4.5 Content Abstraction: Elements of mathematical/scientific abstraction were used appropriately (e.g., multiple forms of representation in science and mathematics classes include verbal, graphic, symbolic, visualizations, simulations, models of systems and structures that are not directly observable in real time or by the naked eye, etc.).
This indicator captures how well the teacher facilitates conceptual understanding by representing relationships or patterns in abstract or symbolic ways. Moving toward abstraction can assist students in understanding the content as a coherent and integrated whole, as opposed to a set of facts, procedures, or vocabulary terms. Abstraction can lead students to see the “big picture” and connections between important concepts in the discipline. In science, abstraction is often represented by the modeling of complex systems or simulations that synthesize complex interactions from the molecular to ecological levels. In a mathematics lesson about linear relationships between variables, after focusing on several cases where the variables have fixed values, the teacher might scaffold the students to generalize their understanding of the relationship by writing the linear equation using symbols.
A rating of 3 is the default score for this indicator, if you notice nothing especially good or especially poor about the use of abstraction. It is important when awarding a high score (4 or 5) on this indicator is to consider whether the abstraction is being used for a relevant and useful purpose; for example, are students writing an equation because it’s part of a school exercise, or are they writing an equation to help them accomplish some larger, more authentic goal?
An NA is an appropriate rating for lessons where abstraction of or generalization to complex systems does not arise for appropriate reasons related to the lesson purposes; for example, if the class is focused on data collection for a lab activity, it is unlikely at that point in the learning sequence that abstraction would be appropriate. Thus if abstractions were not included in the lesson, but you feel this lack of inclusion was an appropriate instructional decision, rate this indicator NA.
Math-Specific Instructions
This indicator captures how well the teacher facilitates conceptual understanding by representing relationships or patterns in abstract or symbolic ways. The teacher may use multiple representations—such as verbal, tabular, graphical, and symbolic—to better allow students to understand concepts and connections between multiple representations. Finally, the teacher should, if appropriate, ensure that students understand what symbols and other abstract representations really “mean” through explicit discussion. In middle school lessons, abstraction can arise in a variety of ways. For example, any formula that has variables in it (such as area = length * width) is considered an element of abstraction, because you can plug concrete, specific values (like L = 2 and W = 3) into this general equation. This indicator is considered applicable to the observed lesson when (1) a letter or another representation (like an icon) is used to stand for an unknown value, or (2) a general relationship (like an equation) is shown for which many specific cases would hold true.
Science-Specific Instructions
This indicator captures how well the teacher facilitates deeper understanding by choosing tasks or lab inquiries that prompt students to make connections between important concepts beyond the immediate scope of the lesson. For example, a teacher might use an interactive computer simulation to facilitate student understanding of the relationship between the hydrologic cycle and the phase changes and physical properties of H2O. With such an activity, it is important that the teacher make explicit that this is a model of a complex process in time and space (i.e., that the changes depicted in this water cycle occur in extremely short periods of time on the molecular level but also happen over the course of geologic time on the macroscopic scale).
General Rubric
- This item should be rated a 1 if there was a major issue with the teacher’s use of abstraction that had a negative impact on student learning during the class period.
- This item should be rated a 2 if the teacher neglected important explanation and discussion of abstraction that was being used during the class period, and this missed opportunity had a negative impact on student learning.
- This item should be rated a 3 if the teacher’s use of abstraction was adequate—the teacher allowed for some discussion or explanation and did not use abstraction inappropriately.
- This item should be rated a 4 if abstraction was used during the class period for a relevant and useful purpose. The teacher explicitly engaged students in some discussion of the meaning of the representation and/or successfully connected different representational forms. Perhaps there was a small missed opportunity with respect to facilitating some students’ understanding of abstraction.
- This item should be rated a 5 if abstraction was being used for a relevant and useful purpose, like modeling, supporting an argument for a scientific theory or mathematical proof, or progressively generalizing important ideas, AND if the teacher engaged students in a discussion of the meaning and purpose of the representation. The abstractions were presented in a way such that they were understandable and accessible to all students in the class.
Specific Examples of Supporting Evidence
- The teacher introduced the students to several new procedures that were written as symbolic equations. There was no discussion of why the procedures worked or what the numbers and symbols meant. The students were confused and repeatedly made mistakes applying the procedures.
- This was an algebra lesson on comparing different cell phone plans. Students were explicitly asked to generate symbolic equations representing each cell phone plan; however, the teacher told the class to skip this part of the activity.
- Abstraction seemed to be adequately used in this lesson. The notes about the different representations of functions clearly connected the symbolic equation to its meaning in terms of graphs, tables, and verbal descriptions of independent and dependent variables.
- Students were collecting data and then using their graphing calculator to determine a line of best fit. Using the equation for the line of best fit, students would then make predictions for different values of the independent and dependent variables. The students were using their symbolic representations for a very practical and relevant purpose, and the teacher briefly discussed with students what the slope and intercept of their line meant. However, this discussion of the symbolic representation could have been more accentuated.
- Students were working on minimizing a function given a system of linear and quadratic inequalities as constraints, and the symbolic expressions were well integrated, with the students coming up with the inequalities symbolically from verbal information and graphing them. Symbolism was also very prevalent and appropriately used during the warm up and wrap up. At all points of the lesson, the abstractions were connected well to what they meant and were being employed for the realistic purpose of mathematical modeling.