A colleague recently shared the below TedxTalk. As I near completion of my Master’s program in Childhood Education and start to think about student teaching and, beyond that, my future classroom and school, this issue returns again and again. Integrating the (yes, capital “A”) Arts, as I have seen all too often in my limited experience, means that students can choose to present their learning through a song or a poem or a drawing – our attempts to include those modalities and learning styles. However, I see these modes of expression and learning used merely in the service of the academic content. The value of the Arts, as used in these circumstances, is secondary to the “more important” (and more useful and profitable) academics. It amazes me when I hear teachers, parents, or adults bemoan our young people’s command of the written word (“Kids today can’t write”), but we rarely stop to consider the idea that powerful writing is not an exclusive function of practical grammar, that art and reflection and self-expression are intrinsic to powerful writing. The Arts remain and are thought of more and more as extra-curricular. Is it better that these modes of communication are at least somewhat included? I think, yes. But the Arts are indentured servants to our obsession with measurable standards and learning outcomes and academic growth. Throughout our brief history on this planet, we’ve strived for expression through the Arts, an attempt to make sense of our world and our experience of it, to create new understandings and perspectives, and to communicate the essence of our individual and collective humanity.
A few days ago, I was driving to campus for class and I decided to take a route I had never taken before. We moved here over a year ago, but I still don’t know my way around town very well. I rarely go out and when I do, I usually use the GPS for directions. I decided to try my internal navigation systems that day – a system that was fairly well-developed in the past. But it seems use of GPS devices and cell phones for driving directions has diminished my personal navigation systems. I assume that when we drive following the directions a device calls out for us, we pay more attention to just the street we need and we pay less and less attention to where we are in relation to the larger context. Well, my atrophied navigation skills got me lost. Not lost lost, just turned around a bit – and I most certainly didn’t drive into a lake. So I did the unthinkable: I stopped and asked someone for directions. It was obvious to me that the man I asked was a bit taken aback and surprised. I can just hear his internal monologue: “Uhm, what the? Doesn’t this guy have a GPS or a cell phone he can check? Weirdo.”
And this simple interaction, which actually was the second time in a month that I had stopped to ask a living being for driving directions, got me thinking about our reliance on digital and internet media. How does reliance on our gadgets affect our self-reliance, creativity, flexibility, and problem-solving skills. Skills we rarely practice will atrophy. With the accessibility of vast sources of information (search YouTube for Do-It-yourself videos and you’ll see what I am referring to), do we sacrifice some of our creativity and perseverance in trying things ourselves before searching for the answer online? I’m not a technology Luddite and I believe that the power of accessibility to information is one of the greatest benefits of the internet. But if we find ourselves faced with a question or a problem, how quickly do we simply “Google it” instead of problem-solving first? What are the implications of our unthinking reliance on our gadgets? I’ve learned more from my mistakes, failures, or repeated attempts than from my successes. If we do not allow ourselves – and our students/children – the multiple opportunities to fail, what do we learn? If we take away instant access to answers, do we simply redefine the problem in question as irrelevant, unimportant? or do we persevere and try and fail and try again?
Read this. Then read it again.
Strive for it. Every day, work toward it.
This math methods course is my fourth methods course at Nazareth College. I have previously taken a social studies, a literacy, and science methods course. There are a few ideas that each of these courses – as well as one or two courses where the ideas were briefly engaged – have in common. Two of these ideas that I would like explore are: (1) the idea that each content area specialist advocates strongly for an emphasis on their particular content area; and (2) the idea of the importance of integration across content areas.
It is not surprising that professors who teach methods courses in a given content area and who were, generally, specialized teachers in that content area, often emphasize the importance of their specific content areas. And I realize that, like younger students, adult learners have talents, abilities, and proclivities for certain content areas – I certainly enjoy and have talents in the creative arts more so than for some of the other content areas. It will be likely that in my future classrooms, I will use the creative arts more so than teachers who might not be so comfortable with them in their own lives – though I can almost guarantee that music (or at least “music” that comes out of my mouth) will not have a prominent place in my classroom because of my utter lack of ability in that art form. This emphasis is reassuring in that it allows for teachers to bring their talents and interests into the classroom. It also reinforces the idea that when we teach what we love – and students are able to see and appreciate our enthusiasm – students are not only drawn in by our interest, but they are also exposed to an adult member of a community of practice and initiated into that community through the practice of the norms and standards of that community. There’s a cliché that states that students do not care what teachers know, until they know that teachers care. I assume this generally refers to students being aware of teachers caring about them, but I believe that this would also apply to content areas. A teacher who disdains reading – or any other content area or field of study – cannot engage students in that content area or get students excited about that field of study.
