apprentice teaching in secondary mathematics and … · web viewcategory 2 asked students to write...

Apprentice Teaching in Secondary Mathematics and Science

Zack Brenneman

Formative Assessment Assignment and Analysis

Directions: 1. Select four formative assessments from 75 Practical Strategies for Linking

Assessment, Instruction and Learning. Describe the assessments that you selected and attach the assessment.

(4 pts)

1. Magic Square: The magic square formative assessment was a puzzle like activity. Students were given a 5x5 square where only one of the squares (tiles) was given. This tile was located in the dead center of the 5x5 square. All of the other tiles were blank. Students were also given a separate handout which had all the tiles with information. Each side (4 sides) of the tile had either an equation or a solution. The students had to match the equation to the solution from tile to tile and glue the tiles on the Magic Square paper. At the center of each tile was a unique symbol, and the placement of the symbol was how I checked to see if the students placed the tiles in the correct spot. Six versions/answer keys were given per class. The topic for the magic square activity was solving linear equations.


2. Column Worksheet: The column worksheet consisted of two columns where there were 11 questions in each column. The solution to question 1 in the left column matched the solution to question 1 in the right column. The same set up applied for questions 2 - 11. Students were to work in pairs, and the students decided who was doing which column. The checked their answers after completing each question (they didn’t answer all 11 by themselves, then compared answers. They did question 1, checked it, then moved to question 2). The topic for the column worksheet was writing the equation of the line in slope intercept form given either the slope and y-intercept, the slope and a point, or two points.


3. Trivia Thursday Assignment: The Trivia Thursday Assignment was a worksheet consisting of four categories. Categories 1 through 3 had 5 questions. The students had to choose three to do an answer them. Category 1 asked students to write the equation of the line in slope intercept form given the slope and the y-intercept. Category 2 asked students to write the equation of the line in slope intercept form given a point and the slope. Category 3 asked students to write the equation of the line in slope intercept form given two points. Finally, Category 4 asked students to choose any three of the 9 questions they already answered, and re write them in to standard form.


4. Hiking the Mountain Worksheet: The “Hiking the Mountain Worksheet” had students analyze three different guys (Adam, Chad, and Jason) and their statistics as they were hiking their mountain. There were three categories of information (Hiking Speed, Height of the Mountain, and Number of Water Breaks), so there were a total of nine graphs students had to analyze (Three graphs each for Adam, Chad, and Jason). Students had to answer one question: “Which of these guys worked the hardest when hiking their mountain?” Students had to justify their answer on a separate piece of paper.


How did you choose the assessment technique and the content assessed? (4 pts)

1. Magic Square: Mrs. Wargo gave me the idea of this assessment technique. I was looking for something that the students would like, but would be challenging. Turns out though, the students despised this activity. They said it was difficult and complicated, and took too much time to complete.

The content I assessed was a review on how to solve linear equations covering all categories (one step, multi-step, fractions, distributive property).

2. Column Worksheet: Mrs. Wargo/Mrs Weber gave me the idea of this assessment technique. Based on the magic square activity, and how most of my students hated how many options there were, it encouraged me to follow through with the column worksheet because the students only had to worry about one answer per each question (where as the magic square activity had multiple answers). And since each solution in one column matched the solution to the other column (per number), the students could easily see if they did the question correctly or incorrectly.

The content I assessed was a review for students on how to write the equation of a line given the slope and the y-intercept, the slope and a point, or a point and a point. This was Wednesday’s review for their upcoming test on Friday.

3. Trivia Thursday Assignment: Fred Jones gave me the idea of doing this assessment technique. When I was reading one chapter in the “Tools for Teaching” book, Fred mentioned how he loved his Friday math class because he had to complete this assignment called ‘Freaky Friday’ where students were given different categories of questions, but only had to do a set amount of problems in each category.

The content I assessed was another review for students for their upcoming test about writing the equation of the line given specific information (slope and y intercept, point and a slope, and two points). From the title, this was handed out to students on Thursday.

4. Hiking the Mountain Worksheet: Mrs. Weber gave me the idea of doing this assessment technique. During one of our meetings, we were to place the “typical student responses” in to each blank. This rubric was based on Mrs. Weber’s own assignment (Which band had the most successful concert tour). I liked this type of assignment, so I took her idea and made it my own.

Instead of using bar graphs (the graphs Mrs. Weber used were bar graphs), I decided to make line graphs since my students are currently learning how to graph lines. The hiking assignment has students analyzing lines (increasing lines, decreasing lines, horizontal lines, curved lines), so I thought it would be good to make a real life example out of something they were learning.


What standards did the assessments address? (4 pts)

1. Magic Square: MAFS.912.A-REI.2.3: “Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.”

2. Column Worksheet: MA.912.A.3.10: “Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line. ”

3. Trivia Thursday Assignment: MA.912.A.3.10: “Write an equation of a line given any of the following information: two points on the line, its slope and one point on the line, or its graph. Also, find an equation of a new line parallel to a given line, or perpendicular to a given line, through a given point on the new line. ”

4. Hiking the Mountain Worksheet: MAFS.K12.MP.1.1: Make sense of problems and persevere in solving them.

2. Administer the assessments and collect student artifacts. How many students and which grade level did you assess? (1 pt)

22 students were assessed between my 4th period and 6th period classes. (The reason this number is so low is because I administered this formative assessment after they took their quiz on Friday, and many of them took it home thinking it was homework). Some students finished the quiz early; some students took the entire period.

