Mathematics: Compare decimals using the language "is greater than", "is less than", "is equal to" and symbols <,>, and =.

Directions: For problems 1-10 please use the symbols >,<,= to compare the decimals. For problems 11-20 please write a decimal on the line provided to make each statement true.

Slide 3Q: The student's source of knowledge came from the curriculum I taught them. The students relied on a supplemental video, my modeled instruction, and small group instruction to learn how to complete this assignment. Our administration team, curriculum leaders and grade level leaders decided that this content was important. At this point I don't feel that anything was missing from this lesson due to my students meeting the objective.

Slide 4Q: My students employed to learn using the following dimensions of behavior:

• Language

• Reason

• Memory

Slide 5Q: Students were able to draw from Mainstream Academic Knowledge to complete the task due to the fact that this curriculum is based on facts and what is true about arithmetic. Human interests do not have an impact on this lesson.

Slide 3: the students combined knowledge from a variety of sources. They are able to apply their own prior experiences to frame how the interpret the text and what they deem important, but also they are limited to the text that we are discussing and the examples availabile within it. In looking at the text rather than the assignment (as they are so closely related) this is definitely Mainstream Academic Knowledge (old, white, male authors) but I try to frame it in a way to make it more universal in our discussions. So I think it's a combination of factors to determine what, how, and why this is important.

Slide 4: there are many dimensions that can be applied to this assigment, depending on a couple factors. All students will emply language, as they are reading a novel to construct their central ideas. Students can employ emotion, reason, and/or faith depending upon which theme they are tracking and how it fits into their personal perspective. Some students may not be able to build upon personal experience but others may be able to very much tap into their own experiences to frame how they interpret the narrative. Students may also have a rich background in faith which can guide their view of sin and morality or forgivenss, or maybe they don't have that religious perspective. So it's very individualized to the student and also gives students a rich opportunity to explore and share their interpretations with other students and see how they correspond and differ.

Slide 5: I already sort of addressed that the choice of the text itself, Scarlet Letter, is definitely founded in Mainstream Academic Knowledge, as this taps into "old white guy" literature, but I think the assignment allows students to include their personal experiences through Personal and Cultural Knowledge. What examples they choose to include and how they choose to interpret them provide some creative and independent thought on the part of the student.

SW name and label the school location

Directions: Students were instructed to orally express then illustrate and fill in the blanks to complete a sentence frame to identify their favorite school activity they are doing and the school location they are doing that activity. Students are utilizing the visual word bank to complete the sentence frame.

Slide 3: Students were employing knowledge of the vocabulary I instructed in the previous week while connecting the vocabulary to their own experience and sharing their opinion on their favorite school activity in order to make the vocabulary more relatable. Students are also employing the knowledge of prewriting skills such as illustrating their thoughts first, orally explaining their response, and then filling in the sentence frame.

Based on the experience of myself and my team in collaboration with the ESL department and our supervisors, we have created resources to assist in teaching our multilingual learners key vocabulary and social language as well as important cultural norms of the school such as making sure toilet paper is flushed instead of thrown in a trash can as it is in some countries. We assess our program and reflect on what we need to add or take out as we survey the needs of our students.

Slide 4: Students used memory to recall the practice of utilizing a visual word bank as well as the practice of illustrating their idea first as part of prewriting. In addition, students used language! That was the main goal! Students also used a sense of perception. For instance, when a student stated, "I am eating in the classroom," that is because they eat breakfast in the classroom or "I am learning in the nurse's office," they explained because they just had their dental lesson. As long as they could explain their reasoning behind their sentence, I was very pleased!

Slide 5:

I believe students draw from a mix of personal/cultural and mainstream academic knowledge when completing most of our activities like one in our newcomer group as we discuss the language, culture, and practices they do in their native country and compare and learn about what we do in our school.

As you go through the store, you find the store is running deals on the items you need. Using what you know about finding unit rates, determine which deal is better. Be sure to show all work on your shopping list.

Include appropriate labels, when needed.

