Final Exam
Planning Scenario
You have been hired as a 6th grade teacher at Bearcat Middle School. Your middle school is
located in a district at the edge of an urban city and its suburbs. Bearcat Middle School is a
typical middle school and classrooms have a wide range of learners, from novice to talented in
the various content areas. The students of Bearcat Middle School have different ethnic
backgrounds and are from diverse home situations and socio-economic levels. In addition,
eighteen percent of the students have been identified with a learning disability, several with
significant concerns in mathematics. Their Individual Education Programs (IEPs) have been
placed in a secure area in your team planning center.
Faculty and administrators discussed results of the Ohio Achievement Assessment and other
school wide assessments during a recent 6th grade teacher meeting. Based on student
performance and teacher observation, teachers and administrators decided that improvements
were needed. The math teachers gathered and decided that decimal and fraction concepts would
be taught to foster understanding for all students.
As a new teacher and first year resident, your assigned mentor encouraged you to share your
recommendations regarding fraction and decimal instruction. Noting this instructional concern
your mentor has asked you to reflect on a number of prompts.
Please respond to each of the following:

What are some overall concepts and big ideas you would keep in mind when planning
instruction for decimals and fractions for your students?
Topic-related:
There is not one single meaning of a fraction. Fractions can represent a part of a whole, but they
may also stand for an operator, a measure, a ratio or a quotient. Some of these ideas are more
complicated for students than others; operators tend to be difficult to understand, for example.
With decimals, middle grades students often do not grasp that numbers after a decimal point are
parts of a unit, and may mistakenly believe that a “longer” number is always bigger (though this
is clearly not true: 6>5.555555555555555). They often do not realize that any whole unit is also
“a decimal,” just without the infinite zeros to the right of the ones place written in. I would also
keep in mind that there are many state standards related to fractions and decimals. Particular
emphasis is placed on flexibly transitioning between the two (Numbers and Operations Grades 57 B, C and D).
Planning-related:
In planning lessons, I would remember principles of backward design. We need to have the
indicator and the assessment that we intend students to fulfill before we plan instruction to get us
to that destination. I would also consider how to use a good mix of levels of cognitive demand in
the fraction and decimal instruction. If we teach only procedural knowledge—how to, for
example, divide fractions by “invert and multiply”—our questions will naturally rely heavily on
memorization and procedures without connections, the lower-demand tasks. To develop deeper
conceptual knowledge, we will also ask students to apply procedures with connections. If
Final Exam
students understand why they are finding a common denominator before adding and how this
looks in a diagram, they can see the connections of multiplying (finding the least common
multiple to get equally-sized pieces) to adding. One way of measuring this understanding would
be to take a performance assessment that involves adding pieces of different sizes (e.g. pattern
blocks) and calculating the new part of a whole represented. The highest level of “doing
mathematics” with fractions and decimals requires a non-routine problem without a well-known
algorithmic solution. (An example is the open-ended question where students are given four
numbers and need to find the arrangement into two fractions that gives the highest sum, lowest
product, etc.)
Finally, I would consider how to use sometimes neglected process standards to increase the
depth and breadth of student understanding. To make full use of communication and reasoning
and proof, students need to collaborate with their peers, explaining why their strategies work to
the teacher and their group. Accordingly, I would include problem-solving activities where the
students would benefit from discussing their strategies. The connected math project, for instance,
includes a problem about selling brownies at a fair for $12 a pan, where customers can buy
whatever fraction of the brownies remaining in the pan they would like. Students could discuss
how to solve and model the problem together. I would also ask students questions individually.
Some questions would be designed to scaffold and assess understanding. Others would
specifically encourage my students to generalize, explaining why or whether their strategies will
always work. For instance, if one student divides across to get the result of a division by
fractions while another inverts and multiplies, they would both benefit from reasoning through
whether these methods will both work and how they are similar or different. Again,
understanding the connections between fractions and decimals as well as how these relate to the
lives of the American worker/consumer is crucial. A problem that involves purchasing building
or baking materials (measured with fractions) with money at a certain sales tax rates (involving
decimals) such as the skateboard ramp problem, could help tie all this together at the end of a
unit. Similarly, students should be able to alternate between the different ways to represent the
same quantity, including words, numbers (fractions, decimals), diagrams and some types of
manipulatives. Finally, I would remember that students gain and retain the most useful
understanding of fractions and decimals while they are developing problem-solving skills.
