Resources To Accelerate High Algebra Achievement

I ran into an LDT alumnus named Dan Gilbert on Saturday who helped facilitate Stanford's GSE Energizer, and he asked me a great question, 

'My son is high achieving in algebra, what resources do you recommend for him?'

Depending on the learner, there are a plethora of options out there. So a more specific request or richer context might produce better resources. However, I will do my best here to answer the general question,

What resources are available for a high algebra achiever to accelerate their growth in understanding?

This is relevant to parents and educators alike. This is a draft list, and I would love your feedback on it as well as additions you might suggest in the comments.

1. POW (Problems of the Day/Week/Month #Problemoftheweek)

GOOD for motivated learners who want a challenge.
BAD for reluctant learners who would rather do something else.

  

Find problem banks
    https://www.nctm.org/pows/ (NCTM)
    http://mathforum.org/library/problems/ (List)
    http://www.moems.org/zinger.htm (Middle School)

    http://pleacher.com/handley/probweek/ (teacher archive)
    http://mathcounts.org/resources/problem-of-the-week
    http://krazydad.com/ (If you like Sudoku/Ken-ken...)
    http://www.math.purdue.edu/pow/ (Universities have higher difficulty)

I really liked the NCTM Calendar problems when I was ~7th grade. Unfortunately, they require NCTM membership to access. I also think that my math club in high school, involvement in tournaments, and acadeca helped to support my mathematical development. Below I list two of the current social options to support learning (Communities and Competitions) as well as two newer supports (Practice Engines and Games).

2. Communities (Clubs, Makerspaces, MOOCs, social media...)             

GOOD for social learners whose experience a supportive environment.
BAD when the learner and community have different goals or there is insufficient support.

Find communities
    School clubs (Academic decathlon, math, robotics,
    cryptography...)
    Hacker spaces
    Makerspaces
    Afterschool programs
    Reddit, youtube or other forums
    MOOCS (coursera, novoed, edx, udacity, udemy,
    iTunesU, MIT OCW...)
    Robotics clubs
    Coding clubs or programs
    Math Circles - http://www.mathcircles.org/
    Cyber Patriots - https://www.uscyberpatriot.org/
    MESA - http://mesa.ucop.edu/

3. Competitions (AMC, Mathcounts, University competitions...)

GOOD for motivated and competitive learners who want to push their limits.
BAD for social learners who dislike competition or underachievers who need more support.

Find competitions
    http://en.wikipedia.org/wiki/List_of_mathematics_competitions
    http://www.artofproblemsolving.com/Wiki/index.php/List_of_United_States_middle_school_mathematics_competitions (Or high school if you'd like)
Look at past exams to determine difficulty
    http://www.artofproblemsolving.com/Wiki/index.php/AMC_Problems_and_Solutions
    http://mathcounts.org/past-competitions
    http://www.moems.org/sample.htm
    https://sumo.stanford.edu/smt/
    http://www.imo-official.org/problems.aspx

I participated in the AHSME and a few local university competitions while in high school and enjoyed being part of a team in many of the school competitions. The next categories are things I wish existed when I was a kid, and areas where innovation is creating opportunity.

4. Practice Engines (Khan, IXL, Alcumus...)

GOOD for tracking and individualization of procedural skill practice.
BAD for conceptual understanding and higher order thinking skills.

Find Practice engines

    http://www.artofproblemsolving.com/liz/Alcumus/index.php (Art of Problem Solving is a much larger effort addressing the overall question)

    https://www.khanacademy.org has mastery missions and can be quite fun

    http://www.sumdog.com/ (Gamified practice elementary skills)
    https://www.sokikom.com/ (Gamified practice elementary skills)
    IXL, ALEKS, EnableMath, Pearson MyLab are paid products (Tons of these...)

I really hope Duolingo make a move into the math problem space, because what they did for language learning is admirable and perhaps applicable.

5. Virtual Games

GOOD if the game is well designed (intrinsic integration) and aligns with learner goals.
BAD if the game is poorly designed (Extrinsic integration) or doesn't align with goals.

