Planning Commentary Template - mATT cRICHTON



TASK 1: PLANNING COMMENTARY

Respond to the prompts below (no more than 9 single-spaced pages, including prompts) by typing your responses within the brackets. Do not delete or alter the prompts. Pages exceeding the maximum will not be scored.

1. Central Focus

a. Describe the central focus and purpose of the content you will teach in the learning segment.

[ The central focus on the learning segment I will teach centers around systems of equations—what they mean, how to graph them, and different ways to algebraically solve them. The key idea is that sometimes we need more than one equation to find the answer to a problem. A system of equations can have one solution, no solutions, or an infinite number of solutions depending on the situation. ]

b. Given the central focus, describe how the standards and learning objectives within your learning segment address

← conceptual understanding,

← procedural fluency, AND

← mathematical reasoning and/or problem-solving skills.

[ The standard that relates to the central focus of this learning segment states, "Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables." Solving systems of linear equations require the conceptual understanding of what the final answer (ordered pair) means, the procedural fluency to actually carry out the algebraic manipulations, and the mathematical reasoning to see that the answer is accurate and makes sense when checked. As students learn more about the combination method that we will be using, they will need to be more adept at problem solving. This is because there is usually more than one way to attack and solve a system of equations.

The National Council of Teachers of Mathematics (NCTM) has eight practices that teachers should seek to develop in their students. The specific practices included in this learning segment are: Make sense of problems and persevere in solving them, reason abstractly and quantitatively, and construct viable arguments and critique the reasoning of others. Students will engage in solving rather lengthy systems of equations. This requires perseverance and also when an answer is reached, the ability to ask: Does it make sense? This means asking, "does the ordered pair obtained match the solution when the two equations are graphed?" Students also will be reasoning using symbols of algebra, and I will also ask students to get into groups and do a problem. After we will be discussing what different groups arrived at and why they got those answers. Students must defend the strategy they used and their answer.

Lesson 7.1 learning objective states: "Estimate the solution of a system of linear equations by graphing and check with algebra." This is the introduction to systems of equations and requires students to use a graphing calculator to visually estimate the solution to a system of equations. Students then need to know the procedure and understand how to check their estimate using algebra to get a more exact answer. This lesson discusses the big picture of what a system of equations is about and starts to go into the nuts and bolts of how to deal with the situation of equations in slope-intercept form (y = mx + b).

The second lesson combines two lessons from the book: we will touch on Lesson 7.2 (the substitution method), and spend most of the time on Lesson 7.3. The objectives for 7.3 state "Examine and solve systems of linear equations using combinations (and other methods if called for)." To examine a system means to look it over and take some time to determine the best approach to solving the system. Students will learn the mathematical reasoning needed to problem solve and choose the best method—whether it be graph and check, substitution, or linear combinations. Students will be solving systems of linear equations in a variety of settings, including systems that have one solution, no solutions, and an infinite number of solutions, and what those situations mean in real life. Students will gain the confidence and conceptual understanding to know the difference between each kind of system and how and when to choose the correct problem solving approach. ]

c. Explain how your plans build on each other to help students make connections between concepts, computations/procedures, AND mathematical reasoning or problem-solving strategies to build understanding of mathematics.

[ There are many ways the lessons in the learning segment build on each other and help students make connections between concepts, procedures, and mathematical reasoning. The first lesson is the introduction into the world of system of equations. It includes a "big picture" discussion and use of graphing and graphing calculators to visually represent what we are talking about when we say a system of equations has one solution, or no solutions, or infinite solutions. Students see the graphs of each line and can make the connections between the symbolic version and visual representation. For many students, starting with this visual graphing method is the easiest way to introduce the material. This is the first of the three methods with which we will solve systems of equations: graph and check, substitution, and combinations.

The second lesson in this learning segment teaches students how to use the method of substitution to solve systems of equations. This lesson builds upon the first lesson by expanding the "toolbox" of methods to solve problems. This lesson also relies on the more abstract and symbolic use of algebra and not the graphing method. As students gain more practice and confidence in solving the systems of equations, they will create deeper connections between the problems and how to solve them. In this lesson we also introduce the concept of combinations and discuss how to solve basic examples (less steps required to solve) of using combinations with equations in the standard format (Ax + By = C).

