This lesson focuses on graphing linear equations. It is a review of a unit just completed on this topic. The lesson is 44 minutes long. There are 36 students in the class.
| 00:03–02:35 | (Whole class) The teacher calls on students to identify methods for graphing equations. Method 1 uses the slope with the y-intercept. Method 2 makes a table for the equation and then plots the points from the table. |
| 02:35–20:03 | As students work on the first page of a worksheet in groups of three or four, the teacher circulates, engaging in private conversation with individual groups. |
| 20:03–20:52 | (Whole class) The teacher assigns additional problems. |
| 20:52–30:35 | As students work in groups on additional problems, the teacher circulates, engaging in private conversation with individual groups. |
| 30:35–32:38 | (Whole class) The teacher assigns a “quick write,” saying, “Write a minimum of two paragraphs on what we’ve learned today.” |
| 32:38–44:12 | As students work in groups on assigned problems and individually on the “quick write,” the teacher circulates, engaging in private conversation with individual groups. |
The lesson is meant to be a review of prior learning. The flow of the lesson seems to indicate that the students had learned very little about linear relations, their behavior (shape and rate of change), and their tabular, graphical, and symbolic representations. The teacher spends considerable time re-teaching most of the content and provides students with what he wants them to be able to do on their own in the review. The students seem not to be sure about how to make tables or graphs and the meaning of positive and negative slopes.
Although this was a review lesson, many students struggled with the problems. Most groups seem to succeed in graphing equations 1–10, but it is not clear if many groups progressed beyond this point. The equations are all in the slope-intercept form. Many students ignore the slope-intercept form and make a table. Another approach to teaching students about graphing linear equations is described in Susanna Epp’s report on her use of HK4.
To encourage more effective group interaction, the teacher could have shown students explicitly how to work cooperatively. Compare US1 (1:13–1:41, 2:54–3:25, 6:39–7:20, 11:37–13:12, 30:42–31:26) with S2 (20:00–23:02). The students in US1 had worked in pairs before, but had no clear instruction on how to work in groups of four, which the teacher had assigned them for the first time in this lesson. In S2, the whole class works as a group, which gives the teacher the opportunity to facilitate whole group interactions.
Group work in U.S. classrooms often functions as a tutorial, as it does in this lesson; the teacher does not present new content and tells students who are struggling to ask for help from students who know the material.
(1:49–2:21) This segment reveals a missed opportunity to make a connection between two strategies. To go further, viewers could explore different strategies (i.e., table versus slope-intercept) that would develop student understanding and fluency with these approaches.
(20:03–20:53) The teacher warns the students of the change from positive to negative and states how the resulting graphs should appear, rather than initiating a discussion to compare the two equations, or consider how the graphs might differ.
(35:44–35:50) The teacher could have developed a third method to graph lines: the point-slope method, which comes from the reason why a line can be represented by a linear equation using similarity of triangles. Viewers could use this as a starting point for a discussion of lesson design and goal issues, for example, whether the teacher only recognizes two “official” methods.
(40:40–41:15) The teacher tries to connect the graphs with perpendicularity. If there had been more time, and if students knew enough about the Pythagorean theorem, similarity, and congruence, the teacher could have asked students to speculate on the angles formed by the chosen pairs (1 and 9, 2 and 6, etc.), to justify their claims by measuring or via geometric constructions and make connections with the slopes. Viewers could discuss and evaluate matching the teaching strategy to content and evaluate their own knowledge of the connection between perpendicular lines and their slopes.
The lesson did not offer an opportunity to make mathematical connections within and among the different aspects of an equation, such as x- and y-intercept, slope and a table. The teacher treated these pieces independently from how they contribute to the make up of a graph.
The following are examples of questions from the teacher:
These questions should stimulate student thinking, but the teacher often does not allow students the time or encourage them to think about a question all the way through. Instead he completes the thoughts for them or immediately indicates the next step. Most are private communications; so the whole class does not have an opportunity to learn from them.
(6:29–34:24) The teacher uses the word “timesing” (6:29) rather than “multiplying”, which does not model for students the use of precise and correct mathematical language.
