Below, Davis Parker (Mississippi ’15, Cleveland High School), shares how he has used performance tasks to build student metacognition – how students think about their mathematical thinking. Davis talks about the intentionality behind his design, execution, and feedback around a recent performance task. I love the feedback activity that he does with students which allows them to not only understand their own rubric rating, but to also use peer responses to grow in their understanding of what great mathematical thinking and communication looks like.  Read on below!

(Note – Davis refers to the QUASAR rubric throughout his post)

The assignment:

  • Students were given a performance task to gauge their ability to work creatively with absolute value functions.
  • In approaching the assignment, I wanted to create a task that gave students significant room for creativity, strategic thinking, and communication. I wanted it to be Math-lite, requiring as little number crunching and slick computation as possible, but allowing students to showcase their reasoning and problem solving skills.
  • The first question set the table for the entire assessment and allowed a low floor for students to enter. It read as follows: “Drake and Meek Mill decide to climb a mountain to settle their differences. Draw a graph that could represent their altitude (y) as a function of time (x). Explain your reasoning in complete sentences.”
    • On this question, students received points for drawing a hill/mountain looking graph and articulating the reasoning behind their graph. While it was hard to differentiate between math and strategic knowledge on this question, I gave students math knowledge credit if they properly interpreted the meaning of x and y-intercepts. With regards to strategic knowledge, students received credit for drawing a graph that could be interpreted (in some way or another) as the trajectory of people climbing a mountain. Ultimately, a student’s strategy was only as good as his/her ability to explain it.
    • If students drew a linear equation but gave sufficient explanation that Drake and Meek Mill never came down (or that they didn’t return on the graph), I gave them points.
    • Students were docked for graphs that didn’t begin at the origin or somewhere along the x-axis.
  • The second question gave students a specific function (y= -2|x-4|+8) to model their path and asked, “how high is [Drake and Meek Mill’s] peak? How long does it take them to reach the peak? How long does it take for them to climb the mountain? Explain your reasoning in complete sentences.”
    • Points were given for an accurate graphical representation of the function as well as correctly identifying the peak height (8), peak time (4), and total climb time (8).
  • The third question forced students to transition from mathematical knowledge (graphing, vertex, etc.) to more strategic knowledge by asking, “At altitudes greater than 6km, Drake needs to use his inhaler 3x/hour. Meanwhile, at altitudes greater than 7km, Meek Mill needs to use his inhaler 5x/hour. Who uses his inhaler more during the hike?”
    • This question forces students to navigate through two distinct measurements (time and distance) on their graph. While the altitude deals the y-axis of the graph, the real question lies on the x-axis with the time. An ideal student would recognize that the number of inhaler uses is dependent on hours, not necessarily kilometers
  • The last question was meant to showcase students communicative and strategic understanding by asking them whose asthma is worse, Drake or Meek Mill?
    • The key aspect of this question is that there is no correct answer. While Drake has to use his inhaler at lower altitudes, MM has to use his inhaler more often. It allows students to approach the question from either side and still come to a strong, evidence based conclusion.

The execution:

  • The most common complaint from students was, “I’ve never done anything like this before! Is this even math?” It seems they’d been trained to think math was 100% worksheets and computations.
    • On a surprising yet positive note, the students outperformed their average quiz score on this assessment, which strikes me as proof that they have underutilized mathematical skills in the way of strategy and communication.
  • The key was keeping students engaged in the problem and having them work through the whole sheet (1 page front-and-back). As the bookend questions required zero mathematical knowledge, even the most struggling student could gain considerable credit if he/she simply finished the quiz. Lastly, including popular culture figures allowed them to feel as if they had an actual opinion on the question at hand. You’d be surprised how many students took the time to confirm that Drake was the superior MC to MM.

The grading:

  • First question (MK: 1, SK: 1, C: 1)
    • The most important aspect of grading question #1 was making sure whatever the student graphed and whatever the student wrote reinforced each other. In the case of student #1 (part A), I docked communication points for a weak explanation. While the slope does in fact increase then decrease, the student failed to connect his reasoning to the original problem. Meanwhile, student #2 (part A) gives sufficient explanation for his graph by relating the vertex to the mountain peak and identifying the variables x and y.
  • Second question (MK: 2, SK: 1, C: 1)
    • This was the most math intensive question of the quiz as it forced students to graph an absolute value function accurately and interpret the results of the graph. The key aspects of the graph were the vertex and x-intercepts. Most students were able to accurately place the vertex, yet many struggled to properly slope their function such that it intersected the x-axis at 0 and 8. Students who failed to do this (student #4, A) lost significant mathematical knowledge credit. From a strategic perspective, students were rewarded for first identifying the vertex then graphing from that point. Additionally, they received credit for properly interpreting the zeros and vertex of the function.
  • Third question: (MK: 1, SK: 1, C: 1)
    • Students received mathematical knowledge points for relating the x-axis on the previous graph to the y-axis. The question forced them to take a y-axis measurement (altitude) and convert it to an x-axis measurement (time). Students who properly did this and performed computations correctly received full points for their work (student #2, B). From a strategic perspective, students received credit for understanding the need to multiply each individual rapper’s rate by the time above a certain altitude. Even if they incorrectly identified these values, they received points for their strategy (student #4, B). Some students had great strategy and math but had weak communication (student #3, B). These students received points for their thinking and computation but very little for communication, as they didn’t effectively explain their thinking or answer the question. Student #1 had a very interesting answer here because he knew to multiply the given rates by some fixed value, but he incorrectly used the altitude instead of the time. Here, he received ½ credit for strategy and close to full credit for communication, as his reasoning is quite clear.
  • Fourth question: (SK: 1, C: 1)
    • Students received credit for clearly articulating an opinion based on results from Q3. Some students received little to credit (student #3, B), while others (student #1, B) received close to full credit, even if they had incorrect conclusions for Q3. The key to Q4 was having a logic-based opinion that was built off the answer to Q3. Even if a student didn’t answer Q3, they received credit for saying that Drake has to begin using his inhaler earlier or that MM has to use his more. There was no right or wrong answer, just good and bad reasoning.

Student responses

  • Student #1
  • Student 1's Task Responses
    Student 1’s Task Responses
    • Across all classes, student #1 was considered to be the worst of the example. What stood out was his poor reasoning and especially poor communication. A common response was, “do people really talk like that?” Fortunately, student #1 didn’t exist and was an answer sheet I filled out with common mistakes. I would strongly encourage other teachers to do something similar if they are trying to show common mistakes.
      • Score: MK: 1.5, SK: 2, C: 1.5
  • Student #2
  • Student 2's Task Responses
    Student 2’s Task Responses
    • Just as student #1 was considered the worst, student #2 was generally considered the cream of the crop. Students appreciated his clear writing style and mathematical computation.
      • Score: MK: 4, SK: 4, C: 4
  • Student #3
  • Student 3's Task Responses
    Student 3’s Task Responses
    • Student #3 was by far the most interesting case. Most students gave him low marks, as they were put off by his poor communication style. What they didn’t recognize was the strength of his reasoning and math skills. For instance, on Q3 he does a terrific job strategizing and drawing a model of when each rapper would use his inhaler (as well as the total # of uses). Student #3 clearly has solid mathematical reasoning, yet it is hidden behind a veneer of sloppy language and organization. He’s like a raw athlete: great potential but very little polish. In fact, this assessment helped identify him as a student who could very well be pushed to not only become a good mathematician but an outstanding one.
      • Score: MK: 3.5, SK: 3, C: 1.5
  • Student #4
  • Student 4's Task Responses
    Student 4’s Task Responses
    • Student #4 received better reviews from her peers than her teacher. Her organized graphs and clear writing were convincing indicators of her knowledge, yet she had significant struggles with her graphs. On all 3, her graphs incorrectly intersect the x-axis, and she misinterprets her previous graph on Q3. To be fair, her errors on Q1 and Q2 were more than likely a result of sloppiness than ignorance, yet she ended up receiving roughly the same score as student #3. What makes that result so interesting is that students #3 and #4 received the score for almost the exact opposite reasons. While one was an effective communicator, the other struggled mightily. While #3 had accurate graphs and strategy, #4 missed the mark. The key is that they both have terrific potential in mathematics, but that they need distinctly different practice going forward.
      • Score: MK: 2, SK: 3, C: 4

Performance Tasks are extremely valuable, if only because they break the monotony that Math class can become. They also allow every student to show off what they can do, as opposed to primarily punishing them for what they can’t. Going forward, I want to keep the PT structure of bookending the assignment with more qualitative, opinion based questions that provide students with low barriers to entry. Additionally, I think exposing the students to the QUASAR rubric is productive because it allows them to see behind the curtain and understand what exactly I am look for as a teacher.  In fact, I don’t plan on taking my upcoming PT up for a grade but rather have students grade their peers and give them valuable feedback. Lastly, I’d encourage other teachers (and myself) to do their best to keep students engaged with the assessment through its entirety. Students often surprise themselves (and their teachers!) with how much they do in fact know.

I hope this review has been helpful, and please leave comments below if you have any questions.

Best,

Davis

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