EDUB 5220 Assignment 1
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Transcript of EDUB 5220 Assignment 1
EDUB 5220: K-9 Mathematics Education—Manitoba’s New CurriculumAssignment 1—Critical Discussion of Your Practice
Critical Discussion of My PracticeBy Samuel Jerema
Preface
It is said that art is its own reward; math then must be an art. For me, the pursuit of
math in the classroom has always been to not answer the ‘how’ but the ‘why’. When
students are taught only how to solve limited sets of problems they miss the
opportunity to learn a bit of truth. The conjecture used as the catalyst for the
following discussion is, as David Long would say, “a search for beauty and truth”.
My goal with this assignment is that I will be able to communicate a bit about what
my practice looked like this past year. While doing this I will aim for growth by
engaging with the course readings and the curricular assumptions as they pertain to
this narrative.
Introduction to Part 1
In the old grade nine math curriculum one of the more challenging topics for
students is right angle trigonometry. The topic of this critical discussion resides
within an introduction to trigonometric functions. SOH CAH TOA is often seen as the
most important piece in teaching sine, cosine and tangent functions. While the
mnemonic, often repeated at exhaustion, can be very useful it can also, if used
haphazardly, lead to misunderstanding (Kilpatrick, Swafford, & Findell, 2001, p.
119). When I first approached teaching trigonometry in grade nine I did a brief intro
using ratios but had little connect to similar triangles. Once I realized the students
struggled with algebra rote memorization became a life raft. This year I knew that I
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had to build the conceptual understanding first before I breathed a mnemonic. I had
to lead in very carefully with the algebraic reasoning and background in similar
triangles.
Part 1: The Description of the Activity/Practice
While right angle trigonometry is the main targeted concept in this examination of
practice, similar triangles along with ratio, proportion and function concepts are
inseparable.
An Anecdote from class
“This guy here”, on the white board I’m referring to the sine ratio of angle theta
equal to thirty degrees, “is like the guy that always gets fifty percent on his test; no
matter what the test is out of, he gets 50 percent.” A few puzzled looks ring out with
no sound.
“So if the test is out of 10 what did he score?”
“Five…five…” many students blurt out the answer in broken chorus; some are still
hesitant.
“In the case of this triangle,” I frequently speak in metaphor in an attempt to connect
new concepts with things students are familiar with, “which side would represent
what the test is out of?” I’m reaching a bit here but hoping that the Pythagoras
review will help the students piece together the hidden dots of my train of thought
—no response, I take a step back…
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“How about if the test was out of 21?” Nearly all students know the answer and are
engaged; many however, are not confident enough to offer a response.
“He would score ten and a half.” Thankfully, the students do understand ratios, at
least within the 50% realm— a benchmark I was counting on before beginning the
lesson. I wanted to begin with something that they could get their minds around, the
next phase is to ask about the girl who gets 71% on every single test. What will she
get on a test out of 27? And then I still have to tie all the pieces together, not to
mention having to have the students discover the reverse, i.e., what the test would
have been out of if the 71% girl scored 11.
Part 1 Continued
What I just described was the beginning of a lesson on the sine ratio. Prior to this
culminating activity I had gone through similar triangles and reviewed many
algebraic concepts studied earlier in the year. The lessons I’ve learned through
teaching this topic are much like the first lesson I learned about teaching grade nine
math; that is, at the beginning of the year I tried to introduce polynomials and
algebra within the context of surface area and volume of a cylinder. I believed that
the connections with the grade eight curricula were vast and that a clear image of
how math is interconnected would emerge from my leading. What I found was that
it was all too much to manage at once and that the dread of my teaching was to come
true—that is, I resorted to teaching concepts in isolation because broken up they
seemed more digestible to my students. The critique of practice (2.2) below will
further illuminate the learning sequence, Pythagoras to similar triangles to right
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angle trig, as it lends itself to evaluation alongside and against the assumptions
found in Manitoba’s math curriculum documents. Firstly, however, I will outline and
briefly describe, those assumptions used in the critique of my practice.
The Critical Discussion 2.1
A metaphor is only as good as the test we put to it. By pursuing a metaphor we
discover its limitations and from these limitations we are able to learn a little more
about the tenor. In this spirit I will begin by introducing a metaphor and testing each
component against the research and theory provided in the course readings.
[Metaphor originally derived for graphic organizer July 2, 2009]
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Goals For Students
We want success for our students. While monetary goals and landing a job after high
school may be important, becoming a pilot can be seen as a measure of success,
math classrooms aim for broader goals. Like all fields of education, math should
enrich our lives and those of our students. We want students to appreciate math, as
Kilpatrick, Swafford, and Findell (2001) suggest the goals of the math classroom are
to have students perceive math as useful and worthwhile—productive disposition.