 

Find Games
FREE
http://www.wuzzit-trouble.com/ (My favorite)
http://www.mathsgames.com/fraction-games_refraction.html (Fraction operations)
https://www.mathbreakers.com/

PAY
http://www.dragonboxapp.com/
http://motionmathgames.com/parents/
http://www.greenglobs.net/

Find Simulations and Interactives
https://teacher.desmos.com/
http://nlvm.usu.edu/en/nav/vlibrary.html
http://illuminations.nctm.org/Games-Puzzles.aspx


As you can imagine, this only scratches the surface. My hope is that I have uncovered some nuggets to get you started. For each broader category you can follow-up with your own search (google, blogstalking, twitter, forumcrawling...) and find an infinitude of cognitive overload. If you come across something great, please share it!

Thank you for your work.

Peace be with you,
Evan

Free Common Core Curricula

The CCSS are a set of desired results, not a curriculum. However, we still need strong curricular materials if we are to deliver instruction that will allow our students to exceed the standards.

I have been pleasantly surprised by the material developed in

New York, https://www.engageny.org/common-core-curriculum 

Exploratory introduction to new concepts from eNY

and Utah, http://www.mathematicsvisionproject.org/

Practice Understanding Task from MVP, which follows teaching and learning cycles

Are there some I have missed? Where are the other states at?

Enjoy work,
Evan

The Chocolate Head Space Promotes Recognition of “Human Ingenuity” through “Play and Work”

The instruction of Aleta Hayes and the Chocolate Heads Movement Band (Chocolate Heads) increased my confidence to learn. This community encourages public exclamations of recognition, or what Erving Goffman refers to as “Response Cries”.

Just as most public arrangements oblige and induce us to be silent, and many other arrangements to talk, so a third set allows and obliges us momentarily to open up our thoughts and feelings and ourselves, through sound, to whoever is present. Response cries, do not mark a flooding of emotion outward, but a flooding of relevance in.

- Erving Goffman, “Response Cries.”(1978, p. 814) [Formatting added]

I sit in the Graduate Community Center laughing out loud while reading, “chuckling aloud to ourselves in response to what we are reading is suspect; this can imply that we are too freely immersed in the printed scene to retain dissociated concern for the scene in which our reading occurs.” (Goffman p. 791) I make more “exclamatory interjections” (Goffman p. 800) since becoming a Chocolate Head and I feel less obliged to be silent because expressive outbursts are encouraged and appreciated among us. I approach life with more of a willingness to learn and celebrate my mistakes in hopes of inspiring others to join us. Aleta Hayes builds a company in which every artist (dancer, musician, lyricist, technologist, director…) plays a part in the overall construction of the piece. The style of collaboration used to choreograph our movement is daringly complex, and wonderfully simple. It cultivates confidence, learning to learn, and creative construction in a superior manner than I have encountered in traditional learning environments.

Ray McDermott and Jason Raley made the claim that, “the social world is built by people working together, and by their work we can know them.” (2011, p. 388) and John Dewey warns that, “It is not enough just to introduce plays and games, hand work and manual exercises. Everything depends upon the way in which they are employed.” (1916, p. 3) There are myriad activities that Aleta Hayes has our group perform, and this analysis is an attempt to connect my learning in the Chocolate Heads Space to Dewey’s ideas around “Play and Work in the Curriculum” (1916) and McDermott & Raley’s insights around “Human Ingenuity” (2011). In their piece entitled, “Looking Closely: Toward a Natural History of Human Ingenuity”, McDermott & Raley explain, “To fashion a natural history approach, we state a problem, make a claim, and promise a better way to proceed.” (p. 374)

Problem Statement

Often when teams are working together, there is a lack of trust needed to have effective collaborative interaction. It is easy to doubt oneself or others. It is difficult to allow oneself to be vulnerable. Teacher-student interactions are a prime example of work between collaborators who need to develop trust. McDermott and Raley describe the current extent of that relationship (p. 381):

From the child’s point of view, the day is spent arranging to not get caught not knowing something and/or getting caught knowing something at just the right time. From the teacher’s point of view, the day is spent finding failure while, at the same time, preaching the availability of success for all and trying not to degrade those who look less able.”

As far as ingenuity is concerned, McDermott and Raley assert that “official school environments either make ingenuity appear scarce, make ingenuity a refugee phenomenon, or bend the purposes of ingenuity toward the pursuit of being seen being able.” (p. 387) How might we increase the level of professional trust between collaborators, and will they perform with more human ingenuity once it is raised?

