Mathematics in Perspective

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I have two engineering degrees, each with a mathematics minor. Wish I could say that I chose that minor because “there’s a great deal of mathematics in engineering.” Not really… I took mathematics because generally it was the easy “gut” course any semester; mathematics learning came very easy – at that time, much easier than history or philosophy or similar.

Move forward in time fifty years or so: I’m now an emeritus engineering professor after almost thirty years at the University of Connecticut, having eleven years earlier in industry, and sprinkled with a variety of consulting opportunities. How many differential equations did I solve in this time? ONE that I can recall. How many integrals did I do? Very few unless you consider the easy ones (such as the integral of Sin(x)dx or xdx). BUT how many times did I use the “basics” of mathematics (calculus)? I.e., “integral of a function equals area under the curve.” Very often, routinely… FIRST PERSPECTIVE: Unless one is teaching mathematics, only the fundamentals of calculus will be important. Too bad the math departments, textbook authors, and faculty don’t think about that…

Now, thinking back on my faculty period. For students in my classes or for my advisees, I was always eager to facilitate their learning. With my mathematics minors, I was frequently able to assist with clearing up confusion – in their math courses or for the math considerations in their engineering courses. The most common difficulty that I identified: When their were multiple paths to use, they had no idea how to decide which one to use! They knew how to use one if selected for them; but no clue how to choose. Learning this, my facilitating was quite straightforward: help them figure out what to look for in order to optimize the likelihood of choosing correctly. SECOND PERSPECTIVE: It’s not enough to facilitate the learning of procedures; optimizing their use must include selecting the best procedure to use. Too bad math departments, textbook authors, and faculty don’t think about that.

My final thoughts for this blog relates to the context of mathematics use. Consider the components WITHIN mathematics: There are the math facts (e.g., 1+1=2 or the previously noted “integral equals area under the curve”), the math procedures, choosing the best procedures, using those procedures correctly, and assessing the outcome for correctness. I’ve already noted the issue with choosing the best procedures. And assessing seems to be expected routinely but not really common among students. The other three components are staples of most math courses and textbooks.

BUT, too bad most mathematics courses, even so-called APPLIED mathematics courses, don’t contextualize the mathematics. Before “doing the math,” the situation encountered has to be understood and MODELED – developing the appropriate mathematics equations, etc. to enable analysis. After “doing the math” then, one has to use the math outcomes to answer the questions identified in understanding the situation. THIRD PERSPECTIVE: There is very little real-world situations included in mathematics courses to assist with gaining experience as well as motivation by the students.Too bad math departments, textbook authors, and faculty don’t think about that.

So the subsequent courses (such as engineering or economics) then should build on robust mathematics learning. In dealing with situations, the student, and later the employee, seeks an overall outcome that is SUCCESSFUL – the efforts are USEFUL in dealing with the situation. Note that I personally never use “correct” as because of uncertainty, assumptions, approximations, etc., we don’t ever know the right answer; the best we can hope for is useful input to dealing with the situation. FOURTH PERSPECTIVE: Too often, the source of “non-usefulness” of efforts are the non-math components of those efforts: the modeling and/or the interpreting. Too bad mathematics departments, textbook authors, and faculty (AS WELL AS THE OTHER “USER” DEPARTMENTS) don’t think of that!!!

A New a Learning Opportunity – for Me at Least

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I just learned of a new learning opportunity, thanks to Charlene Doland.  It’s the Connected Learning Community (on Google+). They have a terrific Website that includes what they call Maker Cycles – an opportunity to practice skills associated with connected learning. They facilitate Google Hangouts and a #CLMOOC Twitter chat (this Thursday the 10th beginning at 7:00 PM ET). As I said, I’m very new to these opportunities and wanted to share with you. As I understand, I missed their CLMOOC facilitated in June.

i gained and continue to gain so much from the opportunities related to the DLMOOC earlier this year that I’m excited to promote these new opportunities – through this short post to this blog!

