Another posting of an analysis I wrote over the summer while starting my teacher education program.
Analysis of a Secondary Mathematics Teaching Lesson
The TIMSS video depicts the use of complex instruction within a heterogeneous secondary mathematics classroom. The teacher conducts a lesson that reviews how to graph a linear equation where students use one of two methods: the slope-intercept and (x,y) coordinate T-table. Throughout the video, the teacher moves from group to group checking for understanding, asking probing questions, clarifying, re-teaching and answering student questions. Students appear to benefit from the pedagogical approach used in this lesson, however, it is difficult to determine that all students within a group truly understood the mathematics being taught. While the teacher is obviously well prepared for the lesson, there seem to be many missed opportunities for deeper contextual understanding by his students; having said that, we do not know the specific constraints the teacher was operating under so it is a bit presumptuous on our part to judge without having better insight into the specifics of his situation.
In this lesson, students, in groups of 3-4 students per group, need to graph several equations on the same sheet of graph paper. Examples of the equations include:
1) y = 2/3x + 8
2) y = 3/5x – 10
3) y = 3x + 7
4) y = 1/4x – 9
5) y = x – 5
While students do appear to be learning for understanding by developing procedural fluency in the group activity, as well as enhancing their productive disposition towards mathematics, the lesson could be enriched to enable students to deepen their level of strategic competence, conceptual understanding and adaptive reasoning – the five dimensions of mathematical proficiency espoused by the National Research Council (NRC, 2001).
For example, students are not asked to formulate the math problems here, the teacher presents it too them in a very simplified form as a list of linear equations in symbolic form. While they are asked to represent the equations in graphical form, this falls short of the potential for them to “formulate math problems, represent them & solve them” and “use mental representations, detect math relationships & devise methods to solve” as outlined by the NRC for establishing strategic competence.
Student learning, in aggregate, occurs throughout the video. However, there are several students who were struggling with the concepts and needing additional help. In particular, the teacher identifies a couple of students who do are not involved with the groupʼs work or who have arrived at answers that deviate from their teammateʼs solution.
Most of the students opt for the more operationally intensive procedure of computing various (x,y) coordinates to plot for a particular equation. The teacher wanders from group to group ensuring students select x values for the equations containing fractional slopes that are whole numbers in order to make the problem easier to compute since the fractions are removed (e.g. choosing x = 3 for 2/3x + 8 by 3 to end up with y = 2+8 = 10).
While this is meant to simplify the process of computing points to plot, the students are not allowed to struggle with the challenge of computing with fractions or to discover, on their own, with minimal intervention, the process the teacher suggests. Additionally, the teacher recommends to many students that they use three points to draw their line, instead of two, so they can check that their computations are correct.
Throughout the video, only a couple of students chose to use the slope-intercept method. This seems to be a missed opportunity to reinforce how both methods can be used to check solutions arrived at using the other and for the students to build connections between these two methods. So while the students are reinforcing how to graph linear equations, they are limiting themselves by mainly implementing one method.
The teacher created this situation by allowing students to choose which method they wish to use in doing the activity. I believe he should have required both, perhaps with one student in a pair responsible for one method each. Afterwards, they can compare solutions and help each other understand the specifics of the other process to the extent necessary to ensure understanding.
The lesson appears to do a great job at reinforcing procedural fluency of creating an (x,y) coordinate T-table to graph linear equations. It also appears to enhance their productive disposition towards mathematics by helping them build confidence in their ability to construct graphs of linear equations successfully.
The lesson connects to the Mathematics Content Standards for California Public Schools (1997), specifically Algebra I standards 5.0 and 6.0, depicted below. However, the connections could have been stronger and more richer by requiring students to utilize both known methods for graphing linear equations allowing students to compare and contrast the two methods.
Mathematics Content Standards for CA Public Schools
Algebra I Grades Eight Through Twelve – Mathematics Content Standards
Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.
5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the re- gion defined by 2x + 6y < 4).
