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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EDUB 5220 Assignment 1 Samuel Jerema

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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http://blogs.wsd1.org/elmwood-math/

http://blogs.wsd1.org/elmwood-math/


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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EDUB 5220 Assignment 1

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