T1 Classroom Climate Hay McBer Research Study: A...

©Routledge/Taylor & Francis 2014

T1 Classroom Climate Hay McBer Research Study: A Summary

Tony Brown (2013) An interesting piece of research was completed by Hey McBer (DfEE (2000) Research into Teacher

Effectiveness: A Model of Teacher Effectiveness) and buried by the department in a deep online archive. It

built a picture of teacher effectiveness on qualitative and quantitative data that studies what effective

teachers do in their everyday practice, as part of the proposals following a Green Paper: Teachers, meeting

the challenge of change. The researchers collected statements from children:

Year 8 pupils said “a good teacher...”:

is kind;

is generous;

listens to you;

has faith in you;

keeps confidences;

likes teaching children;

likes teaching their subject;

takes time to explain things;

helps you when you’re stuck;

tells you how you are doing;

allows you to have your say;

doesn’t give up on you;

cares for your opinion;

makes you feel clever;

treats people equally;

stands up for you;

makes allowances;

tells the truth;

is forgiving.

The argument is that effective teachers make a difference to their pupils. The research identified three

main factors that are under teachers’ control, which can influence to enhance children’s learning and

development:

teaching skills;

professional characteristics;

classroom climate.

Teaching skills are seen as “micro-skills” learned and applied to maths and other subjects and dependent

over time by the broader professional characteristics that the teacher possesses: e.g. wanting to

incorporate more discursive teaching in the classroom; wanting to develop the use of Silent Way and other

strategies; analysing the new NC for opportunities to teach maths more effectively; …. the kind of decisions

that we all make at different points in our career to improve the quality of what we do. Teaching skills are


seen to develop as a result of these broader decisions – not by chance – but as a micro-set of skills within

“professional characteristics”.

One piece of evidence that emerged was the clear link between effective teaching of maths and “knowing

the subject”. It should be interpreted as: “some effective teachers may not know the subject well but, by

deepening their understanding of the subject, they become much more effective in developing children’s

learning and progress”.

Overall, effective teachers observed in the study had well over 90% of the pupils on tasks through their

lessons, and these lessons flowed naturally to achieve an appropriate balance between:

whole class interactive;

whole class lecture;

individual work;

collaborative group work;

classroom management;

testing and assessment.

The research developed a cluster of observations that provided evidence on what was called “classroom

climate” – “the collective perceptions by pupils of what it feels like to be a pupil in a particular teacher’s

class. In effective teachers’ classrooms, the students’ perceptions influenced every student’s motivation to

learn and perform to the best of their ability” (1.4.1). From the students’ perspective, effective teachers

create classroom environments where the pupils:

feel secure;

recognise there is order and routine;

feel they always have an opportunity to participate actively;

feel a strong sense of “belonging”;

are excited and interested in what takes place.

The classroom climate was measured on several dimensions:

1. Clarity

2. Order

3. Standards

4. Fairness

5. Participation

6. Support

7. Safety

8. Interest

9. Environment

References

DfEE (2000) Research into Teacher Effectiveness: A Model of Teacher Effectiveness. Research report 216 by

Hey McBer for the DfEE.


T2 Classroom Dialogue

Tony Brown (2013) Children need to talk and teachers need to listen.

The Five Nations Study is a large scale research study of primary classroom dialogue. Its focus was dialogic

teaching, which is a classroom teaching approach where both children and teachers make substantial

contributions to classroom talk, not only in terms of quantity but also, more importantly, in terms of

quality. The result is that children’s thinking is significantly developed (Mercer & Littleton, 2007).

(Alexander, 2005) identifies a number of elements. Dialogic teaching:

is collective and collaborative;

is a supportive and genuinely reciprocal interchange between all those in the classroom;

involves carefully structured verbal interactions that are gradually extended in length by teacher

intervention;

is promoted by the teacher to build understanding through accumulated experience;

makes children’s own words prominent;

ensures that children’s thoughts, ideas, speculations and arguments feature strongly.

Discussion in dialogic classrooms is very different from the more common classroom routines created by

teachers who seek to encourage question and answer sessions. In the most common classroom routines,

the teacher asks the questions – mostly closed – and expects students to bid competitively for the teacher’s

attention and the opportunity to give a single response. This is intended to result in relatively brief answers,

so that the teacher can test individual children’s responses to questions based on a particular focus.

In contrast, dialogic teaching is characterised by fairly long interactions. Some of these will be between a

teacher and a child or a group of children. More significantly, perhaps, the teacher creates opportunities for

extended dialogue between members of a group without much adult intervention. This means that groups

of children are very familiar with working collaboratively and supportively. It is the children, as well as the

teacher, who take responsibility for ensuring everyone is involved in decision-making, sharing viewpoints

and being heard in discussions.

Dialogic interactions occur in a variety of ways: during whole class, group or one-on-one settings. The

purpose is to help the individual child to work with ideas, practise thinking, present their thoughts and

ideas to an audience and build understanding through externalised talk that describes the concepts they

are working with.

Teachers who adopt this approach do so as a permanent feature of the classroom, not as a single lesson or

a short-practice ‘training’ slot. Dialogic teaching demands full commitment. During these interactions

teachers take on a meta-role, monitoring the extent to which children need dialogic practices to be

modelled for them. Teachers combine the teaching of the maths topic with explicitly taught strategies that

children can adopt to support reasoning, enquiry and negotiation.

The theoretical base for dialogic teaching is the argument that human intelligence develops primarily

through listening and speaking. Our ability to communicate and discuss our thoughts, feelings and ideas is

in large part, seen as determining the quality of our lives. Speech (both internal and external) is intrinsic to


literacy and to the human relationships we form with others. According to (Fisher, 2007) dialogue is the

basis of verbal and emotional intelligence.

The Five Nations Study was conducted in five countries – England, the US, France, India and Russia

(Alexander, 2000). The study focused on primary classroom practice and had a special focus on classroom

talk. The study showed that there were many points of similarity across the different participants. Dialogic

teaching methods featured most strongly in French and Russian classrooms and the children in these

classrooms clearly benefited in terms of classroom behaviour and learning and social development.

Vygotsky (1962) theorises that when children participate in guided interactions with more experienced

peers and adults, they more quickly acquire the “mental tools” required by their culture. Lave and

Wenger’s (1991) work on the benefits of communities of enquiry also suggests gains for children’s learning

in language-rich environments. Vygotsky argues that language tools begin as social products (i.e., external

to the young child) but are internalised as the child gains language skills and then uses language to create

thought.

