Math Bytes

Units of Measure in the 21st Century

It's always been a source of great consternation to me, that mathematics benchmarks around the world still appear to be completely oblivious of the implications of the impact of digital technologies in the world of mathematics.

To see explicit evidence of this, you need look no further than the Mathematics benchmarks that are currently used for the teaching of measure, which are still confined purely to units which, while still useful, are no longer the most common units of measure that are are significant in the lives of people who rely upon digital technologies in their daily lives. That means most of us, especially if you're reading this.

Why is it that in schools that are blessed with the ubiquitous provision of digital technologies, one-to-one laptops and iPads, the students never learn anything about how file sizes and the measurements of file sizes work?

There are two main reasons for this as far as I understand it:

• Almost universal ignorance on the part of teachers, more of whom are still largley completely oblivious of the difference between a megabyte and gigabyte. 
• An assumption that a generation of digital natives just automatically get this stuff, trust me, they don't. 
• A somewhat naive assumption that if there is a subject area that will remain unaffected by the influence of digital technologies it has to be mathematics. 

Teach Maths for your present, and their future not your past... 

Ask yourself, how can it make any sense in this day and age for students to be able to complete their mathematics education, competent in the calculations relating to kilograms and kilometres and millimetres and litres, but not have the slightest clue about the units of measure that are fundamental to the devices that they rely on every day?

The problem is caused by this... lack of awareness are profound. I don't know of any students who find themselves in difficulty is because they were confused about the difference between a metre and a kilometre, but I regularly encounter students and teachers who are flummoxed by their their inability to understand how big a megabyte is and why they can't email that 200MB video that was ‘rejected by the server'.

Fortunately, the solution is obvious; educate the teachers and they will educate the students, but unfortunately it looks like these things will not change until the ‘official, mandatory' benchmarks change. If anything this post is a desperate plea for just a bit of common sense; do you really need a mandatory benchmark for you to realise that in this day and age it is absolutely essential that our teachers, and their students, are as familiar with kilobytes and megabytes, as they are with metres and kilometres?

One extremely rare example of digital measures in Maths - Khan Academy

Trust me, it's not that difficult, if anything it’s actually easier to understand than traditional units of measure. When teaching measurements of length students have two differentiate between tens and hundreds and thousands, but when dealing with units of measure for file size, it's much simpler, everything is 1000 times bigger than anything before. That's it.

Put your practice where your pedagogy is...

So, inspired by Khan Academy, I put my my practice where my pedagogy is, I ran a lesson with a grade 5 class, with the single goal of demonstrating how easy it is to enable students to get to grips with this fundamental unit of measure. Essential to this was simplicity, and I know of no technoogy that is simpler to use, and as transformative in application as an effectively used 'wiki' space. In this case, as a GApps school, we used a Google Site, but any kind of online fora will suffice, more on the power of online 'interthinking' via wiki spaces here.

As with all of these kinds of lessons, most of the work goes into formulating a decent provocation, one that is not too narrow, otherwise all you get is 22 responses that are pretty much identical, as every student just imitates or duplicates the previous response. The prompt below shows how this can be avoided. In terms of 'doability' this task took me about 15 minutes to set up, and while I was able to complete the activity within one lesson, and one homework, to be honest it could have run for a week if I'd let it.


Prompt:

It was immediately apparent how natural this online environment is, and as can be seen in the clip below, far from being a screen dominated task, it stimulated a great deal of (on task) discussion and collaboration.  Remember these kids wrestle (without any help from teachers or textbooks) with file size every day, so asking them to envisage scenarios that use these quandaries as problem solving situation is not a big deal.

Just one example of interthinking in this lesson...

Interthinking - online and offline

It's important to note that, while the activity is screen centred, the learning is not confined to the screen, the students were just an animated (I reckon more so) than they would have been if this has been presented to them as a worksheet (which it could). But the transformative difference here is that moving it to a web based interactivity (not just an activity—see what I did there ;)?) facilitates the leverage of the transformative elements of SAMMS:

Access: They can easily search for clarification on specific elements they find confusing, such as units they want to consider that are not in the prompt (petabytes anyone?) particular vocabulary, or clarification about the amount of storage associated with particular contexts they wish to consider, eg the capacity of their smartphone, their games console, memory card, tablet etc.

Mutability: In response to feedback (from peers and their teacher), students can easily duplicate and revise their posts and post a second post (by effectively 'replying' to themselves) that shows clear evidence that relevant criticisms have been resolved.

Multimodality: Of course this entire medium is multimodal, if we had had another lesson, this activity could have been expanded to allow students to incorporate image, audio and video.

Socially Networked & Situated: As it is online, we can facilitate a P2P homelearn (as opposed to homework) activity. Assign assessment buddies to feedback on each others posts at home, of course the teacher can now easily monitor the quality of these online interactions, and interject, support, clarify, redirect as necessary.

I've included some examples below that illustrate how powerful this task was, and how easy it is to set up. 