Philosophy-related:
Since fractions and decimals are an area students traditionally struggle with at Bearcat Middle
School, I will also need to consider the atmosphere of my classroom. I will need to foster a
classroom culture of raising expectations, design a safe environment in which students are
intrinsically motivated, and help students feel pride in struggling with (and eventually solving)
challenging new problems.
Accordingly, we want to help all students develop perseverance in problem-solving—whether
they are accustomed to struggling with math or that prospect seems unimaginable. On the other
hand, we also want them to be successful. Leaving students constantly frustrated is a way to
foster learned helplessness, not a productive disposition towards math. Modification is, as
Professor Stephen Kroeger put it, “an act of compassion.” It is not designed to make the task
“easier,” but rather to increase independence in the same way that a curb cut allows people with
a physical disability to cross the street without assistance. Students develop and grow, and many
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reach the point that they do not need instructional support like manipulatives to solve problems.
Others always need the “training wheels” to be successful, and that is okay. Our classrooms
should be set up with the flexibility to support this diversity of learners.
We may need to work together with our special educators to select alternative assessments for
students whose growth would not be captured by a traditional assessment. These will have to be
relevant to the students’ needs; for instance, some students might be using concrete fractional
measures that they would need in cooking, slicing a literal pizza into the correct number of
fractional parts or calculating a tip or sales tax. Likewise, we will develop performance
assessments that give a more complete picture of what our students of many different levels, with
and without IEPs, can accomplish.
Whenever we introduce a new manipulative, be it pattern blocks or Cuisenaire rods, we need to
remember that students need some time for free exploration first. We also should structure our
lessons around exploration; a launch-explore-summarize model fits well with constructivism. I
would also be constantly reviewing Bruner’s Stages of Representation. Though some students
who are gifted in mathematics may be able to begin instruction in fractions with pictures and
quickly learn to compute with abstract symbols, most students will need solid instruction and
exploration with fraction tools that they can move around and touch before pictures or symbols
become the centerpiece of a lesson. What’s more, students will not necessarily construct a
working understanding of fractions at the same rate, so manipulatives should remain available
and encouraged for those who are not yet prepared to work with fractions in the abstract.

What specific tools would help students learn about decimals and fractions? How would
those tools be used or modified for your diverse classroom?
Concrete manipulatives like Cuisenaire rods, pattern blocks, fraction strips and fraction circles as
well as egg cartons (for discrete sets) could all be useful for helping students become
comfortable with fractions. Measuring experience using student-made meter sticks could help
build a concept of adding, subtracting and even multiplying and dividing with decimals. In my
diverse classroom, I would make these tools easily accessible for problem-solving and
explaining reasoning. Wherever possible, I would have translucent versions of the manipulative
for an overhead projector, large versions at the front of the room, or (if available at Bearcat
Middle) virtual versions on the Smartboard. If Bearcat Middle provides sufficient access to
technology, there are also some computer applets and simulations that can help students
understand “re-grouping” and adding decimals and fractions better. (Here is a virtual decimals
addition and subtraction applet:
http://nlvm.usu.edu/en/nav/frames_asid_264_g_2_t_1.html?from=topic_t_1.html.) I would also
use a clothesline number line and calculator investigations with decimals to help students keep
track of “how big” decimals are in comparison to each other.
Where possible, I would have the students make their own manipulatives, because this
sometimes helps them understand better how they work. Some students have more difficulty
with fine motor skills that are traditionally challenging for students in the middle grades, e.g.,
dividing a “pizza” into equal thirds. To avoid unnecessary frustration, perhaps a rectangular
Final Exam
“pizza” could be used. Students could be taught to fold rectangles in thirds as a scaffold to
learning to “eye-ball” these divisions.
Some other tools that I would use could perhaps be described as strategies; they are instructional
decisions and types of tasks that are known to increase the effectiveness of classroom design for
learners with exceptionalities. Parallel tasks, for instance, allow the teacher to discretely
differentiate the scope and complexity of a task without limiting the opportunity of all students to
share their reasoning and relate to one another. Because the tasks look the same and involve the
same scenarios and the same “big idea” as Marian Small and Amy Lin put it, students with
cognitive delays or learning disabilities are positioned to be valued contributors to class
discussion. Use of an expository organizer, which outlines in advance the structure of the day’s
learning, could help many students including learners with attention problems. If students
become distracted during the fractions and decimals lesson or trying to execute an algorithm they
have just learned the organizer helps them find their place again. Gifted learners among others
enjoy the challenge of open-ended questions; they can make the task more difficult as desired to
keep it interesting and fulfilling to them.