Claim

People want to build trust and be a source of ingenuity. People enjoy spending time with their friends, family, and acquaintances because it is more comfortable to be around others with whom we share reciprocal trust. We often behave more playfully in trusting settings. “Persons who play are not just doing something (pure physical movement); they are trying to do or effect something“ (Dewey, p. 10). Work and play are not disjoint.

When fairly remote results of a definite character are foreseen and enlist persistent effort for their accomplishment, play passes into work. Like play, it signifies purposeful activity and differs not in that activity is subordinated to an external result, but in the fact that a longer course of activity is occasioned by the idea of a result.  (Dewey, pp. 11-12)

Promise

There are better ways to develop trusting relationships in learning environments. Great teachers organize events to scrutinize people’s activities and gain what McDermott & Raley describe as ability to, “see accomplishments, critiques, and frustrations where others have seen only disorder and stupidity.” (p.375) Aleta Hayes and the Chocolate Heads incorporate methods that approach this promise and arrange events to promote human ingenuity. “Given the materials and persons and moments at hand, what a person does is ‘ingenious’ if it transforms those materials into something interesting, fun, or new.” (McDermott & Raley, p. 387)

Observation

Collaboration plays a pivotal role for dancers. If each individual movement doesn’t synch with the whole group, the effect can be emotionally discordant. However, when the movements are harmonious, the resulting expression becomes sublime. There is a lot of pressure on choreographers because dancers need rehearsal time to perfect a performance. This time constraint causes many choreographers to develop whole pieces before working with the dancers, and leaves little time to make adjustments based on the dancers’ abilities. However, that culture divides “dancers” and “choreographers”. This division makes the role of the dancer mimicry and memorization. Subsequently, most dancers aren’t allowed the freedom to contribute to or alter the performance.

The notion that a pupil operating with such material will somehow absorb the intelligence that went originally to its shaping is fallacious. Only by starting with crude material and subjecting it to purposeful handling will he gain the intelligence embodied in finished material. (Dewey p. 5)

Aleta agrees. The resulting choreography, music, and visual design of the Chocolate Heads Movement Band become an amalgam of the “crude material” that each individual’s particular talents helps mold. They develop individual parts that change with time. They mix and combine different elements from each artist's part to build whole movements. There is an overarching structure of a story, but it is loosely constructed and freely changes day to day. Aleta emphasizes the feeling of a movement over the idea of a narrative. Dancers practice and iterate on their choreography while the instructor circulates the floor and critiques each routine. Aleta recognizes what looks good; and when she sees it, she has dancers teach each other the appropriate way to perform a movement. It constantly develops through this “purposeful handling” as an artistic production being orchestrated by a great teacher. It begs the question, what is the underlying mechanism that allows Aleta to create such a creative culture among artists who have often just met?

Of course there are numerous confounds: Stanford students are motivated, artists and dancers self-identify as creative, the Bay Area is full of unrealistic optimism and positivity, but there is something different about the Chocolate Heads even when compared to other dance groups on Stanford’s campus.

Fig. 1

Chocolate Heads have a culture of connectedness.

On my second night dancing with the Chocolate Heads (and for many of them, it was anywhere from their 10-20^th rehearsal), I found it interesting that when Aleta came to help this pair of dancers (Fig. 1), she noted that they were not doing as well in her presence than when she was glancing at them from across the floor. It was a very astute observation and an example of the inviting, honest, and loving way that Aleta nurtures a relationship with us. My interpretation of this was that the connection between the students was stronger than their connection to the instructor at that time. She was building trust, and after making the imbalance explicit, the dancers embraced the confidence she instilled in them. Aleta then arranged the whole group to focus on our connectedness. For inspiration, we watched a clip with professional dancers who were so connected that they moved as if they were "one body". Imagine a group of twelve dancers making uninhibited “response cries” when the relevance of connected movement hits them all at once. To help foster this among us, she had everyone move in sync through different activities: traversing the floor in groups of four with the “same foot-fall”, having each member create a movement and asking each group of four to “copy dancer ____” while maintaining the “same foot-fall”. Near the end of our rehearsal session, and in the beginning of subsequent sessions, we perform synchronous movements to get in-tune with each other’s body. We make "waves", create “seas”, and develop trust for each other.