Note that a great deal of the conversation (on screen and next to it) centred on wider mathematical concepts that would be relevant to the teaching of traditional units of measure, eg appropriate use of units, conversion of units, and that old favourite, explain your thinking... Now that's what I call a #winwin

Below you can find a PDF of the original discussion in it's entirety, bear in mind that this is not the actual discussion, which continues to develop following this capture.

Transforming Talk in the 21st Century

Does this have to Facebook? No. Does it have to be asynchronous and social?Yes.

Many skills are touted as '21st century' in nature, but the reality is that many, if not most of all them, are little more than refinements of abilities that have always been intrinsic to humanity than reflective of any particular era in history.

That said, there is at least one skill which strikes me an unique, maybe not unique to the 21st Century as it has been in existence since at least the late 90s, but it is unique, it is digital, and it is easily one of the most powerful, transformational applications of digital technology I have encountered, what is it? It's the ability for groups of people to interact, and interthink, online, or to frame this in the words of the title of a thesis that is a little more academic:

The Construction of Shared Knowledge through Asynchronous Dialogue

The above title is taken from a 362 page thesis written by a doctoral student called Rebecca Ferguson (Ferguson, 2009) that I have recently finished reading, and that is the inspiration for this post. This is not a post about the actual pedagogical practice of using this medium for learning, that is the subject of another post here. This post focuses on the big picture, the WHY, why this medium is not only unique but actually essential, critical, pivotal, and in my experience, one of the most genuinely transformative applications of ICT that I have ever encountered. 

All quotations that follow, unless specifically cited otherwise are taken from Ferguson's thesis (ibid). The original thesis is available online, here, all other works cited can be referenced from Ferguson's thesis.

What follows is a summary of the thesis, curated through the lens of the way this looks in a TELE (technology enhanced learning environment), like those that reflect UWCSEA since at least 2010.

Critical to this entire context is the uniqueness of this medium, it is essentially unlike any form of human communication that has ever preceded it, in that "asynchronous exchanges are enriched by the use of textual affordances that are not available in speech." (p4) These online asynchronous environments are ubiquitous, mainly thanks to the unprecedented rise in society of social networks, (and to a large extent email, another asynchronous social medium, albeit not as instant) that are in their very essence 'fora', online communities where text based dialogue, often enriched by image and video are commonplace, and yet, despite it's profound presence in society as a whole, it is very rarely leveraged as tool to enhance learning, even within TELEs.

This post is in many ways a plea to change that; because I believe that in the classrooms of the 21st century, asynchronous dialogue should be the norm, in every subject, every day, just like it is in the social lives of arguably most students and teachers.

A fascinating exchange between Grade 5 students - look at the learning!

--- Page 18 ---

In the first five years of this century, the number of Internet users in the UK almost doubled, rising to 37.8 million in 2005 (CIA, 2001, 2006). For the first time, the majority of the population was able to interact, and to learn together, in virtual settings (Dutton, di Gennaro, & Millwood Hargrave, 2005). As a result, some learners and teachers who have spent years developing the skills necessary to communicate, work and build knowledge together in face-to-face settings face the challenge of adapting those skills for use in an asynchronous, textual environment.

What is asynchronous dialogue?

'Asynchronous dialogue' describes the ways in which online learners construct knowledge together in both the short and the long term without [necessarily] being co-present at the same time or in the same place."


--- Page 21 ---

Asynchronous dialogue is defined here as a sustained discussion, involving two or more people whose contributions are not expected to be produced in temporal proximity, in which language is one of the elements used to convey meaning.

--- Page 23 ---

Users of asynchronous forms of dialogue such as email and online fora make use of various structures and patterns, referred to here as discursive devices, within their communication in order to increase both coherence and comprehension.

--- Page 32 ---

Learning is fundamentally social in nature, even when individuals are separated by time and space

--- Page 35 ---

Vygotsky regarded language as the most powerful of these mediating tools and as the primary tool for thinking

--- Page 42 ---

Different media bring different resources to, and also impose constraints. Relevant factors include copresence (the ability to see the same things), cotemporality (the ability to access messages as soon as they are sent), simultaneity (whether participants can communicate at the same time or must take turns) and sequentiality (the possibility of turns being accessed out of sequence) (Baker, et al., 1999).
...
In an educational context, an important form of language is dialogue: a sustained discussion, carried out through speech or online, in which language is used to convey meaning. Participants in an effective dialogue are both contributors and active listeners (Moore, 1993). Through dialogue, they share knowledge and jointly construct understandings of shared experience that support learning (Crook, 1994)

--- Page 43 ---

One of the earliest recorded forms of educational dialogue is the dialectic employed by Socrates (470-399 BCE) and therefore described as ‘the Socratic method'. This was originally an open-ended dialogue, which was gradually formalised, eventually being codified by Hegel (1770-1831) as a form of logic that proceeds from thesis to antithesis and thence, eventually, to synthesis. In educational settings, dialectic is employed when people need to combine their knowledge by sharing, comparing and combining contrasting views ...