Slide 3: During this activity, students were to use their prior knowledge from their on-level math teacher's instruction and my instruction as their interventionist. The necessity of this topic is defined by the Grade 7 Math Standard CC.2.1.7.D.1: Analyze proportional relationships and use them to model and solve real-world and mathematical problems (Pennsylvania Department of Education, 2024). In addition, this skill was identified as a targeted area for math intervention through the i-Ready diagnostic assessment. I feel as if this activity was soundly established based on the student's prerequisite learning and conceptual understanding during class.

Slide 4: There are two dimensions utilized during this lesson/activity. The first dimension employed is reasoning. Students are using their knowledge of finding unit rates and applying it to real-world situations. In this case, they're having to make decisions on what the "better buy" is on a variety of grocery items. The second dimension used is memory. Many students have been shopping with their families or on their own at one time or another. By tying a math concept into experiences they already carry, the connection may be made that demonstrates the usefulness and purpose of learning the concept.

Slide 5: The Banks Knowledge Typology students had drawn from during this assignment is personal/cultural. Specifically, the personal aspect of this knowledge type. As I mentioned above, students in this class have experience shopping in a variety of settings. Students are able to take their experience and build connections between mathematical concepts and real-world scenarios. For many, these are decisions students have already made or have witnessed adults make.

Students employed prior knowledge from real-life experiences (of eating out, tipping, getting discounts) and previous instruction related to these topics. This knowledge is essential according to both Common Core and state standards. In addition, it is an important life skill. The knowledge that might have been missing was personal experience with eating out, taxes, and tips. Because of this, we discussed typical tip percentages, tipping on the subtotal versus the total with tax, etc.

Students employed Reason: utilizing prior knowledge to synthesize new experiences. Many students have not previously been responsible for figuring out how much a meal will cost or how much tip they should leave when eating out with friends or family. Students also employed Memory: recalling prior knowledge based on ideas or experiences. All students received instruction and have practiced these concepts both individually and collectively. Some students also have prior knowledge based on their shopping and/or eating out experiences.

This particular assignment is largely based on School (or Pedagogical) knowledge as defined in the Banks Knowledge Typology. These facts and concepts were presented through instruction and defined through collaboration and discussion. Personal Knowledge also comes into play, as noted above.

Theme Text Dependent Analysis - Reflect upon your reading of the four pieces of text in Reading class.  Choose two texts and prove how/why they share a common theme.  Use evidence from the texts to support your claim.

Slide 3:

In this assignment, the students needed to use their prior knowledge to complete the essay prompt.  In their Reading class, they read four different texts and were responsible for creating an organizer discussing each texts’ plot and theme.  In their writing class, they took those organizers, decided on a common theme between two of the texts, and created a boxes and bullets (organizer) in preparation to write an essay.  Once they completed their organizer, they wrote a four paragraph Text Dependent Analysis to compare and contrast the common theme between both texts.  As a team of English and Reading teachers, we decided this knowledge was important because of the focus of theme and TDA writing on the PSSA for 6^th Grade.  It also directly connects with State Standards for ELA.  Knowledge that could have been missing could be understanding the theme and plot, as well as relatable connections students made with the original reading pieces.  If students struggled relating to the material, it could be challenging for them to draw connections and analyze the stories.

Slide 4:

Students would have needed to use reason - utilizing prior knowledge to synthesize new experiences.  Using their prior knowledge from our writing lessons that built up to writing this essay, students were able to effectively complete the prompt.  Since we created an example essay together as a class with two different stories, students needed to activate their prior knowledge from constructing that essay to mirror those concepts and skills on their own.  In addition, students needed to use memory - recalling prior knowledge based on ideas or experiences.  The TDA concept is one that has been discussed in prior years to entering 6^th grade.  As a result, students could also draw on those experiences in elementary school to produce this essay.

Slide 5:

When looking at Banks’ Knowledge Typologies, students are drawing from Pedagogical Knowledge to complete this assignment.  Students are drawing from multiple forms of instructional materials in order to produce the final assignment.  Because students are also analyzing the material and drawing their own conclusions, I feel as though there could also be a form of personal/cultural knowledge in this assignment as well.

Read the sentences from the passage.

“I’m not like Pa. People won’t like what I play. They might laugh .”

Write an essay analyzing how the sentences show a theme of the passage. Use evidence from the passage to support your response.

