PEMDAS? Is order of operations the only way?

This year I have had quite a few students who are pushing themselves mathematically and doing a lot of great thinking.  This also equates to making me think as well, and I really need that.  Now a student has me thinking about PEMDAS, how it is taught, and how it is a coveted norm of mathematics.

One thing I have always struggled with when talking with students about PEMDAS is how to make it meaningful for students without pulling out the “This is the way it is” card.  I don’t want it to become a memorization practice only, I really want students to understand why we do operations in a particular order.

One thing I have used and students understand (not surprising because I adopted this because of a student) is that multiplication implies “groups of.”  I’m sure many pure mathematicians might frown upon this, or the fact that it also leads to repeated addition, but students really grasp the concept and start visualizing order of operations.

3×4+2

3 groups of 4 plus 2

4+4+4+2

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When students think of it this way, they don’t want to add 4 and 2 and then multiply that by 3.  This is also true when they see 2+3×4, the “groups of” thinking prevents them from adding 2 and 3.  Is this just replacing a trick with a trick?  It took me a few years to even accept this type of thinking for my students- I don’t want to provide a new crutch for students.

This all leads me to the other day, when a student was solving the equation 3X + 2 = 8.  He was unsure of what to do, so I asked him what step he would try.  He repled: “I would divide by 3 since I want to solve for X.”  I immediately balked, that answer went against all of my mathematical fiber, but I tried to not show it visibly.  I told him to go ahead and try his idea.  This is what was produced:

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Wait now!  Hold on, that is the answer I expected, but not attained in the way I assumed it would be.  I can’t tell you how many times I have used the last in- first out type of thinking with students to solve equations, and I expected seeing something like this:

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But all of these provide the same solution!  Is that a fluke?  Some weird oddity that can’t be reproduced?  It had me thinking of possible counter-examples and what else it could imply.  Does order of operation last in-first out thinking absolute in solving equations?  Or is it something like the standard addition algorithm that we accept as the best method and ignore others?  My student gave me another example to consider later that hour.

This problem involved the area of a trapezoid.  They wanted him to algebraically solve it for b2.  I was very curious what he was going to do, and instead of starting with the parenthesis, so subtracting b1 (which is what I assumed he would do because of his previous reply), he told me he needed to distribute before solving.  His work was this:

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Which again produced the correct solution after he cleaned up the compound fraction.  Once again his work produced the right answer without doing the expected procedure:

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So currently I am rethinking approaches to solving equations.  How will not using the standard approach effect him later on?  I can see how trying some sort of variation on this with powers will be messy.  I’m totally open to suggestions and comments on this, as it has my brain working on overload ATM.

Where our youth go…

I haven’t been on the blog horn or even Twitter that much other than #MSMathChat, and that is because right now I am struggling a bit to keep my head above water this year.  There is a lot going on (and taking Grad classes on top of everything wasn’t a good call).

This post is gonna be short, but it just hit me yesterday and I wanted to get it on my blog so I have it and remember.

This week has been fairly slow, I have had 6 new students come in and 6 leave.  I’m getting used to that, it really makes trying to keep a cohesive classroom hard- but I have been managing.  That’s not the factor that hit me, it’s this one:

70% of my current students are return students.

Normally high school teachers don’t bat an eye at that statistic, in fact they expect a lot higher one, but not when you are teaching in a Juvenile Center.  That number means that even though these students get a grip on their lives while they exist within these walls, they can’t maintain that when they go back home.

As a parent of a 5 and 8 year olds, I am getting more and more sensitive to factors that influence their lives, their behaviors, their choices.  When I talk to my “returning” students, they have ALL told me that they go back home fully intending to keep out of trouble, yet they fall into the same group negatives or they can’t cope with the bad family environment at home.  As a teacher and parent, this makes me immensely sad.

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(this is a random picture taken from the WWW, it is not any student that is enrolled  in my school or staff that works at my school)

Our youth make their own choices, but many are too unsure of themselves to be confident to walk their own path individually.  They need their friends and family, and if those happen to be a negative influence on them- returning them to that environment is setting them up for failure.  I don’t know what the correct answers are for this issue, they go beyond my scope of expertise or experience, but I am beginning to believe that in order to truly change things for these students then those outside factors also need to go through the “treatment” processes these children face.

We can’t keep placing them in the same situation and expecting different results.

All children deserve to be loved, and have a safe positive environment to learn and grow in.

What exactly do you mean, MY students?!