This process is an ongoing acculturation. I produce movement with this group part-time, and started this school year late (in winter quarter). Nonetheless, I have been embraced by the group’s loving energy, and feel myself growing both as a dancer and team member. My continued involvement has exposed me to some of the ways that Aleta pushes each individual dancer to pull the desired expression out of themselves, and the qualities that make her an excellent teacher:

·    Public Recognition with Repetition: “Whoa! Look how ____ did it! Do it again, everyone watch.”

·         Fostering Creativity with our Bodies: “What else can you do? No… what else? YES!”

·      Attention to Detail, Noticing Bad Habits and Targeting Mismovements: (“Stay in your heels”, “sexy is an inside job”, “from the outside in”, “from the inside out”, “five-pointed star”, “out your toes”, “out your hands”, “flex your feet”, “point your toes”, “get up like a dancer”, “be the master of all that you survey”, “honey hands”…) 

These are important points, because it is Aleta’s expertise and ability to transform us as dancers that solidifies our trust in her and each other. She demonstrates these qualities as a teacher and successfully leads a group of artists to organically synthesize an endless array of new ideas. Her orchestration is an act of ingenuity, and as a result of it, she enables her students to behave ingeniously in turn.

References

Dewey, John (1916). “Play and work in the curriculum,” Democracy in Education. Pp. 194-206.

Goffman, Erving (1978). “Response Cries.” Language, 54: Pp. 787-815.

McDermott, Ray & Raley, Jason (2011). “Looking closely: Toward a natural history of human ingenuity.” In E. Margolis & L. Pauwels (eds.), Handbook of Visual Research Methods. Pp. 272-291. Sage.

Appendix A: Theory Drawings

Finite or Infinite Games?

Who is "Yourself"?

Let’s Assess Better

A critique by Evan Rushton of

Looking Closely: Toward a Natural History of Human Ingenuity

by Ray McDermott and Jason Raley

Ray and Jason, you had me at “We agree.” (p. 1)

“…people are usually ingenious, both locally in their most personal circumstances and collectively in their most distributed consequences. In coordinating with each other, people show themselves, to those who would look carefully, to be orderly, knowledgeable, and precise. Given the demands of necessity, they do well what has to be done even if under limiting, or worse, pathological conditions.” (p 1)

I agree in the contrapositive of the final statement regarding a lack of ingenuity among spoiled children: “They do poorly what has to be done, without the demands of necessity.”  I say this because I see people performing poorly (wasteful, inconsiderate, uncreative…), and draw the conclusion that we are not demanding these values (conservation, consideration, creativity…) from them. This is an argument against predominant forms of assessment, and a call to “look closely” at how we can support the development of skills our students need in the 21^st century.

We demand something from our children, and I agree that “[g]iven the demands of necessity, they do well what has to be done…” perhaps too well. Using mathematics as an example, students in 8^th grade honors geometry (“High Achievers”) who took Silicon Valley Mathematics Initiative’s Mathematics Assessment Collaborative’s (MAC) performance assessment exam in 2012 performed well on standardized multiple choice tests (96.1% CST at or above Proficient), but poorly on free response questions (45.6% MAC above proficient) http://youtu.be/MOSS04seBF8?t=57s. In 2^nd grade, 76% of the students were scoring above proficient in both the MAC and CST exams, but over time students were trained to perform well on the only measured outcome, the multiple choice questions.  So much so that by 8^th grade, only 45.6% of our high achievers perform proficiently on critical thinking tasks.

