In pursuit of powerful authentic learning environments in the mathematics classroom and the technology to instigate it
The emphasis on proficiently demonstrating procedural knowledge has been a prominent feature of mathematics classroom for decades. Despite efforts to introduce hands-on manipulative resources and diversified instruction, the prevailing view of success is still measured “in terms of speed, accuracy, and automaticity of performance” which has inadvertently promoted the acquisition of strategies that “lack flexibility and adaptability to new problems” (Hatano, 1984, p. 31). Every year, I inherit a new group of grade 7 students who are determined to reinforce their routine expertise because it is valued in our educational system. Contrary to this common phenomenon, I believe students need to view mathematics as a way of solving real-world problems developing a broader conceptual understanding “that goes beyond the specific situation in which it was first acquired” (Corte, 2007, p. 25). Adaptive expertise enables students to apply what they have learned flexibly and creatively to new situations; however, facilitating the creation of knowledge for understanding requires a shift in pedagogy where the primary focus for learning is process-oriented, not product-dependent. Moving mathematics instruction towards a constructivist perspective cultivates a community of learning and inquiry. It welcomes the idea that learning is enhanced by the social construction of knowledge to help “students be more objective about their own view and realize that even with the same information other people may come to different conclusions or solutions” (Pellegrino & Brophy, 2008, p. 285); therefore, designing classroom experiences around authentic tasks create opportunities for students to make meaningful and relevant connections with their learning and the learning of others.
Students today are immersed in a digital world. With its collaborative and networking potential, technology affords teachers engaging, authentic, and creative ways to design learning environments that are knowledge, learner, community, and assessment-centered. Educators need to embrace technology; a reality that was shared by both the teacher and administrator during my initial interview task that brought issues in math and science education to light, which was somewhat surprising as their ability to currently integrate technology ranges from very hesitant to adventurous, respectively. Opportunities to use technology in math to situate learning and promote multiple pathways for problem-solving are continuing to increase as demonstrated in the development of the Jasper Series by the Cognition and Technology Group at Vanderbilt (Pellegrino & Brophy, 2008). The linear nature of the technology often used to encourage mathematical skills was seen, by one interviewee, as “archaic” and although it might hit outcomes, it was “rigid in the way you could solve problems … it didn’t give you a lot of room for solving it with a different approach.” As this is a reality with many online options designed for supporting math skills, educators need to begin envisioning innovative ways of using technology in the classroom to support learning through social mechanisms” (Pellegrino & Brophy, 2008, p. 285). The goal of integrating technology is not to find an alternate way for students to learn in isolation or be encouraged to develop routine expertise, we need to shift the way we view learning and how technology can be used to support it if we intend to find a viable answer to the question: How can technology be used to create powerful authentic learning environments in the mathematics classroom?
The articles within this annotated bibliography were selected based on their attention to key issues that influence technology use in the classroom and impact teachers’ abilities to harness it to create authentic learning experiences in the mathematics classroom. In addition to CiteULike, online article searches were principally carried out within the UBC Library Collections and Google Scholar using a VPN connection to gain additional access to online databases. Preliminary discoveries led to refined searches within Wiley Online Library, SpringerLink, and JSTOR. In various combinations, search terms included: technology, authentic learning, authenticity, reform, math, learner-centered, curriculum, constructivism, instructional design, elementary, and K-12 education. Where possible, articles published earlier than 2000 were excluded as recent trends regarding technology integration were most desirable.
