Evaluating Assessments

As part of my education class, I conducted classroom observations at a precalculus class and a calculus class once a week. The following post is a reflection on one of my classroom observations. Note: The names used are pseudonyms.

A common form of formative assessment is the Do Now activity. Students come into the classroom, and once they’ve taken their seats, they are expected to solve the problems on the board. The problems usually cover material that students have already learned and are expected to know. During my last observation, there were three Do Now problems on the board for the precalculus class:

  1. cos x = 3/5
  2. cos x = -2
  3. 5x2 + 7x = 6

Some of the students were struggling to arrive at the right answers, so Mr. Brown posed some guiding questions to help them along. For instance, for problem 1, he asked students to look at the unit circle and use their intuition to decide how many answers they thought there were. When a student offered a solution, Mr. Brown asked the class if they agreed, disagreed, or were unsure. This strategy of asking students to reflect upon their own thinking is valuable because it trains them to be metacognitive. There was an article I read about teaching math (I forgot the title of the article) that brought up a really good point: If students truly understand the math they are doing, they should be able to argue the correctness of their answers rather than rely on an answer key to tell them that they are right. Having students defend their answers is also a good way to promote “math talk” between students. Becoming more fluent with the math terminology is beneficial for students and helps them view math problems in a different light.

After the students arrived at the correct answers for the three Do Now problems, Mr. Brown revealed a fourth problem: 5 cos2 θ + 7 cos θ = 6. By solving the Do Now problems, the students had actually completed almost all the steps necessary to solve the fourth problem. I thought this was a clever way of building off the students’ prior knowledge and using scaffolding to introduce a new type of math problem.

That same day, the precalculus class also started a new project. This weather project would serve as their summative assessment on sinusoidal functions. The goal of the project was to generate a model to predict average monthly temperatures for different cities. Each student chose a city, looked up the monthly average temperatures for a given year, and was asked to derive a sinusoidal function that best represented the data. The final product comprised a chart on a poster and a write-up describing how closely the model lined up with the actual data. The project successfully focused on a simplified version of a problem that people face in real life: weather prediction. Students would learn how sinusoidal functions are useful for modeling and predicting real life phenomena, and the write-up portion of the project would help students understand the limitations of using mathematical models.

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Multiple Representations

As part of my education class, I conducted classroom observations at a precalculus class and a calculus class once a week. The following post is a reflection on one of my classroom observations. Note: The names used are pseudonyms.

The precalculus students had spent the previous few weeks studying sinusoidal functions, and last Friday they spent the class period deriving the tangent function graph. Mr. Brown started by having the students use their prior knowledge of sine and cosine functions to find the values of several expressions involving the tangent function.

For the first “derivation” of the tangent function graph, he had students recall tangent values for the most common angles on the unit circle. To fill in the gaps between those angles, he asked students to use their graphing calculators to find tangent values for those angles. Using the table of angles and corresponding tangent values, Mr. Brown asked the students to predict what the graph of the tangent function would look like.

The second derivation also involved the unit circle, but relied on software called The Geometer’s Sketchpad. Mr. Brown projected the Geometer’s Sketchpad document at the front of the classroom and drew a unit circle with the sine/cosine triangle and the tangent triangle.

tangent

Mr. Brown had the students recall their prior knowledge about similar triangles from their geometry class in order to convince them that the larger triangle in fact had a leg with length equal to the tangent of the opposite angle. Using Geometer’s Sketchpad, Mr. Brown animated the motion of a point following the circular path and simultaneously graphed the height of the tangent line. In the end, the students saw the graph of the tangent function.

Overall, I thought using the two derivations was a thoughtful way of introducing the students to the tangent graph. It gave the students better insight into why the tangent function looks the way it does. They would have missed that part had Mr. Brown simply told the students to graph the tangent function on their calculators.

That same day, the calculus students learned how to use their graphing calculators to find the indefinite integral of various functions. Previously they had learned how to find the area by hand. Mr. Brown had a program on his computer that projected the graphing calculator onto the screen, along with the most recently pressed sequence of buttons. As with many demos where students are expected to follow along in real time, many students fell behind, so Mr. Brown ended up helping quite a few students individually.

Eventually though, all the students figured out how to use their calculators to find the indefinite integral and were asked to demonstrate their knowledge and understanding by completing a worksheet. Mr. Brown stressed the importance of doing each problem by hand first and then using the graphing calculator to check their answers. Taking advantage of the immediate feedback from their calculators allowed students to practice metacognition and re-evaluate answers that they got incorrect by hand. Again, this was an effective way of using technology to advance student understanding by providing an additional mode of representation.

Substitute Teacher and Khan Academy

As part of my education class, I conducted classroom observations at a precalculus class and a calculus class once a week. The following post is a reflection on one of my classroom observations. Note: The names used are pseudonyms.