“Scaffolding” is a widely used term for describing the activities of the teacher and more experienced peers

in supporting learning. The more experienced person can use language, activities and emotional warmth to

guide the child’s persona, linguistic and cultural construction of the ideas, skills and processes that are the

focus of learning. Alexander’s research shows the extent to which the teacher’s purposeful classroom

dialogue promotes this process of sharing knowledge and skills with the “apprentice”. Dialogue creates new

ideas and conditions for learners, whereby they have new thoughts that they might not have been able to

create at the time, through independent working and thinking.

Talk is the main mode for instruction in schools. In mathematics, as in many other subjects, much of the

content is abstract and needs description, explanation and metaphor to help make the abstract concepts

more tangible. However, in contrast to many subjects taught in school, maths lends itself to showing as well

as telling, but much of the telling is procedural and connected to “showing-types” of activity: this is how

you complete an addition calculation, this is how to use the number line for subtraction, this is how to use

an angle measure, etc.

Evidence for the essential role of dialogue comes not just from large-scale classroom studies like

Alexander’s, but also from recent brain research. There is evidence of critical changes in the brains of young

children, which take place on a scale that declines significantly as the child approaches adulthood.

Language-rich environments:

have a physical impact on the young child’s brain;

modify the shape of the brain and the neural connections;

cause an expansion of brain power;

stimulate the development of cells;

stimulate new neural connections;

enhance the capacity for learning, memorisation and emotional responses to learning;

stimulate new language acquisition.

There is evidence from psychological studies that language and thought are intimately related. The forms

and contexts of language to which children are exposed have a powerful influence on their cognitive

development. Childhood learning is essentially a process whereby children learn about social, cognitive and

emotional interactions. Children construct meaning most effectively in language-rich contexts where their


contribution is “scaffolded” to ensure full participation. They make sense of their world most effectively

when what they already know is challenged and modified by what they newly encounter. Discourse

provides the most effective medium for making connections between the two.

From a political perspective, the core values of dynamic, democratic societies include the freedom to

engage in listening, questioning, debating, arguing and justifying one’s thinking, tolerating the views of

others, presenting and evaluating ideas, etc. These are the crucial skills that children need to develop

through their daily practice in classrooms that emphasise dialogic teaching. They are also the activities that

nurture learning. Democracies are weakened when their citizens are denied opportunities to exercise these

skills, where listening, debating and freedom of thought are substituted for compliance.

Despite the research evidence that demonstrates the crucial role of classroom discourse, in too many

classrooms it is the teachers who still do most of the talking and children who do most of the listening. For

many children, their contribution to classroom talk occupies no more than a few seconds in every hour.

Only rarely do teachers’ questions help children to complete or elaborate their ideas. Most of the children’s

exchanges in classrooms are very short, and verbal responses in many classrooms last on average no more

than 5 seconds, often limited to three words or fewer.

The Five Nations Study shows that where teachers supported dialogue in their classrooms, they also

expressly taught children a number of important strategies to help them engage fully in extended dialogue.

Teachers in these classrooms deliberately train the children in dialogic techniques as the lessons progress.

They don’t just hope that dialogue will somehow break out in the classroom.

Children need guidance on improving expressiveness, the volume and clarity of voice, the need for

precision in vocabulary, grammar and syntax, and the acquisition of the distinctive terminology needed in

maths and other subjects. Children need to have more than one register: not just a friendly conversational

or colloquial style. There is a need for children to have access to a full dialogic style. They develop this best

when it is modelled by the teacher, and where they have opportunities in every lesson to practice refining

different registers for different purposes.

Dialogic discussion does not a lessening of teacher involvement: far from it. It is not a laissez-faire

approach. It is the role of the teacher that changes to ensure:

the teacher focuses on the enhancement of dialogic techniques;

teacher talk is reduced and student talk increased;

teacher talk includes guidance to children on the use of dialogic techniques;

the teacher works hard to develop the learning process by skilful interactions and interventions

that promote dialogue;

children are not left to discover important ideas, concepts and ways of interacting by chance.

Lave and Wenger’s (1991) legitimate peripheral participation is a useful tool for analysing children’s

behaviour in the dialogic classroom. The Five Nations Study shows that children do not have to be directly

involved in all dialogic exchanges to benefit from them. They need to be taught that active observation akin

to LPP is a legitimate activity, provided that they are intellectually engaged with what is happening, by:

watching another child participating in a dialogue with a teacher;

closely observing a more knowledgeable peer;

hearing peers discuss their thoughts about a concept, expressed in their own language;


watching their peers discussing and coming to know new ideas.

The teacher’s role is to assert the value of LPP, to suggest ways in which a child can observe legitimately

from a peripheral position, and to encourage this practice. When children have the opportunity to work as

observers on the edge of other children’s activity, they learn to internalise the tools that their peers are

using, and make them a part of their own knowledge and understanding.

Evidence from research (Nuthall, 2007) is clear that competitive bidding between children for a few

seconds of time in which to answer the teacher’s questions does little to develop learning. But this state of

affairs characterises traditional classroom question and answer routines, which are usually dominated by

children who Nuthall describes as “key students”.

When children’s experiences are dominated by competitive attempts to participate in classroom talk, they

learn:

that there is little opportunity for most children to become involved;

learning is a competitive activity in which they need to beat their peers;

that learning is about being seen to have the right answer;

that learning is an individual activity, not a collaborative, mutually supportive one.

By contrast, children who are fortunate to work in classrooms where dialogic discussion is effectively

managed by skilful teachers are shown that learning is not a competitive task, but a collaborative one,

where they are expected to receive and give peer support. Alexander (2006) emphasises this by describing

dialogic teaching as; collective, reciprocal, supportive, cumulative and purposeful.

T This time we are going to be sorting numbers. So that’s our objective – sorting numbers.

[Teacher takes on role of child with a grumpy expression] I’m going to work with Donal and Alan today and in my group I’ve decided I’m going to sort the numbers by multiples of three, and I don’t care what they think.

What’s the matter, Maya?

M You should, um, decide as a group.

T Oh super. There’s one of our ground rules already, “Decide as a group”. OK, how am I going to do that? Because I want to sort my numbers by multiples of three. How am I going to make sure that we decide as a group?