Nicaise, M., Gibney, T. & Crane, M. (2000). Toward an understanding of authentic learning: student perception of an authentic classroom. Journal of Science Education and Technology, 9(1), p. 79-94, doi: 10.1023/A:1009477008671
In an effort to shift the view of authentic learning from a presumed theoretical viewpoint to one grounded in empirical data, Nicaise, Gibney and Crane’s qualitative research design aims to establish a basis for understanding authentic learning environments from the perspective of students within three secondary elective courses at one large Midwest high school in the US that were based on a single instructional model taught by the same instructors within the same physical space. They shared common learning materials as well as mutual educational goals of situated learning grounded in an authentic task, and were thus amalgamated into one single unit of analysis. Over the course of one academic year (nine months), multiple data collection methods and case study techniques were drawn on to help the researchers develop an understanding of the learning environment and how it was perceived by the 59 students enrolled in them; this included classroom observations, informal interviews, and document collection, and during the final two months of the study, researchers conducted formal interviews with a representational group of 20 students chosen through non-probability sampling. Within a grounded theory framework, data was analyzed using a constant comparative method to conceptualize the data based on categories that emerged and their relationship to each other. From this, researchers developed theoretical models that drove further investigation leading to further theory revisions as data continued to be collected and compared.
This study provides an alternate and necessary perspective regarding the impact of authentic learning environments on students. Although the authors caution against generalizing their findings due to the high proportion of Caucasian males enrolled in the courses, student perceptions revealed valid criticisms of the learning environment that pertain to a wider population of students. Feedback from both successful and unsuccessful students in the courses suggested that their situated learning context could be improved on by implementing “clearer learning objectives and pre-established assessment methods …. elements that should be standard practices in any classroom” (p. 92). Their investigation also uncovered insights that conflicted with teacher perceptions of the instructional design embedded in the courses. The complex culminating event, a week-long mock space shuttle mission, did not stimulate near as much student interest or engagement as the individual and small group projects that offered greater student autonomy, relevance and choice. Many students felt that teachers had directed their learning towards a pre-conceived path limiting opportunities for students to drive the process and several questioned the authenticity of the activities; however, the origins of student perceptions were not adequately investigated to rule out student misconceptions about the structure of situated learning, or any misunderstandings regarding the professional context of the vocations embedded within it. Moreover, when discussing the collection of data, the authors frequently used descriptors such as “many”, “several”, “most”, “a handful”, and “a number” to illustrate the scale of student responses, but without specific values attached to these statements the potential validity of their conclusions is undermined. Despite the student-teacher ratio hovering around 20:1 (considerably less than an average Canadian secondary classroom), students also reported difficulty accessing the teacher due to the high volume of independent projects, which could challenge the viability of integrating this level of situated learning in core curricular courses. Overall, slightly less than one third of the students considered themselves unsuccessful in this alternate environment and with subsequent comparative analysis a, critical element was brought to light that can be a catalyst for authentic learning or constrain it – the need to scaffold students’ transition from a passive to active learning environment. From this article, educators and researchers can begin to better comprehend the complexity of creating authentic learning environments and the importance of considering student perspective and how it can transform practice (p. 82) as well as bridging the gap in inquiry skills and self-regulation that can stem from prior immersion in traditionally teacher-driven classroom practices.
Franklin, T. & Peng, L. (2008). Mobile math: math educators and students engage in mobile learning. Journal of Computing in Higher Education, 20(2), 69-80, doi: 10.1007/s12528-008-9005-0
Technology offers new means of engaging students in their learning within a familiar context as “the digital world is ubiquitous to their very being” (p. 70). When the connectivity and portability offered by mobile devices is introduced into the math classroom, opportunities for new learning emerge that have the potential to meet diverse student needs and improve educational practice. Franklin & Peng’s investigation into using the iPod Touch to create math videos with grade eight students questioned whether or not this mobile technology can be used to improve students’ understanding of process as well as mathematical content within formal and informal settings. This study included 39 students in two classrooms and their 2 respective teachers within a middle school in rural Appalachian Ohio, a predominantly low socio-economic region. Academic potential ranged from low achieving and students receiving special education support to gifted learners. Student-teacher ratio was approximately 20:1, which again is significantly lower than most grade 8 classrooms in Canada. Selection of these two classrooms was based on the two math educators’ previous experience with Palm technology and the researcher’s prior connection to the school faculty and students through previous professional interactions. Twelve hours of hands-on professional development to become familiar with the iPod Touch and iTunes was provided as both educators had no direct previous experience with this technology.
