In last week’s blog I repurposed the game Stratego and my Squishy Circuit makers’ kit to create an activity centre that relates to the outcomes in the Grade 7 Mathematics Program of studies for the Province of Alberta. Specifically I am going to link this activity centre to Specific Outcome 4-Achievment Indicators 2 and 3; and Specific Outcome 5-Achievment Indicators 2, 3, and 4.
This week I am going to walk you through the learning theories that support the use of this activity centre for student development, what some possible uses for this game board are, including the affordances and constraints (Watson, 2004) this activity center offers.
Squishy Circuits, Stratego, and Embodied Mathematics
Everything we learn, including what we learn about mathematics, is learned through our experiences in the world, or related back to previous experiences. The English language is full of metaphors that map one domain of experience onto another. For example, I received a warm hello, maps the domain of temperature onto the domain of social interaction, the hello isn’t warm in temperature but positive feelings are warm because of their relationship to being warm when held. Feeling being perceived as warm could also be due to activation of neural circuits associated with warmth. Mathematics, like the English language, also uses conceptual metaphors to link understanding of concepts to already known understandings (Lakoff & Núñez, 2000). Take the number line, for example, numbers do not actually exist in a line, however, thinking of them in this manners makes some of the more advanced mathematics easier to understand. The arithmetic grounding metaphor of motion along a line allows understanding of positive and negative integers even though numbers do not occur in lines.
This embodiment of mathematics does not just extend to the actually doing of something but also to using the body to highlight what your mind is thinking about while doing mathematics. A study out of University of Rochester conducted by Susan Wagner Cook (University of Rochester EurekaAlert, 2007) looked at the use of gestures in teaching. Her study looked at teaching the same lesson using speech cues, using speech and gesture cues, and just using gesture cues. The retention of students with gestures alone was ninety percent as opposed to only thirty-three percent from the group with speech cues alone. Interesting to note was that the group who were taught with gestures alone had retention of ninety percent as well. This seems to reinforce the idea of Confucius; “I hear and I forget. I see and I remember. I do and I understand. (n.d.)”
While this activity centre will not be the first exposure students will have to these concepts, it will be a new example of the concept to help to occasion a “firm foundation of factual knowledge” (Bransford, Brown, & Cocking, 2000, p. 20). . Students learn best when there is enough redundancy that a pattern emerges so that they can construct and generalize their own pattern. It encourages capability not specific ability (Ernest, 2004). It is also important that students are the doers in their quest for understanding and the activities need to take into consideration what makes each learner unique (Bransford, Brown, & Cocking, 2000). By having students use the wand them are actively locating the points and during activity three they are physically embodying the tranformations in the movement.
Game Board Information
The game board has two sides, one red, and one blue. I punched holes into it to create a 4 quadrant Cartesian plane as in grade 7 students need to work with in all four quadrants. I marked the x and y axes in the center of the board respectively so that each quadrant consists of four lights horizontally and three lights vertically, not including the lights along the axes. I have not numbered the axes for two reasons. The first reason is by not numbering the axes it is possible to use the game board from both directions affording me the opportunity to create two versions of the same activity. The second reason that I have not numbered the board as this creates the opportunity for students to figure out how the axes would need to be numbered based on their present frame of reference, either from the red or blue side.
Activity One is based on Specific Outcome 4- Achievement Indicator 3. This achievement indicator says that students need to be able to identify a point given its coordinates in any of the four quadrants. Activity One and Activity Two are not sequential and may be done in the reverse order. In Activity One, I will create two sets of cards (one for the blue orientation and one for the red orientation, that give students a coordinate pair that corresponds to a single LED on the board. Students will need to locate that specific LED and touch it with the wand so that the Squishy Circuit is completed and the LED lights up. Students will then record the colour of the LED at that location.
The LEDs are arranged in such a fashion to help me provide feedback as to what challenges students are having. That is to say, none of the other possible errors have the same colour LEDs so that I can determine if they are having trouble with the direction of movement along the x axis, the direction of movement along the y axis, or direction of movement along both axes.
The cards will start with points in the first quadrant to tie the new learning back to their understanding of graphing in quadrant I from grade 6. It is important that we tie learning back to prior knowledge in order to help students connect new learning to old conceptual understandings. (Pirie & Kieren, 1994). Moreover, the difficulty of the cards will increase from points in the first quadrant, to points in the other three quadrants, to points along the axes. By increasing difficulty as the student progresses, the student’s learning is scaffolded (Vygotsky, 1978) by allowing students to build their confidence before tackling more difficult questions.
Activity Two is based on Specific Outcome 4 –Achievement indicator 2. This achievement indicator involves students identifying the location of a given point. In this activity students will locate a LED of a specific colour and record the location of the LED using an integral ordered pair. By allowing the students to choose which of the many lights of a specific colour to identify, it allows students to have agency (Gee, 2005) in this activity. Agency is feeling like you have the power to accomplish your goals. The proceeding principles help students to foster this feeling of agency. Agency brings motivation to achieve more as the belief that success is possible is there (Walshaw, 2001). Gee (2013) also talks about providing ways for people to feel that sense of agency in what they are doing to allow them to use digital tools smartly.