Slide 3: In this assignment, students used prior knowledge of the theme from their reading lessons as well as their own experiences to determine the theme of the passage. Students were given a graphic organizer with the acronym RACE to help structure their essays. This connects with the PA standards for analyzing themes. As a curriculum department, we decided to focus on these TDA prompts and released items to help prepare the students for what they will see on the upcoming tests as TDAs account for 25% of the test scores. The reading level of students could account for struggles as it would be difficult for them to read and analyze the passage. Scaffolds were put in place with the graphic organizer and modeling done by the teacher in small groups to help overcome this struggle.

Slide 4: Students used reasoning and memory to help them complete the assignment. In the passage, the character struggles with self-confidence. The theme of the passage was determination. Students would need to use their prior knowledge as well as their own experiences to determine the theme of the passage.

Slide 5: Using Banks' typologies, students would have used mainstream academic knowledge and personal knowledge.

Research Project: You will make a visual like the one found on pg. 36. Make a copy of this slide show to get started. Follow all of your teacher’s directions and answer each of the questions in the slide show.

Some organism options: Arctic Fox, Monarch Butterfly, Orca, Sea Otter, Cheetah, American Bison, Komodo Dragon, Sand Cat, Three-toed Sloth, Horned Puffin, Peregrine Falcon, Emperor Penguin

Slide 3: This assignment draws upon several knowledge sources for student learning. The textbook chapter provides the foundation with concepts like ecosystem organization, while the teacher's additional instructions and chosen organism list guide specific research directions. Students need to research skills to gather information and apply visual design principles to create their presentations.

Slide 4: The assigned research project on ecosystems encourages students to engage in various dimensions of learning. Cognitively, they'll acquire information from the textbook, understand ecosystem concepts, analyze their chosen organism's role, and apply this knowledge to create the presentation. Students will need to plan their research, monitor their understanding, and evaluate their own learning.

Slide 5:

Mainstream Academic Knowledge: While not explicitly stated, the textbook chapter likely falls under this category, providing foundational scientific concepts about ecosystems.

Transformative Academic Knowledge: This category encourages questioning and challenging traditional perspectives. Depending on the teacher's approach and how students engage with the material, there's a possibility of exploring environmental issues or questioning human impact on ecosystems, potentially aligning with transformative knowledge.

Popular Knowledge: While not the main focus, students might incorporate popular understanding of their chosen organism gleaned from documentaries, nature shows, or even children's literature.

Personal/Cultural Knowledge: If the assignment allows students to connect their chosen organism to their personal experiences or cultural backgrounds, this category could become relevant. For example, students with rural backgrounds might relate to information about farm animals or local ecosystems.

Directions: When we experience big emotions we can sometimes feel them in our body. Where in your body do your feel anger? Pick a color or colors to represent anger and mark the picture of the body where you feel anger.

This assignment draws upon the individual's personal experiences with anger. Each student must assess individual episodes of anger and identify where the anger was felt in his/her body. The colors selected to represent anger could be influenced from information provided in art class or English metaphors.

To complete the task assigned, each student must use memory to recall a prior experience. Emotion and senses are employed to identify where anger is experienced in the body. Imagination is employed to represent anger with a color or colors.

According to the Knowledge Typologies identified by Banks, the following types of knowledge were used to complete the task: personal knowledge, mainstream academic knowledge, and school knowledge.

Directions:

• FIRST: Get a paper copy from the teacher and carefully cut out the 18 boxes identifying the steps of the Engineering Design Process, the definitions of each step in the EDP, and their application to the Black Box Investigation.  ***Note: the cards ARE NOT in correct order***

• SECOND: Put the steps of the Engineering Design Process in order from beginning to end.

• THIRD: Match the definition to each step to the Engineering Design Process

• FOURTH: Match the Black Box application to the EDP

• FIFTH: Photograph the completed card sort and submit the photo to Schoology.

• SIXTH: Check your photograph with the answers at the front of the classroom.

Slide 3: In this assignment students were creating their own knowledge to to fit within the scaffolded information presented by myself of the Engineering Design Process (EDP) and the FOSS program. The EDP was reviewed by teachers several years prior to assignment in 5th grade and six steps of the EDP was selected by the teaching team.

