Many students have misconceptions about mathematics, so preservice teachers should be developing the skills to notice mathematical misconceptions. This qualitative study analyzed preservice teachers' skills in noticing student misconceptions about algebra, according to three aspects of noticing found in the literature: attending, interpreting and responding. Participants in this study were seven preservice teachers from one university in the capital of Aceh province, Indonesia, who were in their eighth semester and had participated in teaching practicums. Data was collected through questionnaires and interviews, which were analyzed descriptively. The results revealed the preservice teachers had varying levels of skill for the three aspects of noticing. Overall, the seven preservice teachers' noticing skills were fair, but many needed further development of their skills in interpreting and responding in particular. This university’s mathematics teacher education program should design appropriate assessment for preservice teachers’ noticing skills, as well as design and implement learning activities targeted at the varying needs of individual preservice teachers regarding noticing student misconceptions, in order to improve their overall teaching skills.

This research aims to describe secondary school students' functional thinking in generating patterns in learning algebra, particularly in solving mathematical word problems. In addressing this aim, a phenomenological approach was conducted to investigate the meaning of functional relationships provided by students. The data were collected from 39 ninth graders (13-14 years old) through a written test about generating patterns in linear functions. The following steps were conducting interviews with ten representative students to get detailed information about their answers to the written test. All students' responses were then analyzed using the thematic analysis software ATLAS.ti. The findings illustrate that students employed two types of approaches in solving the problem: recursive patterns and correspondence. Students favored the recursive patterns approach in identifying the pattern. They provided arithmetic computation by counting term-to-term but could not represent generalities with algebraic symbols. Meanwhile, students evidenced for correspondence managed to observe the relation between two variables and create the symbolic representation to express the generality. The study concludes that these differences exist due to their focus on identifying patterns: the recursive pattern students tend to see the changes in one variable, whereas the correspondence ones relate to the corresponding pair of variables.

Research on conceptual understanding is one of the first steps in designing materials to improve learning. Literature reports that students have difficulties analyzing and describing phenomena in electric circuits. This report contributes to students' conceptual difficulties regarding simple electrical circuits by systematically analyzing an open conceptual test answered by 531 first-year engineering students. We found students' reasoning that has not yet been reported in the literature as misconceptions or difficulties. To deepen our understanding of students' difficulties, we chose five students by convenience to interview. We present evidence that there are two main contributions to the taxonomy in this study: the Series Circuit Misconception, which is when students convey that the current through bulbs is the same because they are in series, using that as a mnemonic ignoring any change in the circuit; and the Inverse Parallel Circuit Misconception, that is when students mention that the resistance of the circuit decreases when disconnecting bulbs in parallel, neither are reported in the literature. The results of this study have implications for physics education research in electric circuits and educational practice in the classroom.

This study aims to produce a product to evaluate students' critical thinking skills that departs from physics content where students often have misconceptions. This research is a development research with research stages covering a) research and review literature; (b) planning chapter objectives; (c) developing a preliminary form; (d) field-testing the preliminary form; (e) Revise the preliminary form; (f) conducting a main field-test. The Waves Critical Thinking (WCT) test developed consists of 7 questions with 15 specific domains. Total percentage of content validity test was obtained 87.98% with appropriate criteria and based on the construct validity WCT test, the Goodness of Fit criteria were obtained which were classified as fit. The test instrument being tested consists of 15 objective items. The reliability of WCT test results 0.597 as a Cronbach's alpha score with the medium category and all the components have a good level of composite reliability. The outcome of the study was the WCT test with a valid state for measuring students' CT in a specific domain of physics wave material.

In this article, we present the results of empirical research using a combination of quantitative and qualitative methodology, in which we examined the achievements and difficulties of sixth-grade Slovenian primary school students in decimal numbers at the conceptual and procedural knowledge level. The achievements of the students (N = 100) showed that they statistically significantly (z = -7,53, p < .001) better mastered procedural knowledge (M = 0.60, SD = 0.22) than conceptual knowledge (M = 0.37, SD = 0.17) of decimal numbers. Difficulties are related to both procedural and conceptual knowledge, but significantly more students have difficulties at the level of conceptual knowledge. At the level of procedural knowledge, or in the execution of arithmetic operations with decimal numbers, we observed difficulties in transforming text notation into numerical expressions, difficulties in placing the decimal point in multiplication and division, and insufficient automation of mathematical operations with decimal numbers. At the level of conceptual knowledge of decimal numbers, the results indicate difficulties for students in understanding the place values of decimal numbers, in estimating the sum, product and quotient of decimals with reflection and in mathematical justification. In relation to difficulties in justification, we observed an insufficient understanding of the size relationship between decimal numbers and difficulties in expressing them in mathematical language. The results indicate that to overcome such difficulties in the learning and teaching of mathematics, more balance between procedural and conceptual knowledge is needed.

Understanding graphs in the dynamics of market (DM) is a challenge to learners; its teaching demands a specific kind of teacher’s knowledge. This study aims to examine the topic-specific pedagogical content knowledge (TSPCK) of experienced economics teachers in teaching graphs in DM to enhance learners’ understanding of the topic. It reports using a qualitative approach underpinned by the TSPCK framework for teaching specific topics developed by Mavhunga. Data were collected through classroom observations and analyzed thematically using a case study of two economics teachers. The study revealed that adopting a step-by-step approach and the use of worked graphical examples promote an understanding of graphs in DM. It also established that active learning is preferable to the predominant chalk-and-talk (lecture) method of teaching graphs in DM. The study proposed a Dynamics of Market Graphical Framework (DMG-Framework) to enable teachers, particularly pre-service teachers in lesson delivery, to enhance learners’ understanding of graphs in DM. The result of this study will broaden the international view in the teaching of graphs in DM.

<p style="text-align:justify">This study aimed to assess the geometric knowledge of student teachers from a university in the Eastern Cape province of South Africa. The study used a sample of 225 first-year student teachers who completed school mathematics baseline assessments on a computer- aided mathematics instruction (CAMI) software. The study adopted a descriptive cross-sectional research design, using quantitative data to measure student teachers&rsquo; geometry achievement level, and qualitative data to explain the challenges encountered. The results show that student teachers exhibited a low level of understanding of school-level geometry. The low achievement levels were linked to various factors, such as insufficient grasp of geometry concepts in their secondary school education, difficulty in remembering what was done years ago, low self-confidence, and lack of Information and Communications Technology (ICT) skills along with the limited time for the baseline tests. These results suggest that appropriate measures should be taken to ensure that student teachers acquire the necessary subject-matter knowledge to teach effectively in their future classrooms.</p>