Elementary Mathematics
Planning Commentary
Planning Commentary Directions: Respond to the prompts below (no more than 9 singlespaced pages, including prompts) by typing your responses within the brackets following each prompt.
Do not delete or alter the prompts; both the prompts and your responses are included in the total page
count allowed. Refer to the evidence chart in the handbook to ensure that this document complies with all
format specifications. Pages exceeding the maximum will not be scored.
1. Central Focus
a. Describe the central focus and purpose for the content you will teach in this learning
segment.
[The central focus and purpose for the content I will be teaching in this learning segment
focuses on students being able to avoid ambiguities in the interpretation of number sentences
by becoming familiar with parentheses and the order of operations.]
b. Given the central focus, describe how the standards and learning objectives within your
learning segment address:
¡ Conceptual understanding
¡ Procedural fluency
¡ Mathematical reasoning OR problem solving skills
[
Conceptual understanding will be addressed through my learning objectives and
standards by introducing students to ambiguous number sentences or expressions and the
importance of parentheses in the first lesson. Students will gain conceptual understanding by
being able to identify ambiguous number sentences or expressions and will be able to explain
why the number sentence or expression is considered ambiguous. Procedural fluency will be
developed in the second lesson with the order of operations. The well-defined conventions for
the order in which operations are performed will allow students to solve ambiguous number
sentences. It is a set of defined rules in which the students will be able to practice their
procedural fluency by solving problems. Students will gain mathematical reasoning by being
able to explain their thought process for solving the expression using the order of operations.
This skill is further enhanced by taking their knowledge of ambiguous number sentences or
expressions and the rules of the order of operations and applying it by making their own number
stories involving these concepts. ]
c. Explain how your plans build on each other to help students make connections between
facts, concepts, computations/procedures, and reasoning/problem solving strategies to
deepen their learning of mathematics.
[
First, students will make the connection that numerical expressions and equations use
parentheses to ensure number sentences or expressions are not ambiguous. Even though the
students are introduced to the rules of the order of operations in the second lesson they build
upon the previous lesson’s concepts and continue to use parentheses as a way of making
explicit the order in which operations are performed when evaluating expressions. The final
lesson, will allow students to use their knowledge of ambiguous number sentences or
expressions to connect their understanding and form their own number sentences bringing
together their conceptual understanding, procedural fluency, and reasoning skills.]
2. Knowledge of Students to Inform Teaching
For each of the prompts below (2a–c), describe what you know about your students with
respect to the central focus of the learning segment.
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Elementary Mathematics
Planning Commentary
Consider the variety of learners in your class who may require different strategies/support
(e.g., students with IEPs, English language learners, struggling readers, underperforming
students or those with gaps in academic knowledge, and/or gifted students).
a. Prior academic learning and prerequisite skills related to the central focus—What do
students know, what can they do, and what are they learning to do?
[
As a class we have finished a unit on exponents and it has led us to begin work on
parentheses for the order of operations.For the majority of the students in the class, this will not
be the first time they will be introduced to the order of operations. However, it is not my goal just
to have them solve problems using the rules of the order of operations. My focus for these
students is to push them further by looking at what language they use when discussing the
problems with their classmates and in class discussion. They will also be assessed by how they
can explain their reasoning to me through a series of informal and formal assessments. To
challenge my my higher level students I will provide expressions that involve nested
parentheses and exponents something I know many students have not been subjected to.
There is a group of seven students who might struggle with this concept and need
further intervention. They struggle with multiplcation and division facts as well as exponents. I
have support during our math block from the special education teacher who helps guide 4 of
these students through the lessons and problems. In the first lesson, I plan to assist these
students with extra one-on-one support during tiered support. Tiered support is a thirty minute
period at the end of the day allocated to teachers to offer any additional support to students who
may need extra help. I use this allocated time to assist students with their math skills. I will also
use the concept of gradual release to help guide students learning. In my second lesson, I have
also differentiated the kinds of questions each group will receive. Even though these students
may not solve a problem with nested parentheses in their own group they are still being
exposed when the other students present their work to the rest of the class. In our final lesson, I
will be using mixed groups and encouraging all levels of students to work together to create
their own number story and explain their thinking to the rest of the group.