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High stakes standardized testing, how we all dread those words.  Most of our students aren’t overly fond of them either, no matter what incentives or snacks you throw at them, it is a time where they are evaluated on what they know in conditions that are so unlike their classroom it’s laughable to think that we attempt to measure student knowledge in this manner.

Sitting on committees of these tests and seeing proposed test questions and data from piloted questions is an interesting experience.  If you haven’t had the opportunity to do so, please do- your state and testing company needs to hear your input on what students are exposed to.  These tests mold students’ personalities over the years, it tells students if they are smart, or dumb, and it even tells them specifically what areas they fail in.  I realize I am being overly dramatic and negative right now, but I want that to sink in for those of you reading this.  No matter what we tell them about those scores, no matter how positive anyone is about testing, constant reminders of shortcomings and failures build up.

Most state education websites have a place for you to sign up for these advisory committees, and I would strongly suggest participating in at least one during your teaching career.  One reason I suggest this is: you get a great idea of beliefs of teachers, testing officials, your department of education, and how students are viewed at these meetings.  Most of that stuff is not pretty, at all as long as you don’t have your rose colored glasses on (like I did the first time I attended one).  I have been on three of these so far, and for some reason they keep calling me back- which is good for the students I work with because they have no voice at these meetings and are in no way represented or considered if I don’t.

This is where the problem happens.  For these committees, there are typically a small number of teachers representing different demographics of our state.  I represent a small school (less than 350 students) and teach a large number of Native American students.  Typically I find that other than small school connections, the other teachers really have no idea how my students approach or think about tests or test items.

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When we look at data, we get the problem, the answers, data about how students performed on that pilot problem, and rationale about why the answers were chosen for the problem (on multiple choice questions).  The biggest data they look at is a value that indicates how difficult a problem is. They also have an indicator on how “relevant” that difficulty score is- meaning is it just guessing or not.  The problem I have is that many of the other teachers in the group were fine with a lot of these questions, but I was not… here’s why.

  1. Many of the teachers I worked with were from privileged schools: middle to upper class white schools OR charter schools where enrollment is screened.  Coming from a background of teaching students of poverty or withing a juvenile justice center, viewpoints on students norms are vastly different.
  2. Many of the questions they considered “hard” or “cognitively difficult” were word problems.  This becomes a test question on a student’s language skills, not math skills or mathematical thinking.  When I stripped all of the language barriers out of these problems, they were not mathematically challenging to solve and I would estimate 80% of my students could easily find the correct solution.  So my question every time was: is there a better way we could ask this question?

I questioned many of these pilot questions for our testing.  They were not especially challenging math problems for students to contemplate; they were problems testing a student’s vocabulary skills, what background knowledge they had compared to the writer’s, and their ability to recognize what facts they needed for the problem and what they didn’t. 

That last statement I do feel is important, students need to be able to determine what they need for a problem and what is irrelevant.  That is a very crucial part of mathematical thinking and problem solving.  The problem I have is when it is shrouded in context that is not familiar or attainable for students.  There was never a time that all students lived in pleasantville, yet those contexts have been are are used for word problems that “relate to all students.”  This thinking needs to stop.  Many of my students don’t have the privilege of their own space to call home, a bed or even a meal to look forward to.  They can’t relate to problems based in context that is a fantasy world for them.

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We, as a math community, need to figure out a way to present mathematically challenging problems to students without providing a language or reading barrier, because that is what we ultimately end up measuring- not their mathematical ability.  I kept getting the the statement thrown at me, “Well Bryan, how many of your students would actually see this problem?”  MY students?  Are you kidding?  They are OUR students, our schools, our communities, our country, our world.  We need to change our thinking about what experiences our students have, how we present problems, and what language we use.

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They are my kids, and I need to stand up for them so they have the same opportunity to demonstrate their mathematical mastery as any other student that is taught in our country.

 

Summer is closing…

I know my twitter feed and blog has collected some cobwebs and dust this summer, and that’s a norm for me.  Summer is a time where I immerse myself in my favorite pass time- my family.  This summer has been extra crammed with the Master’s courses I have taken- just one more year, just one more year…

Things will be firing up here again soon, I hit the classroom the day after labor day.

See you all again soon.

#MTBoS30- Day 22

What’s surprised you the most about your life or life in general?

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The thing that has surprised me the most is that I still live in Minnesota, and have only moved 30 miles from my hometown.

 

It’s odd, when I was growing up I wanted nothing more that to get away from Blackduck, out of the state of Minnesota- and perhaps out of the country.  I wanted to have so many different experiences and go places far from where I grew up, but it has never happened.  It’s hard to pinpoint the how, when or why of the situation but here I am, so many years later living in the same area.

