EDUB 5220: K-9 Mathematics Education—Manitoba’s New Curriculum Assignment 1—Critical Discussion of Your Practice Critical Discussion of My Practice By 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. 1
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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.