Activity Three is linear as it build on the knowledge of Activity One and Two. This activity focuses on the Specific Outcome 5- Achievement Indicators two, three, and four. It pushes students to inventise (Pirie & Kieren, 1994) their understanding of coordinate geometry in new ways that occasion the possibility of pushing the zone of proximal development (Vygotsky, 1978)for each student, thus deepening the understanding of the mathematical concept. While this activity meets all the criteria for achievement indicators two and four, it only meets the initial criteria for achievement outcome three as it focuses on a single point instead of a 2D shape. In this activity student begin to investigate the transformational concepts of translation and reflection. The concept of rotation is not included as while it is possible, rotation of the objects is more difficult than could be attempted by students alone.
Students will be asked to choose pairs of LEDs and then determine the horizontal and vertical distance between them. This allows students agency to be able to choose a pair of LEDs that they feel they will be successful at obtaining the distance between them.
Students choose an LED to start at and then select a card that states a translation to perform. Students use the want to navigate the grid and then record the location of the location of the LED after translating the wand tip according to the card selected. As the starting point of the LED translation is random some of the translations given will move the students off the grid. Students need to be reassured that that could happen and encouraged to write why the translation is impossible on the current grid instead of the final location. If students can extrapolate the location of the point that is off the grid, that should also be encouraged as it shows greater facility with the understanding of a Cartesian plane and moves their understanding from Enactive to more Symbolic in nature (Bruner, 1966).
Students again choose an LED to start, and then using a MIRA students reflect the point across the y axis, x axis, the line y=x and the line y=-x. These choices are arranged in difficulty from least to greatest and the progression will be student dependent. Once students understand how the reflection works using the MIRA they will be encourage to replicate that understanding without using the MIRA.
NEW approach to Cartesian Geometry
For the last five years, I have been teaching this unit using dot-to-dot puzzles that the students have told me they enjoy. In the past, my idea of integrating technology into this unit took the form of computer versions of the same activity or computer games built on the same premise. After watching the TED talk by Richard Culatta (2013), I feel that there needs to be new versions of activities created by leveraging the power of technology not just digital version of the old ones. This ties into the video by Mishra and Koehler (2008)in which they talk about how creativity makes things NEW all in capitals which stands for ideas that are Novel, Effective, and Whole. I feel that this activity is certainly novel, which I hope will spark motivation in my students. Play testing on my family showed that it has the potential to be effective. This activity also meets the definition of whole in that the technology is an integral part of the activity and not just an add-on. The activity would not be complete without the Squishy Circuits, nor the Squishy Circuit lights without the activity to give it a reason to be useful.
Bransford, J., Brown, A., & Cocking, R. (Eds.). (2000). How People Learn: Brain, Mind, Experience, and School: Expanded Edition. Washington D.C.: NAtional Adademy Press. Retrieved from http://www.nap.edu/openbook.php?isbn=0309070368
Bruner, J. (1966). Towards a Theory of Instruction. Cambridge: Harvard University Press.
Confucius. (n.d.). Retrieved from http://www.goodreads.com/quotes/3213-i-hear-and-i-forget-i-see-and-i-remember
Culatta, R. (2013, January 10). Reimagining Learning: Richard Culatta at TEDxBeaconStreet. [Video File]TEDxTalks. Retrieved from http://www.youtube.com/watch?feature=player_embedded&v=Z0uAuonMXrg
Ernest, P. (2004). Postmodernism and the subject of mathematics. In M. Walshaw (Ed.), Mathematics education within the postmodern (pp. 15-34). Charlotte, N.C.: Information Age.
Gee, J. P. (2005). Good Video Games and Good Learning. Phi Kappa Phi Forum, 85(2), 33-37.
Gee, J. P. (2013). The Anti-Education Era: Creating Smarter Students Through Digital Learning (IBooks ed.). New York, NY: Palgrave Macmillan.
Koehler, M., & Mishra, P. (2008). Teaching Creatively: Teachers as Designers of Technology, Content and Pedagogy. [Video File] SITE 2008 conference. Las Vegas. Retrieved from http://vimeo.com/39539571
Lakoff, G., & Núñez, R. (2000). Where Mathematics Comes From How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Pirie, S., & Kieren, T. (1994). Growth in Mathematical Understanding: How can we Characterise it and How Can We Represent It? Educational Studies in Mathematics, 26, 165-190.
University of Rochester EurekaAlert. (2007, July 28). Hand Gestures Dramatically Improve Learning. Retrieved from ScienceDaily: http://www.sciencedaily.com/releases/2007/07/070725105957.htm
Vygotsky, L. (1978). Interactions between Learning and Development. In Mind In Society (M. Cole, Trans., pp. 79-91). Cambridge, MA: Harvard University Press.
Walshaw, M. (2001). A Foucauldian Gaze on Gender Research: What Do You Do When Confronted with the Tunnel. Journal for Research in Mathematics Education, 32(5), 471-492. Retrieved from http://www.jstor.org/stable/749802
Watson, A. (2004). Affordances, Constraints, and Attunements in Mathematical Activity. Research in Mathematics Education, 6(1), 23-34. doi:10.1080/14794800008520128