Education as a Force for Equality

The title of this week’s blog is a quote from James Paul Gee’s book Anti-Education Era (2013, preface par 34) .  The title, however, is not enough for you to glean the importance and to follow my train of thought so I am giving you the full quote and will be referring back to it as this post continues.

We have forgotten education as a force for equality in the sense of making everyone count and enabling everyone to fully participate in our society.  We have forgotten education as a force for drawing out of each of us our best selves in the service of an intellectually and morally good life and good society. (Gee 2013, preface par 34)

This week’s challenge in CEP811 is to redesign the activity we designed in week 2 and then added to in week 3 , to ensure it is a force for equality and that everyone can count and participate.  To accomplish this goal we were challenged to use the Universal Design for Learning guidelines (CAST, 2011a) to ensure our activity allows the most number of students to be successful. When students with special needs were integrated into classrooms, the students with special needs were expected to adapt to the regular classroom.  However, this was often not successful and many students not identified with special needs were also not successful.  Universal Design for Learning (UDL) focuses on how making changes to how the curriculum is taught can benefit all students(CAST, 2011a).  Here is a quick video to help you get the basics of UDL before we move on.

When using UDL to plan a lesson you must focus on three areas, the What, the Why, and the How of learning(CAST, 2012).  It talks about how you can you have more students be successful, by accessing more areas of the brain.  But in looking at the info graphic that follows (CAST, 2012), I was a little lost as to where to begin.  How could I provide multiple means of representation, of action and expression and of engagement.  What did that exactly mean?

brain UDLLuckily enough CAST provided me another chart that highlighted what they meant by each category.  I have provided it for you here as well (CAST, 2008).

UDL guidelinesI especially liked the fact that at the bottom(CAST, 2008), they provided a reason for what implementing those changes could do for students, a why this should matter for teachers.  I find it is always easier to look at change when you realize the benefits that come from it.  If we have students who are resourceful, knowledgeable, strategic, goal-directed, purposeful, motivated learners then we have learners that are capable of “drawing out of each of [them their] best selves in the service of an intellectually and morally good life and good society”(Gee, 2013).  In that vein education would really become the force for equality that Gee envisions.

While this new chart provided some direction in what I should be trying to change in my lesson plan it left me wonder what exactly these changes would look like.  At this point I dove into the UDL Guidelines (CAST, 2011a).  While this document is 36 pages long it takes you through each section.  It not only provides an explanation of why this change is important and which students it could benefit, but provides you with examples to help you visualize what the change could look like in a classroom.  I used this guide to help me fill out a google doc (CAST, 2011b) that is set up to make sure you look at all the sections (you can access your own copy here).

Before I talk about what changes I made to my activity, I want to give one caveat.  While, I believe this process is important in order to improve our education system, it is not a process I would use for a single activity.  This is a process I would use when planning out a unit, to ensure that the unit was a all encompassing as it could be.  While I will keep in mind the guidelines as I plan future activities, there is no possible way I could plan every single activity in this much detail.  Honestly, I spent four hours working on filling out the google doc due to the reading, thinking, reflecting on the activity as it is and then deciding on what could change.  While I believe I will get faster at it with practice, expecting a teacher to do this for every activity individually will lead to more teachers leaving the profession due to burnout.  That being said, I believe that every unit should be planned with UDL to ensure that we as teachers are working towards creating resourceful, knowledgeable, strategic, goal-directed, purposeful, motivated learners.

UDL Redesign of my Cartesian Plane Activity

In week 2’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.

grade 7 program of studies

In Week 3’s blog  I walked 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.  I have reprinted the whole activity from Week 3 and I will be adding in the new UDL pieces in purple, blue, and green to match the section to which the changes relate.  I will only comment on the link to UDL the first time it occurs in the activity , in order to reduce the redundancy that I have built into the activity.

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 idea of having two choices for the activity ties into section 7.1 to allow them some individual choice.   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. The fact that this activity is a physical version of a previous task it pairs the written and physical to increase retention as talked about in section 3.4.  Additionally, if a student with visual difficulties was paired with a student to scribe their colours or to video tape their effort, a student with visual difficulties could use the fact that that the LEDs are raised off the board in much the same way that they use Braille as was talked about in section 1.3. Video taping responses would also tie into section 4.1.  This video taping could also be used with students who have difficulties turning their thoughts into written work, or just as a way choice for students who desire a new way to record their answers such as was talked about in section 5.1.

photo (4)

Activity One

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. Looking at section 1.1, I need to ensure the size and the font choice make the cards easy to read for a wider range of students.   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.  Looking more closely obstacles to this activity and the ideas of section 1.1, I need to choose either red or green LEDs as using both will cause issues for any students with red/green colour blindness.  In section 1.2, I realized that I need to have the instructions available to students in three ways.  They need to be able to review instructions in auditory form, in demonstration form, and in step by step instruction form with pictures that capture the meaning of the words in question.  Having pictures with step by step instructions also ties into section 3.3.If I ensure that the reading level is low enough to be understood by my English Language Learners and other students with reading difficulties this ties into section 2.4.  Having step by step instructions, with prompts to help students continue helps to support planning and strategic development discussed in section 6.2.

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 lights are arranged in a way that over time through exploration students will see patterns as discussed in section 3.2. Having the lights arranged so that specific skill feedback can be given helps to support the monitoring of progress discussed in section 6.4, as well as the mastery feedback discussed in section 8.4.

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. By having the students choose which questions they complete, and allowing multiple opportunities to work with the activity, students would be able to optimize their challenge in reference to section 8.2.  By having the questions increase in difficulty it helps to build fluencies as discussed in section 5.3.

Activity Two

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.  Having the ability to make choices is the idea behind section 7.1.

Activity Three

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.

Part A

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.

Part B

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).

Part C

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.

What I have added to my activity due to UDL

I was quite surprised over all how many sections my initial activity covered.  One thing that I think is lacking in my activity, is a pre/post survey to go along with the activity.  This survey will help students to capture what they feel they know well, what they are unsure of and what they feel they do not know at all.  This ties into section 6.1 on goal setting, section 6.4 on capacity to monitor progress.  With a post survey that also highlights the same skills in the same ways students will be able to see growth and know what exactly they have learned through the activity.  This would tie into section 8.1 and 9.3.  I am even more excited to use this activty with my students this year.  As school starts next week, I do not have long to wait. 

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

Bruner, J. (1966). Towards a Theory of Instruction. Cambridge: Harvard University Press.

CAST (2008). Universal design for learning guidelines version 1.0. Wakefield, MA Retrieved from

CAST (2011a).  Universal Design for Learning Guidelines version 2.0 Wakefield, MA Retrieved from

CAST (2011b) UDL Guidelines-Educator’s Worksheet [Google Doc] Retrieved from

CAST (2012). About UDL. Retrieved from CAST:

Confucius. (n.d.). Retrieved from

Culatta, R. (2013, January 10). Reimagining Learning: Richard Culatta at TEDxBeaconStreet. [Video File]TEDxTalks. Retrieved from

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

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.

UDLCAST (2010, January 6) UDL at a Glance [Video file] Retrieved from YouTube

University of Rochester EurekaAlert. (2007, July 28). Hand Gestures Dramatically Improve Learning. Retrieved from ScienceDaily:

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

Watson, A. (2004). Affordances, Constraints, and Attunements in Mathematical Activity. Research in Mathematics Education, 6(1), 23-34. doi:10.1080/14794800008520128


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