Solving Problems as a Means of Teaching Algebraic Thinking

The National Council of Teachers of Mathematics is one organization that offers evidence-based tactics and top recommendations for improving the efficacy of math instructors globally (NCTM). Hiebert discovered (as stated in Walle, Kark, & Bay-Williams, 2013) that nations with teaching methods most in line with NCTM recommendations had students succeed more in mathematics. Hiebert discovered that, specifically, arithmetic was taught via problem solving in better scoring nations, with an emphasis on conceptual knowledge rather than particular problem-solving techniques (Walle et al., 2013). The NCTM divides arithmetic into five content standards, which specify the requirements for material mastery at each grade level, and five process standards, which specify how students should be taught and applied the content standards’ ideas (NCTM, n.d.-a). This essay will assess how the video lesson “Teaching Math: Staircase Problem” satisfies the algebra content requirement and the process criteria for problem solving and representations using the NCTM framework (Roche, 1996).

The NCTM states that “represent and analyze mathematical situations using algebraic symbols; understand patterns, relations, and functions; use mathematical models to represent and understand quantitative relationships; and to analyze change in various contexts” are the key elements of the algebra content standard (NCTM, n.d.-b, para. 1). With the exception of analyzing change, the staircase problem video tackles every aspect of the mathematics content standard for grades 9–12 in the context of problem-solving. Students were shown a graphic illustration of the construction pattern for a staircase with one, two, or three steps in the movie. The next task required the students to collaborate in small groups to calculate the number of blocks required to create a staircase with 50, 100, or N steps (Roche, 1996). The fact that the staircase problem exercise includes several entry points and enables students of various abilities to tackle the issue in a variety of ways is one factor contributing to its effectiveness (Walle et al., 2013). Students were challenged to make decisions that needed algebraic thinking after being presented with a scenario that was reasonably straightforward (Roche, 1996). The general layout of this task makes it clear that students are picking up algebraic reasoning skills while using the NCTM process standard for problem solving (NCTM, n.d.-a). Finding the number of blocks necessary for a staircase with N steps was a task that required pupils to utilize algebraic reasoning. Communication was the second process standard that was used in this video. The instructor in the video on the stairs, Jesse Solomon, urged pupils to assess one other’s ideas by purposefully declining to provide criticism or provide answers (Roche, 1996, 6:25). Solomon encouraged students to assess one other’s arguments by asking “why” inquiries as opposed to correcting their errors (Roche, 1996, 3:30). Solomon wraps off the session by directing pupils to record their justification for the day’s activities at the conclusion of the video (Roche, 1996, 14:25). He encourages communication about mathematics via writing by allowing the students time to think about and write about their justifications.

Analyzing patterns and relations was the first NCTM algebra content requirement that was covered in the stair problem exercise. Students first understood that they would need to recognize a pattern in order to be able to forecast how many blocks the subsequent step would have in order to determine how many blocks would be required for a staircase with 50 steps. The pattern was found by the students using a variety of techniques and several mathematical representations. The majority of students began by sketching a staircase (or simulating a staircase with square tiles) at each level, then organized the data into a table to search for patterns. In order to recognize the pattern and derive a rule about how many blocks would be needed for a staircase with N steps, students employed a number of tactics with varying degrees of success. One group began with the table instead of producing the figure, utilizing the information from the first three stages specified in the instructions (Roche, 1996, 2:45). This group made the mistaken assumption that the change in blocks was rising at a consistent pace of three blocks for each step by focusing only on the change in blocks from the second and third steps. The group misapplied the rule they came up with because they failed to consider the fact that the initial step was one block, not three blocks. Sadly, the camera shifts to another group just as the third group member starts to dissent (Roche, 1996, 3:05).

This assignment addressed the second NCTM algebra requirement, which was employing several mathematical models to clarify the issue (NCTM, n.d.-b). Students were encouraged to express the staircase issue using drawings, tiles, tables, and finally an equation (Roche, 1996). Solomon observed that although most groups understood conceptually how a rule connects to a pattern, students were having trouble figuring out how to select a rule for the particular circumstance (Roche, 1996, 3:55). Solomon emphasized that students should gather data to search for patterns before trying to identify rules (Roche, 1996). The majority of groups constructed or sketched stairs and counted the number of blocks added each time. Other groups, however, saw the stairway as a square of N by N blocks without the triangular component (Roche, 1996, 12:20). Each group made use of the tables, tiles, and drawings to help them develop a rule and defend it against criticism.

The video’s final NCTM algebra content standard dealt with patterns, relations, and functions (NCTM, n.d.-b). The entire number of blocks is equal to the total number of blocks from the previous step plus the number of current steps, according to a recursive formula that at least one group was able to develop (Roche, 1996, 10:55). Groups were tasked with determining if the relation could be stated without requiring knowledge of the number of blocks in the preceding step after discovering the recursive function for the staircase issue (Roche, 1996, 11:50). Because they were anticipating a one- or two-step openly described rule, the majority of groups found it difficult to identify the simpler recursively constructed rule at the beginning of the film. Students were able to recognize and finally explain the pattern after they were able to stop thinking about what the rule would look like and instead start concentrating on the facts. Many groups were inspired by this learning exercise to reconsider how they examine patterns to ascertain a function.

The problem-solving and communication process criteria were used in the context of the stair learning exercise to enhance algebraic thinking. Students were encouraged to utilize different models to depict connections, to represent situations using symbols, and to grasp how to generate recursively and clearly specified functions from patterns in this problem-based course (NCTM, n.d.-b, para. 1). This session served as a superb illustration of how problem-based learning that is in line with the relevant subject standards may get students to participate in fruitful conflict and deepen their conceptual knowledge of algebraic reasoning.