The third lesson in the learning segment continues with the method of combinations for solving systems of equations. We will look at more examples, where more steps are required (using multiplication of terms), do one problem with pair and share, and look at another homework problem to get students started. By this point, students will have made the necessary connections and now have three different methods at their disposal for solving systems of equations. ]

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.

Consider the variety of learners in your class who may require different strategies/support (e.g., students with IEPs or 504 plans, 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—Cite evidence of what students know, what they can do, and what they are still learning to do.

[ One of the most important skills regarding systems of equations is graphing and also plugging (substituting) the "X" and "Y" values into equations to produce the ordered pair that is the solution to the system of equations. Students have shown me through questioning they can do this fairly well for the most part. Some students need extra practice. They can also find the Y-intercept given an equation, and analyze graphs, and also estimate where two lines cross. Some students are still struggling with manipulating the terms in an equation. This puts them a bit behind because so much of what we are doing now, and what will be coming later uses these skills of algebraic manipulation.

My classroom teacher told me this is new material for the students—they have never dealt with the concept of system of equations. That is why I chose to not have a pre-assessment. I thought it would be a waste of time.

Another point to note is that from my observations, the students in this class do not regularly get into groups to discuss math. From the math problems I have done in class and as part of their homework, I see they get almost no work on real-world problem solving. Many students just wait to be told the answer or how to solve a problem by the teacher. So we have a lot of work to do in the areas of critical thinking and real world problem solving.]

b. Personal, cultural, and community assets related to the central focus—What do you know about your students’ everyday experiences, cultural and language backgrounds and practices, and interests?

[ My students come from a rural, low-income community that a large economic base from the logging and fishing industries. Our class is a low level Algebra class, where we take more time to cover the same material compared to the other Algebra classes. From what I have observed and brief conversations, most students in our class are there because they need the math credits to graduate. For the most part, I would say these are not highly motivated future math scholars, but who knows. These students like to play sports—football, basketball, baseball—and at least three students receive extra outside class tutoring. A few students are into dancing; others are into cars; some are into computers and video games. Some students also come from a farm environment. Most of the students come from middle or low-income families where math and/or education in general may not be the highest priority in the family. Sometimes, getting a paying job as soon as possible is the highest priority. Dealing with poverty and learning effectively in a school environment can be challenging.

Some students may not have the best English skills. There are no students in my class for whom English is a second language. Many of the math terms we are covering are probably new for students who have no concept of what these terms mean. ]

c. Mathematical dispositions—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

← believe in their own ability to learn mathematics

[ Because many of my students come from low-income families, they may not see mathematics as useful, sensible, and worthwhile. This would explain some of their behaviors in class and towards class and homework—not wanting to do it and many times spacing out in class and not focusing. Other students in class come from "better to do" families, or from families where education is important. I see in these students more interested in math, a cognitive ability above the other students, more focus, and they are usually the ones who raise their hands or help me in class.

Many students in my class who have a low valuation of math in their life also have a low "math frustration tolerance." This means that they give up easily when presented with a problem that is difficult, whereas other students who possess a high "math frustration tolerance" keep working at a problem even when it is challenging and the answer is not immediately evident. In real life, students will encounter many problems where the answer is not obvious, and they must endure through some "hard stuff" and pondering and struggling to find the answer. So I believe that working on the skills of persistence and patience on the platform of math will be of benefit, even if students will never touch this type of math again. Systems of equations can be a lot of work, and my students who have struggled in math may find this chapter very challenging and or frustrating. I'm pretty sure the student with the IEP will be lost, given what I know about his math ability.