I also have a student with a 504 plan who is unable to hand write the majority of their
work. As an accommodation, we allow this student to type all of their notes and assignments on
a computer unless they are completing a worksheet in class. I also have a student with an
Individualized Education Plan who has a behavioral disorder. This student is provided with
breaks or additional time to complete assignments or assessments. However, this student
excels in mathematics and it is unusual for this student not to complete his work during the
allocated time. I am sensitive to this students needs by making sure the student is in a group
they feel comfortable with. I tend to place this particular student with a few different groups who
are accommodating to the students behaviors.]
b. Personal/cultural/community assets related to the central focus—What do you know
about your students’ everyday experiences, cultural backgrounds and practices, and
interests?
[
The school I am teaching at has a mix of different cultures and socio-economic
backgrounds. Most of the students in my class receive a lot of support from their parents but
there are some students who do not. Two students in particular who are struggling in math do
not receive extra support at home. I always try to provide extra support during class from the
special education teacher or myself and provide extra assistance during tiered support. Some of
the vocabulary related to my central focus might be difficult for some students to understand
since they have not been exposed to it before. This is why I will implement different vocabulary
strategies we have learned in reading like context clues and antonyms or synonyms to help
students better understand during mathematics. Also, for this lesson I would like students to
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Elementary Mathematics
Planning Commentary
explain in writing how they found their answer. I think it will be difficult for many students to
articulate the steps they took to solve the problem using the order of operations because they
have not been asked to use this kind of high order thinking skill before. Through modeling
techniques I plan to help students explain their reasoning when solving ambiguous numerical
expressions.]
c. Mathematical dispositions related to the central focus—What do you know about the
extent to which your students
¡ perceive mathematics as “sensible, useful, and worthwhile,”1
¡ persist in applying mathematics to solve problems, and
¡ believe in their ability to learn mathematics?
[
Many students in my classroom are very driven and those who are already thinking
about their academic future in the 5th grade are the students who perceive mathematics as
worthwhile. They are persistent at obtaining and applying their knowledge to solve problems.
The majority of the class believes in their ability to learn mathematics. This is why when
planning for my unit, I wanted to challenge students through the use of language and further
apply their own knowledge to make connections to the material and create their own
understanding by working with their peers. In terms of the order of operations, I presented the
students with the rules but they then applied the concept on their own. I also gave them the
opportunity to create their own number stories that involve higher order thinking skills. For a
select few, mathematics is very difficult for them and they struggle not just in mathematics but in
other subjects as well. For these students, I find it is important to provide additional support and
encouragement. It helps to focus on the mathematical concepts that these students can do to
instill them with a sense of achievement. This is why I differentiate my instruction and while still
using the order of operations, give them problems that play to their strengths like addition and
subtraction, simple multiplication and exponential problems while still solving number stories or
expressions. ]
3. Supporting Students’ Mathematics Learning
Respond to prompts below (3a–c). As needed, refer to the instructional materials and lesson
plans you have included to support your explanations. Use principles from research
and/or theory to support your explanations, where appropriate.
a. Explain how your understanding of your students’ prior academic learning and
personal/cultural/community assets (from prompts 2a–b above) guided your choice or
adaptation of learning tasks and materials.
[
I have used a variety of learning tasks and materials to help adapt my instruction to the
various learners in the classroom. First from my own experiences as a student, I know students
learn in many different ways. Gardner’s theory of multiple intelligences argues that children
learn in different ways, and teachers can create and adapt lessons to help students learn. When
giving instruction I try to incorporate auditory and visual cues to help students. This means
speaking clearly and directly to students and also having students take notes so they can have
examples to see and refer back to. I also incorporate activities where students can practice
material as a group because kinesthetic learners do not learn from auditory or visual cues but
by actively engaged in solving a problem.
1
From the Common Core State Standards for Mathematics
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Elementary Mathematics
Planning Commentary
In one of my classes my professor, Dr. Bean, discussed simple strategies that can help
students easily remember a difficult multi-step problem by using mnemonic devices. I will use a
mnemonic device to help students remember the order of operations. Please Excuse My Dear
Aunt Sally (PEMDAS) is an easy way for students to remember parentheses, exponents,
multiplication, division, addition, and subtraction.