K Ask them what they think. Also, when you ask them what they think, don’t turn your back on them because that is not positive body language.

T You mentioned positive body language. What other type of language do we need to make sure is positive? Not just our body language.

C The way we talk.

T The way we talk! Am I going to say “I’m going to sort these in multiples of three!”?

C No

T Maya, what would you say if you were in my situation?

M Um, “I want to sort them by multiples of three. What do you think about it?”

T OK. I am wandering around the classroom . . . I wonder what I might hear you saying.


D What do you think?

T What do you think? Brilliant.

E Why do you think that?

T Why do you think that? That’s another good one, not just what but why you think that. Brilliant!

From: (Mercer & Littleton, 2007)

Where children are offered opportunities to make substantial contributions to classroom talk, they are able

to develop and practise a range of important speaking and thinking skills. These include the ability to:

narrate;

explain;

instruct;

ask different kinds of question;

receive, act and build upon answers;

analyse and solve problems;

speculate and imagine new possibilities;

explore and evaluate ideas;

discuss, argue, reason and justify;

negotiate.

In addition children develop four essential abilities for interacting productively:

listening;

being receptive to alternative viewpoints;

thinking about what they hear;

giving others time to think.

The following excerpt demonstrates the communication skills of children in fifth grade working as a maths

group:

S1 Five, seven and five equals twelve. So put five.

S2 Do you agree?

S3 Yes, and then we need to sort this out. [A little later]

S1 I know, why don’t we use the seven again?

S3 What do we do now?

S1 What do you think we should do now?

S2 I don’t know, it’s too hard. I have never done this before.

S3 I haven’t done this before.

S1 What can we remember? A blank square. All I remember is numbers. Eight plus one is nine.

From: (Mercer & Littleton, 2007)


References

Alexander, R. (2000) Culture and Pedagogy: International Comparisons in Primary Education. Malden, MA:

Blackwell.

Alexander, R. (2005) Culture, dialogue and learning: Notes on an emerging pedagogy. Paper delivered at the

Conference of the International Association for Cognitive Education and Psychology, University of Durham,

UK, 10–14 July, 2005. Retrieved November 18, 2013 from:

http://www.robinalexander.org.uk/docs/IACEP_paper_050612.pdf

Alexander, R. (2006) Towards Dialogic Teaching: Rethinking Classroom Talk. (3rd ed.) Cambridge, UK:

Dialogos.

Mercer, N. & Littleton, K. (2007) Dialogue and the Development of Children’s Thinking. Routledge: London

Fisher, R. (2007) “Dialogic teaching: Developing thinking and metacognition through philosophical

discussion.” Early Childhood Development and Care, 177(6–7), 615–631.

Lave, J. & Wenger, E. (1991) Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge

University Press.

Nuthall, G. (2007) The Hidden Lives of Learners. Wellington: NZCER Press.

Vygotsky, L. S. (1962) Thought and Language. New York: Wiley.

http://www.robinalexander.org.uk/docs/IACEP_paper_050612.pdf


T3 Daniel Kahneman’s Thinking, Fast and Slow: A Review for Teaching

Tony Brown (2013) Rationality and intuition essential for learning maths.

The starting point for The Really Useful Maths Book is that knowing more about how the mind functions

brings us closer to an understanding of children’s learning: something we argue is useful to teachers.

Kahneman’s approach is unusual: he is a psychologist interested in economics and economic decision-

making. His work conclusively demonstrates that economists are no more rational than any other people

when making decisions, and Kahneman’s evidence shows that we aren’t very rational at all! Looking back

over bankers’ behaviour in the years from 2005, we could probably work this out for ourselves – but

Kahneman demonstrates how and why we think the way we do.

Our two modes of thought are what Kahneman calls System 1 and System 2. System 1 is fast, often working

at a level below our consciousness threshold, meaning we are unaware of the processes we use and the

basis of the decisions that System 1 promotes. It is instinctive and linked to emotion, using association and

metaphor to produce quick and dirty solutions.

System 2 is deliberative, more inclined to make use of logic, slow in comparison to System 1 and inherently

lazy – out for the easy life. Kahneman provides evidence that suggests we place too much confidence in

human judgment. He identifies two thinking selves: one that organises judgments based on current

experience and another that makes judgments based on what can be remembered from the past. His

research shows that even a ghastly holiday will be remembered more affectionately if the final hours were

slightly more enjoyable (or less painful) than the earlier part of the holiday. It’s not the duration of pain that

the remembering self cares about: it’s the major peaks and troughs and the ending that matter.

Obviously System 1 and System 2 don’t exist, but we need some metaphors to help us grasp some

understanding of our thinking processes. Our experiencing self can be faced with the most painful period of

our lives, but this is dismissed by our enduring remembering self when looking back, weighing up the

experience and making a judgment in terms of how happy we are about it now. “Odd as it may seem,”

Kahneman writes, “I am my remembering self, and the experiencing self, who does my living, is like a

stranger to me.” Kahneman is emphatic that all of us, and especially experts, are prone to exaggerate their

professional decision-making ability and how well they understand the world.

He argues that much of our decision making is based on irrational thinking. According to evidence gathered

from decades of experimentation, this is how it works. System 1 takes the lead because it responds so

much more swiftly than System 2. System 1 is quick, uses intuitive links and associations to throw up the

best solution to a problem that it can find.

The second problem with our two stage thinking process is that we don’t realise what System 1 has done.

We fail to realise that a two stage process has happened. Instead we only “notice” System 2.

System 1 has supplied some wacky off the cuff suggestions to System 2;

System 2 doesn’t recognise that the options were generated by System 1;

System 2 takes its starting point from the options generated by System 1;

System 2 is convinced it has started with a clean sheet, been entirely rational, done all the (careful

and logical) thinking all by itself.


System 2 is lazy – it can do rational thinking but knows that thinking carefully is tiresome, hard

work and it prefers not to do that if it can find an easy way out.

Kahneman writes that his work challenges the view that people are generally rational. Instead, his research

shows “systematic errors in the thinking of normal people”, which are built into the way our brains operate.

The challenge for teachers of mathematics is to recognise that rational thought, separate and pure, does

not exist. Instead, we all need to recognise as far as possible where our ideas come from, where our

intuitions arise and how to use them without being over-confident. The objective, if it is achievable, is to

catch System 2 before it concludes its decision-making process and enquire as to how influential System 1

has been in selecting the options.