To achieve, students need to appreciate and value math, make connections, commit
themselves to lifelong learning and eventually become mathematically literate. In
order to come to the point where these types of goals are realistic students must
first see math as something that makes sense; their hard work needs to lead to
feelings of competency. If students, as described in Adding it Up, have strategic
competency and are able to apply this knowledge to new problems, ones they create
or encounter in the real world, then they will gain intrinsic motivation to learn. The
fusion of strategic competency with creative applications is adaptive reasoning,
which in turn leads to reasoning and solving problems with confidence—the first
two goals for student in the math curriculum. The goals we have for students in
math are inseparable from our beliefs about students and the affective domain.
Beliefs about Students and Mathematical Learning (we begin with what we believe and this gets us off the ground—provides lift)
Helping kids to fly requires that we understand a bit about how their wings work.
Students learn when they can relate new knowledge with what they have already
learned. A useful metaphor was provided by Fish is Fish. The fish like any student
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understands concepts through his or her own experiential lens. They make new
information fit in with what they already know—a process that requires creativity.
If you understand something, even if its not conceptually perfect, you will see how it
relates to other things and this in turn plants seeds for further learning and
clarification. This works as long as we are careful to identify and flush out
misconceptions—as seen in the video where students were encouraged to share
what they thought regardless whether it was right or wrong.
Affective Domain (Looking out the window—the environment to learn)
Like peering at a city or farmland from the window of a plane, looking at familiar
concepts from new perspectives can capture the imagination. Students are
motivated to learn when they see something as valuable or interesting. If we can
create an environment, where students are intrigued by math then they will put in
the effort required to reach conceptual understanding or any goals for students. The
following description is found in the curricular document “students with a positive
attitude toward learning mathematics are likely to be motivated and prepared to
learn”. Beyond this positive attitude perspective, Adding it Up refers to a productive
disposition as valuing math and finding utility in its concepts. To achieve this
domain students need to feel comfortable and safe—willing to take risks, i.e., try
something even it if might be wrong. Making Sense suggests that a social climate
where students discuss math with their peers is another essential piece to the
affective environment or domain. Essentially we need to take students up in a scary
plane and talk to them about how the wings will provide lift, what powers that lift,
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and where math may lead them. The affective domain puts math in context and its
inclusion in the curricular speaks to human side of all learning.
The Critical Discussion 2.2
In order to explore the assumptions in the curriculum a little further, they will be
examined against three specific learning activities. While the discussion will hardly
be exhaustive of all assumptions, the three activities chosen provide an opportunity
to critique and hopefully create a launching pad of ideas for refinement of
subsequent practice. You will find that I am more critical of phases 2 and 3 while
within the first phase I mainly discuss how the lessons can be seen to align with
curricular assumptions.
Learning Phase 1 (Inquiry and creation)
The first phase is essentially two learning activities however they will be examined
together here. Before beginning the unit on trigonometry I wanted to assess where
the students we coming from in terms of exposure to triangles. The following
snapshot from the blog post describes how I chose to go about this:
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This activity was aimed at engagement; I wanted the students to care about what we
were learning so I gave them authorship. By allowing the students to choose the
focus I was tapping into the affective domain. Further, this activity tapped into the
innate curiosity of students as discussed in, although not discussed at length in this
paper, the Early Childhood section of the curricular document—“Curiosity about
mathematics is fostered when children are actively engaged in their environment”.
In Learning and Transfer, the authors key in on an example where students are able
to solve a distance-rate-time problem in the specific context of a boat trip. The
findings indicated that having mastered a specific context does not mean students
will able to apply their understanding to new ones. With the Pythagoras creation
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problem below I was hoping to have students create their own context for problems
and then solve problems created by their peers. This activity gave the students an
opportunity to both target their adaptive reasoning by relating their lives and
interests to a specific math concepts and also to see a myriad of contexts by
engaging in discussion and problem solving using peer created questions. Adaptive
reasoning is a key component in establishing a positive learning environment.
Further, adaptive reasoning is a key part of the affective domain in that if students
can see how math relates to a wide range of topics then its becomes less about
specific concepts and more about ‘their world’.
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An Example of student work complete with student solution solved in class by
students:
More examples can be found at http://blogs.wsd1.org/elmwood-math/ .
Learning Phase 2 (looking at similar triangles)
Concepts vs. Procedure
The first thing that strikes me in the solution below is that I abandon the algebraic
reasoning very quickly. I jump from 0.45=5/a to a=5/0.45 in one line. I spent time
trying to reason through the algebra with the students prior to this example but
before the students could master the concepts I gave them something to memorize.
