Reflection: my personal experiences as a math student

      My most inspiring mathematics teacher wasn't a member of the county school board nor were they a family member.  Instead, he was a Professor Emeritus who volunteered his time during evenings at the local University to introduce more complex problems & problem solving methods to young students who were keen and/or were not feeling challenged enough by the standard curriculum.  Although the goal was to prepare for the annual Waterloo math contests (of varying difficulty based on your grade level) what I (and I believe most students) got out the experience was a better understanding of the grand scope of math and its sheer power.  I attended these free extra-curricular sessions from ages 12-16; I definitely would not have studied math as passionately/intensely in the years that followed without his invaluable guidance.  Looking back, I can say that he also influenced how I went about explaining math to other students in various environments (tutoring, learning centre groups, tutorials, etc...). 
      One the flip side of the coin, any teachers that were responsible for perpetuating the stigma that mathematics is difficult to understand and is only for the select few through banal, uninspired lessons and/or discouragement all fall into the least favourite category for me.       

TPI Reflection

      Prior to last Wednesday, I had not heard of the Teaching Perspectives Inventory (TPI) test.  My responses to each question yielded scores of 36, 32, 39, 37, and 16 in the teaching perspective categories of Transmission, Apprenticeship, Developmental, Nurturing, and Social Reform, respectively.  The (downward-biased sample) standard deviation was computed as follows:

x bar = (36 + 32 + 39 + 37 + 16) / 5 = 32,
s^2 = [(36-32)^2 + (32-32)^2 + (39-32)^2 + (37-32)^2 + (16-32)^2]/5 = 69.20,
s = 8.32

      The TPI summary page deem any perspective score exceeding [x bar + s =] 40.32 are dominant, while those less than [x bar - s =] 23.68 are recessive.  I did not have any dominant perspectives, but I did have one (strongly) recessive one, namely Social Reform.  I was not surprised to see these results; I believe that most mathematics taught at the secondary level is intended to improve the individual first, before they move onto more advanced work that may affect our society as a whole sometime down the line (pending impact).  Two internal inconsistencies of note are the Belief subsection of the Developmental perspective and the Action subsection of the Nurturing perspective.  Both sub-scores were lower than the other subsections within each category.  Also, the highest subsection in 3 of the 5 categories was Intention.  These observations could be explained by the short amount of time I spent taking the test or the fact that I have yet to establish consistent sub-perspectives as a teacher candidate.  
      Do the TPI test results perfectly describe me as a teacher?  Certainly not, but it displays a fairly accurate portrayal of my pedagogical tendencies.  It would be interesting to retake this test sometime after I finish my practicum and then again 3-5 years down the road to see how these stats evolve.

Reflection: Instrumental vs. Relational debate (Richard Skemp)

      Asking everyone to take a stance on a topic and then defending that stance was something that I hadn’t experienced in a classroom setting for many years.  Initially, I was surprised how many people were on the ‘instrumental’ side of the coin (approximately one third of my colleagues).  It was entirely possible, however, that those that (in class) were on the instrumental side of the debate didn’t actually have that belief (and vice-versa).   

      During our discussion, which I thoroughly enjoyed, many different viewpoints and counterpoints were presented from the groups’ brainstorming lists. Many items were subsequently argued accurately from both sides, but (for brevity) I’ll focus on just one.  The ‘achieves that results’ argument (pro-instrumental) involved teaching the Pythagorean Theorem via simple exercises involving right triangles yielding immediate results and thus fluency.  The 'branch out to other problems' pro-relational rebuttal hypothesized: what if a student turned the paper over only to encounter non-right triangles and then proceeded to apply the PT incorrectly (where the Cosine Law is necessary)?  Motivating PT by introducing the more general setting and explaining how PT is a degenerate case of CL¹ gives the learner a better understanding of the meaning behind the raw calculations.        

      This brings us back to the original question that Skemp posed: should instrumental and relational mathematics be considered two different subjects?  The jury is still out…

           

¹ For a triangle with sides a,b,c>0 and corresponding angles A,B,C we have:

            c² = a² + b² -2abcos(C)        [CL]

            c² = a² + b²                         [PT]

   Hence, [CL] = [PT] <=> C = (pi)/2. 

# of squares on a chessboard

Many (seemingly daunting) problems in mathematics can be broken down into smaller, more manageable ones where the results can then be collected to solve the original question.  Here, we partitioned the original question into 8 smaller ones and then compiled the results in an efficient manner by employing the sum of squares formula.  Some possible extensions for this problem are:
  - number of squares in a larger board (9x9, 10x10, 100x100, NxN)
  - number of squares in a rectangular board (MxN)
  - number of non-square (PxQ) rectangles instead of squares, P<Q.
  - number of cubes in 3 dimensional board (8x8x8, NxNxN)
    

Richard R. Skemp - Relational Understanding and Instrumental Understanding

3 things that made me stop during my read-through:

1.     “There is this important difference, that one side at least cannot refuse to play.” (pg. 4)
       - a wonderful analogy describing the compulsory teacher-student bond that is forged over the years

2.     “But his relational understanding, by knowing not only what method worked but why, would have enabled him to relate the method to the problem, and possibly to adapt the method to new problems.” (pg. 9)

       - possibly the best statement supporting relational understanding from the entire article

3.     “There is more to learn – the connections as well as the separate rules – but the result, once learnt, is more lasting.” (pg. 9)

      - I've seen too many students try to 'memorize' math, it doesn't survive the test of time

      My stance is that relational understanding is superior to instrumental understanding.  Why? Much of our world consists of quick fixes and people demanding (near) immediate gratification, with minimum effort.  Instrumental mathematical understanding caters to this by deadening the topic - only skimming the surface and omitting the overall schema of knowledge in favour of individual (often detached) components.  This practice can result is errors on an exam by students who otherwise are quite competent (the area units when l is given in cm and b is given in yards).  Moreover, a base in instrumental understanding could yield errors resulting in much larger consequences later in the workforce (e.g. confusing 1% with 1bp when dealing with large sums of money).  I’ve seen many circumstances where students churn out an answer (involving many correctly executed steps to get there) and assume they are correct overall only to find out later that they made a small error somewhere along the way.  Everyone is prone to mistakes, (we are human) but if the learner had a more complete ‘mental map’ they would realize that although the steps seem correct, their final answer is nonsensical (e.g. a negative length).  It’s not worth the risk to only teach instrumental understanding/‘plug and play’, because so many math problems can only be solved by straying off the beaten path from A to B; without the proper map you could end up lost.  Relational understanding, on the other hand, enables you to traverse most landscapes since it connects (seemingly) disjoint components together to create a web of knowledge.   

Math is amazing!