Hybrid Pedagogy

Earlier today I stumbled onto hybrid pedagogy, a website for the eponymous open access journal, while reading a post from Rebecca Schuman, a blogger at Slate who covers education. The director, Jesse Stommel, is an assistant professor at UW-Madison. Give their site a whirl if you're into hybrid learning and pedagogy.

Text Set for NSA Mass Surveillance

Text Sets are a tool teachers can use to scaffold instruction for struggling learners. They generally focus on one topic or theme and are a very useful tool for improving comprehension and literacy. Text sets popped into my head when I found this article about the NSA and its mass surveillance. The author does not use the term "text set," but that is what he's proposing to provide context for the journalistic coverage of the NSA following the revelations by Edward Snowden last summer.

Anyone familiar with the Snowden/NSA story has no doubt read an article in which a journalist has compared the NSA to Big Brother in George Orwell's novel "1984." It's a metaphor with legs (is that a meta-metaphor?). Mr. Berlatsky points out that 1984 is not a book that paints the most relevant picture of the current government-sanctioned surveillance of the citizenry. He argues that other works do. He even goes so far as to say that 1984 can "enslave thought." I think he may be referring to the phenomenon of journalists who repeat what other journalists have already tweeted or written, even after a statement has been found to be unsupported by the facts. On The Media is all over this case.

On to the text set, culled from the article. A caveat: this text set might not be appropriate for scaffolding.

• Phillip K. Dick, Do Androids Dream of Electric Sheep?
• Franz Kafka, The Trial
• Arun Kundnani, The Muslims are Coming
• Maureen F. McHugh, China Mountain Zhang

to which I would add these nonfiction works:

• Patrick Radden Keefe, Chatter
• Bruce Schneier, Secrets & Lies
• James Bamford, The Puzzle Palace

Your Inner Fish

It begins in the city of Chicago, with a room full of human cadavers.

That's how the story begins.

We are descended from fish. It's a scientific fact supported by mountains of evidence. Much of the evidence is written in our bodies. I'm talking about evolution, of course. There's a fascinating look at your inner fish hosted on pbs.org. It's a three-part video series hosted by Dr. Neil Shubin. I'm watching it via the PBS app on an apple TV. You can watch it on your computer in the web browser, or with the PBS app on a tablet.

Dr. Shubin discovered tiktaalik rosea on Ellesmere Island. Tiktaalik is a creationist's nightmare. He wrote a book about it, and now he's doing this great series on PBS. I can't wait to see episodes 2 and 3.

added 4/20/2014: Your Inner Fish passes the Bechdel Test. There are female scientists in this documentary, in one of the scenes we have two women who talk to each other about something other than a man. Also, there are some great visualizations, like when Dr. Shubin is standing on some devonian river sediments on Ellesmere Island and we watch as it is transformed into a stream bed teeming with life some 380 million years in the past. I could see how video like this would be great in a classroom teaching evolution or geology. The pbs.org website for Your Inner Fish even has a classroom guide. This is a great resource for science teaching.

Flipped Learning Guidelines

The Flipped Learning Network (FLN) has announced [pdf] a formal definition of the term "flipped learning" on March 12, 2014, Two things immediately catch my attention here.  First, they do not say "flipped classroom." The other attention-getter is who is the FLN? It is interesting that they do not use the more common "flipped classroom" term. They take care to draw a distinction between flipped learning and the flipped classroom. Until I chanced upon this announcement, I was unaware of any controversy or distinction that involved a definition of the concept.

If you are unfamiliar with the flipped classroom concept, head over to youtube and do a search. Their are many videos on this topic.

The FLN bill themselves as a group of experienced flipped learning educators. Their website is attractive and well-organized. Aaron Sams is at the top of their board members page. I believe he is the creator of the flipped learning model of instruction.

The flipped learning definition is published under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, from the non-profit Creative Commons organization. Whoever the FLN is, they are clearly interested in openly sharing their ideas and standards. This is commendable. They do not want others creating derivative works of their standard, so they want to keep control of it. They also do not want others deriving works and selling them for profit.  Having looked and worked in the education field, I cannot help but notice all the for-profit enterprises (ETS, MetaMetrics, textbook publishers, Pearson Education, Charter Schools, ad infinitum). But, as I scrutinize the FLN website, I notice ads at the bottom of the page. There's Pearson, Sophia Learning, Cisco, Adobe, etc. This non-profit organization has some deep-pocketed corporations backing them.

A tip of the hat to Casting out Nines, a blog that posted on this new definition.

