In their investigation into the effectiveness of computer simulations, Finkelstein et al. (2005) concluded that “the conventional wisdom that students learn more via hands-on experience is not borne out by measures of student performance on assessment of conceptual understanding” due to their findings that “properly designed simulations used in the right contexts can be more effective educational tools than real laboratory equipment, both in developing student facility with real equipment and at fostering student conceptual understanding”. Providing students with opportunities to explore concepts in a variety of contexts enriches the learning environment and diversifies instruction to better meet student needs. In the mathematics lesson below, the concept of fractions is embedded in inquiry-based activities to help students visualize this concept while exploring selected interactive applets. Reflection plays a key role in the intended abstraction as a means of guiding students through a process of self-assessment to better understand their conceptual understanding.
| Illuminations Lesson for Fractions (click link above for lesson) |
Rationale
By Grade 7, students are expected to have a basic, but solid, understanding of fractions so that they can proceed with more in depth explorations of relationship comparisons and eventually addition and subtraction operations. Unfortunately, this is often not the case and measures need to be taken to assist students’ conceptualization of fractions in preparation of their extensive use in math strands in successive grades. Much of the problem seems to lay with students’ misconception and simplification of fractions down to sets of rules to be committed to memory creating inaccurate mental models. It’s not surprising then that they are frequently unable to adapt strategies they have used in one context to fit a new situation. Their knowledge of fractions remains superficial and does not lead them to a deeper understanding of what fraction symbols communicate as a representation of a whole. When learning abstract concepts, such as fractions, students must understand the fallacy of focusing on memorization as it “leads to ‘inert knowledge’ that cannot be called upon when it’s useful” (as cited in Edelson, 2001) resulting in a poor or non-existent transfer of skills.
Using Illuminations activities provides students with a “variety of visual cues in the computer simulations [to] make concepts visible that are otherwise invisible” (Finkelstein et al., 2005) or at least more difficult to visualize. When integrated into an inquiry-based framework, they can be used to enhance students’ abstraction of fraction concepts while promoting the acquisition of adaptive expertise and thinking skills.
Intertwining the constructivist principles of the Learning for Use framework and T-GEM instructional model provides and impressive foundation for math explorations. The GEM cycle stages of Generate – Evaluate – Modify are complemented by the 6 tenets of LfU, motivate, elicit curiosity, observations, knowledge construction, refine and apply. While collecting information and generating ideas, curiosity and motivation are provoked as students realize what they do not yet know, but need to in order to be able to complete the task. Through key observations, students construct knowledge as they begin to evaluate their assumptions around relationships between variables. As students work to modify their original theories, they need to refine and apply new understandings that have arisen from their investigation. Applications of the LfU and T-GEM frameworks to instructional design presume that overlaps in each of the stages will occur as they both involve a cyclical process of exploration and inquiry. In fact, several cycles may be needed due to the incremental nature of learning; however, the order of the stages remains a critical factor. In the lesson outlined above, two complete cycles of T-GEM and LfU can be observed.
British Columbia Grade 7 Math Learning Outcome (Number – A7)
- compare and order positive fractions, positive decimals (to thousandths) and whole numbers by using
- benchmarks
- place value
- equivalent fractions and/or decimals
Comparing percent to fractions and decimals is a Grade 6 outcome, but by Grade 7 this is consistently not understood well so it needs to be re-taught in preparation for Grade 8 expectations with percent (greater than 100% and fractions of percent between 0 and 1) providing students with a more substantive opportunity to understand the overriding relationships between all three values; therefore, in this activity, this Grade 6 learning outcome will be reinforced as an integral component of the task.
Grade 6 Math Learning Outcome (A6): demonstrate an understanding of percent (limited to whole numbers) concretely, pictorially, and symbolically.
Before beginning lessons involving self-directed exploration of Illuminations activities, students must possess sufficient background knowledge to prepare them for success with the simulation activity. If the expectations for student learning are high given their current context, they will have difficulty navigating the activity (Kalyuga in Srinivasan, S. et al, 2006) and finding the necessary motivation to learn what they need to know to be successful. In this scenario, essential prior knowledge includes an understanding of:
- numerator, denominator, common/proper fraction, improper fraction, mixed number, whole number, simplest form, equivalent fractions, multiple, factor, benchmark fractions/decimals/percents, addition equations equaling 1 whole, decimal place value (tenths, hundredths), parts of one, relating fractions to decimal place value, percent
Along with having prior knowledge, students must be able to access and activate it; therefore, the initial introductory task is intended as a revision of fraction concepts and relationships, which become essential elements, within the subsequent scaffolded activities.