--- Page 44 ---

In schools, dialogic has been used to emphasise collaborative group work and the uptake of children's ideas, to encourage pupils to recreate accounts in their own words and to emphasise a collective, reciprocal and cumulative approach to learning (Skidmore, 2006).

Less IRF, more IDRF

Less Initiation, Response and Follow-up (IRF) [Also IRE, Initiation, Response, Evaluation] exchanges and more Initiation – Dialogue – Response – Feedback (IDRF) (Wegerif & Mercer, 1996) does not focus on the teacher's input but incorporates dialogue between students, allowing learners a more active role and supporting them in working together. (p45)

--- Page 46 ---

[asynchronous discussions help] students to participate in collaborative and critical argumentation, rather than being too embarrassed to criticise others or to state their opinions directly. Other studies drew attention to students overcoming emotional barriers such as shyness, discomfort or a lack of confidence (Ravenscroft, 2007).

--- Page 47 ---

These forms of communication may be synchronous or non-synchronous, or they may offer both possibilities; they may be text-based, audio-based or graphics-based; they may be used for group communication, for two-way discussion or for personal reflection

--- Page 48 ---

Until recently, it was rarely necessary to distinguish between physical and virtual settings and so the dictionary definitions of many English words take a physical setting for granted. The words ‘talk' and ‘dialogue', with their assumptions of synchronicity and co-presence, both make an uneasy transition to an online setting. [...] there are major differences between oral and written modes of expression

--- Page 52 ---

It is an important tool that teachers and learners employ in a variety of ways: to build social relationships, to mediate collaboration, to construct online learning environments, to supplement face-to-face interaction and to support distance learners who are working individually.

Asynchronous dialogue: distinct from writing and speech

Asynchronous dialogue should be considered as a new form of communication, rather than as a variant form of speech or writing.

--- Page 54 ---

Asynchronous dialogue, being a new language tool, has the potential to produce far-reaching changes. It is a complex blend of inner, oral and written speech, and Mercer's view (2000) is that its combination of characteristics of speech and writing makes it a welcome and valuable addition to the toolbox of language. Although it has some characteristics of both talk and writing, its chronology and its use of layout and typography mean that it has emergent properties that belong to neither. These emergent properties are not the same as those of synchronous online dialogue, which is characterised by immediacy and fast responses (O'Connor & Madge, 2001).

Five techniques

--- Page 58 ---

Build the future on the foundations of the past by eliciting knowledge from learners, responding to what learners say and describing significant aspects of shared experience:


• Literal recap: Recounting past events.

• Reconstructive recap: Aligning accounts of the past with current pedagogic concerns.

• Elicitation: Prompting the recall of relevant information.

• Repetition: Repeating responses in an evaluative fashion

• Reformulation: Presenting responses in a clearer form.

• Exhortation: Asking others to recall relevant past experiences.

Affordances of asynchronous dialogue

--- Page 63 ---

Asynchronous dialogue is distinct from other forms of communication not only because of its textual nature but also because of its unique chronology.

--- Page 69 ---

Researchers studying online learning have identified a variety of relevant affordances of asynchronous dialogue. These can be broadly classified in three groups: affordances of the technology, affordances of the medium and affordances of the dialogue.

--- Page 70 ---

Asynchronous dialogue appears to provides the classic components of cooperation and collaboration – discussion, dialogue and community – without the traditional constraints of time and place.

Martini affordances' of the technology – any time, any place, anywhere

--- Page 71 ---


Learners and educators may use a variety of technologies to encounter each other face to face or online

[Unlike face to face dialogue, online there is always a transcript of the dialogue] that can be consulted, edited and reworked

The dialogue can contain hyperlinks to other resources and dialogue, and may also have documents, pictures, sound files or videos attached to it.

Other affordances of the medium include the ability to link messages through threading


The time commitment is minimal as all of these forums use conventions that are common across online working spaces [ubiquitous in social media].

Students' asynchronous dialogue

--- Page 72 ---

Analysis of students' asynchronous dialogue (Blanchette, 2001) demonstrated that having time to consult sources and check references meant they were able to provide each other with very accurate information. Blanchette also observed that the learners in her study were more likely to ask questions, ask for clarification and seek feedback than those studied in a face-to-face environment.

When learners have time to deliberate, their responses are more likely to be focused and purposeful

--- Page 74 ---

Cooperation versus collaboration

Co-operation is a goal-centred activity (Panitz, 1996) in which different things are done by different actors in order to achieve their goal (Van Oers & Hännikäinen, 2001). It involves splitting work, solving sub-tasks individually and then assembling the partial results to produce a final output. Because the majority of the work is done individually, this way of working makes limited use of the affordances of asynchronous dialogue.

