Background/Context: After-school programs that focus on integrating computer programming and mathematics in authentic environments are seldomly accessible to students from culturally and linguistically diverse backgrounds, particularly bilingual Latina students in rural contexts. Providing a context that broadens Latina students’ participation in mathematics and computer programming requires educators to carefully examine how verbal and nonverbal language is used to interact and to position students as they learn new concepts in middle school. This is also an important stage for adolescents because they are likely to make decisions about their future careers in STEM. Having access to discourse and teaching practices that invite students to participate in mathematics and computer programming affords them opportunities to engage with these fields. Purpose/Focus of Study: This case study analyzes how small-group interactions mediated the positionings of Cindy, a bilingual Latina, as she learned binary numbers in an after-school program that integrated computer programming and mathematics (CPM). Setting: The Advancing Out-of-School Learning in Mathematics and Engineering (AOLME) program was held in a rural bilingual (Spanish and English) middle school in the Southwest. The after-school program was designed to provide experiences for primarily Latinx students to learn how to integrate mathematics with computer programming using Raspberry Pi and Python as a platform. Our case study explores how Cindy was positioned as she interacted with two undergraduate engineering students who served as facilitators while learning binary numbers with a group of three middle school students. Research Design: This single intrinsic case focused on exploring how small-group interactions among four students mediated Cindy’s positionings as she learned binary numbers through her participation in AOLME. Data sources included twelve 90-minute video sessions and Cindy’s journal and curriculum binder. Video logs were created, and transcripts were coded to describe verbal and nonverbal interactions among the facilitators and Cindy. Analysis of select episodes was conducted using systemic functional linguistics (SFL), specifically language modality, to identify how positioning took place. These episodes and positioning analysis describe how Cindy, with others, navigated the process of learning binary numbers under the stereotype that female students are not as good at mathematics as male students. Findings: From our analysis, three themes that emerged from the data portray Cindy’s experiences learning binary numbers. The major themes are: (1) Cindy’s struggle to reveal her understanding of binary numbers in a competitive context, (2) Cindy’s use of “fake it until you make it” to hide her cognitive dissonance, and (3) the use of Spanish and peers’ support to resolve Cindy’s understanding of binary numbers. The positioning patterns observed help us learn how, when Cindy’s bilingualism was viewed and promoted as an asset, this social context worked as a generative axis that addressed the challenges of learning binary numbers. The contrasting episodes highlight the facilitators’ productive teaching strategies and relations that nurtured Cindy’s social and intellectual participation in CPM. Conclusions/Recommendations: Cindy’s case demonstrates how the facilitator’s teaching, and participants’ interactions and discourse practices contributed to her qualitatively different positionings while she learned binary numbers, and how she persevered in this process. Analysis of communication acts supported our understanding of how Cindy’s positionings underpinned the discourse; how the facilitators’ and students’ discourse formed, shaped, or shifted Cindy’s positioning; and how discourse was larger than gender storylines that went beyond classroom interactions. Cindy’s case reveals the danger of placing students in “struggle” instead of a “productive struggle.” The findings illustrated that when Cindy was placed in struggle when confronting responding moves by the facilitator, her “safe” reaction was hiding and avoiding. In contrast, we also learned about the importance of empathetic, nurturing supporting responses that encourage students’ productive struggle to do better. We invite instructors to notice students’ hiding or avoiding and consider Cindy’s case. Furthermore, we recommend that teachers notice their choice of language because this is important in terms of positioning students. We also highlight Cindy’s agency as she chose to take up her friend’s suggestion to “fake it” rather than give up.
more »
« less
Cognitive agency and computer-based tasks
As computer-focused policies and trends become more popular in schools, more students access math curriculum online. While computer-based programs may be responsive to some student input, their algorithmic basis can make it more difficult for them to be prepared for divergent student thinking, especially in comparison to a teacher. Consider programs that assess student work by judging how well it matches pre-set answers. Unless designed and enacted in classrooms with care, computer-based curriculum materials might encourage students to think about mathematics in pre-determined ways. How do students approach the process of mathematics while using online materials, especially in terms of engaging in original thought?