The students were mostly sophomores (a couple juniors and a couple seniors).

What accommodations did you make for students with different learning styles and needs? (1 pt)

I asked the students: “Based on the information provided, which guy worked the hardest when hiking their mountain. Justify your answer.” I have some ESOL students in my class, and writing sentences in English is not easy for them. I told them to just try their best and if needed, write their answer in Spanish. (I have conversational Spanish language ability, so I would have understood most if not all of what they were saying).

Also, many students never bring paper to class, so for this assignment, if the students needed paper, I provided them with a sheet. (The instructions told them to justify their answer on a separate piece of paper).

3. Select 1 of the assessments and develop a detailed rubric for scoring the assessment. Use the appropriate content rubric to help you develop the rubric for your

assessment. (6 pts)


Levels Explanation of the Levels Typical student responses at each level of reasoning on the guy who worked the hardest climbing the mountain.

Extended Abstract

Students generalize the structure to make it new and more abstract

Based on the graphs, I would say Adam worked the hardest since he was going uphill during the entire hiking trip. Assuming all three of these guys were in the same physical condition, clearly Jason did not work the hardest since he was going at an average speed of 1 mph for the 60 minute hike, his mountain was fairly level, and he only took 2 water breaks. Now, Chad did run the fastest on average: (at 8 mph for about 15 minutes, then 4 mph for roughly 10 minutes, and the rest of the time slowly approaching 0); however, he was going downhill for 40 of the 60 minutes he was hiking. Not to mention Chad took one less water break than Adam. But in order to answer this question of who worked the hardest accurately, I would need to know the level of Adam’s, Chad’s, and Jason’s physique before the climb, if any of these guys hiked mountains as a hobby or are just doing it as a one-time thing, and what their condition was after the 60 minutes.

Analytical Students integrate the ideas to create a meaningful structure

I calculated the average speed and the average incline for each guy. Adam’s average speed was 2.8 mph for the 60 minute hiking trip, Chad’s was 4.5 mph, and Jason’s was 1 mph. Adam’s average incline was 12.5 feet per 10 minutes, Chad’s average decline was 5 feet per 10 minutes, and Jason’s average decline was 0.2 feet per 10 minutes. So I’d go with Adam as the guy who worked the hardest, but there are other things I would want to know like the physical condition of each guy, and the landscape of each mountain.

Quantitative Students can identify mathematical ideas in a quantitative way but cannot integrate these mathematical ideas during the task.

Adam was traveling mostly at 3 mph, Chad at 8 mph, and Jason at 2 mph. Adam was going uphill the entire time, Chad was progressively going downhill, and Jason was half and half. Since Chad was going at 8 mph for 15 minutes of the trip (greater than any of the other guys), Chad would have worked the hardest because he is likely to be out of breath.

Transitional Students focus on only one aspect of the solution.

Adam took one more water break compared to Chad, and took twice as many water breaks as Jason. Since Adam took the most water breaks, he worked the hardest.


Idiosyncratic This level is based on subjective reasoning with unrelated data and is affected by subjective beliefs and personal experiences.

If you took all these lines [for each person], straightened them out, and then stacked them on top of one another, Chad’s line would be the tallest and hence, he would be the guy who worked the hardest.

4. Analyze student results using the rubric. Describe in detail how the students responded to the assessments and

how they performed. (3 pts)

Of the 22 students who completed and turned in this hiking assessment, 6 students scored in the idiosyncratic category, 9 students scored in the transitional category, 6 students scored in the quantitative category, and 1 student scored in the analytical category. No student scored in the extended abstract cateogory.

Idiosyncratic Category: The six students in this category had various answers. Some students claimed that Chad worked the hardest without giving any reasoning or justifications. Some students claimed Adam worked the hardest because his lines were steeper. Some students just wrote down “Adam” or “Chad”. It was unclear to know what they really meant, but I took it as them telling me that the name they wrote down was the guy who worked the hardest.

Transitional Category: This category is where the bulk of students scored. Every student in this category said that Adam worked the hardest because of one aspect of the data they were shown. Four students claimed he worked the hardest because he climbed the steepest/ highest mountain, three students said he worked the hardest because his speed was constant, and two students claimed he worked the hardest because he took the most water breaks out of the three guys.

Quantitative Category: This is where the student’s answers started to get a little more interesting. Some students claimed Adam worked the hardest because he climbed the steepest mountain, factoring in the fact that his speed was relatively slow compared to Chads. These students then progressed by stating the reason why Adam took the most water breaks was because Adam climbed the steepest mountain and needed to stay more hydrated. Additionally, some students claimed Adam worked the hardest because he scored the highest on two of the graphs compared to the other two guys. They claimed that Adam almost made it to 70 feet, and had the most water breaks of the three guys. All the other students claimed Adam worked the hardest because he was going at a steady 3 mph the majority of the time. These students also factored in that Adam had the most water breaks because he was going uphill.