Slide 4: In this assignment I recognize students were using several dimensions of behavior to learn the material. First, they were employing "Language" as they were chronologically organizing the steps of the EDP and then matching the correlating definitions. Second, the students were using "Reason" by applying their prior knowledge from Black Box Investigation and EDP structure connections. Finally, students also used "Memory" to recall their prior knowledge of the Black Box Investigation to remember the sequence of events of the investigation.

Slide 5: For this assignment students reflected upon their prior experiences working through the Card Tower Challenge of which the steps were not explicitly taught, rather discussed as "what should we do next?" to make connections with their experiences through the Black Box Investigation and the 6 steps of the Engineering Design Process outlined more subtly. Students were using their Pedagogical Knowledge; drawing from the more subtle presentation with key words guiding through the exploration and their Personal/Cultural Knowledge; using their personal experiences from the Card Tower Challenge and Black Box Investigation, both inquiry based and student driven.

Station 1 - The Roaring 20s: Students would read through a powerpoint presentation and record answers to questions which would relate to the setting of the novel.

Station 2 - The American Dream: Students read opinions from other high school students around the country about how they felt about the American Dream/if they feel they have experienced it/if it exists or has changed. My students would then record their own definition of what the American Dream means to them, and if they believe it is still attainable or not.

Station 3 - Students would watch a video about F. Scott Fitzgerald, the author of The Great Gatsby, and fill out a question/fill in the blank worksheet.

Directions: Students had 15 minutes at each station, after which they would get up and rotate to the next table. At the end, we regrouped and discussed our takeaways from each station.

Slide 3: Students employed knowledge from different sources for each station. The American Dream activity, for example, asked students to read about experiences from other students around the country as well as their own experience to create a definition of the American Dream. They were then asked to decide for themselves, based on testimonies provided as well as from conversations had with their classmates, how they felt about the American Dream. The other two stations used expert sources which I curated specifically to give them the best possible background knowledge prior to reading.

Slide 4: Emotion and memory for station 2, because it was largely opinion based. Students testimonies in the article were largely based on their own experiences as well, therefore justifying their thoughts on what the American Dream means to them. The other two stations were largely based on reason. Students would have to pinpoint the most logical answers based on the information provided.

Slide 5: Personal knowledge (station 2) as well as mainstream academic knowledge (stations 1 and 3). Station 2 was something that most students related very strongly to due to their family backgrounds. Many already had strong opinions on the American Dream, while others were able to relate it to their lives in ways they had not thought of before. The other 2 stations required students to hunt for information from curated academic sources.

Directions: Choose a digital template (document, presentation, video, jamboard, website) or physical poster and create a creative way to present your argumentative essay topic and position. Your source sheet from your essay should be included and will be your citations. In your project, clearly present your topic and claim, at least two pieces of accurate evidence, and an example of a counterclaim or call to action. Your project should feature images and a design that captures your audience. Reference my examples for ideas and support when creating your project. 

Instructions and Rubric

Slide 3: The students employed their knowledge of persuasive techniques, research and presentation of credible sources and evidence, and development of a clear and concise topic and claim. The ELA curriculum called for the exploration of argumentative writing as well as the processes that are used to develop persuasive arguments. While I do believe there are opportunities that prepare students for this material in earlier grades, I noticed there was a lack of prior knowledge in regard to what a social issue was and how it affects many people. Social issues were the topics students were choosing and developing a claim about for their essay and project. That being said, once we distinguished what a social issue is, students were able to access their prior knowledge and reflect on their personal experiences that impact their position on the issue. 

Slide 4: In this assignment, my students utilized a variety of behavior dimensions including reason, memory, intuition, and emotion. The reason behavior dimension was implemented when they had to compare and synthesize their personal perspectives about their topic with the potentially different or deepened perspectives of experts in their research. The memory behavior dimension was used similarly, as students were urged to reflect on their personal experiences and knowledge and consider how both impacted their position on the topic they were arguing. Finally, both the intuition and emotion behavior dimensions work together to develop and establish a student’s position or claim about their chosen topic. Their position was likely based on a combination of their personal experiences and understanding of the world, as well as their initial instinct about what they perceive about right and wrong in the world. 