Collaboration, on the other hand, involves partners carrying out work together (Dillenbourg, 1999). It is a co-ordinated activity, the result of a continued attempt to construct and maintain a shared conception of a problem (Lipponen, 2002); an interaction in which participants are focused on co-ordinating shared meaning (Crook, 1999). It requires more than the effective division of labour that constitutes cooperative work. Participants must negotiate mutually shared or common knowledge in order to work together to solve a problem or perform a task together (Littleton & Häkkinen, 1999).
...

If learners are to make effective use of these affordances to support the co-construction of knowledge, they need appropriate skills and resources to engage effectively in online educational dialogue but may lack these if they have limited experience of online learning (Kreijns, et al., 2003). The importance of learning how to interact should not be underestimated.

Sticking to the point 

--- Page 83 ---

In face-to-face situations, learners modify their talk over time in relation to their context and their understanding of what they are doing. Because they use speech, which is ‘not simply perishable but essentially evanescent' (Ong, 1982, p32) they face the double challenge of pursuing a line of thought systematically and then preserving their understanding of what has been achieved. They employ discursive devices to overcome these challenges, shaping transient speech into shared knowledge (Mercer, 2000).

Discursive: of speech or writing, Tending to digress from the main point; rambling.

In an asynchronous setting, learners do not need to employ these devices [chit chat] to help them to remember what they have said or done, because they have access to the complete text of their past dialogue in the transcript automatically generated by the software (Kaye, 1989).

--- Page 85 ---

Argumentation, for example, can be described as ‘a reasoned debate between people, an extended conversation focusing on a specific theme which aims to establish “the truth” about some contentious issue' (Mercer, 2000, p96) and is thus task focused. Collaborative reasoning, which includes challenges, evidence and evaluation, is specifically a taught approach (Anderson, Chinn, Waggoner, & Nguyen, 1998). ‘Effective discourse' is also teacher led, being dependent on the educator to create a situation in which participants can ‘advance beliefs, challenge, defend, explain, assess evidence and judge arguments (Mezirow, 1997, p10). Accountable talk is similar, including elements such as listening, clarification, extension and elaboration (Michaels, et al., 2008; Resnick & Helquist, 1999)

Kinds of talk... 

--- Page 86 ---

Disputational, cumulative and exploratory talk in the classroom have the advantage that they not only deal with the productive interaction that helps the group to extend its understanding and to achieve its goals, but they also deal with the unproductive interaction that makes a group less likely to achieve its goals

These social modes of thinking are therefore referred to as cumulative, disputational and exploratory dialogue, rather than talk, because their characteristic elements can be observed both online and offline.

--- Page 87 ---

Disputation should not be confused with argumentation, in which conflicting views are presented, sometimes forcefully, but the intention is to reach a resolution


Cumulative dialogue is much more constructive. In cumulative exchanges control is shared. Speakers build on each other's contributions, adding their own information and constructing a body of shared knowledge and understanding, but they do not challenge or criticise each other's views.

--- Page 88 ---


Exploratory dialogue is the type considered most educationally desirable by teachers (Wegerif, 2008b). Learners who engage in exploratory dialogue constantly negotiate control, engaging with each other's ideas both critically and constructively

Exploratory talk, by incorporating both conflicting perspectives and the open sharing of ideas, represents the more visible pursuit of rational consensus through conversations. It is a speech situation in which everyone is free to express their views and in which the most reasonable views gain acceptance.

--- Page 89 ---


They are working to develop ‘a new understanding that everyone involved agrees is superior to their own previous understanding.


Conflict between learners is not necessarily unproductive because the challenges and counter-challenges of socio-cognitive conflict are important for the development of knowledge. Exploratory dialogue is likely to involve the confrontation of different approaches in socio-cognitive conflict. Such conflict may result in different views being coordinated to form a new approach, more complex and better adapted to solving the problem (Doise, 1985). In such cases, conflict and difference between individuals are brought into productive play to support learning

Improvable objects [Ongoingatives]

--- Page 93 ---

The improvable object is an analytical construct that was developed to help explain how learners are able to develop ideas over time. It is associated with the use of ‘progressive discourse'

Progressive discourse is associated with the sustained development of improvable objects over time (Wells, 1999). They also need to be able to identify, augment and maintain common ground as their work progresses (Baker, et al., 1999) and improvable objects offer a way of achieving this.

Resources available to groups can be characterised as improvable objects if they meet certain criteria (Wells, 1999). They must be knowledge artefacts that participants work collaboratively to improve because they involve a problem that requires discussion. They must act as a focus for the application of information and experience by the group. Unlike many assessed assignments, an improvable object must provide a means to an end, rather than being an end in itself. Finally, an improvable object must be both the inspiration and the focus for progressive discourse.


The key elements of improvable objects – a problem that provides as a means to an end, inspires progressive discourse and acts as a focus for the application of experience and information – can therefore all be assembled in asynchronous settings to support the shared construction of knowledge.

Improvable objects are dynamic representations of a changing situation.

Improvable objects are more akin to the rough drafts that Vygotsky (1987b) described as a powerful means of reflecting on work. Not only do they support the development of understanding by groups of learners, but they are also developed as part of that progress. They are a means of sharing and building ideas over time; sites not only for the display and comparison of different understandings but also for their manipulation and development.