We attempt to condense the learned experiences of children onto a ballot used to measure student growth. I see a parallel between standardized tests and the way Ray and Jason speak of 20^th century social sciences’ objectivity, “Even personal developments and events − even desire − get described and managed as if intelligible to a cold and calculating eye that looks on activities not as they are performed, but by their symptoms − their droppings − lined up in patterns only after they have run their course.”  (p. 2) And I believe that we can do a better job, with assessment that gets at what William James recommended in 1897, “’a more radical empiricism’ that seeks things in the full variety of their connections in experience” (p. 3)

When I drive alone, I usually make good lane change and turn decisions. But I always warn my passengers, that when other people are in the car I rely on the shared knowledge to direct us to our location. This often leads to missed streets and U-turns. I can’t control the huge shift in my personality between an empty vehicle and the “ones-with-others” vessel. But I know the latter is more difficult to manage, and is closer to the stuff of human interaction. The authors claim, “[A] natural history analysis examines organisms and environments interwoven in real time in situations consequential to their participants and beyond.”  (p 2) Ray and Jason’s goal in using a natural history approach is also what I see as the window through which we can accurately measure learning: “[I]t is ‘not the point of view of one toward the other’ that we seek, but ‘the very processing itself of the ones-with-others.’” (p 3)

How do we assess that?

Quotes:

Ray McDermott & Jason Raley (2011). “Looking closely: Toward a natural history of human ingenuity.” In E. Margolis & L. Pauwels (eds.), Handbook of Visual Research Methods. Pp. 272-291. Sage.

Test Data:

David Foster. (2012) Silicon Valley Mathematics Initiative. http://www.svmimac.org/

Googling "rails video tutorials" aka Learning Rails

I took CS 106A at Stanford, and "Mehran Sahami change my life!" as I'm sure he has done for many Stanford-crafted software developers.

So I have decided to learn enough Ruby on Rails to finish a rubric-based assessment tool that a friend was building. I will reflect here and will be updating this post regularly.

I tried on my own* - to little avail. So I took CS 142 Web Development at Stanford, and learned that higher level CS courses at Stanford have a DIY feel. A bit tricky when you are a grad student with few friends in the class. So I became an office hours inhabitant and made some friends, the TAs were accessible and made the class very manageable.

<!-- To be continued -->







*Preliminary Musings
I'll be starting with some reading of http://ruby.railstutorial.org/ruby-on-rails-tutorial-book

http://railsforzombies.org/ has been fun.
http://tryruby.org/ was a bit glitchy... but taught some syntactical rules.

I wanted to check the usual MOOC suspects and iTunesU for a short listing, but figured someone's blog contains this search. So I gBlog-searched, "best ways to learn ruby on rails"

I then read:

http://blog.appfog.com/quite-possibly-the-best-rails-tutorial-in-existence/

http://astonj.com/tech/best-way-to-learn-ruby-rails/

http://www.readysetrails.com/index.php/1951/the-very-best-way-to-learn-ruby-on-rails/

suggested codeschool

Started to watch nettuts+

Talked to a web-developer who wrote the code I am attempting to add to, she said:

1) Google MVC framework - first chapter or two of book or something similar (conceptual)
understand js, html, css. Make sure its Rails 3.2 and ruby 1.9

2) Think of an action that a user can perform. And can you trace it through the code?

3) Get everything working from my machine. Installing ruby, installing rails, requisite gems,

Once comfortable use (rails s) to localhost and test changes to the code.

I might use free video tutorials at  ShowMeDo

There are responses in a forum thread on stackoverflow and that led to stackoverflow's learning threads

Pretty sure I can youtube it for free.
Rating Filter
View Count

Design Behind Wuzzit Trouble

A medium as interactive as a tablet should be doing a lot more than spoon-feeding worksheet problems to children.

Game Design

Innertube Games is breaking the mold. They have a super team of individuals developing learning experiences to serve as grounding metaphors for mathematical ideas. They bypass the traditional education system to address the quantitative skills gap directly. Their product, Wuzzit Trouble (2013), is a prime example of this. I demonstrate the design below.

Wuzzit progression from Evan Rushton

The game can easily be learned through play, but if you’re curious and press the ? key you’ll see the instructions screen.

Your world is a numbered gear with modulus 65. You have the ability to scale given integers with small gears that turn the large gear. You can scale to the left or to the right. By playing this game, a learner can develop a schema for integers and integer modular arithmetic.

The game levels are like lessons planned by an experienced math instructor. They provide an adequate challenge, give immediate feedback, and encourage mistakes.

You may be thinking, "So I just turn these little gears and they move the big gear to free the cute creature called a Wuzzit? Easy enough, what’s the catch Rushton?"

First check out the list of advisors. No catch.