Kahneman writes, “Although System 2 believes itself to be where the action is … the automatic System 1 is

the hero of this book.” System 2 is especially likely to roll over and be bossed about when our mood is a

happy one.

References

Kahneman, D. (2011) Thinking, Fast and Slow. London: Allen Lane.


T4 Discussion of TIMSS and PISA:

Tony Brown (2013)

International performance comparators and their political use

One of the opportunities afforded by international testing is to ask the question: Is there anything we can

learn from the way other countries organise the teaching of mathematics? However, because the testing

process and the results have become politicised, it is difficult for educators to consider the question in a

disinterested way. This is partly because of the way the media report information – often siding with one or

other political trend or viewpoint.

Where can we look for detailed but (politically) unbiased comment? The Mathematical Association (MA)

has a non-searchable website with a section, “We Say”, containing a “Documents Archive” that does not

reveal anything covering TIMMS or PISA.

The Association of Teachers of Mathematics (ATM) website has a section dedicated to “opinion” which

contains, for example, a discussion of vedic maths.

Typing TIMMS in the search box returns a discussion that includes TIMMS and PISA data, by John White

(Hon Sec).

If testing data could be used by educationalists to indicate which countries were doing something right in

their education processes, it would offer opportunities for practitioners and researchers to study exactly

what that could mean for another country. It’s naïve to assume that many teachers will have the resources

to carry out investigations abroad, or that international comparisons could be culture-free, where one

simply takes a Japanese or Hungarian schooling practice and copies it as closely as one can in ways that are

exactly similar.

Apart from the element of competition, (certainly better than going to war to prove superiority), there isn’t

an obvious way in which classroom teachers can benefit from international datasets, unless one finds ways

to go behind the data and into the professional practices of teaching colleagues from other countries,

which could stimulate curiosity about new practices that one could trial in one’s own school as a “research

project” – perhaps using the Lesson Study [Link to pdf T7] approach over time to identify whether adopting

practices from other countries leads to improvements in one’s own classroom.

Then there is the data itself. What exactly is it? How does it vary over time? Which countries participate? Is

this a stable group? What do we mean when we think about successful education systems? Where do we

look for evidence of quality? How would we judge:

a country where a moderate proportion of students do very well, and a largish proportion do quite

badly with a widening gap between the best and worst performing students?;

a school system where there is a narrowing of the gap and a general upward mobility of

attainment?;

a country where PISA scores are not outstanding but a moderate proportion of students continue

to be involved in both formal and informal education?

a society where children appear to enjoy school, seek jobs involving maths, and where the

population appears to be successful and happy in general?

Both Hong Kong and Finland do quite well in the international tests. Is it likely:

http://www.atm.org.uk/community/previous/ATM-Opinion-Forsythe.pdf

http://www.atm.org.uk/journal/archive/mt221files/ATM-MT221-46-47.pdf


that they achieve their scores for the same reasons?;

that they achieve their scores for very different reasons and with very different school practices?;

that trying to copy one or the other would be for entirely neutral reasons, or would cultural

attitudes be likely to play a part in our choice of which to copy and what bits to duplicate?

Maybe we should take notice of a derogatory acronym – GERM – from a Finnish scholar, which stands for

Global Educational Reform Movement. Maybe in reality this bit of globalisation is not such a good idea. Can

we really claim that we can copy another school system with complete success? Perhaps it’s just PISA envy?

There are six nations that scored at the top of the 1995 TIMSS rankings in eighth-grade math. They are

Belgium (Flemish community), the Czech Republic, Hong Kong, Japan, Korea and Singapore.

In 2008 they were held up as examples of who to imitate, but the Czech Republic left the TIMSS study after

2007 when its TIMSS score had fallen 42 points compared to its score twelve years before.

Belgium (Flemish community) has not participated in TIMSS since 2003 at which time its TIMSS score had

dropped 13 points. The other four countries all took TIMSS in 2011. Hong Kong saw a rise of +17 points, and

Korea’s score rose by +32, which is a significant gain. Japan experienced a decline of -11 points and the

score for Singapore was little changed at +2. Of these “top” six, there were statistically significant increases

for two and significant losses for three, with one scoring pretty much the same.

The tendency is that we focus on top performers when test scores are released. We have been trained to

assume that they are doing something secret or mysterious that causes magical results, and that something

like a school system in a country can be reduced to a list of independent factors that we can observe,

measure and import. Causality is difficult to assess. There’s the curriculum of course, but we only have to

think about the UK changes to curriculum in the last 20 years to realise that curriculum isn’t a static single

entity; it’s a fluid, changing set of complex activities, rather than a stack of pieces of paper.

Culturally, different nations prize education in different ways, ascribe status in different ways, value

parenting practices in different ways, value individual effort and stamina in different ways, view

collaboration in different ways, value and reward teachers differently, treat the children of migrants in

different ways, use state and private education in different ways, provide access to university in different

ways, …


Links

OECD Education at a Glance (2012)

OECD Education at a Glance: United Kingdom (2012)

Trends in International Mathematics and Science Study (2011)

Programme for International Student Assessment

http://www.oecd.org/edu/eag.htm

http://www.oecd.org/edu/United%20Kingdom_EAG2013%20Country%20Note.pdf

http://timss.bc.edu/timss2011/index.html

http://www.oecd.org/pisa/


T5 Gattegno’s Key Ideas

Tony Brown (2013) Caleb Gattegno was an Egyptian educator whose work extended around the world from the 1950s until the

1980s. He is known by language teachers as a major contributor to teaching languages through his Silent

Way approach. This involved groups of language learners working with a chart in one of many languages.

Gattegno could speak fluently and teach in Spanish, Chinese, English, French and other languages.

Following Gattegno’s Silent Way language work, the teacher would ensure the students could see the chart

and get to it if necessary. The teacher would utter a short phrase (I am David; Je m’appelle David …) in the

target language that students wished to learn and would encourage the audience to repeat the phrase,

with individuals inserting their own name.

When students’ phrasing, accent or intonation was repeated inaccurately, the teacher directed the

students’ attention to the key letter or character symbols on the chart which represented the appropriate

phoneme. With a finger or stick the teacher taps the chart whilst repeating the spoken phrase thus making

a link between each spoken unit of speech and the corresponding written symbol.

