Chanllenging Students, Challenging Educators

This summer Apple offered a number of short webinars for educators in their Apple Summer Semester Series.  One of the sessions focused on Mathematics and featured Dan Meyer, a high school math teacher who is attempting to turn mathematics education upside down.  His main focus is selling math to students in a way that is more engaging than the traditional mathematics text.  To do this he poses traditional math problems in non-traditional ways.  Typically by staging videos that get students questioning and inspiring them to look for answers on their own. He also proposes that this format for introducing problems increases the level of challenge for students.

As I think about what I want my future classroom to look like, a number of the themes Dan emphasizes keep coming up.  For example, I want students to learn to question and know what information is necessary to answer their question, and then know how to go about finding the answers to their questions.  I want students to feel confident in their answers, specifically, to recognize when their answers are reasonable and when they clearly are not reasonable.  I want students to understand that math is powerful and can really help them understand how the world works.

This style of teaching, however, I find very unsettling.  If the level of challenge for students is increased, by default, the level of challenge is also increased for teachers. I was not taught in this manner, as a student the instruction I received was overwhelmingly of the skill and drill variety.  I can write beautiful mathematical proofs, and I can solve equations, but I do not feel confident as I watch these videos that I would know how to start some of these problems.  If I don't know how to begin how can I help students see where to begin?  

With that concern in mind, I remembered reading about Thinking Mathematically by Mason, Burton and Stacy.  The intended purpose of this book is to help develop problem solving skills.  So I have ordered this book and in between my regular coursework, I am going to begin working through this book to strengthen these skills. Disclaimer: the problems in Thinking Mathematically are not presented in the raw way that Dan recommends for true exploration, however, I figure as long as I am interested that is more than can be said of high school students faced with a traditional mathematics textbook.  I am hoping that if I can learn to be more confident in approaching this type of problem then maybe I will be able to introduce these sorts of problems into my classroom successfully. 

Supressing Genius

I have been reading "Journey Through Genius" by William Dunham and a disturbing thought occurred to me while reading about Archimedes.  Arguably the greatest mathematician of all time, Archimedes would become so absorbed in a problem that he wouldn't eat or sleep for days.  His family would have to drag him kicking and screaming from his work to bathe and nourish himself.

This is not the only example of madness in this book.  Many of the greatest mathematical minds of all time were what we would today diagnose as OCD, ADHD, Autistic, etc. In their time these individuals were allowed to live out their existence as they were biologically or naturally programmed to behave.  Through the course of living their reality, they were able to develop and discover mathematics far beyond the current mathematical thinking of the day. 

Here is my concern, if we insist on judging individuals who behave differently than we deem "normal" as having a disorder that must be "cured" (ie. drugged) are we in fact suppressing some of the greatest thinkers of our time?

Philosophy of Teaching Mathematics

This semester I will complete the second of two Math Methods courses.  In middle school math methods we began discussing the idea of developing a "philosophy of teaching mathematics" statement.  Here is where I left this process at the end of Spring semester 2011...

·      Mathematics is about learning to ask questions, formulating interesting and relevant questions leads to the drive to find solutions. It is impossible to get a meaningful answer if you have a meaningless question.  So many times all students care about is getting the “right” answer, however, if the questions was pointless or not interesting then the answer does not matter. When searching for a solution to a problem, students should be taught to ask many questions and then evaluate those questions to determine which answers are most likely to lead to a solution to the problem. 

·      Mistakes are not only acceptable but are to be celebrated; the only way to advance mathematical understanding is by making mistakes.  This includes struggling with concepts and ideas, looking at a problem from one direction and not making any progress and having to start again from another side.  Too often students are given the impression that there is always a fast answer, or that there is always a formula ready at hand to use to solve any problem.  When the reality is that many problems don’t fit into a tidy mold with a quick solution.  The more students are able to see that mistakes in math lead to better solutions the more likely they will be to stick with difficult problems and persevere in solving them.