Hyperreal Numbers and Calculus without Limits

Once upon a time, I stumbled into a hyperreal number field on the internet when searching for something related to calculus. Maybe I was calculating the volume of a Steinmetz Solid? The so-called hyperreals can be used to develop the calculus without limits. The claim is made that, perhaps, the use of "infinitesimals" is less confusing than the concept of the limit for struggling calculus students. In fact, there is a calculus text using this approach that one can download for free from this page. The author, H. Jerome Keisler, is a professor emeritus of mathematics at UW-Madison.

What are the hyperreals? They are an extension R* of the field of real numbers, which is usually denoted by R. R is a subset of R*. Without going in to too much detail, R* is obtained from R by adding in infinitesimals and infinitely large numbers. An infinitesimal number is infinitely small. What does this mean? For every real number a, a number ε > 0 is infinitesimal if -a < ε < a. That's it. In contrast, 1/ε is infinitely large (a "hyperinteger"). Conceptually, hyperreals are similar to complex numbers. However, unlike the complex numbers, they are depicted in one linear dimension, not a two-dimensional plane. The nitty gritty is all in Keisler's book. With the concept of the infinitesimal available, we can have the hyperreal quantity x + ε that is infinitely near x. This hyperreal x + ε can then be used to formulate the derivative, which won't be surprising to anyone who has studied calculus.

Here's figure 1.4.3 from Keisler's book. An "infinitesimal microscope" has been used to zoom in on the hyperreal number line:

The infinitesimal microscope is a captivating pedagogical concept. I claim that it's easier to grasp than the concept of a limit. Children learn about microscopes in grade school. Maybe they learn about it nowadays by zooming in on their iPads. In any case, R* is a useful mathematical concept that can be used to formulate the calculus without that ungainly Σ thing. All you need is a little ε guy. :-)  Keisler's book provides more than 800 pages of evidence.

The history of infinitesimals is interesting. The epilogue [pdf] to the aforementioned calculus text lays out the story. Archimedes anticipated both infinitesimals and the ε, δ approach of limits in some of his proofs. In the 17th century, the modern calculus was developed independently by Newton and Leibniz. In his reasoning, Newton used both infinitesimals and the concept of a limit as well as the so-called velocity method. Leibniz used infinitesimals. So there were three competing methods for doing calculus: infinitesimals, limits, and the velocity method. Jumping ahead (see Keisler's epilogue for whole story), the first truly rigorous treatment of calculus was formulated by Karl Weierstrass in the 1870's. I believe it was Weierstrass who introduced the  ε, δ notation that we still use today.

There can be little doubt that the limit approach is firmly entrenched in the current pedagogy. Other than Keisler's book, I am unaware of any other textbook that uses this approach. I never heard it in Altgeld Hall or Van Vleck. Mathematics teachers should be aware of this alternative approach and should consider using it to introduce students to derivatives and integrals.

Randomness and literacy

The generation of random numbers is too important to be left to chance. Robert Coveyou

I banged my shin the other day. The bruise on my left tibia just below the knee is a painful reminder of this event. A distraction was provided by the phone ringing. I stood up, thinking about answering, and I smacked my leg right into the coffee table. The ringing of a phone is a perfect example of a random event. If it had not rung, would there be a bruise on my shin? Probably not.

Random numbers, randomness, and the generation of random numbers are important topics. Randomness is particularly relevant to current events because of its essential use in modern cryptography, which has been in the news lately with articles about Edward Snowden and the NSA. Embedded within a larger frame of mathematics, science, and current events, these topics can provide plenty of impetus for interesting conversation and mathematical diversion.

More to the point, I intend to discuss literacy in the mathematics and science classroom. What can we do to motivate students to learn mathematics? One technique I have used and will continue to use is critical literacy. The teacher can display, e.g., the text of a newspaper or magazine article that gets the math wrong, that provides an example of innumeracy. This text can then serve as a jumping-off point for a discussion of a proper mathematical analysis.

Another technique I find intriguing is the discrepant event. A discrepant event is a demonstration or a question with a surprising or startling conclusion.  An attention-grabbing event can be used to initiate the process. The discrepancy creates a cognitive springboard and forces the students to think about the subject matter. An example of a discrepant event was provided to me recently. A question was posed about fish bladders, an organ possessed by ray-finned fish such as the largemouth bass, Micropterus salmoides. I intend to expand on this topic in a later post.

A closely related technique is the thought-provoking question. How many years is one billion seconds? How many cells do you have in your body? These questions can provide a nice stimulus for a lesson and get the gears turning in the student's heads. Each of which has hundreds of thousands of hair follicles, of course.

Literacy can be used in the classroom to motivate, to captivate, and to initiate discussions. Mathematics is a complex topic, and motivating young students to learn can be challenging. It behooves us as teachers to have many arrows in our educational quivers.