Stage 5: This assignment encouraged students to draw from their personal knowledge, popular knowledge, and transformative academic knowledge. Their personal knowledge established their original argument about their topic, but popular and transformative academic knowledge both played a role in the development of a strong argument that featured a variety of perspectives and information from varied types of sources. Transformative academic knowledge also played a role when students were asked to consider counterclaims or opposing arguments to their personal position on the topic. In a lot of ways, this was the most challenging aspect of the project and encouraged students to not only consider what opposing arguments exist, but also dive deep into why others might view a social issue from a completely different lens.

Slide 3: In this assignment students needed to use knowledge from the textbook and notes gathered through teacher instruction. Students also need to use their research skills to design the experiment.

Slide 4: In this assignment students used several dimensions of behavior. Students needed to us language skills as they were working in groups and needed to communicate and work together to accomplish a common goal. They also needed to use sense perception to describe what they can see, feel and hear happening during their demonstration that would lead them to believe a chemical reaction has occurred. Students also had to use memory and reason in the creation of their demonstration.

Slide 5: Students main use of knowledge was mainstream academic knowledge and school knowledge to create their experiment. Students used mainstream academic knowledge as the concepts we are learning about are rooted in established scientific findings. They used school knowledge through the use of their textbook and teacher given notes to support the explanation of their demonstration. Students also used their personal/cultural knowledge when the decided upon their demonstration. Each student was asked to research and was inspired by a different reaction, which influenced what they chose to present to the class.

Students were then give independent warm-up time to practice their scale before playing for their assessment.

The students were directed to employ their own knowledge of their scale as well as proper playing technique, which they know and have been practicing. I decided that knowledge is important because playing with good technique makes for better sound and musician health. The knowledge of the scale is important because we play songs in that key so learning and practicing the scale makes it easier for the students to play those songs (it is developmentally appropriate and in our curriculum).

For this assignment, students learned with sense perception, emotion, reason, intuition, and memory.

The students drew from school and personal knowledge for this assignment.

Slide 3: Students needed to employ their own knowledge from their review combined with the knowledge of the teacher to practice identifying and comparing types of sentence structure from a given sentence. They then had to employ their own knowledge of what makes them happy and implmement the strategies given to identify types of sentence structure and create these sentences. The ELA curriculum calls for this instruction to be implemented. We explain to our students the importance of being able to create unique sentence structures in their creating to become stronger and more engaging writers. We provide evidence of this in the various novels they read adnd how this can affect their writing credibility.

Slide 4: Students had to implement language in order to identify the requirments of each sentence structure type and to create their own complicated sentences. They had to implement emotion to come up with 5 things that made them happy. This is being utilized as a practice that was recommended in my other graduate course that I am taking that provides evidence that completing this exercise benefits the mental and emotional well being of students. They also had to implement reason to create new sentences. This is the first time that students will be intentionally generating their own complicated sentences. Lastly, students used memory to recall the requirements of sentence structure and all prior lessons to generate their own sentences.

Slide 5: Students were able to draw from Pedagolical School Knowledge. They are using the notes from class to generate their own ideas of happiness. This could also dip into personal knowledge since they are expressing their own personal preferences within an academic practice.

Directions: Pick 4 of the 8 cartoons and analyze them. Think about some of our recent classroom discussions about economy and spending and how they may relate to these cartoons. Please submit your answers using Google docs

Slide 3: Students needed to employ their own knowledge from recent lessons and discussions that were done in class about the economy and spending. They needed to use their prior knowledge to help them figure out what was happening in the political cartoon or why it was important. The importance of this activity was to ensure students understood the significance of what was happening in the cartoons and of the discussions we had in class and how they related to history. It helps lead to a deeper understanding of historical content in a different modality.

Slide 4: I think the students were using Emotional, Reason and Memory the most. Emotional because students are relying on thoughts and experiences they had learned about the world around them.

Reason because students were using prior knowledge to help them understand what was going on in the cartoon and connecting it to history and also current situations.