Analytical Category: Only one student scored in this category. This student wrote a detailed description of all three guys, comparing each statistic/graph with one another. Not only that, but this student also thought outside of the box, questioning other factors and making meaningful structures out of the data she was given (She stated, “They all worked very hard to climb to the top of the mountain, taking in account that they were different mountains”). She somewhat touched on the topic that one mountain could have been an easier climb than another mountain. In addition, this student was the only student who referenced Jason. She stated, “Jason limited his water breaks probably to make up for his speed”. Perhaps she was thinking this was a race?


Nonetheless, she was able to connect that less water breaks translated in to less speed. She ended her response claiming Adam worked the hardest because “he climbed 80 feet in only one hour, had a good amount of stops for water, and kept a good pace.”

Include actual samples of student work. (3 pts)

Handing in hard copies class (turning in November 10th at 4:05)

5. Reflect on student understanding based on results obtained using the rubric. Were you surprised at how the students performed on your assessments?

(2 pts)

First off, this type of assignment was something my students had never done with me. I wasn’t sure how well it would go, but I thought it went very well all things considered. Of the twenty two students who turned in this assessment, I was not surprised with where they were placed in terms of the grading rubric (Extended Abstract, Analytical, Quantitative, Transitional, Idiosyncratic). There was a direct correlation between the overall grade of the students in my class, and their score on the grading rubric. The students who have an A or B in my class were placed in the analytical and quantitative categories. The students who have a C or a D were placed in the transitional or idiosyncratic categories. So it wasn’t too surprising to see where each student placed for the different levels on the grading rubric.

What do you know about the thinking patterns of the students in your class after having performed the assessment that you didn't know beforehand? (6 pts)

To give a little background: my students are usually given a problem in class where the instructions tell them to solve for x or write the equation of the line in slope intercept form. This was the first time my students had to critically think and analyze data charts to make a meaningful structure of the data that was presented. When I told Mrs. Wargo about this assignment, she told me that the students wouldn’t do too hot on this, and to expect a lot of blanks. At first, I was going to change the assessment, but then I thought, it would be good to see if the students could handle an assignment like this. Based on the results, I think they did a wonderful job.

Now, to answer the question about the thinking patterns, I now know that my students can be given line graphs with labels on the x and y axis, and make sense of the information. The students realized that a horizontal line meant there wasn’t any change as time progressed, an increasing slope (for the height of the mountain) meant that the mountain was inclining, and that faster the mountain reached a new height, the steeper the slope was. It certainly surprised me and Mrs. Wargo to see the students making sense of all these graphs.

Twenty one of the twenty two students gave responses which were pretty good for their level. These students could make sense of the data, and almost all of them said that Adam worked the hardest when hiking the mountain (I would have also said that Adam worked the hardest). However, there was one student who went above and beyond my expectations. She was the student who scored in the analytical category: She made a reference to the idea that not all the


mountains were the same. Although these are my words, she could have been thinking that the landscape of one mountain was icier/ muddier than another mountain, thereby making it harder to hike up. She was also the one student to make a reference to all three boys in her response. Some students explained a little bit about Chad (the runner up to Adam for working the hardest), and completely leaving out Jason (I purposely made his stats lower to see if the students could see he wasn’t even in the running for who worked the hardest [they saw it]). But the student who scored in the analytical category had a valid argument, analyzed all three boys, and made a decision on who worked the hardest using all of the information provided. I would not have thought any of my students would be capable of that, but I was proven wrong.

6. Address how the assessment will inform your instruction and address student needs. Discuss the instructional implications stemming from the assessment (i.e.,

how will the results alter the way that you teach the class now that you have more information about how they think)? (6 pts)

Based on the results, I think it is important to not just practice solving problems, but also making real world situations out of them. By doing real world examples, I feel students will be able to remember how the math applies, and make the whole learning of it more meaningful. I started incorporating real world examples in my most recent engagements: We were learning about the x and y intercepts, and I made a basketball/ football reference to show the students that intercepts can apply to something in which they are interested.

Second, I want to try and have the students do more independent work. The reasoning is: this assignment asked the students to think for themselves and answer the question: “Which of these guys worked the hardest” without collaborating with any of their peers. Every day during the two periods I teach, the students are doing some kind of cooperative learning assignment. I saw how effective this was when I was observing Mrs. Wargo’s class for the first week of my apprenticeship; I did not want to change this technique up. But for future classes, I am planning on doing more individual work so students are given the opportunity to critically think by themselves. Actually now that I am thinking about it, I think a “Think - Pair - Share” activity would be perfect for my situation. During Think-Pair-Share, students would be given the opportunity to solve all the questions by themselves, and then compare their answers with a partner after and only after they have completed all the questions. This way, students can initially think independently, and then compare answers with their classmates.Due Date: November 10, 2014 ____/40 points

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