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)

Neighborhood Boundary Project

I tried to use some Buck Institute Project-Based Learning materials and methods for a few days with Algebra Project Cohort students.  If you are not familiar with their free DIY guide, I recommend it.
http://www.bie.org/diy/

Fig. 1

This came after we spent a good deal of time with students on the idea that when "solving" a linear inequality in one variable, we were putting an equals sign in place of the inequality symbol in order to identify the boundary points (Fig 1). After finding the boundary points we were then able to find the solution set by testing left/right of the boundary point on the number line.

The Neighborhood Boundary Project was an attempt to extend the idea for two variables.

Hook

The original hook involved overlaying multiple definitions of our neighborhood (zip codes, city council districts, community member-made street line drawing) and asking who was right?

Who decides the neighborhood boundary lines?

Fig. 2 Neighborhood Boundary deck. See the .key for the actual animation

Fig. 3 Eagle Rock product

Purpose

The idea was to teach a geometric interpretation of linear inequalities as boundary lines with a shaded region above/below the boundary line in the y-direction or left/right of the boundary line in the x-direction.

Students had to define their own system of linear inequalities such that the overlapping solution areas would generate the community they were assigned to.

Fig. 4 Highland Park product

Learning Outcomes

Students did a good job of explaining why they chose particular inequality symbols to get the correct shading. However, not all students were able to produce the symbolic representations of the boundary lines, which is the major skill on standardized tests. Therefore, in future iterations it would be important to provide an opportunity for students to practice and repeat this skill in a way that does not detract from the engagement of the project.

Extensions

We did not put together a composite graph due to time constraints, even though the students were all using the same coordinate system.

I would have loved to have students argue about the boundary lines that were defining adjacent regions. We alluded to gerrymandering of district lines, and this could have made that idea more relevant.

There need to be other ways to get students reflecting on their abstract symbolic inequalities.

Grab the files and adapt as you please. Let me know how it goes.
https://app.box.com/s/km6sgf2h1oybrc4fp1hw

Standards-Based and Rubric-Based Grading

JumpRope

I would be using JumpRope if I were teaching this semester. I would save so much time not having to do purposeless record-keeping. I'm a fan. Try it.

http://www.jumpro.pe/features/standards-based-gradebook

This is definitely in the correct direction when it comes to a Growth Mindset replacing a Fixed Mindset. We should be focusing student attention away from scores/numbers and toward goals/mastery. A 67% doesn't tell me what I need to improve. Feedback needs to be self-explanatory and timely.

Example for teachers:
Do you truly believe every one of our students can achieve mastery?

I do. I believe there is a distribution of the amount of support and intervention required to get every child to mathematical literacy. We aren't successfully serving our youth yet. I argue that our progress in defining the appropriate interventions is muddled by assessment practices that damage curiosity and confidence in our youth. New assessment practices will help put an end to the shaming our defamatory culture brings to the people.
Enjoy Work.


Image: Nigel Holmes / Graph Content: Carol Dweck / Blog Source: QED Foundation

ASAPScience

It was to my great amusement when a student of mine told me how his sleeping in class would increase his productivity. 

 

 I am a fan of ASAPScience. 

http://www.youtube.com/watch?v=NJ_f9onTTQE

Resources: Assessing Mathematical Proficiency for Common Core

What questions do you have about this?

http://map.mathshell.org.uk/materials/background.php?subpage=formative

I had the honor of attending Critical Issues in Mathematics Education (CIME) at the Mathematical Sciences Research Institute on April 3-5. There was an impressive gathering of mathematicians, educators, education researchers, and other interested parties. We worked collaboratively to build mathematical tasks, and felt unguided and uncertain of our products. It is critically important that tasks are implemented, iterated, reshaped, and adapted to the needs of the particular students. While we don't all have time to go write tasks, we do have the time to meet with our departments and create realistic goals for ourselves. I recommend for anyone who isn't already incorporating weekly or monthly common group tasks in their classrooms, to do so. Once there is momentum in the direction of this change, it will become increasingly easier to implement these ideas. I hope to quickly share some of the resources I was introduced to.

CIME 2013 Resources for Item Writers

• A livebinder with a collection of great resources (some of which are repeated):

http://www.livebinders.com/play/play/187117

• PARCC sample items

http://www.parcconline.org/samples/item-task-prototypes#7

• Smarter Balance Sample Items

http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm

• New York City Common Core math task exemplars developed by “Common Core Fellows”

http://schools.nyc.gov/Academics/CommonCoreLibrary/TasksUnitsStudentWork/default.htm

• New York State Common Core Sample Assessment Questions

http://www.p12.nysed.gov/assessment/common-core-sample-questions/

• The Mathematics Assessment Project

          http://map.mathshell.org/materials/index.php

"The team uses its well-established engineering research [PDF,100K] methods involving: input from prior research; design skills to produce draft materials; iterative systematic development through trials in US classrooms, with revision informed by structured feedback data from the observer teams."

• Illustrative Mathematics, 700+ tasks aligned to the CCSSM intended for a variety of purposes:

http://www.illustrativemathematics.org/

• The Shell Centre:

http://www.mathshell.com/

• East Side Community High School Math Portfolio

https://sites.google.com/a/eschs.net/www/mathportfolios@eastside

There is a clandestine implementation of the Common Core State Standards for Mathematics in LAUSD. Everybody expects there to be a mastermind pulling the strings, but in the end, it is our collective effort and effective collaboration that will eventually result in changing the way we teach.

A large district like ours seems very much like a game of telephone, and the messages get morphed as they are passed from vessel to vessel. Luckily, we all have the freedom to access materials at their source. Unfortunately, many parties use this freedom as an opportunity to spread ideas/practices, and this creates the added job of data filtration and discovery.

For all skeptics of the Common Core, be mindful of the research and work that has been put toward these efforts. As for the image above, I emailed the Shell Center requesting the research paper it is in. I could not find it on their publications site.

Enjoy work.

Creating Relevance to Accelerate Algebra Instruction

I have been trying to get my 9th graders to use the Singapore math 8-step process using unit bars to approach word problems. Every so often I feel the need to start with an image from the real world instead of the cut and dry word problem. The keynote presentation is linked below:
http://www.box.com/s/bca82131ea358bdc8083

Finding Intriguing Data for Students

Borrowing a teaching technique that I have learned from Dan Meyer; I will be taking the graphs from this article to create an introductory problem for my students to wrestle with. Relevant systems of linear equations are much more engaging than textbook word problems.

http://public.iwork.com/document/?a=p1090410687&d=Comcast_Vs_Netflix_47_hulu.key

http://flowingdata.com/2012/02/09/why-i-want-to-quit-cable/