I also differentiate my instruction to help meet the varying academic levels in the
classroom. In Literacy for the 21st Century, Gail E. Tompkins states the expectation that all
students are to meet the same standards at each grade level implies that all students should
receive the same instructional program but this is not the case because some students might be
working at grade level while others are struggling or advancing. Tompkins argues to differentiate
an activity it should be rigorous by providing challenging instruction, relevant so that it still
addresses the key concepts and strategies, flexible by implementing different strategies and
grouping techniques, and complex where students are thinking deeply about the concept
involved. I want to challenge my students who have already been exposed to the order of
operations and have a good understanding of the basic concept. To challenge them, I would
give them higher order problems involving nested parentheses and exponents within
parentheses. For my struggling learners, I would give problems that still are important to the
order of operations but are aimed at the students strengths like addition and subtraction, simple
multiplication and division where they need to identify that multiplication and division take
precedence over addition and subtraction and they should work from left to right if their is more
than one specific operation in the problem. This will address rigor where I am challenging my
students on different levels, relevant because we are addressing the standards used for this
lesson, flexible by grouping students according to level, and complex since thing students are
thinking about the order of operations but within their zone of proximal development.
Teachers should also be aware of Vygotsky’s zone of proximal development which
assigns students tasks between their actual developmental level and their potential
developmental level. Vygotsky argues that students learn very little when they perform tasks
they can already do independently. While I am sure my more advanced thinkers may be able to
solve simple number sentences using multiplication, division, addition, and subtraction, I am
trying to challange them with other variables like exponents and nested parentheses so they are
still working between the actual development level and their potential development level.
Tompkins also emphasizes the concept of gradual release, which allows teachers to use
the five levels of support: modeled, shared, interactive, guided, and independent. By using the
levels of support teachers are able to move from more to less gradually giving the students
increased responsibility.I also use the concept of gradual release when I introduce a new topic.
When we begin with parentheses my lesson is very structured beginning with explicit instruction
to introduce ambiguous number sentences or expressions. Then followed by structured
guidence working on a problem first in pairs and then as a class. Next students would work in
small groups to complete a serious of problems to be taken up as a class and then released to
practice independently.
I try to give students as many opportunities to work in pairs or small groups so they are
discussing their ideas and opinions. It is important for students to make connections to their
learning and do so in a supportive environment where students help one another. In Anita
Woolfolk’s Educational Psychology she states Vygotsky emphasized participating in a broad
range of activities with others where learner outcomes are produced by working together. Social
relationships help lead to the construction of knowledge by using tools from the culture like
language, maps or music to help guide the child to generate new problems and solutions within
this zone of proximal development.]
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Elementary Mathematics
Planning Commentary
b. Describe and justify why your instructional strategies and planned supports are
appropriate for the whole class and students with similar or specific learning
needs.
Consider students with IEPs, English language learners, struggling readers,
underperforming students or those with gaps in academic knowledge, and/or gifted
students.
[
My teaching aims to support and challenge students at the same time. It is suitable for
the whole class because it caters to a variety of learner needs. By differentiating my instruction
it allows for students to be challenged at the appriopriate level. By providing students with
specific questions of varying ability it allows all students to benefit from my instruction. For the
below-level learners, it is beneficial to provide them with questions where they can feel
empowered by aiming to challenge them at their level and by focusing on what mathematical
concepts they do know. Also, it is beneficial to the higher-level learners to take their knowledge
to the next level and challenge them with more difficult problems.
I always model what I expect students to know, and how to use the vocabulary and
language that is used for this unit. For example, describing how to use the rules for the order of
operations to solve an ambiguous number sentence or expression.
I try various grouping strategies so I do not have the same students working together all
the time. I use differentiated groups for when I want to challenge learners with a specific skill or
mixed groups when I want students to work together for the benefit of the whole group. I want to
empower my above- or at-level learners to help and guide a classmate who may be struggling. I
am also aware of my student with a behaviorial Individualized Education Plan. I try to place this
particular student with a group the student can excel with. Unfortunately there are some
students who might disagree or provoke this student so they are limited by who they can work
with and I try to avoid conflict situations as advised by the special education teacher.
One of my students has difficulty writing and causes this student to fall behind in note
taking so I photocopy my notes so this student will have the same material to refer back to or
the special education teacher will help the student to copy the notes into their notebook.
I also provide extra help during tiered support at the end of the day. I’m grateful that the
school I am student teaching at offers this time with the students. It allows me to analyze the
problem of the day or exit slips during my break and meet one-on-one with students to clarify
any misunderstandings or needed clarifications from the lesson. ]
c. Describe common mathematical preconceptions, errors, or misunderstandings within
your content focus and how you will address them.