Many students mix up numbers in between hearing what the teacher said or looking off the board
and trying to copy it onto their own papers. Although not always practical, for students with
cognitive processing problems, it is helpful to have redundant cues with at least one of them
close at hand to the student (e.g. write the details on the board, say them out loud, have a student
read them aloud and print them on their direction sheets). Allowing for multiple modalities of
demonstrating understanding including student interviews is also helpful because it helps
teachers determine where Student A got the number 23.2—was it the result of a misconception?
Was it just a guess? Or was it simply a digit transposition of the correct integer, 32.2?
Students experiencing memory problems (which nearly all of us encounter at some point) may
benefit from a “trigger” (hint) that references previous activities. A simple question like
“Remember when we…?” could help them remember how they or others have attacked similar
problems in the past. Metacognitive deficits can mean that a lack of with organization and other
tangentially mathematical strategies gets in the way of the actual lesson objective. Some students
might not think of crossing off the rational numbers printed on their worksheet as they prepare a
new, smallest-to-largest-order list for finding the median. This might cause them to miss that the
number 3.45 is repeated three times and 2/3 only appears once. Some students also might not
catch on that the teacher always seems to rewrite orient fractions next to each other instead of on
top of each other for adding and just be stuck when they see them written above each other. The
teacher should use explicit modeling when a procedure is being taught (showing and explaining
together, not in isolation). A “think-aloud” (demonstrating how an experienced problem-solver
works through an unfamiliar task) is also helpful to developing generic strategies and
understanding how to think about math.
For English language learners (ELLs), I would want to make sure that I give directions with
pictures and familiar vocabulary wherever possible. If I had a decimal problem involving a
menu, for instance, I might use an actual photocopy of a menu with pictures and possibly even
dishes in the students’ native language. Also, predictable, well-established routines help students
who are less familiar with English feel secure and prepared.
Final Exam

How could you utilize collaboration and co-teaching with your colleagues at Bearcat Middle
School to enhance the teaching of decimals and fractions?
Several forms of co-teaching could help Bearcat Middle make progress toward its goals. Inservice meetings could allow all the sixth grade math teachers time to jointly develop (or revise
from an existing source) one lesson or short unit focusing on a fraction and decimal indicators
that students have historically struggled with on tests. Each teacher would agree to present this
lesson to his or her class, but not all at the same time. Then the “one teach, one observe” model
could be used to allow a mentor teacher, administrator or other math teacher to monitor student
response to the lesson and suggest improvements to its content or delivery. This lesson, if
successful, could be refined based on this year’s feedback and taught again in upcoming years.
Different teachers tend to have different strengths, and involving more of them in the planning
process can lead to higher effectiveness. Alternatively, half of the teachers can be randomly
selected to use a lesson, new technology or manipulative marketed to improve fraction and
decimal understanding while the other half serves as a control group. This allows comparative
analysis using performance assessments, school-made tests or standardized tests.
The “one teach, one assist” model would be a good way to begin working together with the
special educator to provide individualized instruction to every student. Both the special
education teacher and the math teacher should take the role of primary instructor (and,
conversely, the role of instructional support) about the same number of times so that the class
respects each as a full-fledged teacher. If one teacher is at the front of the room modeling how to
use the fraction strips, the other can be circulating through the room, using proximity to control
behavior and helping students who are having trouble.
I would use station teaching to integrate educational technology or manipulatives of which the
school does not have a large supply into lessons. Since BMS is a typical middle school, we do
not have enough computers for everyone in the class to use them at once. However, one-third of
the class could be engaged independently (or in small groups) in technology-based fraction and
decimal investigations while another third works with the classroom teacher and the final third
works with the special educator. Each group works on different activities that don’t need to be
learned in any particular order and then they move to the next station. I would advise that the two
teachers position themselves so that their voices travel the same direction (and don’t interfere
with each other) and so that they can both monitor the independent group while they teach. I
would also recommend thoroughly explaining the procedures at the independent station (and
having students explain the procedures back) before beginning rotations.
