Special Cases and Testing

[Shaw Qin]

Below is my attempt at a backward lesson plan for Mrs. Talbot’s 8th-grade algebra class on Tuesday.

Objectives:

Identify the two special cases in solving three-step equations: identities and contradictions.

Understand why the two special cases are special.

Understand how to solve the special cases.

Warm-up:

Mrs. Talbot first talked about the homework for Monday, which corresponded to the general solution of three-step equations. However, she asked the class whether they noticed two “weird problems” in which it was hard to tell the solution. She asked students what they thought the solutions were and explained that the two questions foreshadowed Tuesday’s lesson on special cases in solving three-step equations.

Motivational strategies:

When the students worked on a sample problem, Mrs. Talbot asked students who finished to stand up. Standing up might work as an extrinsic motivation for students to be seen (literally) when they have worked out the solution. On the other hand, students could be motivated to avoid the shame of being the last students sitting down and working on the problem. In general, however, Mrs. Talbot did not tie this lesson to any intrinsic or extrinsic motivation except mentioning that they will have a test soon.

Activities:

Warm-up: Mrs. Talbot mentioned the weird problems in homework and asked the students what they thought were the solutions. One student answered. This activity took little time, maybe 3 minutes.

Introduction to identities and contradictions: Mrs. Talbot introduced the two special cases with analogies (e.g., identities and identical twins) and worked through two examples on board, modeling the problem-solving process. This step took approximately 10 minutes.

Individual work, discussion, and modeling: Students worked on the third sample problem independently and then discussed in pairs about the solution. Mrs. Talbot eventually solved this problem on board, modeling the problem-solving process again. This activity took 5 minutes.

Individual work on a (not) special case: Students worked on the fourth problem individually. Mrs. Talbot and the tutors told each student whether they got it right. This was a trick question because it was a regular case and had 0 as its solution. Mrs. Talbot eventually solved this problem on board, emphasizing that students needed to be cautious when identifying special cases. This activity took about 8 minutes.

Homework: As in most class periods, students worked on their homework in class for the rest of the period, with Mrs. Talbot and the tutors available for help. This activity took the last 25 minutes.

Closure:

The closure of this lesson was not very obvious, as were many other periods that ended with homework time. Toward the end of the period (and school day), students gradually stopped working on the homework and waited for the bell to ring.

Follow-up:

Mrs. Talbot indicated that this was the final lesson in the second chapter, and students will have a test on Friday.

I think the structure of the lesson was clear and most likely intentional. The central part of the lesson centered around the four sample problems of special cases. Mrs. Talbot divided them into three parts that gradually gave students more opportunities to solve problems individually and understand the nuanced differences between regular and special cases. However, I don’t think the lesson, especially the warm-up part, took into account the needs and strengths of all students. Mrs. Talbot only asked one student to respond to what she thought about the two weird problems in the homework. This student was, unsurprisingly, one of the quickest in the math class. She answered almost perfectly except not using the correct vocabulary, which means that she understood the principle of the two special cases before learning about them. Then Mrs. Talbot moved on to lecturing about the two special cases. I don’t think this session successfully invited all students to think about the special cases before learning about them (which seems to be influenced by the cognitivist approach to inspire students’ active learning and connecting new materials with previous knowledge). In contrast, Mrs. Talbot could have asked students to discuss the two problems in pairs. Even though most students wouldn’t have arrived at as neat an answer as that one student, they could have more opportunities to reflect on their thought process when doing homework, their prior understanding of the new topic. In general, however, I think the lesson was successful. Many students got the third sample question right, and they seemed to appreciate why they shouldn’t immaturely decide that all “weird” problems are special cases.

This week, I observed tests in both the 7th-grade and 8th-grade classes. The rules for assistance and notes were very different from each other. Mrs. Rose’s 7th graders could ask questions, but the teachers and I could only offer minimal help, including referring students to one of the posters in the back of the classroom with math definitions and skills. Mrs. Talbot’s 8th graders could only work individually, but they could consult the notes they took in class (on structured worksheets, and Mrs. Talbot usually reminded students what and when they should write down). Comparing the two grades, the 8th graders were significantly less scaffolded. The teacher expected more autonomy and individual responsibility from the older students.

As a side note, I noticed again how much class time was occupied by tests in the 7th-grade class. Below is this week’s schedule on the whiteboard in front of the classroom.

For the A class I am tutoring, the last four days have all been related to Thursday’s test. The fact that I miss the opportunity to observe a variety of instructions is perhaps the least of the problem.

[Anastasia Romanova]

Hi Shaw!

I really like your idea of letting the students work out the problems in pairs. I understand how that might be time-consuming in the class, but I also believe that it would hold tremendous benefits for the students’ understanding of the material. In my experience, I learn best when I arrive at the answer myself instead of simply being reveled the answer. I think the though process and being aware of it is critical to education generally and the development of critical thinking more specifically. I am sure that eventually the students would all arrive at the same answer and asking just one student to explain it does save up the precious class time, but I really do agree with you that some students would have benefited greatly from getting the chance to work on the problems with a partner. Also, I really wish Mrs. Talbot worked more on providing some intrinsic motivation for the students. The class I described in my blog entry also lacked that, and I think that focusing only on the extrinsic rewards can be detrimental to learning and retaining the information long-term.

Thank you for sharing your thoughts!

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