Memory: utilizing that prior knowledge to relate past happenings or experiences to the current topics,

Slide 5:

Personal- students could interpret different cartoons based on personal experiences that they or their family may have encountered in life.

Popular- students get information from social media outlets all of the time. It may not all be correct but it helps to make a great discussion in class when filtering through what is correct and what is an opinion.

Pedagogical knowledge- students understanding from what they read in the curriculum

Directions: I will read aloud a scenario. Based on the definition we talked about today, I want you to talk to the people at your table and decide if what I described is work, or it is not work. Once you have agreed on an answer, write it down on your whiteboard and hold it up. Compare with your classmates at different tables, if you disagree have a discussion about why you chose your answer.

Slide 3: In this activity students were asked to take scenarios in which they thought would be considered work before entering this class and implement the new definition we learned. This lesson used a combination of their preexisting knowledge and new information taken from a science textbook. This information was deemed important by the science curriculum.

Slide 4: Students primarily used reason during my lesson. They had prior knowledge that helped them connect to the new concepts we discussed in class.

Slide 5: For this activity, according to Banks' knowledge of typology, students used school knowledge and mainstream academic knowledge. The concept of 'work' in science is an established theory that was introduced using a combination of class notes and references to the class textbook. The scientific definitions we used are accepted universally. Work is dependent on the force on an object and direction of movement. I would consider these two factors objective data, further supporting that it is mainstream knowledge.

Part 2: Once most students have completed the task, I will explain part two of the activity. Instead of working backward, we will have the students take the parents’ genotypes and draw their own Punnett Squares for the parent cross. I will then pass out the parent crosses. Each pair of students should have 3-4 to complete. As students complete this task, I will walk around the room to assess their understanding of the material.

Slide 3:

In this assignment, students were directed to employ their knowledge of Punnett squares and the principles of Mendelian genetics to complete the task. The decision that this knowledge is important comes from standards or learning objectives established for the course. However, students may also see the importance of understanding how Punnett squares can be used to predict the likelihood of specific traits appearing in offspring, as well as how genetic variation arises through processes like independent assortment and genetic recombination.

Slide 4: Reason- Students utilized their prior knowledge of Punnett squares and Mendelian genetics to understand the instructions and complete the task of working backward to determine parental genotypes. Memory- They recalled their understanding of genetic principles and Punnett square construction to effectively complete the assignment. Language- They read and understood the instructions provided by the teacher and communicated with their partner to complete the task.

Slide 5: Students utilized mainstream academic knowledge related to genetics, specifically Punnett squares and Mendelian genetics principles, to complete the task. However, I think this is also transformative academic knowledge because it enables students to understand and manipulate genetic information in new ways, allowing them to apply it to real-world scenarios or to their own inheritance from their parents.

I will ask a few students to share what they currently know about cancer. 

We will watch the video on cancer and I will ask students to fill out the W section of the KWL chart. I will ask a few students will share what they want to know about cancer. 

After the video students will get to choose an article on cancer from What is Cancer, The Truth Behind the Three Natural Cures of Cancer, What Causes Cancer, and How Cancer Starts Grows, and Spreads. Students will be put in groups based on their chosen article. Students will read the article independently and discuss it as a group. Students are expected to individually take notes on their chosen article. Students should underline or highlight anything they think is important, don't know, or have questions about. 

After students have finished the article they will complete the L section of the worksheet with 2-3 things they learned. 

Slide 3: The knowledge that the students were to employ was their own. This whole assignment was about what the students know, want to know and learned. I designed this lesson this way because it is important to incorporate and consider students previous knowledge, desires for learning and actual learning done. Since the students are not experts in the field they might not know much about the topic of cancer but that is ok. Reading and discussing complex topics like cancer allows students the opportunity to ask questions about things they normally wouldn't.

Slide 4: Students mainly utilized memory for this assignment as I asked about their experiences and prior knowledge with cancer. Some students discussed experiences that they have had regarding family members or friends who have gone through cancer. Several students also talked about commercials they had seen.