Using case study methodology within the bounded system of the classrooms, eighteen hours of instruction over three weeks was allotted for this investigation; however, this was eventually stretched into four weeks due to unforeseen circumstances and consequences of bad weather. Data was collected from daily student journals, classroom observations, informal conversations with the faculty-at-large as well as peer reviews of student videos and university faculty assessments of the product . Open-ended survey questions and semi-structured interviews were administered to the two math educators and the students, although only a small sample of students was selected for interviewing. This provided the researcher with a broad range of insight into the effectiveness of the iPod Touch technology from within the classroom and school community combined with educators not directly involved or affected by the study; however, it was unclear if the diverse student population was represented within the interviewees because the selection process was not discussed. Overall results of the study indicate a positive correlation between student engagement and the use of iPods in the classroom. Despite initial skepticism, one educator commented on the reflective potential the study had afforded him as the video process provided him with valuable information about how students were thinking and what he needed to teach better. The video production process presented opportunities to think about the mathematical concepts in new ways, explain their thinking, and teach others through their final product, both formally and informally within the classroom and outside of it. Academic achievement was not assessed in this study, so claims of improved understanding cannot be substantiated although the math educators and external university faculty believed there was evidence to support this within the videos. Without knowing the full extent of support provided to students while making the videos, it remains unclear if the videos represented each group member’s independent understanding. Although iTunes was used by student to add music to their videos, the issue of copyright was not specifically addressed. It was unclear if students were encouraged to download licensed content. While it is evident from this article that integrating mobile technology into classroom has the potential to engage students in their learning through all levels of academic abilities, it merits additional attention for its acknowledgment of external factors that can and will affect technology use in schools. Obstacles that surfaced included lack of adequate and willing technology support, the need for ample charging stations, and educator proficiency with the technology being used.
Lei, J. (2010). Quantity versus quality: a new approach to examine the relationship between technology use and student outcomes. British Journal of Educational Technology, 41(3), 455-472, doi:10.1111/j.1467-8535.2009.00961.x
Contending that empirical studies have not been able to substantiate a consistent link between students using technology and improved academic achievement, Lei chose to target “technology-in-context” (p. 458) as opposed to technology in general. Given that the “same technology can be used differently in various contexts” (p. 457), Lei moved his inquiry beyond “how much” or “how frequently” technology is used to investigate the relationship between the quantity and quality of use relative to student outcomes. Using a mixed-methods research approach, data were collected quantitatively through closed-ended question surveys and student report cards as well as qualitatively through interviews. It is not clear if these methods of data collection were carried out sequentially or simultaneously as the author did not reveal when the interviews took place or if interviewees were selected based on completed surveys. Qualitative data was coded and quantified using regression analysis to triangulate links between the variables.
Research was carried out over the course of one academic year involving seventh and eighth grade students and a selection of their teachers at a middle school in the north-western region of the US. Technology resources were abundant in the school, including 1:1 laptops, and student-teacher ratio was considerably low, 9:1, which are both atypical of most schools and limit potential generalizations arising from this study. Out of the total 237 students surveyed, only 133 surveys could be used after incomplete and ineligible surveys were discarded. Outliers were not addressed, so it is unknown if this impacted the subsequent analysis of data. Although only 56% of the surveys were viable, within these results a reasonably balanced representation of the student population by gender and grade was achieved – 48% male, 52% female, 48% seventh grade, and 52% eighth grade. Survey results relied on students self-reporting so verification was not possible. To gain further insight into how technology was used, nine teachers were chosen to be interview based on the subject they taught and associated grade. For perspective on how their learning was impacted by technology, nine students were also selected for the interviewing process based on their interest in using technology to insure a range of viewpoints was established; though, the selection process for this sample was not explained nor did the author indicate whether these 9 individuals were cross-referenced with completed surveys. When the quantity of technology use was isolated against student outcomes, no significant relationship emerged in terms of student GPA or technology proficiency; however, when quality of technology use was examined, significant positive association was discovered in relation to most student outcomes and technology proficiency. Types of technology use affected student outcomes differently, but out of the five categorized uses not one significantly influenced student GPA, which contradicts other research claims. In accordance with Lei’s interpretation of these results, this doesn’t preclude technology from improving learning. In fact, it could be a critical indicator that traditional assessment methods for measuring student achievement compromise the analysis as they do not complement the learning that is being encouraged through technology use. Lei’s findings provoke further thought and encourage future researchers to delve deeper into effective uses of technology (how, what and why it is being used) and alternate assessment methods to accurately evaluate it.