The lesson incorporates elements of the mathematics process standards outlined by the National Council of Teachers of Mathematics (NCTM, 2000), such as communications and representation as well as provides the conditions for students to explore connections between the two methods (slope-intercept and (x,y) T-table), however, it seemed to fall short of its potential given the nature of the lesson: review. Additionally, with respect to problem solving, the equations were provided directly to the students instead of the students finding the equation from a word problem, as an example, then graphing the equation.
As the teacher walks around the room, he probes students for their reasoning and justification of a procedure but only superficially, and not anything that would allow the students to make conjectures about what is going on in the data or to show deeper understanding of the concepts important to linear equations such as slope and intercepts. He typically suggests an approach to use and prevents the student from elaborating further as in the following example.
Student: How many points do you want us to find?
Teacher: Well, thatʼs a very good question. Obviously, how many points do you need for a line?
Teacher: Two. Right? Well, just to make sure, plot at least three points. That way if theyʼre all in a line, at least you know you did it right. OK?
While he asks a question of the student at the outset, he jumps to a statement next instead of asking another question such as “how could you check to see if those two points are really on the line? or similar. It is my perception that he necessarily makes statements after an initial round of questioning students due to time constraints and the need to cover a large number of student groups in the allotted time.
With this lesson, the teacher appears to allow students to solidify their understanding of a procedure to use in graphing a linear equation. Most students select what on the surface is a more straightforward approach to graphing the equation: creating an (x,y) coordinate T-table. Very few students choose the slope-intercept method which is much simpler computationally and is more efficient, from my perspective. This prevented them from obtaining more insight into the importance of slope and intercept in shorter periods of time, both of which could be very important on standardized tests. Additionally, none of the problems had any context to them. They were simply linear equations. It might have been more instructive for the equations to relate to something with which the students identified or could see in their lives.
There is a high potential in this lesson to provide students with the opportunity to develop a strong contextual understanding of linear equations, their key parameters such as slope and intercepts, multiple methods to graph them, and examples from everyday life that are modeled via linear equations. One step removed from that level of understanding, there is an opportunity to allow students to explore how the (x,y) coordinate T- table values yield the same value for slope as depicted in the slope-intercept form of the linear equation given. And lesser in potential, but still higher than achieved in the lesson, students could be required to graph the equations using both methods, at a minimum to ensure they obtained the correct solution between the two methods, and ideally to relate the parameters in the slope-intercept form to the data in the T-table. However, this lesson fell far short of its potential and simply required students to follow one method to graph the equations without relating the two methods, the parameters within, what the data depicts, or how the equations relate to the world around the students.
Notwithstanding the above, it is apparent that the teacher planned for this lesson and that students benefited from it. The teacher is even a bit nervous at the outset, perhaps due to concern over the outcome (will students solidify their knowledge of graphing linear equations?) and perhaps due to being videotaped or both. And perhaps the teacher is struggling with the tension between supporting his studentsʼ learning of the material with ensuring they receive the necessary instruction to learn. Nonetheless, the teacher composes himself and leads students through a lesson that helps them graph and visualize multiple lines as they are related to the linear equations that spawned them.
According to “The Task Analysis Guide” from Stein (2000), the level of cognitive demand for this lesson appears to be that of “Procedures without Connections Task.” Specifically, students follow a repetitive, algorithmic procedure taught earlier in the course to graph the linear equations. There is no apparent connection to the concepts or meaning underlying the procedure used, even though the potential exists to link the two methods (slope-intercept and (x,y) T-table to each other as well as connect at least one of the equations to a real world context. There are no obvious examples of where students develop greater mathematical understanding. Instead, they are focused on getting correct answers by following the rote procedure. Additionally, the teacher guides the students strongly through the procedure as follows:
The teacher is working with Chris (approx. 8:45 on video) and paraphrased as follows:
Teacher states “The idea is to plug in any values for x to find the corresponding y. But since this is with fractions, like 2/3, what number should you use? That would make it easier for me.
