.  Some problems that we have encountered have been relatively straight-forward, while others have been on the somewhat tricky side.  So, several nights ago, this was one of the problems on the homework:

24 – (8/2) + 6 =

I always do out the answers before starting to correct the problems.  I calculated that the answer should be 26.  Sitting down to correct the work on Wednesday evening, the first student had 26 for an answer.  So far, so good.  But the next five students had 14 for an answer.  Clearly, there was a difference of opinion on the correct answer.  Since, I am always subject to making my own errors, I wasn’t sure.  Since I have been getting ready for an observation, I was recently reading over “22 Verbal Events in a Class that Lets Kids Get Smart.”  One of the suggestions was to for students to take initiative to explain another student’s thinking, including how they might have made an error.  So, I decided not to make that answer right or wrong and put it in front of the class for discussion.

On this rainy Thursday, I started our math class by putting that problem on the ActivBoard and put down 26 and 14 as a possible answer.   I asked for volunteers to explain how someone may have arrived at 26 for an answer.  “You divide 8 by 2, which is 4, then you subtract 4 from 24 which is 20 and add 6 which is 26.  As soon as that student finished, over half the class raised their hands, saying “no, you do addition before subtraction, remember PEMDAS.  Addition comes before subtraction.  So, 8 divided by 2 = 4, 4 + 6 = 10, and 24 – 10 = 14.  Another girl raised her hand and said she had a similar question on this problem the night before and her father had looked it up on the internet, where he said that addition and subtraction are treated equally and one should do the operation from left to right.  More mass confusion rang throughout the room.  But, you taught us PEMDAS, addition is before subtraction.

Quickly, I decided to phone a friend, in this case our math specialist Mrs. Link.  She shortly came downstairs and respectfully listened to the “26” camp and the “14” camp.  She listened to the student who talked about her dad’s research and nodded that she agreed with that theory.  Again, more confusion.  Always diplomatic, Mrs. Link announced to the class that was going to consult with some other math experts and see if she could clarify the confusion.  Off she headed, and we tried to settle down to do some place value work.

After about five minutes, Mrs. Link returned, a book in her hand, to deliver her “ruling.”  Everyone waited for her to open up her “Math on Call” book.  Opening up to section 208, she started to read the definition that multiplication and division are treated equally and are solved from left to right as are addition and subtraction.  A learning moment for us all for sure.  I thanked her for her work in searching out the truth for us.  She headed back upstairs again and we were left to work on some addition and subtraction problems before lunch.

Since lunch was next, I headed upstairs to talk with Mrs. Link.  She commented on the students’ passion for discussing math.  Too often, the students just sit there passively and are not actively engaged.  Not these 20 students (3 were out today).  They were passionate in their views, they challenged each other’s views, and they willingly admitted (as did I) that they were confused.  I felt that this was a learning moment for us all.  Sometimes, it is not about being right, it’s about learning from one another and being passionate about your views.  I don’t think that any one of us will ever think about PEMDAS again without thinking about this great learning experience.

 

1300 – 446 Continued

One of my students was so inspired by yesterday’s conversation in class, he went home and penned an essay on the topic.  Here’s what he has to say about learning math.

I think that the old way of teaching is not as good as the new way.  My thesis is “Thinking is better  than just memorizing.”   My boxes are “moving up to a higher math,” mindful and mindless, and how new math is better.

Moving up to higher math includes algebra, calculus, and trigonometry.  In some algebra, you need to replace numbers with letters.  In the old way of teaching math, the problem is that a letter can be any number, so there is no memorizing.  I think this is important because you need to be free, not held back by memorization.  In the new way of math, you are more free to think, so it is easier to guess.  With trigonometry and calculus, you need to guess with proofs and calculations.   With calculations, they aren’t on pieces of paper so you need to make them yourself and the old way teaches you just to know equations and that is being mindless.  This leads to my next topic.

Mindful and mindless are two opposites.  Mindful is when you’re thinking and fully awake.  The new way supports mindfulness with showing your work.  This is important because if you memorize a fact and then forget it, you wouldn’t know how to do it.  Mindless is like the old way of doing math, just knowing a fact but never really knowing how it works.  Of course it’s more efficient because you don’t have to think, but in school, you are supposed to think.

Lastly, new math is better.  For example now people are learning harder things in lower grades.  I think this is because people can use more ways to solve problems.  Now new ways are always being made because people are encouraged to think freely and be let out by the chains of the old ways.

My conclusion is that even though the old ways are efficient, you can learn more by being free.  Sometimes mixes of old and new can be both mindful and efficient.

Pretty insightful for a nine year old, wouldn’t you say???

1300 – 446

"The Thinker"

Its funny how on some days, it’s hard to figure out who is the teacher and who are the learners in the classroom.  Take today for instance.  We had a discussion about the math problem 1300 – 446.  I had a handful of students come up and demonstrate different strategies on solving this problem.  One student broke the 446 into 400 + 40 + 6 and then did 1300- 400 = 900, 900 – 40 = 860 and 860 – 6 = 854.  I said to the students, see that’s the difference between you and I; I always saw 446 as just that 446, and you see it as 400+ 40 + 6.  I then put up the problem in the only method that I would have ever attempted it, the traditional U.S. algorithm for subtraction.  As I crossed numbers off to make the top row 12/9/10,  and performed the algorithm I’ve used 1000s of times, I turned to the students and said, “this is the only way we were ever taught and the only method that would have been accepted to use.   I didn’t have a clue on what it meant, I just knew how to plug in the numbers”  “But,” one student said, “you clearly know what to do.”  It suddenly dawned on me — do we want students who “know” what to do, or who “think” about what they are doing.  I explained, along with another adult in the room, that we could use the   algorithm very well, but when we both got to Calculus, we fell apart.  Just being students that knew what to do didn’t serve us well when we needed to think about what to do.

This all ties in with my thinking on mindfulness and mindlessness.  Do we want students who are simply going through the motions in a robot-like program?   Or do we want to go with students who are mindful, who think creatively and like to problem solve?   Who think outside the box?   Who can be the teachers in a classroom?

And BTW, after watching my teachers, I put two and two together with my problem — when I did all my “borrowing” I was really writing 1300 as 1200 + 90 + 10.   It’s fun being a learner along with my 20 teachers in Room 305B.  Thanks Coaches!