Students repeat the phrase, those who are closest in articulation are identified by the teacher and

encouraged to repeat, whilst the other students listen and try to adjust their response to more closely

match that of the student who is making the correct utterance. This listening and copying continues until

the majority are speaking the phrase accurately in the target language. The teacher then moves to offering

a response to the initial phrase.

The teacher remains largely silent during this process, only speaking to reinforce sound-letter links by

pointing at a letter and uttering the correct sound: or when introducing a new phase (What’s your name?;

Comment tu t’appelles?; Come ti chiami? …) and tapping the chart to show the sounds as they are being

used.

A polymath, Gattegno: translated Piaget’s work into English; encouraged the study of psychology as a

means to better teaching; gave demonstration lessons in many countries and many languages to encourage

language and maths teaching; established a national association in England that would become the

Association of Teachers of Mathematics (ATM); developed the use of rods (manipulatives) and created the

Cuisenaire company; and developed a science of education intended to inform educators and encourage

teachers to base their work on the psychology of children’s abilities and aptitudes for learning. His writings

and the software produced by Educational Solutions encompassed literacy, language learning and

mathematics. He was especially interested in displaced persons and the need for adult learners to acquire a

new language.

In The Really Useful Maths Book (RUM), we make use of the Silent Way approach to teach the structure of

the Arabic number system – a topic that many teachers ignore, assuming that children will somehow

acquire an understanding of the system merely through acquaintance and absorption.

Gattegno gave many demonstrations and workshops on Silent Way methods in mathematics, which the

authors attended in Bristol and London during the early 1980s. Much of this work was brought together by

Geoff Faux who was a highly active member of the ATM and maths advisor for Cumbria. His collection of

the charts and his continuation of Silent Way workshops helped enormously in consolidating these teaching

practices.


Perhaps the best way to begin is for a group of colleagues in a single school to visit a local branch of the

ATM with the specific intention of learning to use Silent Way and Cuisenaire methods in the classroom.

Working together after the initial support of ATM members, teachers can practise on each other and

introduce the methods into their classrooms in ways that allow them to reflect, discuss and compare the

effectiveness of different classroom techniques, so that teaching can develop directly from enquiries into

practice. This approach has important links with the Lesson Study [Link to pdf T7] – a Japanese school

enhancement method described in RUM and on the CW.

Link

http://www.calebgattegno.org/



T6 Language and word problems

Tony Brown (2013) Improving classroom talk and relationships, questions and questioning.

There are two main considerations to be made. The first concerns quality and the second concerns

quantity.

Teachers need to ensure that their questions support learning: better questioning in this sense means

better quality learning. High quality questions are those which cover a range of levels, from the need to

provide children with basic information to higher order questioning that tests children’s understanding, to

the highest order, where teacher’s questions encourage children to:

investigate mathematical problems;

discuss with peers and explain their reasoning in collaborative activity;

express their thinking and their reasoning using their own language;

refine their language to accommodate the technical and specialist vocabulary of mathematics;

use their language skills to question, review, hypothesise and predict the outcomes of purposeful

activity.

The second issue is about quantity. In many classrooms, the teacher asks many questions and teacher talk

accounts for the largest proportion of classroom talk. What is left is shared between all the children,

meaning that:

a few key children provide most of the pupil talk, while most children say almost nothing of

relevance to the lesson;

children’s responses to questions rarely extend beyond a few words;

children’s responses rarely engage them in higher order thinking;

children’s talk is almost always directed towards answering the teacher, seldom in dialogue with

peers.

Treating maths problems as language problems is a strategy reported by the Russian psychologist, Krutetski

(see page 231 in The Really Useful Maths Book). The teacher who was being observed treated maths word

problems as a special form of language problems, and taught the children in her primary school classroom

to explore the meaning of the language found in a specific maths word problem before attempting to tackle

the mathematics.

Teacher’s questions can be used to encourage children to apply the skills developed in language lessons to

a study of word problems in maths – where maths is seen as an example of a special language genre.

In this clip of a language lesson, the teacher prepares the study of a text by reading it to the children in a

lesson on day 1 and asking them to prepare questions at home that they will ask another child during the

lesson on day 2.

Analysing texts by choosing teacher-prepared strategies

The sequence in a maths lesson prepared in this way could be:

1. Select or create a word problem that refers to mathematical knowledge and skills that have just

been practised.

https://www.teachingchannel.org/videos/teaching-higher-order-thinking-skills


[The school is planning a beach trip in May which will take a whole day. There are 321 children and

teachers who will go to the beach. The bus company has two types of buses. One bus holds 52

people and costs £65 for the day. The other bus holds 38 people and costs £50 for the day. What’s

the cheapest way of arranging the beach trip, making sure that all 321 people can go? Will there be

any seats left over that could be given free of charge to some parents who would like to come too?

2. Give the problem in written form so children can use highlighters or pencils to mark the text. Get

them to read the text and identify important words and unimportant words. Encourage them to ask

questions about the problem and begin to write these down on a display for all to see. Ask them to

take the problem home and to write some questions down – including questions about being stuck,

not being sure, etc.

3. In the next lesson, invite the children to read the question to the class. Ask a few individuals to read

out their questions to the class and encourage other children to provide answers. Then move to

pairs and ask each child to give their question to their partner, who has to think about how to

provide an answer.

4. Work on the text with the whole class. Ask children to work in pairs to identify relevant words and

to explain to their partner why they think a word is important. Do they agree? Then bring them

together to hear some suggestions to ensure everyone understands the task. Then get back into

pairs to complete the task. Work through the text in the same way, with pairs of children

identifying irrelevant words. Use highlighters and colour codes to break up the text.

5. Now we are ready to work with the children on higher order language discussion. What is this

problem about? Can they work in pairs or threes and fours to come up with a good description of

what the problem is? What is it they have to find out? When they’ve worked together, they need to

combine their ideas and produce the best written explanation they can. Next, share some of these

statements with the whole class. Get children to query each other’s statements until a compromise

is reached where most people can explain in their own words what it is they have to find out.

6. Depending on the word problem, there will be some specialist words and phrases that they need to

know, recall, practise or explain to their partner/group.