Alternative teaching with fractions and decimals could be a way of providing enrichment or
remedial help. The pull-out groups should not be static; sometimes the group (or some of its
members) should be chosen at random. It would also be empowering to alternate teach a novel
strategy that will help groups later on to students who don’t always feel that they are essential
parts of the team. For instance, while I am leading the class in an investigation of multiplication
of fractions using discrete sets (looking for a rule), the special educator could be helping a group
of often-overlooked students understand and grow competent in using the area model to multiply
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fractions. Then heterogeneous groups would reconvene and solve real world problems about
multiplying fractions (like cutting a rectangular pizza multiple times). The students with the new
strategy would now be in high demand!
Although it doesn’t really divide the work for the teachers except for behavior management,
parallel teaching would be really useful for challenging topics for which students need more
individual attention and fewer distractions. Using fractions as an operator might represent one of
these topics. Each teacher could set up a “function machine” with his or her small group,
demonstrate how the operator tells you what to do to an input to get an output, and then practice
with the current trading cards fad (e.g. “2 water Pokémon are worth 5 earth Pokémon”) and
exchange rates.
When there is good chemistry between the instructors, team teaching can be a powerful way to
engage today’s “special effects generation.” Teachers can add variety to class by doing skits to
set up problems and share instruction. One may write on the board or model how to do
something while the other explains. Since multitasking is difficult, a lesson often slows down
when one teacher must transition between all of these roles. However, seamlessly sharing the
roles allows the team to minimize direct instruction and maximize exploration, ideally keeping
up a pace of fractions and decimals instruction that students follow well without becoming
bored. If fractions are embedded in a data analysis context, one teacher could help students enter
their data into an Excel spreadsheet while the other helps the class see how the fraction of
interviewees who like rap out of the total number on the big screen changes as more data is
added.

What are your areas of strength with regards to planning/teaching for understanding?
I enjoy researching student needs and engaging in self-critical reflection. I am quite creative and
skilled at adapting a task to make it more exciting and challenging to high-level learners.
Likewise, I can use the objectives to build up simpler, yet similar challenges (with the same big
idea) for learners who need to take smaller steps. I create lessons and assessments that are wellaligned with the indicators. I am good at seeing connections between math and student interests.
Finally, I work hard and am committed to helping students achieve success.

What are your areas of concern with regards to planning/teaching for understanding?
I remain leery of testing the limits of intrinsic motivation— I am more skeptical than my
professors that middle schoolers would choose to challenge themselves given an easier option.
Though I am a self-starter who was known as an overachiever through school for her exuberant
exploration and extension of creative projects, the main reasons I and my peers in gifted or
accelerated classes with me chose a challenge is that: A) the challenge was fun or, B) they had
someone to impress or a reputation to uphold. There are probably more students than I
acknowledge) who do find the mere idea of challenge engaging and would, between two nearly
identical problems, choose the one involving five decimal places instead of the one involving
two decimal places, even if the teacher emphasized that neither problem is any better than the
other. Because “put a little more faith in people” and “raise expectations” are too vague to be
productive goals, I will describe the more empirical procedure I will take in order to stretch my
Final Exam
comfort zone as necessary. First, I will investigate Small’s (2005-2010) research on
differentiated instruction and open questions and also read other researchers who discuss choice
in education. I will seek guidance from teachers who have successfully used choice to encourage
intrinsic motivation, even when neither option was “fun” per se. I will write a set of summary
guidelines for myself on how to establish a classroom culture in which it is good to challenge
yourself without embarrassing students who are challenged by the less complex task. I will work
with the special educator to keep the principles I uncover alive in my class, possibly creating
posters to decorate the classroom that support building and challenging yourself. Then I will do a
few tests of the power of choice in my own classroom through a tiered parallel classroom
experience. My special educator and I will parallel teach to a randomly selected half of our
students. I will give each student in my group a sheet of paper with both (or all three) parallel
tasks and allow them to select one to do, with the caveat that no question is better than the others.
My special educator will discretely hand out different tasks to different students, thus eliminating
the element of choice. We will take note of their feelings about the experience and survey them
using a short Likert scale with space for comments and subsequently repeat the experiment with
variations—i.e. the teachers switch roles, a more or less interesting task is used, etc. Eventually
we would have a class discussion about choices, why they like or do not like them, and what they
pick when given a choice. I would use all this information to help me decide how much I should
use choice when the choice really doesn’t make the subject of the problem more interesting, only
more challenging.