The teacher makes several comments about how he “liked” (6:43, 10:36) certain procedures students used and “did not like” (10:48) others. Viewers should be aware of how these types of comments may influence students to try to please the teacher by solving a problem the “right” way without thinking deeply about the underlying mathematical relationships.
(38:14–38:20) This segment shows the teacher rewarding effort over understanding or accomplishment. This raises the issue of how teacher expectations in this lesson impact student learning.
File name: NB10311107.
There is no commentary from the national research coordinator. The Resources section of the CD does not include copies of the worksheet, but the lesson graph includes the tasks.
The students needed to:
This lesson prepares students to:
(1:49–1:59) Teacher educators can use this segment to discuss and examine the number of problems the teacher assigns to students and whether this matches the content and level of student understanding. The level of complexity across the problems remains the same, does not build learning, and may not reflect the intended goals. Viewers could examine this factor in order to redesign the lesson and recommend problems that would help students be more successful. By contrast, S4 contains a sequence of problems on solving linear equations of the form ax + b = cx + d which builds from the easiest to the general case.
In the entirety of the lesson, the teacher is doing most of the work. Compare this observation with other lessons, namely J3.
(6:39–7:22) This segment reveals cooperative work in mathematics and the need to model and scaffold student attempts at learning mathematics in this manner.
(30:51–31:19) This segment offers an opportunity to discuss the implementation of instructional strategies and whether they align with lesson goals and content. Possible questions for viewers include:
(38:28–39:05 and 41:41–41:52) These segments, when the teacher assigns a two-paragraph “quick write” to review the lesson, offer another opportunity to evaluate and discuss instructional strategies. Viewers might consider:
(40:00–41:19) The teacher works with a small group and introduces the term “perpendicular.” This clip offers an opportunity for viewers to discuss and evaluate matching strategy to content.
(11:36–12:56) Students are working, but many do not appear to understand the mathematical content and basic concepts. This clip provides an opportunity to discuss the difference between student engagement and student understanding and how teachers can support the learning of all students in a diverse classroom.
(1:33–2:35 and 3:25–3:35) These segments show the teacher stating goals. Viewers can use these to explore:
The value of stating a goal in this lesson can be compared with S3, where the teacher did not begin the lesson with a stated goal.
(30:51–30:59) This segment shows how the teacher has not implemented his (stated) goal of cooperative learning and must stop the class to remind students to work together.
(38:14–38:20) This segment can also be used to compare stated goals with their implementation; although the teacher’s stated goal was to get the correct answers, in the implementation all that mattered was effort. (See also the teacher commentary at 38:14.) Viewers should compare the teacher’s expectations with those in the HK2 lesson.
(26:38) The teacher says, “Aren’t all these slopes positive? So all of these should be going up.” Connections like these are not always obvious to students. The Cartesian Connection is a detailed specification of what constitutes understanding of linear equations and graphs of lines in the Cartesian plane (see Schoenfeld, Smith, and Arcavi, 1993, especially fig. 24).
For examples of questions similar to those the teacher asks students in this lesson (e.g., “What do you notice [about this graphs]?”) and related responses (e.g., “The higher number you multiply by x the more upright the line”), see Moschkovich, Schoenfeld, and Arcavi (1993). The article unpacks some of the complexity involved in understanding how to graph equations and some common U.S. students’ perceptions, such as that slope, y-intercept, and x-intercept are needed to characterize a line.
Teacher educators can use this lesson to discuss a number of issues, including:
Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. Romberg, E. Fennema, & T. Carpenter (Eds.), Integrating research on the graphical representation of function (pp. 69–99). Hillsdale, NJ: Lawrence Erlbaum.
Schoenfeld, A., Smith, J. & Arcavi, A. (1993). Learning: The microgenetic analysis of one student's understanding of a complex subject matter domain. In R. Glaser (Ed.), Advances in instructional psychology (vol. 4, pp. 55–175). Hillsdale, NJ: Lawrence Erlbaum.
| School Improvement |
Research and Evaluation |
International Studies |
RBS Publications |
Education Resources |
About RBS |
| Contact Us | Home |