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I essentially said, “Here, this is what you do if the variable is on the bottom…” During
the course readings, I made a note in the margin of Adding it Up page 123 which
said: conceptual understanding = needs less practice. When evaluating this
statement I feel confirmed that the rush I felt to get kids tackling similarity problems
on their own was not necessarily a time saving effort. While some students would
never have achieved the conceptual understanding necessary to solve similar
triangle problems independently, I might have done better to focus on ‘math’ as
opposed to blind procedures. I was resorting an algorithm much like the long
division algorithm in that many concepts were hidden or barely visible especially to
novice students. For many I turned math into this unknowable procedure that you
just need to ‘do’ not ‘get’. Learning And Transfer further confirms this notion against
the hurry (p. 58). As noted before, the assimilation of new concepts is a challenging
and creative process; like the fish, cognitive integration requires context and time.
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Learning Phase 3 (See Anecdote from Class—Part 1 above)
While I felt as though conceptual understanding was the focus in my class, I realize
that the time frame was too short once again. I pulled up notes from the day after
the class described in the anecdote and realized that I didn’t leave much
investigation time before I hit the students with the herculean mnemonic SOH CAH
TOA. The day I introduced sine had many notes on the whiteboard, however this
was the beginning of the following lesson:
Page 1 Page 2
What am I doing here? The answer is I am teaching to test. After having examined
what the students need to know about trig for the final exam, I came up with this
note set. What I missed out on was the “well-timed practice of the skills they are
learning” (Adding it Up, pg. 122). To develop procedural fluency with the concepts
of sides of right triangles and their ratios depending on given angles, I would have
had to follow the intro to sine lesson with rehersal of these skills. In my teaching I
failed to connect procedural fluency and conceptual understanding. What I also
notice about this lesson is that I neglected to allow students the opportunity to
engage in peer discussion. While I’ve set my role as a teacher, as discussed in
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Making Sense, as facilitator of conceptual understanding by engaging in metaphor
and conversation with the students, the essential idea that students must interact
with each other was left out. The social culture of the classroom for this particular
lesson was not about the students’ ideas and thoughts, it was about my ideas.
Making Sense reminds us that students build understanding by talking math with
each other. A lesson where groups of students had some trig ratio problems to work
out similar to those I began discussing, the 71% girl for example, would have
enriched the classroom experience. Fostering a student centered classroom culture
as well as providing time to find procedural fluency would have helped meet my
desired goals for conceptual understanding.
Another aspect, as of yet only vaguely alluded to, is the richness that right angle
trigonometry has in formulating strategic competence along with the goals for
students outlined earlier. One strategy that is blatantly obvious is an algebraic
approach. With an elementary understanding of functions, t-charts or input/output
models, students can create equations from the trig ratios. Another strategy is to
focus on the ratios and solve the problem from a proportion standpoint. In larger
problems the Pytagorean theorem can be used to aid in the solution process. Under
the heading Allowing Multiple Strategies, How Students Learn suggests that if
students believe that any math situation has more than one correct method then
student egagement in developing strategic competence is strengthened. From
sailing to construction the opportunities to teach and solidify concepts in interesting
contexts are unlimited with trigonometry.
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Having made these claims it is also worth noting that the benchmarks for this type
of work are many and this is why the mnemonic is so overused. The goal for
students are centered around confidence and competence with math; when trig
disintegrates to a mnemonic and rote memorization we are not accomplishing any
assumptions set forth in the curriculum—we are merely pretending to achieve the
SLO’s.
Part 3: Reflection
After I got hired and was settled in my classroom, the math department head, whom
was one of my interviewers, told me why she chose me of all the interviewees. She
felt that from my responses and portfolio I appeared to be someone who would help
students understand the ‘why’ of mathematics. The department head went on to
communicate to me her challenge with the department and certain teacher’s
reliance on booklets and worksheets. By this time I was feeling quite affirmed with
regards to my pedagogical methods. Fast forward to July 2nd, 2009 and a new
challenge arises. It is no longer about whether I’m committed to helping students
achieve conceptual understanding but how I go about it. What I’ve been able to find
in the readings are full sentences for my minds fragments; I have a clearer road map
and will have to rely on a little less on the stars for navigation in the classroom.
Professional development demands that we reflect upon our practice in a critical
way. The movement away from personal anecdotes and recollections about what
worked in one classroom and not in the other towards academic discourse steeped
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in theoretical articles backed by research has to be intentional. On our own its easy
to resort to personal experience, which does have a place, however, the insight I
gained into how to pair conceptual activities with constructivist strategies come
formally through engagement with theory and research. In an effort to focus on the
conceptual my practice became somewhat anaemic in procedural fluency.
Reflecting on the rest of my teaching I am already beginning to see a pattern in my
teaching. The lessons I’ve learned analyzing these learning activities can and will be
generalized as I prepare for a new year of teaching.
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