[
When the class encounters problems with nested parentheses students might be
confused about how to insert them into a number sentence or expression. To address this
situation I need to model how to use parentheses in a number sentence and provide guided
practice. Students need to practice solving multiple problems and discuss solutions as a class. I
need to assess how students work by informally observing students and collecting homework as
a way to guide them and correct their mistakes.Students might not realize while solving an
expression inside of parentheses the order of operations still applies. Also, for the order of
operations students might think multiplication has precedence over division and that addition
has precedence over subtraction. I think I will need to be very explicit in my instruction that this
is not the case and students should work from left to right when encountering more than one of
these operations. When encountered with a problem like 7 + 9 * 7 / 3 =? Instead of following the
order of operations students might simply work from left to right to solve the problem instead of
multiplying 9 times 7 then dividing by 3. I will have to make sure this is clearly stated through our
class examples.]
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Elementary Mathematics
Planning Commentary
4. Supporting Mathematics Development Through Language
a. Language Demand: Language Function. Choose one language function essential for
student learning within your central focus. Listed below are some sample language
functions. You may choose one of these or another language function more appropriate
for your learning segment:
Categorize
Compare/contrast
Describe
Interpret
Model
[Students will be able to explain how they use the order of operations to solve numerical
expressions.]
b. Identify a key learning task from your plans that provides students with opportunities to
practice using the language function. In which lesson does the learning task occur?
(Give lesson/day number.)
[
My second lesson addresses the key language function and has students practice using
the language function in small groups. After writing and discussing the rules of the order of
operations students will be broken into differentiated small groups and given a problem of
varying difficulty. For example, my above-level learners might be given an expression with
nested parentheses and exponents, while my below-level learners might be given an expression
with only multiplication, division, subtraction and addition. In their groups students would not
only solve the problem but work together to explain how they decided to answer the question by
writing a step-by-step explanation describing how they used the order of operations. There are
key steps I want to make sure the students are incorporating into their answers like making sure
to first evaluate the problem, to work from left to right on exponents, multiplication and division,
as well as addition and subtraction, and if there were nested parentheses to begin with the inner
most pair.
As students worked in small groups, I would circulate around the classroom checking in
with each group and ask guiding questions to help them thoroughly explain what steps they took
to solve the problem. After about 20 minutes and students have completed the problem and
their explanations a representative from each goup would come to the projector and presnt their
problem to the class.]
c. Additional Language Demands. Given the language function and task identified
above, describe the following associated language demands (written or oral) students
need to understand and/or use.
¡ Vocabulary and/or symbols
¡ Plus at least one of the following:
¡ Syntax
¡ Discourse
Consider the range of students’ understandings of the language function and other
demands—what do students already know, what are they struggling with, and/or what is
new to them?
[
Students will need to articulate their answers in paragraph form. This means they must
be able to communicate their ideas in sentence format. Additionally, students will need to
integrate mathematical vocabulary into their explanations. For example, they will need to use
vocabulary terms like parentheses, nested parentheses, exponents, multiplication, division,
addition, and subtraction.
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Elementary Mathematics
Planning Commentary
In this class, students are already used to presenting their reasoning and explanations
when solving problems to the class. Some students struggle with using the correct vocabulary.
For example, when solving a problem using multiplication a student might say “I times it” instead
of saying “I multiplied.” There will be new vocabulary words like nested parentheses and specific
phrases like working left to right that might present some difficulties while articulating their
thoughts.]
d. Language Supports. Refer to your lesson plans and instructional materials as needed
in your response to the prompt.
¡ Describe the instructional supports (during and/or prior to the learning task) that help
students understand and successfully use the language function and additional
language identified in prompts 4a–c.
[
As we write the process for using the order of operations to solve number stories or
expressions I will be modeling the language I want students to use in their answers. As students
work in small groups they can help one another to help clarify their ideas and put them in
writing. By having the groups present in front of the class, the students will get to hear a variety
of answers and see what their answers should incorporate. Finally, with the problem of the day
students will be able to try writing their own answers allowing for teacher feedback. ]
5. Monitoring Student Learning
Refer to the assessments you will submit as part of the materials for Task 1.
a. Describe how your planned formal and informal assessments will provide direct
evidence of students’ conceptual understanding, computational/procedural fluency, and
mathematical reasoning/problem solving skills throughout the learning segment.