Learning Design

B. F. Skinner wrote, "A machine that teaches by arranging contingencies of reinforcement is a teaching machine.”[1] (p. 110) Many of the game features in Wuzzit Trouble are similar to those found in popular mobile games, like Angry Birds (stars for achievement, high scores, cute creatures expressing emotions...) Our children become increasingly persistent at tackling the obstacle presented by the game to earn the next reward. We demonstrate a behavioral reaction not so different from Skinner's dog jumping to the sound of a clicker.

James Paul Gee wrote, "Good games operate at the outer and growing edge of a player's competence, remaining challenging, but do-able" [2] (p. 2). This sounds like a condition for the state Csikszentmihalyi dubbed "Flow." I was completely absorbed by this game, and feel like it satisfied many of Gee's video game learning principles. The complete discussion of these principles is in his 2003 book, What Video Games Have to Teach Us about Learning and Literacy [3]. This work was heavily referenced in Keith Devlin's 2011 book, Mathematics Education for a New Era: Video Games as a Medium for Learning [4] where Devlin applies Gee's ideas to mathematics education, through a mathematicians lens.

For more immediate content:

There is a useful post/interview of Keith Devlin by Jordan Shapiro.

There is a useful post/interview of James Paul Gee by Henry Jenkins.

James Paul Gee has a short series of youtube clips on this topic.

People love video games. Most readers would probably agree with Gee arguing that all video games are teachers. They are designed to teach the user how to play the game better. The trick lies in making the mechanics of the game near enough to the underlying principle being taught so that transfer of the mental models can occur. Wuzzit Trouble is fun to play and the progression is smooth. In order to get to the later levels, the game designers will have taught a course to you in the form of a game. The fundamental mechanics of a well designed game will transfer to other structures in the world. You will be able to solve systems of linear equations in 4 variables on a mod65 ring, feel accomplished for it, and see number patterns and structures in your mathematics classes as a result of your fun.

Keep 'em coming Innertube Games! Thank you for your considerable contribution to our society.

This is an extraordinary game. Get it for your children.
Peace and Blessings,
Evan

Enhancement - I Wish...

The highest scores were reserved for the least number of moves. (3 stars and all bonus items in 6 moves should not be the same as 3 stars and all bonus items in 5 moves )

There were more levels.

When you overshoot a position it counts against you. (right now the excess is ignored)

There were a way for me to tell the story of the game and identify features of the game, in my own reflective journal. (interact with the progress metrics somehow)

There were multiplayer options to promote collaboration and/or competition.

Appendix - Wuzzit Trouble is "Standards-Based"

I will identify the “standards” that are being taught for the first 5 levels to serve as examples to support this claim, and I encourage the reader to play the rest of the levels to verify the veracity of this claim.

Integer addition, subtraction and multiplication are a subset of the Number & Operations in Base Ten Domain: http://www.corestandards.org/Math/Content/NBT and the fact that there is negative scaling to turn the wheel counterclockwise provides a foundation for the Number System Domain that starts in grade 6: http://www.corestandards.org/Math/Content/NS

Beyond content standards, the game helps to reinforce some of the Standards of Mathematical Practice. It is a process of discovering new properties of numbers in order to beat the progressive levels. It involves persistence, precision, repeated reasoning, and developing an awareness of the underlying mathematical structure.

Level 1-1 – Positive integer addition
The student will demonstrate the ability to tap and turn the small gear clockwise.

Level 1-2 – Positive integer multiplication and addition
The student will demonstrate the ability…
    (2 stars) to tap and turn the small gear in multiple moves
    (3 stars) to tap and turn the small gear a scalar multiple of turns.

Level 1-3 – Positive integer multiplication and addition
The student will demonstrate the ability…
    (2 stars) to tap and turn the small gear in multiple moves
    (3 stars) to tap and turn the small gear a scalar multiple of turns.

CCSS.Math.Practice.MP7: Look for and make use of structure.
The student will be introduced to repeated intervals

Level 1-4 – Negative integer multiplication and addition
The student will demonstrate the ability…
    (2 stars) to tap and turn the small gear in multiple moves
    (3 stars) to tap and turn the small gear a negative scalar multiple of turns.


