Procedural Fluency

This topic skates around the edges of the calculator post. The biggest thing to remember about teaching mathematics at any level is that each topic, concept, chapter, or course is part of a continuing story. The exploration and discovery is rooted in time and never complete. In order to share that story effectively with our students, yes, we must rely on the procedural fluency because it becomes part of the larger topics of number sense and reasoning. If students cannot perform basic operations independently, mentally, then they have an even greater challenge of making sense of the story. How is it that the operations can be boiled down to addition and multiplication? Yes, you need to understand and be effectively using procedural fluency to investigate this. But think  about the kinds of problems you would have students consider as they investigate. Would they be the longest, most-trying, paper and pencil problems so that students get lost in the procedures instead of the concepts? Or would they be carefully arranged problems that allow for all types of exploration and visualization to better develop that number sense? Yes, we are going to need to teach mathematical procedures in our classrooms. But not every one and not every day. Some of them need to be discovered by the students so they can make valuable connections with the teacher's orchestration.

When I think of the eras of testing that I have seen in my career so far, I remember that, for less confident mathematics students, teaching procedure after procedure didn't allow students to remember how to solve math problems. They simply taught a list of steps that students were supposed to memorize. Can you even imagine having to memorize a new list of steps a day, a week? No wonder kids can't keep the "tricks" straight. I was a good student but because I studied really hard, and it didn't make me more knowledgeable about the concepts and connectedness of math. If my father didn't show me exactly the way the teacher had in class, I was lost. I only learned how to teach math, to myself and my students, by planning and teaching math to my students. Through their eyes, I was able to understand my own struggles.

So procedural fluency? Important, but not nearly as important as conceptual understanding. And tools should be provided to aid in the discovery of the wonderful story of mathematics.

Look at all of the connections to other math concepts in this short video.

http://www.youtube.com/watch?v=DK5Z709J2eo

Why can't I use a calculator?

Since we have been focusing on my current graduate class on technology and since I am a math teacher, I thought I would connect the two in another way. I have two elementary school children and one middle school child and they frequently get homework (more on that topic's irrelevance at another time). Often, their assignments ask for long, complicated multiplication and division procedures that have no connection to any other context. So why do this? Well some would say that there is a need for procedural fluency, and I understand that point. But if the focus is on when to perform certain operations in a context and not how to do them, then isn't knowing when enough? Don't we all, in this day and age, carry with us a device that has a built-in calculator? Remember that the calculator is only as smart as its operator. If the student performs the incorrect operation for the context, the error should be easy to determine based on the problem's context. Wouldn't we have the answer to "do the students understand" based on their calculator answers?

Fast forward to my middle school student. Think about skills like percent of a number and proportional reasoning. How many of us carried tip cards or use cell phones in order to save time (and our tired brains) to compute tips? As long as we know what to do with the calculator, it makes sense to use it as a helpful tool. Now, he had very little math homework as a 5th grader (especially in math) and he attends and 5/6 middle school, so maybe it will be in high school that he will again be responsible for math homework. But for now, if a student asks can they use a calculator (in a non-procedural fluency context), or if they are carrying their smart device, why should they be prohibited from using it? We know they are carrying phones, why not let them use them? If they are incorporated in the lessons, students will be much less likely to "play" with them. We would be teaching them a lesson in social ethics for technology as well as mathematics. We want to model appropriate behavior with technology as well as to allow students to incorporate the mathematical practices into their daily lives. Like their phones...


About calculators in elementary classrooms - from Vanderbilt University - Nashville, TN
http://bit.ly/158a6iX
About calculators in an eighth grade class - from University of Nebraska - Lincoln
http://bit.ly/1dbxhfn
Math Apps for iPads - from edtechteacher.org
http://bit.ly/11wssNJ

Comics
http://www.oaknorton.com/comics/1-calculator.jpg
http://bit.ly/12TCPJK
http://bit.ly/11wtY2x

Funnily (and I checked, that is a word) Enough

Just saw this in my email from Education Week

	TODAY'S TOP STORY

 

Free Online Content Forces Publishers to Adjust The growing use of open educational resources is leading publishers to blend free materials into their products and services.

 

Again, why are some of us paying when we should be sharing and collaborating?

WHERE GOOD IDEAS COME FROM by Steven Johnson

This might help remind you that collaboration is key to moving forward. We can be more connected now than ever before. We should be professionally sharing ideas, thoughts, inventions, innovations with each other without caring who "owns" it. The best product we can make might be something we all own.

Funny Maths lesson

Here are the dangers of procedural teaching without concrete understanding. Math is a story that continues to build from grade to grade and topic to topic. Students need a concrete understanding to build to an abstract one that can transfer from year to year, content to content.

Kid Snippets: "Math Class" (Imagined by Kids)

Think about the lesson plan needed for this...

What is a teacher's professional responsibility?

Ok, so I have been reading about a website (there's probably more than one) where teachers can go and pay other teachers for lesson plans and lesson ideas. I am a classroom teacher and I know that teacher's do not make the big bucks. But we all knew that before we went into this profession, and many of us chose our profession because we want to make a difference in the lives of young people. Our district curriculum offices hire teachers during the summer to write curriculum for our courses as it is needed (and with the coming of the Common Core, that's a lot of needed curricula) and these teachers apply and are accepted by the content area offices to write what is needed. So essentially, our best teachers are selected to share their expertise and knowledge about delivering a solid educational experience. Their own ideas, lessons, and activities are willingly given over to the district to be published in the curriculum guides. Yes, the original ideas belonged to the teachers, but most of the teachers that voluntarily apply for curriculum writing understand that they would be willingly giving up their "ownership" of the ideas to have it published and vetted across the district by other teachers.

And isn't that ultimately what we want to happen as teachers? Don't we want our ideas to be recognized and praised and used by our colleagues? Don't we need feedback from others in order to facilitate professional growth? Don't we need to share ideas and work collaboratively as teachers? Isn't that part of the profession? Notice I did not refer to it as a "job." Teaching is a profession and we garner more respect when we refer to it as such.

Think about doctors for a minute (not the pay, it will make you crazy). If you have a health issue, you generally see multiple doctors. You will want those doctors to share their expertise with each other, to gain and accept feedback and suggestions from the other doctors, to be working collaboratively to best benefit your overall recovery. We don't pay doctors to meet and share information and doctors do not pay each other to work together for a patient, it is expected of the profession.

Ultimately, the students (all students, especially in the era of Common Core) should "own" our lessons and plans and ideas. That is how we know they have learned what we wanted them to learn.

Here are some free lesson plan links. Remember, though, a teacher still needs to plan how the lesson is going to work for his/her own students and fit the curriculum as well. Even though you will find a "lesson plan," you will still need to PLAN to make it your own.


Thinkfinity - blogs and lesson plans for free

Here is a link to NCTM illuminations, which has a strict vetting process for teachers that wish to provide math lessons.

http://illuminations.nctm.org/Lessons.aspx

This link is geared more toward supervisors, but may have some helpful information for teachers.

National Council of Supervisors of Mathematics

Here is a link to one of my favorite comics (only the math ones). My favorite one is about a math atheist, but that's for another time.

Calvin and Hobbes math comics site

Here are links to fellow educational bloggers of interest (to me and hopefully to you as well.)

http://rigglesmath.blogspot.com/

http://elementaryschooltech.blogspot.com/

http://tech2013leader.blogspot.com/