Using technology to create powerful learning environments requires a multi-faceted approach. The articles analyzed here demonstrate the interconnectedness of these elements and how they can support authenticity enhanced by technology. Nicaise, Gibney & Crane (2000) demonstrate how designing authentic learning environments requires a constructivist perspective. That alone will require a shift for some educators as well as students. Teachers and students must have a distributed understanding of authentic learning if the goal is to create the most meaningful experiences, with or without technology. Teacher intention may differ from student perception and transitioning students from a passive transmission model into methods of inquiry, independent learning, and critical thought is a process that needs to be explicitly facilitated. Through the integration of mobile devices into classroom learning, Franklin & Peng (2008) illustrate the great potential technology has to enhance authenticity in the classroom if teachers can adopt a broader view of its capacity for engaging students in the learning process. Emphasis on quality over quantity is essential when making decisions about how technology is to be used and constructivist environments value the learning process, not specifically when, where, or how many times an outcome has been demonstrated. Successful integration also necessitates teachers and pre-service programs address the need for increased technology proficiency for facilitators. As Lei (2010) criticized, the relationship between technology use and student achievement has not been consistently established, yet a positive correlation is presumed to exist. Moving beyond questioning “if” or “how often” technology is used in the classroom towards the more specific purpose of identifying the different ways technology can be used will assist educators and researchers in establishing a more constructive framework for analyzing technology use. Evaluating how technology is used within authentic learning environments requires the development of new assessment models that appreciate the complexity of technology-enhanced learner-centered instructional design rather than trying to measure it using methods that are antiquated in light of educational reform. The issue of current assessment practices and their relevancy to learner-centered environments was also mentioned by Nicaise, Gibney & Crane (2000) where they questioned the authenticity of grades within a situated learning task and wondered if this practice did more to upset the momentum of learning rather than encourage it. This poses an issue in education that demands further research and inquiry: if we are changing the way we teach to encourage authenticity, shouldn’t we also be changing the way we assess?
Further research on the issue of what constitutes authenticity in education would help educators better understand how to create effective learning activities and scaffold students’ transition into them. Future investigations focusing on how technology can be used to create authentic powerful learning environments in the mathematics classroom needs to be conducted to corroborate the findings shared in these separate articles. While Nicaise, Gibney, & Crane and Franklin & Peng investigated innovative educational practice using varying degrees of technology, the low student-teacher ratios within these studies make it difficult to generalize their findings for the average Canadian upper intermediate or secondary classroom. Hopefully, additional research can provide further clarification about how the proliferation of authentic learning tasks, enriched by technology, can be encouraged within larger-scaled mathematics classrooms.
References
Corte, E. (2007). Learning from instruction: The case of mathematics. Learning Inquiry, 1, 119–30. doi: 10.1007/s11519-007-0002-4.
Hatano, G. & Inagaki, K. (1984). Two courses of expertise. Research and Clinical Center for Child Development Annual Report, 6, 27-36. Retrieved from http://eprints2008.lib.hokudai.ac.jp/dspace/bitstream/2115/25206/1/6_P27-36.pdf
Pellegrino, J.W. & Brophy, S. (2008). From cognitive theory to instructional practice: Technology and the evolution of anchored instruction. In Ifenthaler, Pirney-Dunner, & J.M. Spector (Eds.) Understanding models for learning and instruction, New York: Springer Science + Business Media, pp. 277-303.