7. Let them work in pairs, threes or fours to come up with a way of recording what they are going to

do, so that another group will understand what they are up to. It could be a storyboard, a sequence

of drawings, diagrams or steps with short phrases or statements. When most people have got

something on paper ask several groups to share and explain what they intend to do. Invite other

groups to comment. The purpose is for each pair or group to have a way of working towards a

solution that they have created and perhaps adapted following comments from others: not looking

for a standard way that everyone has to follow.

8. You can introduce key words into the discussion that you want the children to use – maybe they

can use these words already – hypothesise, calculate, predict, explain, etc. Encourage them to use

these words in their discussions.


9. When they are clear that they have an understanding of the problem and a possible way of

working, which includes drawing a plan, design or a picture that helps their thinking, the children

can begin to work in groups towards a solution. Some may need help if their method of

representation of the steps isn’t effective in guiding them.

There are several video sources that show teachers structuring work with particular texts. It’s worth looking

at teachers working in English and other lessons. We can analyse the steps they take to help children

engage with texts and work with higher order concepts.

In this English lesson video with older students, the teacher has a structure for questioning children – she

wants to move from student experience to content, to big picture. How could this work in maths? The

actual lesson content and the age of the students is not relevant, but her planning is. She starts by wanting

maximum involvement, so she asks questions around what they think, feel, like, dislike or imagine in

relation to the chosen text. She then chooses a content question that helps students to analyse the text: in

the case of a maths word problem this could be the children’s knowledge of specialist words and specialist

meanings (like forwards and backwards on a number line, vertices, edges in relation to shapes and whether

they know what the specialist maths language requires you to do). The proportion of teacher talk is low and

the students get to make extended contributions to debate. Only then does she move to the “big picture” –

what is this asking us to do? What can the children predict might happen and why do they think that?

In the next video, the teacher sets up the problem (write some equations that equal 2.4) working with the

children in a plenary session before they work in pairs. The teacher reminds the children of the ways to

work and reminds them of key strategies: discuss, share, reach consensus, etc. He tells them how he

expects them to work; to be creative and find original ideas and solutions. The children need to work in

pairs to explain what they intend to do.

This video shows a group working on a problem about a trapezoid. The teacher is faced with a dominant or

key child, who is influential in the group, who doesn’t appear to know what to do but is exercising some

influence over the others. The teacher’s style is to ask questions for clarification for herself, to challenge

thinking and to help the group review what they are doing. Her style helps focus the group and promotes

lengthy responses from the children.

This trapezoid is composed of a square and a triangle. The length of one of the trapezoid’s bases is three

times that of the other. Assume that the trapezoid has an area of 32 square units. What is the area of the

square? What is the length of the sides of the square? What is the area of the triangle? What is the length

of the base? Its height?

https://www.teachingchannel.org/videos/structuring-questioning-in-classroom

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

http://bionicteaching.com/?p=1514


T7 Lesson Study

Tony Brown (2013) Lesson Study is a professional development process or programme associated initially with Japanese

schools from the 1990s, see Yoshida (1999). It is Japanese approach to CPD, where teachers jointly plan,

observe lessons and review, and was taken up and explored extensively by US educators, see Fernandez1

(2002).

There appears to be little related activity from 2010 onwards in US colleges and universities so it is unclear

the extent to which Lesson Study has been taken up by US teachers and educators. An extensive and

valuable discussion of Lesson Study in Japan is provided by Makoto Yoshida online.

Lesson study, Jugyokenkyu in Japanese, has played an important role in the development of mathematics

teaching in Japan, though it features as a developmental tool for all curriculum subjects. It is a familiar and

important part of professional development for Japanese teachers, and its purpose is to raise the quality of

the learning experiences they provide to their students.

The form that Lesson Study takes has been adapted to suit the many different schools and professional

development needs of Japanese teachers. The core of the activity is that teachers plan, teach, observe and

review lessons over a long period of time – a year programme on a single Lesson Study theme is typical.

Lesson Study is part of Konaikenshu, or school-based professional development, where school-wide groups

of teachers collaborate. Lesson Study also features across groups of schools and, in Japan, it is common for

Lesson Study to be organised by school clusters in a region with voluntary teacher groups (e.g., math study

groups) or as part of an induction year programme for newly qualified teachers.

Lesson Study has three key stages:

1. identifying the group of participants and the agreed lesson study research theme;

2. conducting a small number of research lessons that explore this theme;

3. meeting to discuss the lessons, reflect on the process and write reports.

A mathematics focus for Lesson Study in a Japanese school will typically follow the same research theme

and the same content area for three to four years, especially if it is associated with Konaikenshu, where a

school or a cluster of schools seek to improve student learning through greater consistency in classroom

practices.

A lot of time has to be made available for Lesson Study. Frequent and regular meetings are needed

throughout. It takes a long time to negotiate the research lesson because each teacher in the study group

needs to teach it, so they need to establish a common basis for discussion, comparison and review of the

research lesson.

Key points for discussion include:

1 Abstract of Fernandez’ paper: This article first describes the Japanese professional development practice of lesson study and its articulation

within the Japanese educational system. Next, insights gained from an empirical study that explored the feasibility of lesson study in a U.S. setting are discussed. More specifically, challenges to lesson study practice are highlighted, with particular attention paid to the difficulties faced by American teachers in trying to adopt the research focus that is inherent in lesson study. The article concludes with reflections about what the study of lesson study can teach us about efforts to improve teaching, which, like lesson study, center on having teachers examine their practice or that of others.

http://www.rbs.org/Special-Topics/Lesson-Study/Lesson-Study-Conference-2002/Overview-of-Lesson-Study-in-Japan/206/


what maths the lesson is intended to teach;

how it is presented in the different textbooks and resources used by the teachers;

how the lesson, and the topic it is associated with, relate to the wider maths curriculum;

identifying the key mathematical concepts to be taught;

the children’s existing knowledge and their current state of understanding;

how the research lesson fits into the rest of the teaching associated with the topic to be taught;

the teachers’ personal goals for the research lesson;

the overall primary goal of the lesson.