Drawing on Pickering’s (1995) dance of agency and Sinclair’s (2001) conception of students as path-finders or track-takers, I define two modes of mathematical behavior: trail-taking and bushwhacking. While trail-taking, students follow an established approach, often relying on Pickering’s (1995) disciplinary agency, wherein the mathematics “leads [them] through a series of manipulations” (p. 115). The series of manipulations can be seen as a trail that a student may choose to follow. Bushwhacking, on the other hand, refers to actions a student takes of their own invention. It is possible that, unknown to the student, these actions have been taken before by others. In bushwhacking, the student possesses agency, which Pickering (1995) describes as active (rather than passive) and as hallmarked by “choice and discretion” (p. 117).
In this study, students worked in several dynamic geometric environments (DGEs) during a geometry lesson about the midline theorem. The lesson was originally recorded as part of a larger study designing mathematically captivating lessons. Students accessed both problems and online addresses for corresponding DGEs via a printed packet. Students interacted with the DGEs on individual laptops, but were seated in groups of three or four. Passages of group conversations in which students transitioned between trail-taking and bushwhacking were selected for closer analysis, which involved identifying evidence of each mode and highlighting the curricular or social forces that may have contributed to shifts between modes.
Of particular interest were episodes in which students asked one another to share results, which led to students reconsidering previously set approaches, and episodes in which students interacted with DGEs containing a relatively high proportion of drag-able components, which corresponded to some students working in bushwhacking mode, spontaneously suggesting and revising approaches for manipulating the DGE (e.g., “unless you make this parallel to the bottom, but I don’t think you... yes you can.”). Both types of episodes were found in multiple groups’ conversations. Further analysis of student interactions with tasks, especially with varying levels of student control and sharing, could serve to inform future computer-based task design aimed to encourage students to productively engage in bushwhacking while problem-solving.
more »
« less
- Award ID(s):
- 1652513
- NSF-PAR ID:
- 10084136
- Date Published:
- Journal Name:
- Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education
- Page Range / eLocation ID:
- 1011
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In March 2020, the global COVID-19 pandemic forced universities across the United States to immediately stop face-to-face activities and transition to virtual instruction. While this transition was not easy for anyone, the shift to online learning was especially difficult for STEM courses, particularly engineering, which has a strong practical/laboratory component. Additionally, underrepresented students (URMs) in engineering experienced a range of difficulties during this transition. The purpose of this paper is to highlight underrepresented engineering students’ experiences as a result of COVID-19. In particular, we aim to highlight stories shared by participants who indicated a desire to share their experience with their instructor. In order to better understand these experiences, research participants were asked to share a story, using the novel data collection platform SenseMaker, based on the following prompt: Imagine you are chatting with a friend or family member about the evolving COVID-19 crisis. Tell them about something you have experienced recently as an engineering student. Conducting a SenseMaker study involves four iterative steps: 1) Initiation is the process of designing signifiers, testing, and deploying the instrument; 2) Story Collection is the process of collecting data through narratives; 3) Sense-making is the process of exploring and analyzing patterns of the collection of narratives; and 4) Response is the process of amplifying positive stories and dampening negative stories to nudge the system to an adjacent possible (Van der Merwe et al. 2019). Unlike traditional surveys or other qualitative data collection methods, SenseMaker encourages participants to think more critically about the stories they share by inviting them to make sense of their story using a series of triads and dyads. After completing their narrative, participants were asked a series of triadic, dyadic, and sentiment-based multiple-choice questions (MCQ) relevant to their story. For one MCQ, in particular, participants were required to answer was “If you could do so without fear of judgment or retaliation, who would you share this story with?” and were given the following options: 1) Family 2) Instructor 3) Peers 4) Prefer not to answer 5) Other. A third of the participants indicated that they would share their story with their instructor. Therefore, we further explored this particular question. Additionally, this paper aims to highlight this subset of students whose primary motivation for their actions were based on Necessity. High-level qualitative findings from the data show that students valued Grit and Perseverance, recent experiences influenced their Sense of Purpose, and their decisions were majorly made based on Intuition. Chi-squared tests showed that there were not any significant differences between race and the desire to share with their instructor, however, there were significant differences when factoring in gender suggesting that gender has a large impact on the complexity of navigating school during this time. Lastly, ~50% of participants reported feeling negative or extremely negative about their experiences, ~30% reported feeling neutral, and ~20% reported feeling positive or extremely positive about their experiences. In the study, a total of 500 micro-narratives from underrepresented engineering students were collected from June – July 2020. Undergraduate and graduate students were recruited for participation through the researchers’ personal networks, social media, and through organizations like NSBE. Participants had the option to indicate who is able to read their stories 1) Everyone 2) Researchers Only, or 3) No one. This work presents qualitative stories of those who granted permission for everyone to read.more » « less
-
more » « less
Background/Context: Computer programming is rarely accessible to K–12 students, especially for those from culturally and linguistically diverse backgrounds. Middle school age is a transitioning time when adolescents are more likely to make long-term decisions regarding their academic choices and interests. Having access to productive and positive knowledge and experiences in computer programming can grant them opportunities to realize their abilities and potential in this field.