·      Explaining concepts to others helps students better understand their own ideas.  Having to clearly state why a particular method works or how an answer was derived ensures that students have a firm understanding of their ideas.  By describing the method used to arrive at an answer students must examine their own thinking and are likely to catch their own mistakes or find areas of misunderstanding.  A dialog back and forth between students provides students the opportunity to learn to question not only their own answers but also the thinking of other students.  Students should know it’s ok to questions someone else’s answer and continue looking for clarification until they are convinced.

·      The mathematics classroom should extend beyond the classroom walls, not simply by providing “real-world” problems in a textbook, but by providing students problems from other areas of their lives.  Although, there is a beauty in solving a mathematical puzzle just for the sake of finding a solution, there are opportunities throughout the other disciplines that lend themselves to being solved with math.  In particular, I think that math and science should be closely linked in the curriculum and

·      All students have something they can contribute in a mathematical conversation. Students will approach problems from different places due to differing background knowledge.  Because math learning builds on previous knowledge many students will be at different places in their understanding of math at any given time.  However, at any level of understanding, each student will have something unique to offer to the conversation.   

This is a living document, over the course of my final semester of coursework at UNI I will continue to examine and refine this belief statement. 

Smart Phones in School

Image Source: Johan Larsson

I am torn between engaging students with worthwhile apps and preventing students from becoming distracted by the toys at hand. On the one hand there are many apps that are not only appropriate for use in a mathematics classroom but genuinely create an opportunity to advance student learning.  Not only do these provide additional resources for learning but could be used in place of other expensive equipment. Why should we purchase graphing calculators for hundreds of dollars when most students can download a free graphing program on their smart phones. 

However, it is just a simple click away to turn a student's attention from honest classroom work to checking texts or playing games.  I spent the last week observing in a seventh grade math classroom where students were allowed to use the calculators on their phones or itouches when working problems.  Despite a firm warning that any use of the phones for other purposes would result in their not being allowed this privilege, I saw a number of students flipping between programs when the teacher wasn't within their immediate zone.

I want to encourage students to use the technology they have to make sense of what is happening in the classroom, however, I don't want the technology to become such a distraction that students are able to tune out during class.  Hmmm?

ICTM Conference

This past Friday I attended the ICTM (Iowa Council of Teachers of Mathematics) Conference in West Des Moines, IA.  Overall the experience was great and I got a bunch of ideas for implementing the Iowa Core Standards in my future classroom.

The one main idea that I am interested in exploring was presented by Elizabeth Tapper.  She recommended using "Notebook Checks" to replace correcting piles of homework.  I have one professor who uses a similar system, and in that class the he only grades a few randomly selected problems out of the assigned homework problems.  My one concern is that I never know if he happened to grade two problems that I got correct while in reality I misunderstood the greater bulk of the material.  While I'd like to think this is not the case, I really have no way to know if I am completely off base on a number of the topics.

So I attended this session with a bias against a system where only some of the assigned problems are graded.  However, I came away from this session ready to think more deeply about how this could work in my classroom.  Elizabeth described to us the features of her system that overcome these concerns:

1)  All student work, including notes and homework assignments are kept together in one spiral notebook.
2) Students are given the "correct" answers to all assigned homework problems, and then given the opportunity to ask questions or get clarification on missed problems.
3) Students then have the chance to correct or rework incorrect problems.
4) On the back of quiz papers students copy from their notebooks solutions to randomly selected homework problems.  (Here is the key: they are only allowed to COPY work that is in their notebook)

Here is how this system addresses my one concern, students are given the opportunity to review their work and pinpoint exactly where they have misunderstood. As a bonus this system encourages students to rework incorrect problems so that they are responsible for their own understanding.

I may decide to tweak this system as I consider how this could work for me as I plan how to organize my future classroom but I think I can certainly find a way to implement Notebook Checks in my classroom.