[
In lesson one, I will be using informal assessments as I walk around the classroom
checking in on pairs and small groups to see how and what progress is being made on the
assigned task. Doing this allows me to guage what areas need further clarification as a whole
group, if I need to help a student right away individually, or if further internvention is needed
during tiered support. I will also be giving students the following exit slip: ‘Explain how you used
parentheses in Problem 6 on journal page 220 to write the expression for the total number of
undamaged cans.’ I will be looking to see if students can refer to the use of nested parentheses
to identify the total number of undamaged cans. This will show me that my students are able to
understand the concept of parentheses and how this enables them to make number stories and
expressions unambiguous. The following day, I will be collecting and checking the students
homework for prodcedural fluency. If they are able to insert parentheses in the correct places
and use parentheses properly then I know they have gained procedural understanding. The
following lesson for the problem of the day I will review ambiguous mathematical expressions. I
will be asking the following question: ‘Robin asked her friends to solve 4 + 5 * 8 =? What
problem might arise from her friends answers?’ I will collect the problem of the day and I will be
looking for students to use the previous lessons contents and vocabulary by writing different
possible answers to solve the expressoion but to also state that the expression cause confusion
amongst Robin’s friends because it is ambiguous.
In lesson two, as students are working their small groups, I will be walking around the
classroom and checking in with each group to see how and what progress is being made on the
assigned task. I will be looking for specific vocabulary and phrases to explain how to solve
number stories or expressions using the order of operations. Also, during the presentations I will
be making sure the correct language is being used so there is no confusion to the rest of the
class. I will collect the class worksheets to see if any responses need further clarification in
class the next day. During group work it can be difficult to distinguish if a student is struggling
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Elementary Mathematics
Planning Commentary
with an idea or concept. For the following lesson during the problem of the day, I will ask
students to solve the followng: ‘Solve the following problem. Explain how you found your answer
using the order of operations. 12 x 2 + 8 / 2 =?’ Students will write their answers on loose-leaf to
be submitted to the teacher. From this I will be able to tell if students can explain how to use the
order of operations to explain numberical expressions in paragraph form. It will also be able to
tell me if they are able to incorporate the vocabulary we have been using in class. By being able
to solve the problem corectly, it will also show me they have procedural fluency in this area by
correcting using the steps to solve the expression. I will also be collecting and reviewing the
homework to check for procedural fluency.
In lesson three, students will be working in small groups, I will be walking around the
classroom and checking in with each group to see how and what progress is being made on the
assigned task. Also I will be collectng the groups’ worksheets to check their progress and see if
they were able to create a number story using the order of operations. This might be a difficult
assignment for many groups because it involves mathematical reasoning and problem solving
skills. I am aware that some groups might not finish the assigned task so for the quiz the
following day I will be reviewing concepts from lessons one through three which will include a
question that checks for students mathematical reasoning and problem solving skills. This
question will pose a probem and have students come up with their own numerical expression
using the order of operations.]
b. Explain how the design or adaptation of your planned assessments allows students with
specific needs to demonstrate their learning.
Consider all students, including students with IEPs, English language learners,
struggling mathematics students, underperforming students or those with gaps in
academic knowledge, and/or gifted students.
[
Since I am including a variety of assessments it allows students to show their knowledge
through a variety of ways orally or written. By walking around the classroom it allows me to hear
what students are saying but having them commit their thoughts to paper allows me to evaluate
their reasoning more thoroughly. My student who has a behavioral IEP likes to participate in
class but sometimes finds it difficult to work in groups. While I try to accommodate this student
by placing this student with classmates the student enjoys working with sometimes this does not
always work. If the student cannot function in a group that day we find other ways for this
student to show me what they know whether it is the student completing the assignment
individually or telling me verbally. Another student of mine struggles with reading and writing, so
for this particular student the special education teacher or myself will read the questions aloud
so they can understand and then this student will verbally say their answer aloud and then write
it down. Also I know two students who are struggling in math and sometimes get overwhelmed
during a unit test where we test a variety of concepts. If these students excel on the problem of
the day and the exit slips it shows me that they know the concept I have taught but have trouble
identifying which skill to use on a larger scale test. If they are unable to complete the problem of
the day or exit slips this tells me that further intervention is required. I also have a student with a
504 plan who is allowed to type all their notes and assignments. When possible I will this
student type his answers unless it is a worksheet where there is minimal writing as specified in
the students 504 plan.]
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