Resources can be found at:

1. Global Education Resources

2. Lesson Study Research Group (archived resources)

3. Lesson Study in Japan

4. Research for Better Schools

5. Centre for Innovation in Mathematics Teaching (CIMT)

Extract from p7: “Many have credited the steady improvement of Japanese elementary mathematics and

science instruction to their teachers undertaking ‘lesson study’. This is a process in which teachers jointly

plan, observe, analyse and refine actual classroom lessons, called ‘research lessons’. It is a widespread

initiative in Japan, and an integral part of their pre-service teacher training programmes. In addition, lesson

study forms the basis of continuing professional development for teachers within their own school and

sometimes across schools within a district.”

Resources

Fernandez, C. (2002) Learning from Japanese approaches to professional development: The case of lesson

study. Journal of Teacher Education, 53(5), 393–405.

Yoshida, M. (1999). Lesson Study: A Case Study of a Japanese Approach to Improving Instruction Through

School-Based Teacher Development. Unpublished doctoral dissertation, The University of Chicago.

http://www.globaledresources.com/events/2007/LSIP/main.html

http://www.tc.columbia.edu/lessonstudy/

http://www.lessonresearch.net/

http://www.rbs.org/Special-Topics/Lesson-Study/Lesson-Study-Conference-2002/Overview-of-Lesson-Study-in-Japan/206/

http://www.cimt.plymouth.ac.uk/papers/lessonstudy.pdf


T8 Locating reality

Tony Brown (2013)

The power of pedagogical documentation to speak ACTION to teachers

According to Ballard (1973), in the past we took the external world as representing reality, however much it

confused us. It contrasted with our inner world, our imagination and fantasies, hopes and fears, which we

took to represent a fictional world. These roles, he argues, have been reversed and now the prudent step

for us is to deal with the world around us as the fiction and our inner world as the reality.

The diet we are fed by external sources – mass media, advertising, big business, politicians – appear less

and less part of a reliable, trustable world. These sources need us so much that they constantly seek to tell

us new fictions about what they imagine we want. So slippery, ungraspable, fleeting and unreliable has the

outer world become that the best strategy for dealing with it is to see it as a complete fiction. In contrast,

what we believe in, know for sure, what we are committed to, comes from the reality of an inner world

that we can rely on with increasing confidence.

There are alternatives to the dominant discourses of schooling and education that are being promoted by

rich individuals who want to run schools, and politicians who see education only in terms of missed revenue

opportunities for big business. What teachers, children and the great majority of parents possess with a

considerable degree of congruity is a powerful inner vision of what education should and can be. This is the

reality that denies the fictional stories from the external world that wish to marketise children, their

childhood and their education.

One way of challenging orthodoxy is to look for alternative discourses and that is what Lenz Taguchi (2010)

does in her book, Going Beyond the Theory/Practice Divide in Early Childhood Education. Taguchi follows a

feminist post-structural approach to educational practices. She writes about embracing reflexive,

deconstructionist collaboration, together with a number of EY practitioner colleagues. Lenz Taguchi seeks

to “trouble and challenge what is going on in the education arena today, where pedagogical practices are

being increasingly mainstreamed and normalised” (p. 4). She uses the Agential Realism theories of Karen

Barad (2007) Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and

Meaning. Barad’s conjecture is that the world is made up of phenomena. Objects do not precede their

interactions. Rather they emerge in space–time as a consequence of particular intra-actions. This

perspective blurs the distinction between human and nonhuman, heralding an important challenge to

individualist metaphysics. As distinct from actor–network theory, Barad’s view is that phenomena are not

gatherings of humans and nonhumans. The world of artefacts emerges as a result of conditions of

possibility following the laws of physics and the speculations of Neils Bohr, among others.

In her book, Lenz Taguchi’s approach is to innovate around discourses of Early Years pedagogy, seeing it as

an emergence that is consequent upon intra-activity. This allows her to develop her discussion of EY

education so as to shift the gaze toward intra-active relationships and away from the dominant discourse

built around interpersonal relationships. Interestingly this may be seen as establishing similar viewpoints to

those held by the young baby, who makes few distinctions between self–other; inside–outside; living–

nonliving, learning–experience. Lenz Taguchi’s approach blurs distinctions between living organisms and

the totality of the physical environment – “things”, artefacts, spaces and places.


This collapsing of independences in favour of agential realism requires a collapse of epistemology and

ontology into an onto-epistemology; a blurred intra-relational space which includes notions of being

(ontology) and notions of knowing (epistemology).

What this preliminary argument does for learning and teaching is the heart of Lenz Taguchi’s book. Gone

are the binary divides; theory–practice, science–philosophy, reality–discourse and nature–culture. In their

place is pedagogy, emerging from practice, from discourse between professionals but crucially also

between people and documentation. From this perspective, both the human and the nonhuman evidential

records of pedagogy demonstrate agency.

The totality of children’s lived experience in formal settings emerges through the agential consequences of

the intra-relational dynamics of human and nonhuman resources, where pedagogical documentation

occupies a significant role because of the powerful condensation of intentionality that it contains. Thus,

pedagogical documentation has performative agency. “[T]he photograph, sketch, or written words of an

observation will also put things in motion by means of its own agentic force and materiality. Thus, new

possibilities for intra-action with other matter and organisms will emerge” (p. 64).

References

Ballard, J. G. (1973) Crash. London: Vintage.

Barad, K. (2007) Meeting the Universe Halfway: Quantum Physics and the Entanglement of Matter and

Meaning. Durham, NC: Duke University Press.

Taguchi, L. H. (2010) Going Beyond the Theory/Practice Divide in Early Childhood Education: Introducing an

Intra-active Pedagogy. London: Routledge.


T9 Neuroscience is a collection of disciplines

Tony Brown (2013)

Can neuroscience talk sense to education and politics?

Hard science – neurology, molecular biology, physiology, genetics – and softer sciences like psychology.

Brain imaging techniques have made an enormous contribution to our understanding of brain function.

Some neuroscientists behave as though their work is divorced from the social world of humans and that

cannot be good news. Neuroscience is a social science, no matter how hard and techie the component

disciplines that supply a lot of the knowledge base.

Education resists science: many teachers preferring to see it as an art, a practice, despite the obvious need

for training to precede qualification. For Lib-Con alliance politicians, education is an expensive

inconvenience. They are busy preparing the complete sell-off to big business who will be encouraged to sell

education to us. Neuroscience needs to speak to educators. An open and frank discussion is overdue, not

helped by the reticence of some educators to face the scientific facts about some of education’s practices.