Learning is fundamentally social in nature, and that knowledge is not a static entity, but is co-constructed by learners with the help of meaning-making tools.

--- Page 242 ---

Postings that are taken up, quoted and re-quoted, commented on and reposted come particularly close to being improvable objects.

--- Page 267 ---

The text-based nature of dialogue supports collation of work and also the direct and detailed comparison of different understandings.


--- Page 300 ---

An important role of the tutor is therefore as a discourse guide (Littleton & Whitelock, 2004) who, by modelling skills and behaviours, can help students to develop appropriate ways of talking, writing and thinking in an asynchronous group environment. Wertsch speculates that the skills called for by online environments will ‘have a major impact on how we define success, intelligence, and other aspects of human functioning in the years ahead' (Wertsch, 2003, p903). If such skills are not identified, modelled, or explicitly taught, learners will find it difficult to make effective use of them.

--- Page 304 ---

Improvable objects allow learners to regain an important element of online study: time independence. Online learning is commonly assumed to allow students freedom to choose when they work.


--- Page 317 ---

Asynchronous dialogue can be far more detailed and complex than face-to-face talk, that to groups of learners offers affordances that are not available in face-to-face situations. [...] 

... prompting groups of learners to share knowledge, challenge ideas, justify opinions, evaluate evidence and consider options in a reasoned and equitable way. 

... increasing understanding of the skills and meaning-making tools that support the shared construction of knowledge.

Ferguson, Rebecca (2009). The Construction of Shared Knowledge through Asynchronous Dialogue. PhD thesis, The Open University.

Classrooms as Social Networks

Interthinking, Interaction & the Internet

Interthinking

Possibly one of the most exciting applications of digital technology in education are the kinds of activities that encourage the unique ‘peer to peer’ interactions between students, and/or the teacher, within a shared online space, a forum, or a ‘wiki’ to use the term in it’s most fundamental sense:

“wi·ki
noun
a website that allows collaborative editing of its content and structure by its users.
(Google)

“[A wiki] differs from a blog or most other such systems in that the content is created without any defined owner or leader, and wikis have little implicit structure, allowing structure to emerge according to the needs of the users.”
http://en.wikipedia.org/wiki/Wiki 

The tendency when this kind of conversation starts, is for people to leap to the assumption that we're going to have to use 'Facebook? FACEBOOK? Are you serious?' or perhaps Twitter. Yes, those could work too, but there are many platforms that are more ... appropriate for a classroom that you can use for this, any blog format, and wiki, but for us as a GAPPS school, Google Sites provide a 'wiki' environment that works magnificently. It is one of those applications of tech that goes deep with educators—if you want to see a teacher embrace and RUN with a tech tool, this is the one; it takes (literally) a few minutes to post a provocation (maybe longer to actually think of a good provocation) then sit back and watch the students take it and transform it.

All you need is a good question... 

As Erickson describes in her chapter on "Different Types of Questions to Support Conceptual Thinking",  the right blend of questions help students construct their own understandings. She recommends designing three types of conceptual question (factual, conceptual and provocative) to guide students thinking,

"students seek resolutions to queries or issues, search patterns and relationships, and bridge specific examples to conceptualise ideas, questions provide the necessary support or nudge. As a result, learning is more memorable and better retained because rather than being told what is important, students construct understandings themselves."
(Erickson et al, 2014, p. 56)

Now the teacher sits back and absorbs the feedback, watching the conversations unfolding, expanding and developing, but their role, is still critical. You see the kind of interactions afforded in a wiki space are very similar, and yet very different to those in a typical face to face classroom discussion. And just like their face to face variety, they have their potentials and pitfalls, which need to be monitored as feedback so they can use the evidence to feed forward, at class, group, or individual student level.

A recent study shows that students who regularly participated in these kinds of activities improved their maths and reading scores by the equivalent of two extra months of teaching, even though the activities were not designed to improve literacy or numeracy. More than 3,000 students in 48 schools across England participated in weekly discussions about concepts such as truth, justice, friendship, and knowledge, with time carved out for silent reflection, question making, question airing, and building on one another’s thoughts and ideas.

This post focuses on the actual pedagogical practice of using this medium for learning. I have written another post that focuses on the big picture, focusing on the WHY—why this medium is not only unique but actually essential, critical, pivotal, and in my experience, one of the most genuinely transformative applications of ICT that I have ever encountered.

What is interthinking?

Karen Littleton and Neil Mercer wrote the book that describes the kind of internet based interaction—interthinking.

“Mainly by using spoken language, people are able to think creatively and productively together. We call this process 'interthinking' to emphasise that people do not use talk only to interact, they interthink.” (my emphasis)

In Interthinking Littleton and Mercer describe three types of talk:

Disputational talk, cumulative talk, and exploratory talk. 

The first is one to avoid, the second is safe ... but ineffective, the last is what we want, that’s where the transformative activity, the interthinking, happens.