Chris answers ʻ3.ʼ
Teacher states “Numbers based on 3, like maybe 0, 3, 6, -3, -6. Do you want my help or do you want to do it yourself first?
Chris asks for help and he helps Chris work through a couple (x= 0, 3) – you got it, you got it. Ben, see what Chris gets there.”
and mentions mistakes he observes to the students without them having to discover these themselves as shown in the following dialog:
“oh, oh, I donʼt like equation 3…its going down, that’s a +3x…it should be going up”
even though the group of four students had the opportunity to do so, especially since it appears that groups included students who clearly understood the procedure and those that might not understand it as well. Lastly, studentsʼ explanations to the teacher did not venture beyond simply describing the arithmetic used in the procedure for computing the (x,y) coordinates in the T-table.
Having said that, the teacher ensured equity by asking questions of most, if not all, students in an equitable manner that probed for understanding such as the following:
a) do you understand what you are doing?
b) let me see how you did one of these
c) tell me how you would graph that?
The problem I have with these questions is they were not followed up by questions that probed for a deeper understanding, they appeared to be aimed at ensuring the student understood the procedure solely. I did not observe a sustained press by the teacher which resulted in many lost opportunities to ask questions addressing the points brought up earlier in the “Student Learning” portion of this paper.
One new point to consider is that of collaboration. The lesson was intentionally structured to enable a degree of collaboration between students within a group. Nonetheless, students appeared to miss reaching consensus through mathematical reasoning; it is possible that this occurred within groups that were not captured by the video.
This instance of teaching and learning in a secondary mathematics classroom offered many real-life teaching situations for me to consider, reflect upon, experiment with and enhance over time. First and foremost, it is clear to me that teaching, in a manner that satisfies all of the proposed frameworks put forth by entities like the National Research Council (NRC) and National Council of Teachers of Mathematics (NCTM), as well as various education researchers and advocates such as Stein, Boaler, etc while addressing the content standards for a subject as required by a state such as California, coupled with the challenges of managing nearly 150 plus adolescents a day, with all of the factors that impact them outside of school, and inside, throughout the year, is no easy task. Simply standing in front of a class and lecturing about following rote procedures is one thing, ensuring all students understand and truly learn so they can recall and reapply the proper mathematics in various situations in life, beyond assessments, is quite another. Teaching surely is a profession when the latter, and all the preconditions necessary to ensure it, are taken into consideration.
Several comments as to how the lesson could be improved have been sprinkled throughout this writing and some consisted of questions about specific situations in the video. On a broader level, the lingering questions I have consist of how often the teacher employs complex instruction albeit he did mention at the outset that he was not going to take out his overhead projector for this lesson which leads me to believe that he normally provides direct instruction and this pedagogical approach was used to reinforce earlier instruction. Another question I have is why he did not require students to use both methods for graphing the linear equation. That one, simple requirement could have enriched the lesson considerably. At the same time, I am all too aware of the investment in time required to produce simple lesson plans, much less robust ones that enable deeper contextual learning for all students. I hate to second guess this teacher since I do not know his world at the time of this lesson.
Furthermore, it is my belief that our educational system places too much of a burden on individual teachers to be expert in content, curricula, lesson planning and creation, assessment, pedagogical diversity, classroom management, etcetera, etcetera. There is little to no support provided by national or state departments of education, districts, or within individual schools themselves, to ease the burden placed on the teacher in our ever changing world. Our educational system appears to be grossly inefficient in its use of resources and hence, mostly ineffective in its primary objective of educating our nationʼs youth to be able to handle our future needs. I firmly believe that much more can and should be done to improve the support systems and services provided to teachers to enable them to focus their energies and attention on delivering world class instruction to all students, which in my opinion, is no simple task. If I have time, in a couple of years after graduation, I hope to start a doctoral program where I could pursue my ideas for improving the effectiveness of our nationʼs educational efforts, primarily by providing teachers with support systems and services that free them to focus their time and effort on creating world class learning environments for all of our nationʼs youth.