Considerations and Memory

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I am a dedicated follower of the Brain Pickings newsletter of Maria Popova.  A recent piece discussed short- and long-term memory.  According to the piece, short-term memory consists of four so-called slave systems. Of interest to me and of relevance to this blog is the fourth slave: the episodic buffer.  This buffer “gathers all of the diverse information in from the other slaves, and maybe other information from elsewhere, and integrates them together into what might be described as a multimedia memory.” if you’ve read other posts to this blog or have discussed “considerations” with me, I trust you will hopefully see that this episodic buffer could also be termed short-term memory considerations. i of course am not a brain-function researcher. But I am a firm believer in the importance of “considerations” to effective, deeper learning. Again to me this is the development of long-term memory. Continuing this thinking, may I suggest a few items relating this transferring knowledge from short-term memory to long-term memory:

  • The transfer is the conscious expansion of the considerations relative to the topic. Whereas short-term memory happens “on the fly” as events raise the topic, the transfer to long-term memory happens by design.  That is, the learner decides the importance and sets out to deliberately build long-term memory. To me this means identifying other information to integrate into the vision of the topic as I’ve chosen to call the outcome. In particular, this additional information should include contrasting viewpoints (e.g., if the initial short-term memory resulted from a cold-weather experience, then effort must be made to find comparable information related to the other seasonal weather experiences).
  • The transfer will improve when these additional sources enable the comparison of these viewpoints. Some outcomes will not vary with contrasting viewpoints, some will. Not only will the vision be more complete (helpful in subsequent use), I am convinced the vision will provide those “hooks” to provide recall from long-term memory.
  • it is important to remember that increasing the viewpoints will also help identify the “outliers” – those that don’t mesh with other viewpoints. The tendency is to immediately dismiss those viewpoints as flawed in some way, a distinct possibility. But it’s best, I suggest, to keep them included in the vision – identified as     appropriate. Subsequent discussion or application of the vision will provide input supporting the validity of the viewpoint or its classification as an outlier.
  • The considerations will enable that vision to reside in long-term memory. But now to what the late Paul Harvey called “the rest of the story.” The vision demands regular assessment, starting with self-assessment of course: based upon additional viewpoints gathered as they are identified. But there is also a need for discussion with others. This discussion has many benefits: identification of inappropriate interpretations, of additional interpretations, and further reinforcement of those “hooks” to name just a few.
  • And of course, there is the ultimate reinforcement of and reason for developing this long-term memory: its application to address situations that make people’s lives better.  It’s reinforcement because not all applications will go as planned of course (maybe very few?) – more input to the refinement of the vision; and of course, this refinement emphasizes the hooks as well.

Summarizing then, there is apparently the most elementary levels of my “considerations” notion in short-term memory. However, it’s the conscious expansion of these considerations resulting in the vision, together with the assessing / refining of that vision through discussions with others and applications, that lead to valuable long-term memory. I trust this posting will prove of some value to you – in term of your understanding of short- and long-term memory. More importantly, possibly, this posting might provide input to the process / plan that I utilize when I “consider” a topic (memory) triggered by an article identified of value to me.

PLAYING, TINKERING, AND EFFECTIVE LEARNING

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Being a proud “Papa” (as my grand children call me), I’ve of course had many opportunities to both observe them, and at time engage with them in their PLAY. It is a fascinating observation, one many if not all of you have experienced as well.

Probably my favorite recollection is the time everyone had gathered on our porch one warm summer day. As they typically did, the five grandchildren (probably about five to nine years old at the time) were off on their playing for that day. After a while, they came over to me and said: “Come with us, Papa. We need your help!” We went to the patio nearby (with an unpaved surface typically used for unpaved driveways). The following scenario was provided: There was an older man trapped below the surface – who the grandkids were working to rescue. Using Grandma’s gardening tools, they had dug a 3-inch diameter hole about six inches deep. [Aside: When our daughters were the same age, the patio surface was completely off limits for disturbances of any kind – as they both reminded us often!] There problem was that while they could hear the man calling out, they could dig no deeper!!! Perfect opportunity for some informal learning … We explored the new material and determined it was clay (as I had remembered was put there when the patio was constructed). We talked about the differences between the materials and other uses for clay. Ruling out their requested use of shovels and picks, they decided to tell the man he’d have to wait for help (?) to arrive – and went off to some other scenario; I went back to the porch and the adults …

In my university classes, I very frequently suggested to my students that they “PLAY” with the notion(s) we were exploring together. I’d briefly explain that this was an informal exploration of those notion(s) – “having some fun with the learning.” Unless questions / concerns were raised (rarely happened unfortunately …), I moved the classroom efforts on; sadly, my suggestion was certainly forgotten I’m sure. While I do believe in use of class time in any course to introduce, discuss, and even practice the habits important to deeper, effective learning in ALL courses, I did not do the same for this “playing.” That would change in any future opportunities I might get.