Over the course of my schooling, I had the impression that being a substitute teacher was incredibly difficult, mostly because students tended to bully the substitute. However, during my last classroom observation, I saw something that surprised me. Mr. Brown, who normally teaches the math classes, was not at school, so the substitute teacher, Mr. Green, took care of the class. I learned that Mr. Green was actually a “permanent” substitute teacher at the school and substituted for about five teachers every day. His goal was to become a high school psychology teacher in the future, but said he would probably end up teaching English instead because there was higher demand for English teachers than psychology teachers. Because Mr. Green substituted for many different classes, all the students already knew him and were comfortable around him.

At the beginning of the period, Mr. Green told a story. It had absolutely nothing to do with mathematics, but the students were all very attentive when he told the story. The moral of the story was open to interpretation, and overall I thought it was a neat little way of getting the students to focus on being in the classroom again right after the lunch break. Classroom management wasn’t a problem at all even though there was a substitute teacher.

The lesson plan that Mr. Brown had left consisted of separating the students into tutor and tutee pairs. About half the students had missed class the previous day due to some academic event off campus, so the students who were absent were tutees. The students who were present the day before were the tutors. The tutor in each pair was given the responsibility of guiding the tutee in completing Khan Academy exercises on sinusoidal functions. Mr. Green wasn’t all too familiar with the material, so I was responsible for answering the students’ questions.

Because the students only had to answer three consecutive questions correctly, many of them ended up rushing through the exercises and focused solely on memorizing procedures. The first topic was finding the midline of a sinusoidal function, and many of the students who asked me for help phrased their questions like this: “How do I know when to add the numbers they give us and when to subtract them?” I did the best I could to bring their attention to the bigger picture. I asked them what the midline of the function represented and what the numbers they provided represented (e.g. max value, min value, amplitude). Some of them gained a better understanding of what they were doing conceptually, but I’m afraid not all of them fully understood why they had to add sometimes and subtract at other times. Students who have a solid conceptual understanding of what the midline is should have no trouble answering questions correctly on the Khan Academy platform. However, it is also possible for students who don’t have a solid conceptual understanding of the material to score well on the exercises. It’s important to realize this distinction and use other methods of assessment to get a more accurate measurement of student understanding.

Precalculus Video Projects

As part of my education class, I conducted classroom observations at a precalculus class and a calculus class once a week. The following post is a reflection on one of my classroom observations. Note: The names used are pseudonyms.

During my classroom observation last week, I got to see some of the final video projects that students made for their precalculus class. The topic for that unit was exponential and logarithmic equations, and the assignment was to create a video demonstrating a real-life application of exponential and logarithmic equations.

Students worked in teams of three and came up with their own scenarios. The first video was on the population boom, and the second video featured a group of girls saving money for a trip by earning interest on bank deposits. This assignment was successful at assessing the students’ ability to use exponential and logarithmic equations to solve real-life problems.

What made the video project assignment especially successful, however, was the community aspect. All the footage for the videos was shot on campus, and several teachers (other than Mr. Brown) were actually featured in the video either as “experts” in the field or as characters in the plot. The best learning environments are community-centered, and this assignment helped create connections between the pre-calculus class and the broader school community. Many teachers only see students in their own classes, so by encouraging other teachers to get involved with the math project, Mr. Brown was also strengthening the community of teachers. The teachers involved probably gained more insight into what their students were working on in other classes, too. The video project also helped develop a sense of community among the students in the class. Students from different teams helped each other film their projects.

Mr. Brown told me later that it took a long time for the students to put together the math projects. Not only did they have to set up the scenario and get footage, but they also had to learn video-editing, which took a significant amount of time. Overall, the students did seem to enjoy the project, and those whose videos were featured during class were proud of their work.

The math video project shared many similarities with the capstone projects for schools like Mission Hill and High Tech High. Both of these schools have non-traditional approaches to assessing students. Students are not graded on a series of exams; instead, students are responsible for developing a portfolio throughout the school year. At the end of the year, students at Mission Hill are responsible for sharing their portfolio in front of a panel of teachers, parents, and in some cases experts in the field. On the other hand, students at High Tech High have a project showcase night where the students’ families are all invited to see their work that term. Both the portfolio presentation and project showcase give students an opportunity to celebrate what they learned that year, and it makes students more accountable for what they “turn in.” Students are much more motivated to do a good job if they know that someone else will see their work. This relates back to the math video project because the precalculus students knew their videos could be shown to the class, so they wanted to make high-quality videos. The next step would be to show the videos to a larger audience.

Fishbowl Discussions

The other day Miss Frizzle had her 10th grade students discuss a memo written to the Internet Engineering Task Force (IETF) entitled “The Internet is for Everyone.” They covered potential challenges and threats to the idea that the Internet is for everyone, including censorship and creating safe spaces online. Miss Frizzle’s computer science classes rarely had class discussions, so it was interesting for me to witness how she led this activity.

Miss Frizzle had her students do a “fishbowl” discussion. For each round, students were separated into four groups of four. Each group took turns sitting at the discussion table in the middle of the room and were given 10 minutes to discuss their challenge in front of the class. The students in the remaining groups sat around the perimeter of the room, and they were each assigned someone at the discussion table to observe and give feedback to on a written handout. The handout had questions like “What was the student’s strongest point?” and “What could the student do to improve?”