Some students have never ever been successful in a math class (they have never received a grade above a C or even above a D. This is devastating to a student's belief and self-confidence in the area of math. The students may have given up trying and do not believe they can learn math any more. These students may be the ones that cause problems in class and display challenging behaviors. Other students may have already had much success in math class (nothing below an A or B). These students have a high belief in themselves, and know they can and will learn much new math. I see both types of students in my current class. ]

3. Supporting Students’ Mathematics Learning

Respond to prompts below (3a–c). To support your justifications, refer to the instructional materials and lesson plans you have included as part of Planning

Task 1. In addition, use principles from research and/or theory to support your justifications.

a. Justify how your understanding of your students’ prior academic learning; personal, cultural, and community assets; and mathematical dispositions (from prompts 2a–c above) guided your choice or adaptation of learning tasks and materials. Be explicit about the connections between the learning tasks and students’ prior academic learning, their assets, their mathematical dispositions, and research/theory.

[ I will bet a few students will not do the homework I assign. Beegle (2003) explains that many low-income families may not have a home environment conducive to "doing" homework. It's not the students' fault, and they should not be labeled as "deviant." So I will try to lessen the homework burden for my students and place more emphasis on in class work. Working in groups is beneficial according to Burke (2011), who says that it is beneficial to students and teachers as well. Employers will place candidates with group work and collaboration skills higher on the pile of resumes. One of my lesson plans will have students working in groups. Beegle also states that poverty is not just an "external" condition, but one that is internalized by the student and negatively affects their self-esteem, educational goals, and ability for the student to ask for help.

I have observed the students waiting for answers, without thinking at all. I have also observed that students almost never get to practice problem solving questions. Given all these factors, I will do my best to allow silence in the room after I ask a question. Menon explains that in the classroom, he never realized how reliant his students were on his thinking, not their own thinking, until he stopped talking. It is good to find ways for me as a teacher to keep silent after asking a question, not giving answers so easily, even agreeing with a wrong answer, reflecting a student's question back at the entire class, being a scribe for a student's thoughts, and fostering a culture where students bounce ideas off one another, not waiting for the teacher to announce the correct idea. These strategies will help students increase their reasoning and sense making skills.

I wanted students to practice posing problems, something that may be new to them—even if they were not solid on the material yet. Problem solving is important, and the problem-posing aspect of this process has been viewed as important and emphasized in curriculum standards and instruction as Leung (2012) states.

I sense there is a lot of Algebra "catch up" these students need, so my lessons will attempt to not extend the learning "umbrella" too far in three lessons.]

b. Describe and justify why your instructional strategies and planned supports are appropriate for the whole class, individuals, and/or groups of students with specific learning needs.

Consider the variety of learners in your class who may require different strategies/support (e.g., students with IEPs or 504 plans, English language learners, struggling readers, underperforming students or those with gaps in academic knowledge, and/or gifted students).

[ There are two main groups I see in this class—students who do well in math and may see math as useful in their personal lives, and those who are not doing well and may not see math as useful in their personal lives. I make an effort to call on every student in class when I teach at least once, and direct specific questions as appropriate to the level of the student being asked. I like to have interactions and engagement with my students, so I ask many questions of them. I also like to have them work in groups, and try to mix the ability levels of students within each group, so one student can help another student. If there is new material we are exploring, I may ask a "what if" question the class, and then direct it to one of my students who has shown a better grasp of past material to see if they can answer. There is one student who has an IEP and I check in with him frequently during class, both by asking and with eye contact and looking at his papers. ]

c. Describe common mathematical preconceptions, errors, or misunderstandings within your central focus and how you will address them.

[ Common misconceptions in this learning segment of systems of equations include: not multiplying all the terms in an equation by a scaling factor, trying to mix the equations (adding the columns) before zeroing out one variable, not handling the "-1" scale factor correctly, assuming an ordered pair answer works and checks out after substituting into only one equation—not checking both equations with the determined ordered pair, and lastly, the meaning of substitution in the context of systems of equations.]

4. Supporting Mathematics Development Through Language

As you respond to prompts 4a–d, consider the range of students’ language assets and needs—what do students already know, what are they struggling with, and/or what is new to them?

a. Language Function. Using information about your students’ language assets and needs, identify one language function essential for students to develop conceptual understanding, procedural fluency, and mathematical reasoning or problem-solving skills 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.

|Compare/Contrast |Justify |Describe |Explain |Prove |

Please see additional examples and non-examples of language functions in the glossary.

