Programming and Mathematics understanding

Logo was the first programming language created for teaching mathematics. Later came programming languages, which can be described as iconic, visual or drag and drop. They are also promoted as ways for children to better understand mathematics concepts. These programming languages include Scratch, Etoys, Turtle Art, Gamemaker and more.

Logo, (by Papert and others), was originally used to control real turtle robots which left pen trails on paper, later the turtle and pen became virtual. Logo was text based rather than visual, this limited it uptake in schools.

image http://museum.mit.edu/150/items/2007.019.001.jpg

Scratch, Etoys, Turtle Art and Gamemaker, which are discussed here, use visual blocks rather than text and are essentially syntax free, giving students more time to concentrate on higher level tasks than syntax error correcting.

It is thought that the use of these programming languages, before or during the formal teaching of mathematics concepts, will transfer to improved mathematics understanding as well as higher order thinking and problem solving skills. This is based on a constructivist understanding of learning, that learning is highly individualised and that multiple representations of a concept are more likely to meet individual learning needs.

Here I look at some basic mathematics ideas: number, equality, addition and functions to see how the visual representations match the mathematics concepts and might aid understanding.

Number line
Number can be visualised as movement along a line. Moving forward, along with turning, also create geometric shapes.

Subtraction on a number line can be visualised by walking forward then back a number of steps. All languages except Game Maker handle this well. Turtle Art has an advantage with negative numbers.

Scratch

Etoys

Turtle Art

By having a "back" block, Turtle Art allows the operations of addition and subtraction of positive and negative numbers such as -1-(-2) =1, forward(-1) back(-2) = forward(1)

Game Maker

Game Maker evolved as a game creation tool rather than being created as an educational tool, it does not easily do pen trails or easily move incrementally forward. It does handle addition of velocities well though.

Equality
The equals sign has 3 meanings
The equation, an assertion of equality 3+5=5+3
Assignment, let x=5
Testing, if x=5

Assignment
The coordinate of the Turtle can be assigned a new value, implicitly as above or explicitly. The Scratch block is a representation of a verbal or written instruction. The Game Maker one is also text based. The Etoys representation does use a back arrow to denote action and is better, though it is a bit crowded with selector hotspots.

The Turtle Art block is the one more likely to be useful to the learner, giving a clear visual cue to assignment.

Scratch

EtoysTurtle Art

Game Maker

Binary arithmetic operator, addition
Addition is the first function which children encounter.
It can be written as infix, 2+2, prefix + 2 2, or postfix 2 2 +
In mathematics it is written in infix notation. Early scientific calculators used postfix. Turtle Art uses prefix, the other three languages infix.

Though infix notation is more consistent with the way equations are written in mathematics, prefix notation gives a better understanding of what is happening, that the + operator combines two values to give a third.

Scratch

Etoys

Turtle Art

Game Maker

More on number
Rational numbers can be represented as fractions or as decimals.

Scratch's "say" displays 2 decimal places

Etoys' watchers can display and edit data at run time, the number of decimal places is alterable. Etoys is the clear winner in this category.


Game Maker's "draw text" and "display message" display to 2 decimal places.

Turtle Art's "show" and "print" display 2 decimal places.

Function
The current understanding of a function is relatively new. A mathematics function is currently understood as a many to one mapping from a domain to a range.

A programming function is more understood as a process which takes an input and produces an output (without overwriting the input). It can be thought of as a machine or a black box with input and output.

Scratch and Etoys have a function box which allows selection from a limited drop down list of functions. Game Maker allows functions to be typed in from a very extensive range of scripting language which is documented in help. Turtle Art allows the use of Python functions which are poorly documented.

Only Turtle Art allows the function to be visualised as an input/output machine.

Scratch
Etoys


Turtle Art

Game Maker

Conclusions
Though a large number of teachers support the use of these languages for learning mathematics concepts and problem solving skills and though it seems likely that the benefits are realised, there is virtually no hard evidence of any of these benefits. This is not because the experimental data fails to support these beliefs but rather because there is no experimental data.

Neither is there any solid theoretical basis for preferring one kind of visual representation over another. There is poor understanding of how maths concepts are learnt and virtually none of how these alternate visual representations interact with this learning, either helping or hindering.

I do expect but cannot offer proof that:

• Turtle Art's ability to do "back" gives the opportunity for more complex number line concepts
• Turtle Art and Etoys' plug in metaphor is a good way to conceptualise assignment
• Turtle Art's prefix notation is a better way to visual binary operators
• Etoys' variable watcher with editing on the fly and variable decimal places is a good way to visualise number and variables
• Turtle Art's visualisation of a function as an input/output machine or black box is a good way to visualise a function but it is limited by poor documentation
  

This comparison of features is not intended as an endorsement of one language over another. Other important factors are the affective (or fun), flow, low entry, wide walls and high ceiling.

Computer programming and the acquisition of critical literacy

I present the following argument for the authoring of interactive or programmable multimedia as an important meta-literacy skill. Though I am not fully decided on this proposition or the role of critical literacy in schools, I think the issue is significant and worthy of consideration.