Reflection on the idea of specific content areas being stressed or focused on – a sort of themed pedagogical style – necessarily leads to the idea of incorporating all content areas into the focused area, as well as the idea of integration of all content areas. Some content areas have a more natural-seeming and obvious association. The pursuit of inquiry-based science without the use of mathematics seems impossible to me – imagine observations and notation and measurement and representation without the use of numbers and math. Likewise, social studies and history are filled with dates (as place markers) and patterns and statistics. Other content area pairs might require a little deeper probing to reveal the associations. Math and art are not typically seen as associated. There are, however, numerous examples of possible connections, e.g., patterns and tessellations, mandalas, fractals, the use of the physical tools and mathematics by artisans/craftsmen to create works of art, etc. The integration of content areas – in opposition to the idea of content areas in isolation – allows students to realize that these academic areas are related and connected, provides a broader context for the study of and in these areas, promotes a deeper understanding of the various concepts, and develops a more internalized realization of the value of the various concepts because they are encountered and explored in different but related ways.
There is a proof on Brainfood website that shows an interesting and intriguing proof for the following: 1 = .999…. This idea and its proof have been troubling me, causing me some minor cognitive dissonance. The proof itself is simple enough. It’s my inquiry into the nature of infinity and number representation that has been troubling. The proof for the statement is:
a) Let x = .999…
b) 10x = 9.999…
c) 10x – x = 9.999… – x
d) 9x = 9
e) x = 1
f) 1 = .999…
The trouble/cognitive dissonance for me comes in when considering the counterintuitive idea that 1 = .999… as the repeating number will simply never reach 1. I think of it in graphing terms, where a curve approaches but never reaches a point or one of the axis). Even more troublesome and confusing (yet strangely engaging) is reading some of the debates and various proofs against the above statement which revolve around the idea of infinity (is it a finite number? is it a mathematical concept or process only?). There’s even a rather long Wiki page dedicated to the proof.
What I found intriguing about this little Brainfood puzzler is the extension, through my personal curiosity, to follow the idea to deeper thinking. Admittedly, the introduction of analytical proof, infinite series and sequences, nested intervals and least upper bounds, Dedkind cuts, Cauchey sequences, etc. left me scratching my head. I suppose I could make sense of it, if I slowed down enough and took them step-by-step. A puzzler of this kind, however, could be useful in a classroom to spark student curiosity, engage an appreciation for the logic of mathematical proofs, and possibly be used as an extension activity for those students who are able to handle the more complex concepts and proofs.
One of our first assignments for the Math Methods course was to write our own “Math Autobiographies.” The assignment, at first glance, was seemingly simple and straightforward: recall your experiences in math through elementary, middle, and high schools and through the present, keeping in mind teachers’ styles, pedagogy, difficulties encountered, etc. As I traced my experience with math back through time, I found that I had very little – if any – recollection of any math teachers or for that matter, any math classes. I could recall general memories of elementary school and high school days, but nothing much to do with math (save for the difficulties I encountered with algebra and calculus and my enjoyment of geometry). My difficulty with remembering my experiences with math was disconcerting.
Upon deeper reflection on my current, adult experience with math, I realized that I have two “versions” of math: purely theoretical or abstract and strictly practical computation. When I think about my adult experience with math, the greatest portion is in computation: how early do I need to wake up to get to work, balancing finances, taxes, the occasional measurements when I decide to build or repair something, etc. etc. etc. The other portion is theoretical but has little practical use (e.g., again, see Numberphile’s YouTube video, “Infinity is Bigger Than You Think”) or purely conjecture and wondering about questions that address quantities (How many blades of grass in my backyard? How many insects does a single bird eat in a year? How many pieces of kibble in my dog’s food bowl per meal?). These musings have little noticeable practical purposes, and it is extremely rare that I attempt to find answers to these questions. But I wonder if these types of inquiry do have a place in classrooms, and, if so, to what purpose? (And again, that practical-theoretical dualism returns).
Almost mid-way through the Educational Technology course (the impetus for this blog, but not necessarily the reason for its continuation), I wrote a post about my idealized future classroom as I had imagined it before undertaking studies at Nazareth College. The technologies and the attendant uses of and ideas related to technology as a learning tool that we explored in the Ed Tech course were a sea-change for me and I consequently reimagined my ideal future classroom.
For my current coursework, we were asked to read Jonathan Safran Foer’s How Not To Be Alone and I was – and still am – moved by the author’s words (both reflected in the ideas behind the words and his prose). I’ll not vitiate either Foer’s ideas or words with my own, but merely direct you to the article. Read it. The read it again, slowly. It’s about balance, about our shared humanity and remaining human, and about the place of emotions and connection in our digitally technological world; and it’s an important read.