Class 1: 9th Grade Algebra I
Classroom Environment: The classroom has approximately 28 students; the desks are arranged in 7 mixed-ability groups of 4 to 5 students each. The classroom is equipped with an interactive whiteboard with graphing software, a bulletin board displaying “Tasks to be completed today,” and posters of equations, tables, graphs, and real-world models on the walls. Each group’s workspace is equipped with manipulatives such as algebra tiles and number lines; if any are missing, students can pick them up near the classroom door. The teacher for this class is a woman with 5 years of teaching experience, and she uses a very unique and engaging teaching style. During group activities, she circulates among the groups, asking students to justify their chosen representations and encouraging them to translate mathematical concepts between different forms of representation. Teaching Content: Linear functions (transforming between tables, equations, graphs, and real-world contexts).
The representation forms in this class are diverse, student-centered and multimodal, including figurative teaching AIDS, numerical tables, symbolic equations, images and textual descriptions. Example 1 (Linear Function Representation) : Solving the problem of “A cup of coffee in a certain cafe costs $2, and an additional $0.5 is required for each extra serving of flavor concentrate.” When dealing with the problem of expressing the total cost as a function of the number of flavor concentration servings, each group used four different forms of representation: Concrete representation: using algebraic blocks to simulate a base cost of $2 (two 1-unit blocks) and a variable cost of $0.5 per concentration serving (each serving corresponds to half a unit block). Numerical representation: Prepare a corresponding table of the number of concentrated flavor servings (x) and the total cost (y). Symbolic representation: Derive the equation y=0.5x+2 based on the table and the building block model. Image representation: Draw a straight line in the coordinate system, marking the Y-axis intercept as “basic coffee cost” and the slope as “extra charge for each serving of flavor concentrate”.
Classroom 2: 10th Grade Geometry
The classroom environment was a very traditional lecture-style classroom, with approximately 31 students; the desks were arranged in straight rows facing the front of the classroom. A projector in the middle of the classroom displayed static diagrams of geometric figures to the students, and there were no teaching aids or visual representation tools provided for the students to use. A ​​”Quiet Study” sign was posted on the wall. The teacher for this class was a male teacher with a long teaching career, approximately 20 years; he used slides to present pre-made diagrams, demonstrated standard proof processes, and then assigned independent practice exercises focusing on triangle congruence criteria, and confirmed the answers. Teaching content: Triangle congruence (SSS, SAS, ASA, AAS congruence theorems) and corresponding side and angle relationships.
The representation form of this class is single, teacher-led and static, only including pre-made two-dimensional geometric diagrams on slides, without integrating numerical, symbolic or textual representation forms. Example 1 (Schematic Diagram of Congruence of Triangles) : When teaching the determination of congruence of sides, angles and sides, the teacher only projects one schematic diagram. In the diagram, the two corresponding sides and the included angles of the two triangles are marked as congruence. He asked the students, “Copy this diagram exactly into your notebook – this is the only correct way to represent congruence on all sides, corners and edges.” Students are not allowed to mark the side lengths or Angle values of triangles, nor can they describe congruence relations in words. Example 2 (Refusal of Concrete Representation) : A student asked if a congruent triangular model (concrete representation) could be built using straws and clay, but the teacher told the student, “Straws are teaching AIDS for primary school students. High school geometry requires standard schematic diagrams.” The student immediately gave up the hands-on attempt and instead copied the graphics on the slide. I think this is a restriction for students.
Classroom 3: 12th Grade Advanced Placement (AP) Calculus
Classroom Environment: The classroom has approximately 24 students; it features group tables and a front-of-the-room presentation area with an interactive smart whiteboard. The walls are covered with student-created “representation connection posters,” linking derivative graphs, function tables, rate of change equations, and physical motion diagrams. “Project posters” created by each group are also displayed. Each table is equipped with graphing calculators and “representation transformation worksheets.” The teacher is a woman with approximately 8 years of teaching experience, and she stated that she holds AP Calculus certification; she facilitates discussions about the advantages and disadvantages of different representations and requires students to use multiple forms of representation to solve complex problems. Teaching Content: Derivatives as rates of change (connecting symbolic, graphical, numerical, and verbal representations).
The representation forms in this class are conceptually profound, interrelated and have clear objectives, including derivative symbol expressions, function-derivative corresponding graphs, numerical tables of rates of change and textual descriptions of physical scenarios. Example 1 (Derivative Representation) : When analyzing the motion problem of the particle position function s(t)=t ² – 4t+3, students use three forms of representation: Symbolic representation: Calculate the velocity function v(t)=s’ (t)= 2t – 4 by applying the rule of differentiation of power functions, and interpret it as the instantaneous velocity of the particle at any time t. Image representation: Draw the graph of the position number s(t) (a parabola with an upward opening) and the velocity function v(t) (a straight line with a slope of 2). Mark the points where v(t)=0 as the “moment of rest of the particle”, and the intervals where v(t)>0 as the “stage of the particle moving to the right”. Text – Physical Characterization: Write a motion analysis report: “The initial position of the particle is 3. It moves to the left (with a negative velocity) before t=2, remains stationary at t=2, and then moves to the right after t>2.” The derivative describes the velocity and direction of motion of a particle at any given moment.
After this classroom observation, I realized that mathematical representation is not merely a “tool for demonstrating problem-solving processes,” but rather a core cognitive tool that shapes students’ understanding, communication, and application of mathematical concepts. However, I still have several questions: 1. For students with learning disabilities or math anxiety, which specific representation strategies (such as teaching aids and visual images) are most helpful in building foundational skills without creating an undue learning burden? 2. In a teaching environment with significant pressure from standardized testing, how can teachers balance cultivating fluency in mathematical representation with teaching the specific representation forms required by standardized tests?