Here are some great examples of Interthinking I've grabbed from some of our class sites, trust me there is a lot more where these came from. Now they didn’t get this good on their own, skilled teachers guided and developed these discussions. Online discussions generally tend to start out as very pleasant but bland ‘cumulative’ talk (more on that in a minute), then, once they become more comfortable with the medium, it can unfortunately easily denigrate into ‘disputational’ argumentative talk. The role of the teacher is to ‘mentor’ these students to interact or ‘talk’ in ways that find a powerful balance between these extremes, talk that is ‘exploratory’.

Online discussions*

Click to see a collection of examples of online interthinking

Now, looking at the conversations here, you could mistakenly assume that all of it happened in the sole confines of an individual screen, students hunting and pecking away in isolation, but that is only half the story—all of these conversations are grounded in good old fashioned classroom ‘face to face’ which iteratively shifts online, and then back to face to face over time. Now that timeframe could be as short as a back and forth in one lesson, to an ongoing back and forth, iterative feedback loop over an entire unit, whatever the teacher feels is needed.

I strongly recommend that you read the book yourself, but in the interest of action, I’ve summarised the points that relate to the context of internet interactivity below. The following is quoted directly from the book, anything inside [square brackets] is mine.

*You can any wiki space that supports a reply option for this, eg Blogger, Google Classroom, Edmodo etc. 

Interthinking elements

Interthinking has been necessary for the development and dissemination of all human knowledge and understanding. However, as we will all know from personal experience, collective thinking is not always productive or successful. Two heads are not always better than one, and we need to understand how and why that is the case. (p 2)

p15

Three kinds of talk in groups (as first reported in Dawes, Fisher and Mercer 1992). They can be summarised as follows.

Disputational talk in which

 • There is a lot of disagreement and everyone just makes their own decisions;

 • There are few attempts to pool resources or to offer constructive criticism;

 • They are often a lot of interactions of the open 'yes it is! – No it's not!' kind;

 • The atmosphere is competitive rather than cooperative.

Cumulative talk, in which

 • Everyone simply accepts and agrees with what other people say;

 • Children use talk to share knowledge, but they do so in an uncritical way;

 • Children repeat and elaborate each others ideas, but they don't evaluate them carefully.

Click to enlarge

Click to enlarge

Exploratory talk, in which

 • Everyone engages critically but constructively with each other's ideas;

 • Everyone offers the relevant information they have;

 • Everyone's ideas are treated as worthy of consideration;

 • Partners ask each other questions and answer them, ask for reasons and give them;

 • Members of the group try to reach agreement at each stage before progressing;

 • To an observer of the group, reasoning is 'visible' in the talk.

Click to enlarge

Click to enlarge

p63

Digital technology and interthinking

Much as been written about the ways that electronic, digital technology can 'transform' or 'revolutionise' interpersonal communications. This is not mere hyperbole. People can now communicate quite easily with other individuals far away, send each other various kinds of multimodal information and organise group events and collective creations without being ever in the same room as other participants.

...

However, there is a danger that the wealth of communicative facilities offered by digital technology distracts us from concerns about the quality of communication. A clear line on the telephone has never ensured that two speakers would have a conversation in which they understood each other well, and the added visual dimension offered by Skype will not do so either. Computers in their various forms, and their software, are cultural tools that we employ well or badly. They can certainly make interthinking possible between people who would otherwise have been separated, and they can provide practical and very useful support for groups of people who are working and learning together.

...

[For some reason these researchers seem to focus exclusively on the IWB as if it is the most ubiquitous form of technology, and the most relevant? I would argue that the context that is considered here would be far more effective if each student had their own screen [situated]and were still able to talk together in the same physical space [located] while interacting in the same virtual space in real time (ie, synchronous not asynchronous), such as perhaps annotating a shared image within Google Drive?]

...

p77

Improvable objects and interthinking [Mutability]

As we saw earlier, the IWB [and surely any situated shared screen environment] is very useful for generating and recording synoptic, written conclusions; it is easy for the whole group to see and comment on what each member writes, and for the final text to be very quickly modified in light of feedback and evaluation of the emerging ideas (see Littleton, Twiner and Gillen 2010). One of our former doctoral students, Alison Twiner, has called the kind of text the children are creating on the IWB a 'digital improvable object' (Twiner 2011; Twiner, Littleton, Coffin and Whitelock in press). Teachers often encourage children to record what has been said in their group discussions. It was the classroom researcher Gordon Wells (1999) who first suggested that if they are treated as 'improvable objects' rather than finished pieces of work, such records can, if used appropriately, provide a cumulative basis of common knowledge upon which future discussions and other activities can draw and progressively build. Of course, such records do not have to be digital—they can also be created on paper—but computer–based technology offers a way of doing so easily, so that modifications can be made, several versions kept and copies distributed. Such digital records can also include other things such as diagrams and drawings that capture ideas created in discussions. They can offer a kind of half-way stage between the ephemerality [temporality] of talk and the permanence of written texts, and represents one way that technology can help people think collectively.