There was also a “hot seat” (an extra seat at the discussion table) for each of the rounds. If a student on the sideline wanted to join the discussion, then she could take a seat on the hot seat, ask a question to the discussion group, and return to her seat on the sidelines as the discussion group answered her question.

I thought that the fishbowl discussion style was a particularly effective practice because students were not only more likely to speak up during their own group’s discussion, but they were also encouraged to listen to what their peers had to say. The handout gave students the opportunity to practice giving constructive feedback to others and to assess the strength of other people’s arguments.

Later that day, Miss Frizzle explained to me that the students originally learned the fishbowl strategy in their English class. She followed up by saying that she was standing on the shoulders of giants (i.e. the English teachers). Since the students were already familiar with fishbowl-style discussions in their English class, Miss Frizzle could easily integrate that learning strategy into her own classroom without having to explain the details behind it. This is an example of how working together and collaborating with other teachers in the community can be beneficial both for student learning and for teachers.

Teaching App Inventor

The students in Miss Frizzle’s 9th grade Introduction to Computer Science have started the MIT App Inventor unit, which has given me some more insight into Miss Frizzle’s considerations when planning the curriculum. The first thing the students did with App Inventor was follow a tutorial from the App Inventor website that walked them through the steps of making the “I have a dream” app (a video player app). The students were able to replicate the app from the tutorial, and they familiarized themselves with the App Inventor interface along the way.

However, Miss Frizzle explained that at this point, many students often don’t think they have the skills to create their own apps because all they’ve done is follow a set of instructions. Therefore, the next part of the curriculum was meant to teach students how to abstract their knowledge from the tutorials.

The day that I observed the class, Miss Frizzle gave the students links to three different sample apps that were based off of the “I have a dream” app. The sample apps used the same concepts introduced in the “I have a dream” app, but played different sounds and videos. Miss Frizzle asked the students to test out each of the sample apps and write down pros and cons of each app. The students then discussed their feedback as a class. The idea was to get the students thinking about what constitutes a good app and give them ideas on how they can create their own.

Having completed the tutorial and seen sample projects based on the tutorial, the students were then ready to design and implement their own video player app. This was the stage where students started to feel more confident in their own app-making skills.

The following week, Miss Frizzle repeated this same process for a drawing app. The students followed a tutorial, gave feedback on sample apps, and designed their own version of the app. The second time around, the students started to get the hang of the routine, which prompted Miss Frizzle to explain to me how there is a fine line between having a routine that students are comfortable with and having a routine that students are bored with. That day, the students were still engaged with the routine and were able to stay relatively focused on the task at hand. Once the students start to get restless again, however, Miss Frizzle says that’s the cue for her to move onto something new.

Rethinking assignments and classroom norms

I was able to speak with Miss Frizzle and her students a bit more during my last two classroom observations. My conversations with them made me rethink assignments and classroom norms.

I had two interactions with students that particularly stood out to me. One student I worked with called me over several times to ask me questions about why her code wasn’t working. When I asked her questions to guide her down the right path, she gave me insightful answers and was eventually able to solve the problem. I was really proud of her accomplishment and her patience when working through the problems. Unfortunately, she had forgotten to log into her account, so all her progress from that class period was lost. She was disappointed, but didn’t seem all too upset about it. I made sure that Miss Frizzle knew that the student had finished the exercises, and Miss Frizzle decided to give her credit anyway.

The second student I worked with also ran into trouble with the same exercise. When I tried asking him questions to lead him down the right path, however, he simply responded with, “I don’t know. I just want to pass this exercise and move onto the next one.” I was surprised by his honesty, and it made me realize that even I sometimes find myself in similar situations. Sometimes I get so caught up with the idea of finishing an assignment that the purpose of the assignment is completely lost on me. This is especially true when I’m working on something that has an impending deadline. My attitude becomes “Just finish it!” when it really should be more like “What am I not understanding correctly? Why is it not working, and how can I fix it?”

I’m not advocating for the elimination of assignments, but I do wish there was some way of making sure that students know that the purpose of assignments is not just to complete it before a certain deadline, but to actually learn the material.

The ninth graders spend most classes working through Code.org exercises on their computer, so I asked Miss Frizzle how she decides when to lecture the students and when to let them learn by following the instructions on Code.org. She responded by telling me that the classroom norm has made it so that everyone expects the teacher to stand up in front of the class for a certain amount of time each class period. Since computer science is something that is best learned by doing (not listening about during lecture), she ends up talking about logistical things like homework expectations during the “lecture” time.

Miss Frizzle and I also talked briefly about the role of assessments in education. Her tenth graders had the opportunity to take apart a computer and learn about the different parts. The activity itself was a learning experience for her students, but Miss Frizzle felt obligated to have the students create Prezi presentations to encapsulate all the material they learned. To quote Miss Frizzle, “It’s school, so of course there has to be some sort of assessment.”

Overall, it was interesting for me to learn how classroom norms place hidden restrictions on teachers and influence the way they teach their students.