When I was a kid my Dad was a military man and we moved quite frequently.  I was not born in Minnesota, I was born in Virginia- a place which still provides me with a comforting feel when I occasionally (OK, hardly ever) return.  All of our moving and such happened before I started school, but I don’t remember ever wishing that we could stay in one spot.  I did not mind moving, at least as far as I can remember.  We moved to Minnesota before my Kindergarten year, we returned to my Dad’s family and his hometown.

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Growing up 6 miles outside a rural town (population less than 500 for most of my life) that consisted of loggers and farmers, life was always simple.  Trips to town were big occasions, and that got tougher on me as I grew up.  I didn’t get to attend birthday parties or do sleep-overs, I typically didn’t have a ride to get to any of those places and I had chores around the house to do.  When I was old enough to play baseball, you would find me pedaling that 6 miles on my 10 speed across gravel roads to and from practice.  I can’t honestly tell you which activity gave me more exercise, baseball or biking.

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When my brother got old enough, he was allowed to drive the old, old pickup.  It was the old 3/4 ton Ford that we used to haul our pickup bed camper and ourselves across the country when Dad was in the military.  It was a stick, beat up, but it got us around.  It was great until- I realized I was the tag-along younger brother.  I was not allowed to go with my bother often, although I knew if I put up enough of a fuss I could have had my parents force my brother into taking me more.

I ended up inheriting that truck when I got older, and things seemed fine.  I loved driving around- it was a freedom from the farm.  I often thought I was driving across the country, traveling from one coast to the other.  Sad thing is- that has never happened.

I applied to many different colleges when the time came, and I was accepted to almost all of them.  I finally could have left this place- it was what I always wanted- but fear kept me back.  Fear kept me from doing the thing I had wanted so much throughout my teen years.  I grew up with no car, no real job outside the farm.  I had no confidence that I could support myself or even get around if I ventured too far.  I planned on going to the University of Minnesota until I found out I was on a long, long, long waiting list for a room.  I ended up attending hometown BSU and the rest is what makes me who I am today.

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One thing I keep telling myself is that there will be a time when I can go to all of those places I dreamed of, but was too scared to visit.  At the very least, I owe it to my younger self to visit one.

 

 

 

#MTBoS30- Day 20

The power of one…

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While I am taking these classes, I keep thinking about what Special Education teachers are expected to do as well as what elementary teachers do daily.  I think about student need, and how those teachers address that.  I am licensed to teach 5th through 12th grade, but I am not totally convinced on my effectiveness at the 5th or 6th grade level.

If we are so concerned about the message of NCLB, how can we expect our elementary and special education teachers to be experts for all core curriculum?  I am not saying that it can’t be done, or isn’t done effectively.  I am saying that we as teachers all have our strengths.  I am acutely aware of this with my classes and learning how to instruct students in reading.  This is definitely not my strong suit, and I am unsure of how comfortable I can be if I find myself in this role.

We have found that students have gaps in their learning- some of those are due to their individual circumstances at home, some are due to personal application in class, but some are also due to the teachers they have throughout their learning career.

I am not saying that some teachers are bad or inept, but I am saying that we as teachers have our strong subjects, those that we understand and feel comfortable with.  My wife teaches 3rd grade and she is also a math person.  Her group consistently scores the highest on the MCAs compared to her team.  She also works hard with her students in language arts, but is consistently behind her coworker who has a passion for that discipline.  What if a students consistently gets assigned homerooms with teachers who are strong in math?  What will that do to their language arts abilities?  Is there a correlation there?

I also know many parents request a certain teacher for their student on the grounds of “best fit” or personality.  What if that too promotes a deficiency in the student’s learning?  Is there a correlation between personality and proficiency within a discipline?

These questions may seem off base or a stretch, but I think it is something to consider.  Our students need to have the best opportunities available to them as learners.  Sometimes that means instead of going with a family friend as a teacher or one who is strong in a student’s interests or skills, that they need to be led in the direction of the teacher that is strong in an area of that student’s need.  I have to fight with Brayden about reading, and I know other than my love for reading there isn’t a lot of home support on that front.  It has become a chore to get him to read, and Brayden doesn’t see, feel or get exposed to the passion that can come from a teacher who truly loves it.  I want Brayden to get teachers who he will get along with socially, but I also want teachers who can best support his needs- which currently lies in reading.

There is a power in one, each teacher has educational strengths that perhaps we need to utilize more.