P78 [paraphrased]

Once saved, these collective creations are then available as a tangible results for discussion by groups of students. For example, a teacher easily project it onto a whiteboard screen for students to refer to, both as a powerful 'aide memoir' for initial reactions and ideas, and as a subsequent focus for collective thinking.

p79

Other kinds of electronic text can also support the collective revision, development and evaluation of ideas. [...]  ...comment boxes enable groups of students to capture, during the data collection phase of their enquiries, important contextual information that would assist them in the interpretation of the data during analysis. Observations of the software in use revealed that as the students moved toward the reporting of the investigation, they also reworked, refined and continually edited and saved the text within the boxes [A process which would have become increasingly and inevitably extremely messy and convoluted if it had been carried out on paper]. In doing so the text became an ongoing work in progress, capturing emerging ideas and (inter)thinking, over time, in respect of the interpretation of data and the key findings. Their initial comments recorded in the boxes provided a base from which to develop and build shared knowledge and understanding. This process of reworking the comments in the boxes also helped students make connections across different phases of the enquiry and so help them maintain the 'thread' of their joint activity (see Littleton and Kerawaller 2010 for more information).

[Interesting to note that the researchers themselves could not consider how this activity could be practised in any other any other software environment, and seem to be fixated on only extremely sophisticated futuristic models], "as a tool for enabling into thinking by a group of, 'tabletop' interactive computers that are sensitive to touch may prove to be more useful.… Another researcher, Stahl (2011), has suggested that tabletops could serve as a 'multimedia tribal fire for the classroom, workplace, or social gathering', though they are currently so costly that it is unlikely that they will soon become as common in classrooms as the IWB]

p82

Collaborative learning at a distance 

[Interesting to note here again that the researchers have a very narrow/fixed view of how the technologies can be used, for example the situated nature of these technologies means that they don't have to be used at a distance the same tool could be used very effectively within the same physical location, just because you can use them to span thousands of miles, doesn't mean that you have to]

There have been many studies of the use of electronic communications in distance education, but few have been studies of how spoken or written communication between students in distant locations might enable their learning or problem-solving.

...

p83

One of the ways that communicating through text [why just text? What about image? Video] online is rather different from talking face-to-face is that it can either take place in real time, so that speakers respond immediately to each other, or people may take some time to respond. An online 'conversation' can be spread out over a period of days, weeks or even months. Ingram and Hathorn, two researchers into the uses of online communication in education, describe the differences between these two modes of computer-mediated communication (CMC) as follows:

"CMC can be divided into synchronous and asynchronous modes. In synchronous communications all disciplines are online at the same time, while asynchronous communications occurs without time constraints. Synchronous discussion involves the use of programs, such as chat rooms, instant messengers or audio and video programs, in which all participants exchange messages in real-time. Messages appear on the screen almost immediately after they are typed, and many threads can occur simultaneously. Those who have experienced these rapid exchanges of information, ideas, and opinions know that even extraordinary typing skill and quick response times do not guarantee that one can keep up with the constantly changing discussion. Hence, synchronous discussion may be best suited for brainstorming and quickly sharing ideas. In asynchronous discussions students can participate at any time and from any location, without regard to what other discussants are doing. Asynchronous CMC allows participants to contribute to the discussion more equally because none of the customary limitations imposed by an instructor or class schedule apply. Full and free expression of ideas is possible. Although these communications are text-based, they have little in common with traditional printed information. Experienced users use a style that is characterised by a abbreviated writing and emoticons (eg smileys). asynchronous discussions, which can occur over email or threaded web discussion, allow more time for considered opinions… And are more effective for deeper discussion of ideas." (Ingram and Hathorn 2004: 220)

p85

[limitations, or why it is good to combine on-screen interactions with face-to-face interactions in real time in the same space, face-to-face]

Distance educators need to make the best use of the affordances of Digital technology to compensate for the loss of some of the most attractive and useful features of more traditional ways of teaching and learning [face-to-face].

And the open University, Rebecca Ferguson (Ferguson 2009; Ferguson, Whitelock and Littleton 2010). She was interested in how students working online managed the task of building knowledge and understanding together, as they pursued assignment in groups. [...] Ferguson also usefully identified some important ways that asynchronous online interactions among a group are different from those among people working face-to-face, in terms of the resources group members have to support their interthinking. For example, they often have digital improvable objects of the kind that we mentioned earlier. As she comments:

They do not need to employ devices that will help them to remember what they have said or done, because they have access to the complete text of their past dialogue in a transcript automatically generated by the software. What they need to replace is the range of tones, expressions and gestures are available to support sense making in a face-to-face setting. They must find a synchronous methods of agreeing on what they have achieved together, and on how they can shape past dialogue to build shared knowledge. At the same time, they need to avoid disagreements and find a way of moving dialogue forward safely when only a subset of the group is online and able to participate. (Ferguson up. cit.: 168)

The temporally extended, and even disjointed nature of online talk creates different kinds of obstacles to interthinking from face-to-face settings. Requests for explanations and checks of understanding are more labourious to make, as are the responses they require; and so any disagreements that arise are harder, to resolve.

p90

Various kinds of digital tools [...] can provide some valuable support for productive discussion. They can resource what Wegerif (2007, 2010) has called a 'dialogic space' in which different ideas, perspectives and understanding can be collectively explored, and material can be modified to record the development of a discussion and capture emerging ideas. Digital communication offers opportunities for students to interthink online, and to do so without the constraints of time and location that arise in more conventional educational settings.