Slide 5: I believe students used both mainstream academic knowledge and personal knowledge to complete the activity. Students looked back on personal experiences and had discussions with other students using prior knowledge. I also believe it to be mainstream academic knowledge because students read numerous articles about the history of cancer, potential cures, causes, and the growth and spread of cancer. This is mainstream academic because what we know about cancer is always changing as more research is done and we aim to improve treatment and prevention of cancer.

Slide 3: This project is a summative assignment on quadratics for students. They are to employ their knowledge of the characteristics of quadratic functions and solving quadratic equations. They are also to research the gravity of planets as well as the rules of basketball to come to their own conclusions about basketball rules on other planets. The knowledge of quadratics is decided as important by the standards. At my placement, basketball is huge among the students, and I wanted them to explore how we can examine the universe through math. Due to the project being around basketball on other planets, the subject matter could be altered to better reflect students’ identities and familiarities in their communities.

Slide 4: The dimensions of behavior students employ are language, reason, imagination, and memory. When answering questions like maximum or minimum height, students are interpreting that they are finding the maximum x value of the quadratic function. In their questions for activities 1-3, they are using their memory of quadratics to find the information. Activity 4 requires their reasoning for their inference about the relationship between acceleration due to gravity and maximum vertical height. It also requires their imagination to pick a planet and make rule changes for the game of basketball due to the information they find.

Slide 5: I would say this assignment requires students to draw from popular (or maybe it is personal) knowledge and mainstream academic acknowledge. In terms of mainstream academic knowledge, they are recalling the properties of quadratic functions (maximum and minimum heights) and how to solve them (time required to reach surface, maximum, and minimum heights). They also must reflect on their own personal knowledge, if applicable, or popular knowledge of basketball through research.

Directions: Work individually or in small groups of up to three to solve three 3-variable systems of equations. When you complete a problem, raise your hand so that I can stamp your paper (I had a physical ink stamp with the message "Taco'bout a great job!".

Slide 3: Students had initially learned how to solve a 3-variable system of equations by adapting techniques they already knew for 2-variable systems. Therefore, I'd argue that the knowledge they employed was a mixture of their own, mine, and the generations of mathematicians who came before us that first invented these techniques. Like Newton said, we stand on the shoulders of giants.

In terms of who decided that this knowledge was important, the dry answer would be either the Pennsylvania State Board of Education or McGraw-Hill, our textbook publisher; this topic is not usually taught by my mentor teacher, but its inclusion in the systems chapter gave us an opportunity to try teaching it to the honors class.

In a wider context, solving systems of linear equations of however many variables is an important aspect of linear programming, a field of applied mathematics invented in the mid 1940's to solve logistical problems for the U.S. military. It currently has many applications in private sector logistics as well. One could argue that the motivation for a strong mathematics education for citizens is not only to make them better critical thinkers as voters, but to ensure that we have citizens who can contribute economically and militarily. I think this context would constitute as missing knowledge, as it is often brushed over or not mentioned in an effort to focus on the content and procedures itself. I'll admit, I haven't had an opportunity to elegantly weave this bit of history into my lessons.

Slide 4:

Dimensions of behavior employed include:

• Language - Mathematics uses both unique symbols like "=", "+", and "-", as well as familiar symbols such as letters for variables and parenthesis to signify grouping. Students routinely read and write in this language in Algebra, including during this assignment.

• Reason - Pretty straight-forward display of this behavior; students had knowledge of 2-variable systems to draw from as well as skills that they had just worked with me to create. They used these experiences and applied them to new problems.

• Imagination - While procedural steps may be imitated, there are countless ways individuals learn to do mental arithmetic, conceptualize algebraic manipulations, or explain their reasoning to others that are unique to them. Math drills may not be flashy, but they do offer a chance for students to refine their own methods of problem-solving, even if it is at a small scale.

• Memory - The act of solving the systems involved reasoning, but the foundation for how to do so involved students recalling previous examples.

Slide 5: This assignment had students draw from school/pedagogical knowledge. We have not yet covered applications or word problems (and may not get to due to time constraints), so students were not able to draw from personal/cultural or popular knowledge. Mathematics is not a behavioral or social science - neither mainstream nor transformative academic knowledge would be appropriate categories. If Banks were to expand his definition of mainstream academic knowledge to include the hard sciences and mathematics, I could say that students drew from this type as well.