Purpose/Focus of Study: This study focuses on the exploration of the kind of relationship that bilingual Latinx students developed with themselves and computer programming and mathematics (CPM) practices through their participation in a CPM after-school program, first as students and then as cofacilitators teaching CPM practices to other middle school peers.
Setting: An after-school program, Advancing Out-of-School Learning in Mathematics and Engineering (AOLME), was held at two middle schools located in rural and urban areas in the Southwest. It was designed to support an inclusive cultural environment that nurtured students’ opportunities to learn CPM practices through the inclusion of languages (Spanish and English), tasks, and participants congruent to students in the program. Students learned how to represent, design, and program digital images and videos using a sequence of 2D arrays of hexadecimal numbers with Python on a Raspberry Pi computer. The six bilingual cofacilitators attended Levels 1 and 2 as students and were offered the opportunity to participate as cofacilitators in the next implementation of Level 1.
Research Design: This longitudinal case study focused on analyzing the experiences and shifts (if any) of students who participated as cofacilitators in AOLME. Their narratives were analyzed collectively, and our analysis describes the experiences of the cofacilitators as a single case study (with embedded units) of what it means to be a bilingual cofacilitator in AOLME. Data included individual exit interviews of the six cofacilitators and their focus groups (30–45 minutes each), an adapted 20-item CPM attitude 5-point Likert scale, and self-report from each of them. Results from attitude scales revealed cofacilitators’ greater initial and posterior connections to CPM practices. The self-reports on CPM included two number lines (0–10) for before and after AOLME for students to self-assess their liking and knowledge of CPM. The numbers were used as interview prompts to converse with students about experiences. The interview data were analyzed qualitatively and coded through a contrast-comparative process regarding students’ description of themselves, their experiences in the program, and their perception of and relationship toward CPM practices.
Findings: Findings indicated that students had continued/increased motivation and confidence in CPM as they engaged in a journey as cofacilitators, described through two thematic categories: (a) shifting views by personally connecting to CPM, and (b) affirming CPM practices through teaching. The shift in connecting to CPM practices evolved as students argued that they found a new way of learning mathematics, in that they used mathematics as a tool to create videos and images that they programmed by using Python while making sense of the process bilingually (Spanish and English). This mathematics was viewed by students as high level, which in turned helped students gain self-confidence in CPM practices. Additionally, students affirmed their knowledge and confidence in CPM practices by teaching them to others, a process in which they had to mediate beyond the understanding of CPM practices. They came up with new ways of explaining CPM practices bilingually to their peers. In this new role, cofacilitators considered the topic and language, and promoted a communal support among the peers they worked with.
Conclusions/Recommendations: Bilingual middle school students can not only program, but also teach bilingually and embrace new roles with nurturing support. Schools can promote new student roles, which can yield new goals and identities. There is a great need to redesign the school mathematics curriculum as a discipline that teenagers can use and connect with by creating and finding things they care about. In this way, school mathematics can support a closer “fit” with students’ identification with the world of mathematics. Cofacilitators learned more about CPM practices by teaching them, extending beyond what was given to them, and constructing new goals that were in line with a sophisticated knowledge and shifts in the practice. Assigned responsibility in a new role can strengthen students’ self-image, agency, and ways of relating to mathematics.