...

more than one way of talking can be productive but discussion is likely to be most productive for learning if participants agree to follow the kinds of ground rules for discussion that will generate an online version of exploratory talk. Therefore, they should be expected to encourage universal participation among group members, seek ideas and clarification of them, challenge ideas and proposals in respectful ways if they have good reason to do so and support their own ideas and proposals with reasons and explanations.

p89

Cooperation versus collaboration

[I believe that for formative assessment to be effective, when students are asked to work in groups, it is better to pursue a cooperative approach rather than a collaborative approach. This way students are accountable for their individual contributions as opposed to all of the contributions being mixed into a melange where it becomes difficult if not impossible to ascertain the accountability or input of specific individuals]

"Cooperation is defined as the style of working, sometimes called "divide-and-conquer," in which students split an assignment into roughly equal pieces to be completed by the individuals, and then stitched together to finish the assignment.

 In contrast, we define collaboration as a more complex working together. Students discuss all parts of the assignment, adding and changing things in conjunction with one another as they come to understand more about the topic.

 At the end, the final product is truly a group product in which it is difficult or impossible to identify individual contributions. There appears to be differences between corporation and collaboration in both the complexity of the interactions and the effectiveness for instruction and education."

(Ingram and Hathorn 2004:216)

SAMMS

When people asking what learning ‘transformed’ by technology looks like, this is the kind of thing I think of, and the SAMMS framework helps here—this kind of activity, and its inherent,

Situation: 

- these kids can continue the conversation anywhere, any place (even international), face to face or any space, or time that works for them. I would argue the context of a wiki (as opposed to a shared IWB in the book) has the advantage of adding the situated affordance of technology, making interthinking much more effective, by augmenting it with a greater focus on the kinds of intrathinking and reflection that can be afforded by reworking or contributing to a group discussion in asynchronous isolation subsequent to a synchronous face to face session.

Accessibility: 

No need to debate minutiae and semantics when clarifying points of fact or fiction are only a click away, got a reference to back that up? Great, link to it. The wealth of online resources offers great potential for learners; but the context of interthinking places greater demands on all parties to evaluate and filter the information they offer.

Multimodality: 

Now for the most part, Interthinking assumes a text only mode, but adding the contact of face to face automatically makes it multimodal, but it actually it isn’t difficult to multiply the modes in a screen context, the examples above  include a video, or image prompt used by the teacher as a provocation, as well as students relating their ‘interthinking’ to multimodal content they have posted, from image to video, to mind maps and presentations.

Mutability: 

The ease with which students can modify their content facilitates learning, but needs to be managed carefully to avoid ‘revisionism’ this is the teachers call, some like to encourage kids to go back and revise their posts in light of their changing position, others see this as potentially dishonest—Did I say that? No I didn’t [quick edit] .. see?

Social Network: 

This activity literally creates a ‘micro social network’ like a Facebook the size of your class, with all of it’s phenomenal benefits, minus the suspicious disingenuous marketing.

References 

Erickson HL, Lanning LA, & French R (2017). Concept-based curriculum and instruction for the thinking classroom.

Ferguson R, Whitelock D, & Littleton K (2010). Improvable objects and attached dialogue: new literacy practices employed by learners to build knowledge together in asynchronous settings. Digital Culture and Education, 2 (1): 103-123

Ferguson R (2009). The Construction of Shared Knowledge through Asynchronous Dialogue, Unpublished PhD Thesis, The Open University. [Downloadable from: http://oro.open.ac.uk/id/eprint/19908]. Accessed January 22 2013.

Ingram A & Hathorn L (2004) 'Methods for Analysing Collaboration and Online Communications', in T Roberts (ed), Online Collaborative Learning: theory and practice, London: Information Science Publishing.

Littleton K, & Mercer N (2013). Interthinking: putting talk to work. Routledge.

Littleton K, & Kerawalla L (2012). Trajectories of inquiry learning, in K Littleton, E Scanlon and M Sharples (eds), Orchestrating Inquiry Learning, Abingdon: Routledge.

Wegerif R (2007). 'Dialogic, Education and Technology: expanding the space of learning, London: Springer Verlag.

Wegerif R (2010). 'Dialogic and teaching thinking with technology: opening, deepening and expanding the interface', in C Howe and K Littleton (eds), Educational Dialogues: understanding and promoting productive interaction, London: Routledge.