Level 1-5 – Negative integer multiplication and addition
The student will demonstrate the ability…
     (2 stars) to tap and turn the small gear in multiple moves
    (3 stars) to tap and turn the small gear a negative scalar multiple of turns.


CCSS.Math.Practice.MP7: Look for and make use of structure.
The student will again see a repeated interval in the negative direction

References

[1] Skinner, B. F. (1986). Programmed Instruction Revisited. The Phi Delta Kappan, 68(2), 103-110.


[2] Gee, J. P. (2003). What video games have to teach us about learning and literacy. Computers in Entertainment 1, 1, pp 20-24. DOI= http://doi.acm.org/10.1145/950566.950595


[3] Gee, J. P. What Video Games Have to Teach Us about Learning and Literacy. New York: Palgrave Macmillan, 2003. Print.


[4] Devlin, K. Mathematics Education for a New Era: Video Games as a Medium for Learning. Natick, Massachussetts: A K Peters, Ltd., 2011. Print.

Interaction Design: 4 Approaches to Balancing Whole Number Expressions

Mathematical Model:

A scale measures equality between two weights, represented by mathematical expressions. By adding opposites that sum to zero, expressions can be simplified. The balance responds to the relative weights in the virtual manipulative.

Problem Statement:

What is the best user experience that stays true to the mathematics, and is engaging to interact with?

1) MathPlayground (http://www.mathplayground.com/AlgebraEquations.html) offers a solution to the problem of Solving Whole Number Linear Equations (Fig. 1).

Fig. 1 Mathplayground's Algebra Equations Practice Mode

I like:

·      Realistic Functionality - balance responds to inequality

·      There is exactness to the equality of the symbols on each side

·      There is a symbolic representation of the equation
·      Tiles are in a convenient place to grab

I wish:           

·      You could scale the number of tiles you were adding/subtracting, and could have more than 20 tiles on a side

·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 
·      That the manipulatives were made into fun items that were less abstract
·      That tutorial mode was not the default user first experience, it is much more restrictive than the practice version of the manipulative. 

2) Utah State University's NLVM (http://nlvm.usu.edu/en/nav/category_g_3_t_2.html) has Algebra Balance Scales (and a Negative Integer version in Fig. 2)

Fig. 2 NLVM's Algebra Balance Scales - Negatives

I like:

·      Realistic functionality - balance responds to inequality

·      Exactness of symbols - equivalent on each side

·      Symbolic representation of the equation
·      Symbolic manipulation of the equation
·      Tiles are in a convenient place to grab, and they drop easily

I wish:           

·      Other forms of manipulation other than symbolic (The objects were still moveable after being placed on the scale)
·      You could scale the number of tiles you were adding/subtracting, and were able to have more than 10 of a given object on a side

·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 
·      That the manipulatives were made into fun items that were less abstract

3) DragonBox Algebra (http://www.dragonboxapp.com/) has a gamified approach to teaching the procedures for solving linear equations. (Fig 3)

Fig. 3 DragonBox's Algebra Equations Progression

I like:

·      The graphics are great and the functionality is intuitive
·      The symbols and operations are consistent
·      The way anti-objects are implemented, and the scaffolding pictures are slowly removed

·      The gamification elements make it an addicting experience
·      That a child will learn all of the procedures for solving linear equations without thinking they are doing math

I wish:

·      There was a relation to the real world, like the balancing of weight

·      There was a conceptual underpinning to things like dividing every term by a value
·      The game had multiple representations

o   Table

o   Graph

o   Abstract symbolic

·      That other representations could be chosen to work with and automatically updated when a move was made 

4)  Curtis Wang designed a Whole Number Linear Equation lesson that can be extended as a game concept. (Fig. 4)

Fig. 4 Curtis Wang's Solving Whole Number Linear Equation Lesson

I like:

·      The graphics and friendly design

·      The animations in the original version (https://app.box.com/s/204p4g5wnk3fr4n2pf5a)
·      That this could be more fun to play than the abstracted form

I wish:           

·      There was exactness to the symbols on each side

·      There wasn’t a change in the moment of inertia when placing objects

·      There were anti-objects like helium balloons to remove weight with

·      We could incorporate scaling (Whole number multiplication/Division)