When I learned of the efforts of Aaron Maurer – including his “Play & Tinkering” Google+ Community, my considerations of the use of PLAY in learning ramped up quickly! So, thinking about the activity of play for youngsters briefly outlined above, what are (OR should be) the features that should be translated from the playground to the classroom? Let me suggest seven that come to mind:

The classroom efforts will or should be less prescriptive. Play is very loosely controlled; the learning is very much student-centered.
The level of minimally acceptable outcomes of play in the classroom will or should be lower. The expectations are much less threatening. Improvement is sought, not completed outcomes.
While real-world questions are indeed important to motivated engagement, reduced real-world insistence for play (following playground – “man trapped …”) can be good: the “model” developed need only enable some contribution to understanding.
More hands-on activity will or should be involved in the classroom. After all, the playground activity is not discussion dominated. This type of activity is very motivating with regard to student engagement in the entire deeper, effective learning effort.
The classroom play will or should be team oriented. Sure, there are times on the playground probably when one or more children on the swings are keeping to themselves. But it’s the interactions that enhance the innovation and excitement.
Because of the looser, more informal play in the classroom, it will or should include effective, deeper learning more naturally. This outcome, likely happening without consciously knowing it is, enables the teacher to mentor / facilitate with far better results. This is so much better than a teacher-presented overview followed by assignments to practice effective, deeper learning.
Because of the informal, laid-back, student-centered nature of classroom play, there will or should be varying paths or plans followed in these classroom activities. Through informal or formal sharing of efforts made by each team, there is an increased likelihood of more students being motivated and learning.

So there are the seven features that I believe translate from playground to classroom. I’m sure there are others I’ve missed. Let me know your thoughts if you would. These and any others will be featured in any explicit attention to habits for lifelong, effective, deeper learning I have the opportunity to facilitate.

An Exciting Opportunity

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For the past eight weeks or so, I have been participating in DLMOOC, an opportunity to explore Deeper Learning with many others similarly interested (and, yes, partially explaining – also had a total hip replacement – the lack of new posts from me).  It has been an incredible experience, grounded in the CONSIDERATIONS approach that I have offered and will continue to offer in this blog.

One of our final “assignments” was to reflect on our experiences. What follows below is my submission.

Deeper Learning & DLMOOC

Deeper Learning has always been important to me. My long-standing thesis is as follows: Effective learning (learning for longterm retention and is of application) is best facilitated when that learning is considered a problem to be solved. To me, this approach to learning thus includes: (1) clearly understanding the topics included and their importance; (2) accepting that there are hurdles and false starts to be overcome in these efforts; (3) there will be ongoing assessment of progress made / directions taken in organizing and understanding the material identified; (4) crucial dedicated efforts to understand the application of this material by others to real world situations AND the personal application of my understanding to other real world situations; and (5) the ongoing reflection, refinement, and documentation of both the understanding and the application of this material – for my personal use and potentially that of others. (Do you detect my personal OSCAR – Objective, Speed-bumps, Considerations, Answers, Reflections – problem solving procedure in this list?)

And I have been excited by this personal joy of deeper learning “because it’s important to me” as well as from the similar joy in others – friends, colleagues, and students. One former student that stands out in this regard is “Andy.” Andy was not only in my classes, he was also my advisee. I always looked forward to his frequent visits to my office – I rarely knew the agenda … “Dr.B, I was just at the dentist; while waiting, I was reading about the solar wind. I’ve got to learn more; are there courses or can you and I work on this together?” And we did … Andy would have been a great engineer; but his career goals evolved into education – including HS science with a determination to add a philosophy course to the electives (because he believes educated people should consider philosophy).

This DLMOOC opportunity has helped me refine my understanding of the importance of Deeper Learning to rewarding careers and personal lives – FOR EVERYONE!!! Will everyone make the effort? Sadly no, but that doesn’t reduce my commitment to help facilitate the efforts of those that buy into that importance. These DLMOOC activities have increased my desire for and capabilities to both encourage serious deeper learning and facilitate the skills enabling it to happen. AND the best part is that I’m promoting such an exciting opportunity (not selling snake oil)!!!

Being older than dirt (so old that when I was young, dirt was clear – not dirty yet), my opportunities are limited. Yes, I’ll continue my personal scholarship of deeper learning, PBL, team dynamics, … Yes, I’ll continue my CONSIDERATIONS blog (http://johncbennettjr.com ). Yes, I’ll seek opportunities to work with learners of all ages face-to-face, limited though they might be. Yes, I’ll continue to explore opportunities to engage in MOOCs similar to DLMOOC – most likely presently being some sort of professional development. Yes, I’ll continue my most recent passion, play and tinkering in effective learning, with e-colleagues such as Aaron Maurer met through DLMOOC. Yes, …

Oh, and yes, one more commitment: I am keenly aware of my advantages being emeritus faculty in enabling my broad participation in DLMOOC. I am determined to work with others to broaden the time available to teachers as well as other individuals for precisely the personal and interactive efforts promoted by DLMOOC events.

