Learning mathematical symbolization: conceptual challenges and instructional strategies in secondary schools

This paper investigates South African 12th Grade students’ conceptual challenges with mathematical symbolization and instructional strategies that teachers use to mitigate mathematical symbolization. The study is motivated by the students’ failure to connect representations between symbolic and mathematical ideas to understand concepts and procedures. The study attempts to gain insight into mathematical symbols as potential barriers to students’ understanding of mathematical concepts and processes. The study consists of 120 randomly selected 12th Grade students and 15 purposefully selected mathematics teachers from Sekhukhune district of Limpopo Province, South Africa. Data was collected through questionnaires and focus group interviews. A mixed-method sequential explanatory design was employed. An SPSS cluster analysis of data produced three (3) clusters consisting of 50 (41.6%), 47 (39.3%) and 23 (19.1%) students with severe, mild, and minor challenges with mathematical symbols. Two themes emerged from the students’ difficulties with mathematical symbols. Firstly, students lack symbol sense for mathematical concepts and algebraic insight for problem-solving. Secondly, students disregard conceptual and contextual uses of symbols. The study therefore suggests that students’ negotiation of discourse between the mathematical symbol and the mathematical concept or procedure is crucial developing symbolic meaning. Therefore, teachers need to use appropriate strategies to engage students in processes that allow them to make meanings of mathematical symbols. The study recommends that concepts should be understood before symbolised.

An Analysis of Errors on Mathematical Symbol as a Metaphor in Linear Programming

Symbol sense is crucial in the understanding of mathematical problems comprising various symbols. The misuses of symbols happen due to misinterpretation, which is considered the constraint to learn algebra more comprehensively, including in linear programming. The term ‘metaphor’ is defined as a means to carry over symbol sense, and is used to improve mathematical understanding. This present research was aimed at analyzing errors on mathematical symbol as a metaphor in linear programming. This research was conducted by means of descriptive qualitative design, with a test and interview as the instruments. The test was made essay, and its results were analyzed qualitatively. The test, further, was administered to five eleventh graders selected according to highest rates of errors committed. This research has shown that the students committed a number of errors in some cases, such as representing symbols as variables, representing numbers, and interpreting symbols as relational operators. In addition, errors which the students committed in constructing mathematical models covered defining the final value, representing numbers, applying inequality system, and interpreting symbols as operation counts. This present research has provided some ways for symbol sense, and thus the errors on mathematical symbol as a metaphor could be lessened. Next, this research can be further followed up by reviewing the effectiveness of remedial instruction according to the committed errors on mathematical symbols.

The relation between graphing formulas by hand and students’ symbol sense

Students in secondary school often struggle with symbol sense, that is, the general ability to deal with symbols and to recognize the structure of algebraic formulas. Fostering symbol sense is an educational challenge. In graphing formulas by hand, defined as graphing using recognition and reasoning without technology, many aspects of symbol sense come to play. In a previous study, we showed how graphing formulas by hand could be learned. The aim of the study we present here is to explore the relationship between students’ graphing abilities and their symbol sense abilities while solving non-routine algebra tasks. A symbol sense test was administered to a group of 114 grade 12 students. The test consisted of eight graphing tasks and twelve non-routine algebra tasks, which could be solved by graphing and reasoning. Six students were asked to think aloud during the test. The findings show a strong positive correlation between the scores on the graphing tasks and the scores on the algebra tasks and the symbol sense used while solving these tasks. The thinking-aloud protocols suggest that the students who scored high on the graphing tasks used similar aspects of symbol sense in both the graphing and algebra tasks, that is, using combinations of recognizing function families and key features, and qualitative reasoning. As an implication for teaching practice, learning to graph formulas by hand might be an approach to promote students’ symbol sense.

Students’ Algebraic Proficiency from the Perspective of Symbol Sense

Algebraic proficiency, including procedural fluency and conceptual understanding, is widely discussed worldwide. Algebraic proficiency refers largely to proficiency in symbolic representations which can be investigated through a framework of symbol sense. This research, therefore, aims to analyze students’ algebraic proficiency in terms of symbol sense. We set up a pilot study, involving 22 Indonesian mathematics education students (18-19 year old), in the form of two weeks teaching that combine a conventional approach and the use of a camera calculator in the learning and teaching of quadratic and related equations. The results showed that more than half number of students lacks of symbol sense in the sense that they tend to use procedural strategies rather than symbol sense strategies in solving equations. From the perspective of symbol sense, we concluded that the students acquired more on procedural fluency than on conceptual understanding.

SPEKTRUM SYMBOL DAN STRUCTURE SENSE MATEMATIKA SISWA MADRASAH TSANAWIYAH

The purpose of this study is to determine the symbol sense and structure of mathematical sense in terms of students' ability to solve algebraic problems or other mathematical problems that require symbol expression or structure. Difficulties experienced by students in solving mathematical or algebraic problems may be due to the ability of symbol sense and low structure sense or didactic design that the teacher conveyed is not in accordance with the category of symbols and structure sense that students have. A student with good symbols and structure sense is able to appreciate the power of symbols, knowing when to use the right symbols and being able to manipulate and understand symbols in various contexts. The method used in the study used a qualitative descriptive method. The population of this study was seventh grade students of the Islamic junior high school, the instruments used were symbol and structure sense tests, questionnaires, and interview forms. The results of the study indicate that the ability of symbol sense and student structure sense is still low because of a lack of conceptual knowledge and algebraic manipulation, for this reason it is necessary to have an appropriate learning model to improve both of these abilities.

Search for Symbol Sense Behavior: Students in Upper Secondary Education Solving Algebraic Physics Problems

Students in upper secondary education encounter difficulties in applying mathematics in physics. To improve our understanding of these difficulties, we examined symbol sense behavior of six grade 10 physics students solving algebraic physic problems. Our data confirmed that students did indeed struggle to apply algebra to physics, mainly because they lacked both sufficient symbol sense behavior and basic algebraic skills. They used ad hoc strategies instead of correct, systematic rule-based procedures involving insight. These ad hoc strategies included the cross-multiplication, the numbering, and the permutation strategy. They worked only for basic formulas containing few variables. In problems with more variables, students got stuck. The latter two strategies substitute numbers for variables. The permutation strategy randomly checks several permutations to guess which one is correct. The numbering strategy substitutes numbers to check algebraic manipulations. Our results indicate insufficient focus on conceptual understanding of algebra in some mathematics textbooks, leading to reliance on poorly understood ad hoc strategies. Effective teaching of algebraic skills should not focus on either basic algebraic skills or on symbol sense behavior. Instead, both aspects should be taught in an integrated manner. Our operationalization of symbol sense behavior turned out to be very useful for analysis. In contrast to earlier qualitative studies, it provided us the opportunity to measure symbol sense behavior quantitatively. This operationalization should also be applicable to other science subjects. Furthermore, we discussed some implications of our results for curricula, teachers, science teacher educators, and textbook publishers aiming at successful application of mathematics in physics.