In summary:

• Critical literacy is an important skill

• Literacy is not just about text and can apply to a range of non-text media

• Non text media are becoming increasingly important means of communication

• Despite their apparent skill with the new media, students still need help developing critical literacy in new media

• Media authoring skills are necessary for developing critical literacy

• Media are increasingly becoming interactive

• Hence authoring skills of interactive or programmable media are an important literacy skill

Meta literacy or critical literacy

Critical analysis is an important literacy skill. When applied to traditional media, it means not just comprehending the words on a page but assessing the truth and reliability of the writing and understanding the motives and biases of the writer.

“Critical literacy is the defining aspect of meta-literacy. The literate person examines texts to determine the currency, accuracy, bias, and comprehensiveness of the information. It means taking our thinking to a deeper level with critical questions.” (Doiron 2005)

Edgar Dale, well known among a earlier generation of educators and researchers for his work related to literacy, discussed the need for critical reading, listening, and observing in contending with the new literacies implied by audiovisual media of the 1940s.

“The new literacies of the Internet and other ICTs include the skills, strategies, and dispositions necessary to successfully use and adapt to the rapidly changing information and communication technologies and contexts that continuously emerge in our world and influence all areas of our personal and professional lives. These new literacies allow us to use the Internet and other ICTs to identify important questions, locate information, critically evaluate the usefulness of that information, synthesize information to answer those questions, and then communicate the answers to others.” (Leu et. al.)

New literacies

Literacy is not, and never was, just about written text. Literacy is about conveying meaning through the use of symbols, these symbols are most commonly text but not always. Even early manuscripts were illustrated and diagrams are an important part of technical texts (Lemke).

Nor is text literacy just about the literal meaning of words. Poetry and drama are used to convey nuances of meaning that prose cannot convey. Understandings of literacy have changed over time, the novel is only about 400 years old and was controversial when first introduced.

Other media greatly enhance the meaning of text, they don't just add to it. “Meanings in multimedia are not fixed and additive (the word-meaning plus the picture-meaning), but multiplicative (word-meaning modified by image-context, image-meaning modified by textual context), making a whole far greater than the simple sum of its parts “ (Lemke)

Nor is prose that simple, meanings can be modified by tone of voice, orthography and calligraphy. Text can be communicated in sign language.

programmable multimedia in Turtle Art

Typological and topological

Lemke draws the distinction between typological and topological symbol systems. Typological systems (language) work more through classification into mutually exclusive categories whereas topological systems including maths and visual media work at the level of variation and relationship. Critical literacy is required with typological and topological media.

Electracy

The word electracy has been coined to describe the 'kind of 'literacy' or skill and facility necessary to exploit the full communicative potential of new electronic media such as multimedia, hypermedia, social software, and virtual worlds.” (Wikipedia)

Social media

Blogs, wikis. Myspace, Facebook, Youtube, Twitter, Skype, Ventrilo, SMS, IM, listservers, Secondlife, etc. are interactive rather than broadcast media. They are increasingly becoming the media through which citizens inform themselves. They are also increasingly channels for business communication.

For example, the most up to date news of the 2004 Tsunami was provided not by the mainstream media but by blog sites.

Social media also played an important role in the 2009 Australian bushfires. “As the worst bushfires in Australia's history raged across Victoria, Twitter, Flickr and Facebook lit up with condolences and horrific first-hand accounts, while many used innovative online mapping tools to assess the risk of the fires reaching their own homes. Mainstream news outlets, battling to provide comprehensive coverage of the tragedy, have incorporated accounts published on the social networking sites extensively in their reports.” The Age February 9 2009,

Political parties are using Youtube and blogs to communicate their messages.

Digital Natives

Digital natives grew up in the age of computers. It is often said that they are much more proficient in digital media than their teachers. How then can they be taught about digital media? Livingstone, among others, questions how deep their apparent proficiency is, quoting Scanlon he says “education ought to be based on assessing students’ individual strengths and weaknesses rather than making glib generalisations that mistake using Facebook for technological savvy”

Leu outlines a role for teachers, “Teachers will increasingly need to orchestrate complex contexts for literacy and learning rather than simply dispense literacy skills, since they will no longer always be the most literate person in the classroom.”

Authoring skills

Authoring is an important part of literacy. If this is true for text, would it not also be true for multimedia?

“Literacies cannot be understood as passive receptivities. Making sense with a printed text is a complex and active process of meaning-making not so different from writing the original of that text “ (Lemke)

The generic literacies of the Information Age will certainly include: multimedia authoring skills, multimedia critical analysis, cyberspace exploration strategies, and cyberspace navigation skills. (Lemke)

“Jamie Myers and colleagues described in 1998 how involving students in creating multimedia hypertexts about literacy and historical figures such as Pocahantas led to a critical stance toward various sources of information”(StateUniversity.com)

"Our proposal for a multimedia literacy that gives the ability to participate freely in the society of the third millennium, and ultimately to transform it, stems from students and teachers authoring multimedia." (Gutiérrez Martin)

Interactive media, simulations and games

Increasingly, interactive media including games are used to inform and persuade. Critical literacy skills also apply to this medium. Some examples of “serious games” include:

Americas Army, “America's Army (also known as AA or Army Game Project) is a series of video games and other media developed by the United States Army and released as a global public relations initiative to help with recruitment”. (Wikipedia)

Food Force is an Educational game published by the United Nations World Food Programme (WFP)

Darfur is Dying is a viral video game for change that provides a window into the experience of the 2.5 million refugees in the Darfur region of Sudan

Ayiti: The Cost of Life , produced through Global Kids and supported by Microsoft's Partners in Learning Mid-Tier Initiative, which “seeks to identify and encourage "pockets of innovation" for increasing digital literacy and career readiness”.