BOTTOM LINE: It is impossible for me to over-exaggerate the improved understanding of Deeper Learning and its importance to meaningful careers AND personal lives – my own and others’ – that comes from my participation. Thanks to all that made it possible.

Getting to Clarity of Purpose

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In his post of January 7, 2014 to the Switch & Shift blog, Shawn Murphy argues that clarity of purpose is critical to any group having success in their work together.  Shawn suggests a series of “mini-workshops” to visit questions such as “What do we do?” and “What do we need to do?” to establish purpose so critical to the subsequent team efforts.

In my thinking, such workshops will be more successful if each group member were to consider her or his personal connections to each of the questions, the vision of what to organization purpose has been and seems appropriate for the immediate future.  Where does each person see their contribution optimized for the best possible sense of organizational purpose.  This would likely include among other information what the sense of accomplishment has been individually, within the group, and across the organization.  It would include feedback from customers as well as information on competitors; it would include information on what has transpired in terms of product revisions and possible revisions.

Such consideration by each member does of course require effort. But I would argue that such efforts are important to subsequent group discussion and to optimization of personal opportunities. As with the referenced blog post, most leadership mentors / coaches argue appropriately that the decision making must include all members working with the leadership – must be transparent.  In the post, Shawn notes that decision making that’s not transparent often descends into chaos, serving mostly as a information delivery system.

With the additional individual consideration efforts, the subsequent honest and open group discussion has so much more information for input: individual input to the status of current efforts and the potential for growth or at least realignment of future efforts.  The considerations are unlikely to lead to any one vision being chosen for implementation; but the development of those visions provide a rich background for considering each person’s input as the group works toward Steven Covey’s better alternative: the outcome that each group member believes honestly is better than their original vision.

Contrast that with the honest group discussions (everyone’s input genuinely sought and considered) but without the preceding individual considerations.  I would argue that such discussion is better than the manager providing the sense of purpose and implementation path for sure.  But think about the impact of the honest discussions that are not fueled by the visions leading the understanding of each participant.  Is it likely that opportunities or potential roadblocks / critical shifts will be missed? Is it possible that concerns about suggestions made by any member might be overlooked?  Is it likely that the emerging sense of purpose and implementation plan will be less robust, less understood, and thus less motivating?  My expectations are that the answer is yes to these and other questions of concern.

There are unfortunately regular examples of promising organizations with good initial results being driven out of business because of lack of sustained efforts that were responsible for the original success.  The organization needs a sense of purpose, a set of goals, and a plan of action that is accepted and acted upon routinely as Shawn suggests.  I love his thought that this is so obvious by the similar stories / messages each person tells.  My additional suggestion is for that employee input to optimized by the expectation and valuing of individual consideration.

Understanding Requires Consideration

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In a recent post from Seth Godin to his blog of 01/05/14, he writes about wanting an opinion without understanding the information relevant to that opinion. This blog is dedicated to the importance (really requirement) of the skill / habit of CONSIDERATION to the gradual development of understanding!

Why do I call it a skill? Probably obvious because there’s no switch that can be thrown to add this or any other capability as a useful one. It takes conscious attention to the process (dare I write consideration of the process – of consideration itself in this case) as one tries different efforts, routinely self-assesses progress and the need for refinement, and adjusts as the skill is developed into a usable and useful one. Lest you think there will be an end to this effort, hope I’m not discouraging you when I suggest this will be lifelong!

Why do I call it a habit? Clearly obvious to me at least in that a habit is sort of like a pair of comfortable boots in a snow storm: the peace of mind from relying on “auto-pilot” of habit enables the confidence in achieving the best outcome. In reaching that best outcome, we are comforted in our judgement and our ability to deal with the inevitable miss-steps.

But this consideration does much more than enable an opinion to be formed. It enables us to discuss the topic and test our opinion with others. It seems clear to me that the best opinion I might develop myself will be better through consideration. BUT to believe it the best possible without “testing” through conversation to me is not possible. Nor is any opinion “set in stone” when clearly it’s an opinion about a dynamic situation!

Most importantly, conversation among parties with opposing opinions BUT ones developed through consideration enable what the late Stephen Covey called the BETTER ALTERNATIVE. Otherwise that “conversation” is reduced to a prelude to “going to the mattresses” and conflict without resolution.

Consideration is YOUR best opportunity to develop a vision regarding any topic that enables you to engage with others in improving lives in general. That, for me at least, is far more important and satisfying than merely defending a possibly poorly developed position.

As always, your feedback and dialogue is honestly sought.