(This post serves as further reflection on the application of knowledge representation and information visualization as it applies to my future personal practice and includes the alternate activity requested in lieu of directly related papers on the use of Illuminations)
References
Edelson, D.C. (2001). Learning-for-use: A framework for the design of technology-supported inquiry activities. Journal of Research in Science Teaching,38(3), 355-385. http://onlinelibrary.wiley.com/doi/10.1002/1098-2736%28200103%2938:3%3C355::AID-TEA1010%3E3.0.CO;2-M/abstract
Finkelstein, N.D., Perkins, K.K., Adams, W., Kohl, P., & Podolefsky, N. (2005). When learning about the real world is better done virtually: A study of substituting computer simulations for laboratory equipment. Physics Education Research,1(1), 1-8. Retrieved April 02, 2006, from: http://phet.colorado.edu/web-pages/research.html
Khan, S. (2007). Model-based inquiries in chemistry. Science Education, 91(6), 877-905.
Srinivasan, S., Perez, L., Palmer, R., Brooks, D., Wilson, K. & Fowler, D. (2006). Reality versus simulation. Journal of Science Education and Technology, 15(2), 1-5
The most prevalent obstacle that impedes inquiry-based learning in educational settings is the instructor’s understanding of inquiry and pedagogical approaches as well as the ability to implement these successfully. This was shared through the expressed frustrations of the Jasper Series designers when teachers did not seem to recognize the value in exposing students to analog problems that were conceived for the purpose of improving transfer and abstraction of concepts and strategies, opting instead for adventures that introduced the need to use different skills overlooking the opportunity to increase adaptive expertise (Hatano, 1984). Within the WISE environment, customizing the platform for successful inquiry-based learning requires a level of competence that designers cannot necessarily assume teachers possess. The inquiry map alone, which directs students through the process, can present a significant challenge in that even Linn, Clark & Slotta (2003) caution that its level of detail affects student engagement. The prescriptive nature of WISE projects provide students with the necessary information to proceed independently, but also provide opportunities for teachers to misinterpret the structure of the investigation. Manipulating the available scaffolding steps along with the limited opportunities for socially constructing knowledge embedded within WISE provide a potential recipe for reinforcing the transmission model, albeit with animations and the technological affordances of accessing past progress. While the Jasper Series was founded on stronger pedagogical principles that provide valuable insight into TELEs and continue to describe essential qualities of powerful and effective learning environments, both it and WISE promote more of a packaged approach to inquiry that does not require teachers to explicitly understand the theory and pedagogy behind them before integrating them. As potent as they could be in bringing inquiry-based learning to the classroom, they could also be used to further entrench traditional instructional approaches that reinforce inert knowledge. It cannot be assumed that teachers possess the aptitude to integrate these TELEs. Just as students require explicit instruction to develop inquiry skills, teachers need to be “explicitly taught about interactions among pedagogy, content, technology, and learners” to develop their Technological Pedagogical Content Knowledge, or TPCK. This conceptualization is critical.
Anchored instruction in the Jasper Series, WISE’s scaffolded knowledge integration framework (SKI), the Learning for Use model when applied to My World, and applying the T-GEM cycle to Chemland explorations showcase the application of pedagogical design in response to ongoing research regarding effective technology-enhanced learning experiences (TELE) in mathematics and science classrooms. All four TELEs are driven by documented discrepancies between theoretical best practice and actual instructional approaches in all levels of education. Although varied in their application, each design is grounded in constructivist principles that focus on inquiry-based learning, mental models, socially constructed knowledge, and reflective conceptualization aimed at integrating both content and process outcomes of science or mathematics education. Reasons for pursuing this common pedagogical design are rooted in substantive conclusions of researchers who assert that “inquiry is associated with an array of positive student outcomes, such as growth in conceptual understanding, increased understanding of the nature of science, and development of research skills” (Khan, 2007, p.877). To achieve this authenticity within TELEs the design must be nourished by activities that “provide the opportunity to ground abstract understanding in concrete experience” (Edelson, 2001, p. 378). Reforming science and mathematics requires a pedagogical shift away from the passive “transmission approach [which] does not acknowledge the importance of the motivation and refinement stages of learning and relies too strongly on communication to support knowledge construction” (Edelson, 2001, p. 377).
The 
Concepts in Earth Science can be challenging for students to grasp as real examples relating to curricular content are often difficult or simply cannot be brought into the classroom to provide students with first-hand experience with them. Size and accessibility are factors which compromise students’ abilities to form mental models that accurately reflect scale, so comparative models are often relied on in place of actual phenomena or their processes. For students to conceptualize these appropriately, spatial-thinking and scale must be understood which requires abstract reasoning that teachers cannot presume is already present. Lack of opportunities to collect first-hand data presents an additional problem, which results in an over-reliance on data banks that detract from the authentic mirroring of processes within the scientific community.