Virtual Worlds

Virtual worlds include Secondlife, World of Warcraft and Quest Atlantis. Virtual worlds are inhabited by the avatars of large numbers of real people, there is a lot of interaction and a lot of communication, some of it language based and some not. It is likely that we will be spending increasing time in virtual worlds for work and play and that a critical literacy will be desirable.

Secondlife is not a game, but it is a game-like virtual world. deWinter and Vie have written on using Second Life to teach critical media literacy.

The multiplayer game and virtual world, World of Warcraft now has over 10 million subscribers making it larger than many countries. “World of Warcraft is rich in new literacy practices as there are so many other players online around the world at the same time” writes a teacher . Beavis discusses how a World of Warcraft trailer was critically analysed by a class of year 8 students.

Quest Atlantis is a learning and teaching project that uses a 3D multi-user environment to immerse children, ages 9-15, in educational tasks. The literacy aspects of this virtual world are many and obvious.

Visual programming languages

Multimedia is becoming increasingly interactive. The media experience changes depending on consumer choices. The webpages we view change their content based on tracking cookies. The authoring of web pages is computer programming. We regularly interact with web aplets, games and simulations. Even entering phone numbers and ring tones to our phones is programming, when we change our computer wallpaper we are programming. When we set up a spreadsheet or word processor template we are programming.

The ability to write computer programs is an important authoring skill and as has been argued above, requires a critical literacy.

Multimedia authoring tools are increasingly interactive and programmable whilst programming languages are increasingly accessible to younger learners and allow the embedding of multimedia: images, video and sound.

For example, Powerpoint, which is primarily a slideshow authoring tool can be used for games and animations. And game programming tools can be used to author presentations.

For students to learn to author interactive media and multimedia, they don't necessarily have to do all that with one package, they could use one package to author multimedia and another for interactive media, but it does seem like a good idea if they can blend both in one package.

Multimedia in a Turtle Art program

The desirable features of an authoring system in which students can develop multimedia critical literacy are

• can present a range of multimedia, sound, images and video,

• can create interactive media

• low entry ( easy for beginners )

• high ceiling* ( no restrictions for high level tasks )

• free**, if students are given authentic and relevant challenges they will spend many hours working at home for each class contact hour

* closely related to high ceiling is the concept of wide walls, that is that students are unrestricted in the breadth of their project as well as being unrestricted in top end complexity

** related to zero cost is software freedom, open source software can be modified or remixed by teachers and students, aside from the question of whether schools should lead by example or reflect the realities of commercial software, there is a case that the ability to inspect and modify is part of developing multimedia authoring skills.

The characteristics of some popular programmable authoring environments are shown below

	Low Entry	High Ceiling	Open Source	Free	Text/graphic
Turtle Art	Y	N	Y	Y	graphic
Etoys	N	Y	Y	Y	graphic
Game Maker	Y	Y	N	Y (lite version)	graphic
Scratch	Y	N	Open source closed development	Y	graphic
Kahootz	Y	N	N	N	graphic
Flash	N	Y	N	N	text
Agent sheets			N	N	graphic
Powerpoint			N	N	graphic

References

Alfonso Gutiérrez Martín PhD. Multimedia authoring as a fundamental principle of literacy and teacher training in the information age University of Valladolid (Spain) edu.of.ru/attach/17/1382.doc

Lemke, J.L. Metamedia Literacy: transforming meanings and media, in Literacy for the 21st Century: Technological Transformation in a Post-typographic World, D. Reinking et al. (Eds.), Erlbaum.

Ray Doiron and Jessie Lees It Takes a Village to Raise a Reader: Building Literacy Across Generations http://www.nald.ca/library/research/village/report.pdf September 2005

Dale, quoted in Literacy - Multimedia Literacy http://education.stateuniversity.com/pages/2186/Literacy-MULTIMEDIA-LITERACY.html

Donald J. Leu, Jr. Charles K. Kinzer Julie L. Coiro Dana W. Cammack Toward a Theory of New Literacies Emerging From the Internet and Other Information and Communication Technologies http://www.readingonline.org/newliteracies/leu/

MMOGCHART.COM “Charting the future of the MMOG industry.” http://www.mmogchart.com/Chart1.html

Jennifer deWinter and Stephanie Vie Press Enter to “Say”: Using Second Life to Teach Critical Media Literacy

Beavis Paying attention to texts http://www.aate.org.au/files/documents/English%20in%20Australia/Beavis%20EinA%2043-1.pdf

Kahootz3 at forefront of research http://www.une.edu.au/creme/uploadedfiles/kahootz_newsletter.pdf