[ The language function for systems of equations that would be most important for students to develop a solid conceptual understanding, procedural fluency, and problem solving skill is "explaining." This function is important since the definition of explain is to make plain or clear. Having a plain and clear understanding of math concepts in important.]

b. Identify a key learning task from your plans that provides students with opportunities to practice using the language function identified above. Identify the lesson in which the learning task occurs. (Give lesson day/number.)

[ Lesson number 3 provides students the opportunity to explain when they are doing the pair and share task. ]

c. Additional Language Demands. Given the language function and learning task identified above, describe the following associated language demands (written or oral) students need to understand and/or use:

← Vocabulary and/or symbols

← Mathematical precision[2] (e.g., using clear definitions, labeling axes, specifying units of measure, stating meaning of symbols), appropriate to your students’ mathematical and language development

← Plus at least one of the following:

← Discourse

← Syntax

[ The vocabulary and or symbols students use during the lessons are as follows. During the pair and share activity, students use the "=" symbol in the equations. The lesson-specific vocabulary students use include "combinations," "substitution," and "ordered pair." Other important words and phrases we will use include "variable." These words would be oral, since the activity doesn't ask students to write their thinking and reasoning on paper. They will explain their thinking and reasoning to their partner, and then if we have time, to the whole class as well.

The precision we use in the activity includes lining up the "=" signs when solving the equations, writing neatly, so no careless mistakes.

The discourse is the class discussions we will have, geared at increasing the students' understanding of the material. I will give very specific line of questioning to guide and lead the students in their thinking. I will look for productive conversations during the pair and share, using lesson specific vocabulary. Also, talk of the process of solving the systems of equations we will do in class. ]

d. Language Supports. Refer to your lesson plans and instructional materials as needed in your response to the prompt.

← Identify and describe the planned instructional supports (during and/or prior to the learning task) to help students understand, develop, and use the identified language demands (function, vocabulary and/or symbols, mathematical precision, discourse, or syntax).

[ The planned language supports I indentified in the lesson plans include using questioning to help lead the students' thinking. I want to open students thinking and make the shift from them being told what to do, to them doing some of their own thinking. ]

5. Monitoring Student Learning

In response to the prompts below, refer to the assessments you will submit as part of the materials for Planning Task 1.

a. Describe how your planned formal and informal assessments will provide direct evidence of students’ conceptual understanding, procedural fluency, AND mathematical reasoning and/or problem-solving skills throughout the learning segment.

[ I have multiple ways for formal and informal assessments that will provide direct evidence of students' conceptual understanding, procedural fluency, and mathematical reasoning and/or problem-solving skills throughout the learning segment. One of these informal assessments, which may be the most flexible, is the "in-the-moment" assessment, where I ask the student(s) with which I am engaged what their thoughts are on a particular problem, and if they can tell me the next set of steps. This tells me where they are in their understanding of the concept, their procedural fluency, and some of their problem solving skills. This is an assessment I can use at any time—throughout the entire learning segment. "Just in time" assessment might be another way to think of it. If I sense I need to ask an understanding question, I can do as needed to which ever students need some help.

The exit tickets students provide me at the end of lesson one and lesson two show me their understanding of systems of equations and how to write and solve problems. I will use a few of their examples in class the next day to illustrate points to clarify misunderstandings.

The quiz at the end of the learning segment is a more formal assessment that captures what students know—their overall conceptual understanding, procedural fluency, and problem solving skills spanning the entire learning segment. ]

b. Explain how the design or adaptation of your planned assessments allows students with specific needs to demonstrate their learning.

Consider the variety of learners in your class who may require different strategies/support (e.g., students with IEPs or 504 plans, English language learners, struggling readers, underperforming students or those with gaps in academic knowledge, and/or gifted students).

[ The adaptation of the planned assessment will help the one student who has an IEP in this math class. The change is that this student will receive a one-on-one tutor to help him with the quiz. ]

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[1] From The Common Core State Standards for Mathematics

[2] For an elaboration of “precision,” refer to the “Standards for Mathematical Practice” from The Common Core State Standards for Mathematics (June 2010), which can be found at .

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