-
In this proposal, we will share some initial findings about how teacher and student engagement in cogenerative dialogues influenced the development of the Culturally Relevant Pedagogical Guidelines for Computational Thinking and Computer Science (CRPG-CSCT). The CRPG-CSCT’s purpose is to provide computer science teachers with tools to enhance their instruction by accurately reflecting students’ diverse cultural resources in the classroom. Additionally, the CRPG-CSCT will provide guidance to non-computer science teachers on how to facilitate the integration of computational thinking skills to a broad spectrum of classes in the arts, humanities, sciences, social sciences, and mathematics. Our initial findings shared here are part of a larger NSF-funded research project (Award No. 2122367) which aims to better understand the barriers to entry and challenges for success faced by underrepresented secondary school students in computer science, through direct engagement with the students themselves. Throughout the 2022-23 academic year, the researchers have been working with a small team of secondary school teachers, students, and instructional designers, as well as university faculty in computer science, secondary education, and sociology to develop the CRPG-CSCT. The CRPG-CSCT is rooted in the tenets of culturally relevant pedagogy (Ladson-Billings, 1995) and borrows from Muhammad’s (2020) work in Cultivating Genius: An Equity Framework for Culturally and Historically Responsive Literacy. The CRPG-CCT is being developed over six day-long workshops held throughout the academic year. At the time of this submission, five of the six workshops had been completed. Each workshop utilized cogenerative dialogues (cogens) as the primary tool for organizing and sustaining participants’ engagement. Through cogens, participants more deeply learn about students’ cultural capital and the value of utilizing that capital within the classroom (Roth, Lawless, & Tobin, 2000). The success of cogens relies on following specific protocols (Emdin, 2016), such as listening attentively, ensuring there are equal opportunities for all participants to share, and affirming the experiences of other participants. The goal of a cogen is to reach a collective decision, based on the dialogue, that will positively impact students by explicitly addressing barriers to their engagement in the classroom. During each workshop, one member of the research team and one undergraduate research assistant observed the interactions among cogen participants and documented these in the form of ethnographic field notes. Another undergraduate research assistant took detailed notes during the workshop to record the content of small and large group discussions, presentations, and questions/responses throughout the workshops. A grounded theory approach was used to analyze the field notes. Additionally, at the conclusion of each workshop, participants completed a Cogen Feedback Survey (CFS) to gather additional information. The CFS were analyzed through open thematic coding, memos, and code frequencies. Our preliminary results demonstrate high levels of engagement from teacher and student participants during the workshops. Students identified that the cogen structure allowed them to participate comfortably, openly, and honestly. Further, students described feeling valued and heard. Students’ ideas and experiences were frequently affirmed, which served as an important step toward dismantling traditional teacher-student boundaries that might otherwise prevent them from sharing freely. Another result from the use of cogens was the shared experience of participants comprehending views from the other group’s perspective in the classroom. Students appreciated the opportunity to learn from teachers about their struggles in keeping students engaged. Teachers appreciated the opportunity to better understand students’ schooling experiences and how these may affirm or deny aspects of their identity. Finally, all participants shared meaningful suggestions and strategies for future workshops and for the collective betterment of the group. Initial findings shared here are important for several reasons. First, our findings suggest that cogens are an effective approach for fostering participants’ commitment to creating the conditions for students’ success in the classroom. Within the context of the workshops, cogens provided teachers, students, and faculty with opportunities to engage in authentic conversations for addressing the recruitment and retention problems in computer science for underrepresented students. These conversations often resulted in the development of tangible pedagogical approaches, examples, metaphors, and other strategies to directly address the recruitment and retention of underrepresented students in computer science. Finally, while we are still developing the CRPG-CSCT, cogens provided us with the opportunity to ensure the voices of teachers and students are well represented in and central to the document.more » « less
-
Inprasitha, Maitree ; Changsri, Narumon ; Boonsena, Nisakorn (Ed.)In mathematics education, much research has focused on studying how students think about the equals sign, but equality is just one example of the larger concept of equivalence, which occurs extensively throughout the K-16 mathematics curriculum. Yet research on how students think about broader notions of equivalence is limited. We present a model of students’ thinking that is informed by Sfard’s theories of the Genesis of Mathematical Objects, in which she distinguishes between operational versus structural thinking (e.g., 1995), which we conceptualize as a continuum rather than a binary categorization. Sfard also describes a pseudostructural conception, in which the objects that a student conceptualizes are not the reification of a process. We combine Sfard’s theory with a categorization of the source of students’ definitions, where stipulated definitions are given a priori and can be explicitly consulted when determining whether something fits the definitions, while extracted definitions are constructed from repeated observation of usage (Edwards & Ward, 2004). We combine these theories with inductive coding of data (open-ended questions, multiple-choice questions, and cognitive interviews) collected from thousands of students enrolled in a range of mathematics classes in college in the US, to generate categories of students’ thinking around equivalence. We see this model as a tool for analysing students’ work to better understand how students conceptualize equivalence. With this model we hope to begin a conversation about how students tend to conceptualize equivalence at various levels, as well as the ways in which equivalence is or is not explicitly addressed currently in curricula and instruction, and what consequences this might have for students’ conceptions of equivalence.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