Neuroscience is prey to those who want to use its findings to justify their own position, even where this is

misogynistic, anti-democratic or elitist. It can be used to reinforce stereotypical views of any group. It is

vulnerable to latter day eugenicists who want to argue that education is wasted on the young, the poor, the

outsiders, the “others”.

Good neuroscience experimentation has debunked the nonsense of “right-brain” means arty and creative

while “left-brain” means scientific and rational. Nevertheless, there are plenty of books out there peddling

the nonsense – because even if it doesn’t make for good education theory and practice, it makes money.

The news from neuroscience is that the brain is plastic – it develops, changes, self-organises, self-modifies

and recovers from damage in amazing ways. It remains plastic for much longer than we anticipated. It

continues to develop and show increasing capability beyond childhood and into adulthood. Intensive, full-

time education can have dramatic effects beyond 16, 18 and even 20, and education needs to talk to

neuroscience about what is possible at all ages from birth onwards.

The political elite are currently busy ignoring the findings of neuroscience in their grab for votes. They want

an education system that models their political ideology. They want to twist neuroscience findings so they

can argue that selection is good for 11-year-olds, that children’s brains are hard-wired by the age of 5 and

that some children from elite families will do well in “grammar schools” with the rest of the nation’s

children dumped into “education factories” that are tied to a national curriculum which prepares them as

factory fodder to receive instructions from their more intelligent betters, rather than encouraged to use

their creative potential to solve problems and contribute directly to the economic and social wealth of the

country. Many top scientists are appalled by the politicians’ views of elitism based on nonsensical

arguments of “innate intelligence” and “fixed potential”.

Neuroscience has shown that brain development is not usefully described by a biological model. Our

development is not just genetically determined. Our brains are made the way they are by experience,

culture and, of course, the quality of education we experience. There are many more similarities between

our brains than there are differences.

Reference

The Royal Society (2011) Brain Waves 2: Neuroscience: Implications for education and lifelong learning.

http://royalsociety.org/policy/projects/brain-waves/education-lifelong-learning/


T10 Situated Learning Theory of Lave and Wenger

Tony Brown (2013)

Legitimate Peripheral Participation: Community of practice

Situated learning was first proposed and described by Jean Lave and Etienne Wenger (1991) as a theoretical

description of learning in a community of practice. Their immediate definitions related to non-school based

learning with adult learners. A situated learning space is one where learning and its application takes place

in the same location. The community implies a group of people willing to work together and prepared to

support each other’s coming to know. Some members will be core, long-term leaders, while others will be

newcomers often with a peripheral role, at least for a time.

The type of learning characterised by Lave and Wenger is highly influenced by socialisation and imitation.

Transmission models of teaching as experienced in schools are not the key ones focused on by the authors.

The community typically comes together to solve a problem (e.g. alcoholism, problems of midwifery and

antenatal care, etc.). Learners who seek communities of enquiry do so because they have shared interests

and seek to benefit from the knowledge of others who may be more experienced or more knowledgeable.

There are connections between Lave and Wenger’s work and earlier observations by Margaret Donaldson

(1978) reported in Children’s Minds. Donaldson critiqued some of the experimental work of Piaget whilst

broadly supporting his constructivist theories. What she argues is that young children are highly sensitive to

social contexts and classroom dynamics, so much so that they will suspend commonsense, logic and “what

they know to be true” in order to comply with what they think teachers want and expect. Even to the point

of absurdity, children will ignore knowledge and experience in order to “fit in” with the classroom culture.

Donaldson skilfully created versions of Piaget’s experiments which he used to establish a theory of

children’s development. By manipulating the social and cultural contexts whilst leaving the essential logic of

the problem unchanged – Donaldson showed that many children were perfectly able to complete tasks that

were supposedly beyond them. What mattered was the context. And it is contexts for learning that are the

core of Lave and Wenger’s work.

Richard Skemp, a mathematical pedagogue, introduced the idea of relational vs. instrumental learning,

arguing from a different perspective from Donaldson, but in a similar way, that learning is more powerful

and leads to more securely held knowledge when it is acquired in terms of relational elements, i.e., in

terms of its associations.

Zoe Evans’s work, much of it with enjoyably tactile fabrics and toys, which children delight in handling,

shows from yet another perspective that very young children can solve complex logic tasks using these

attractive resources, because the children identify with them as toys to play with and to which they (the

children) can ascribe rules. For many children, this freedom for rule-making is distinctly different from the

teacher’s classroom artefacts, which only the teacher knows how to control and to which teacher-laws

apply. Children can quickly solve problems of deciding which (logically different) toys “want” to be together

long before they can perform the same mathematical task with formal classroom objects. Again, context is

the key in shaping the quality of the learning.

Legitimate Peripheral Participation (LPP) is a phrase emerging from the work of Jean Lave and Etienne

Wenger, that helped them to theorise their data on a number of adult groups (groups of non-drinking


members of Alcoholics Anonymous, groups of armed forces personnel, groups of midwives, etc.). LPP seeks

to describe and account for learning in participatory groups.

In his later work on communities of practice, Wenger (1998) replaced LPP with the idea of a yin and yang-

like duality. However, LPP is still widely used in theorising educational settings and practices. There are

connections between Lave and Wenger’s notion of LPP and situated learning, critical pedagogy and

teaching for social justice, all of which we draw on to some extent within the pages of The Really Useful

Maths Book.

We use LPP to make political statements about democracy and justice in the classroom, as well as arguing

from a psychological perspective that LPP is important when considering how to promote effective

learning. Educators have turned the work of Lave and Wenger inside out; a perfectly proper thing to do if

one wants to use it to debate the development of pedagogy.

Whereas Lave and Wenger observed LPP in the groups that they studied, many educators argue that

ensuring LPP is part of children’s experience in school “is a good thing”. From this starting point they then

seek to develop pedagogical strategies to create situations where LPP occurs.

LPP cannot function effectively in authoritarian, teacher-dominated and syllabus-dominated classrooms.

For reasons which will become evident from reading below, it is in EY practice that there is the greatest

opportunity for pedagogical interventions that allow LPP.

1. Children and teachers both need to exercise choice when it comes to joining and not joining

groups.

2. Child-led and teacher-led groups need to respect the newcomer’s wish to remain peripheral, even

though some group members might invite more central involvement in the group’s activity.

3. LPP is about know-how rather than know-what and is therefore most easily supported in group

activities which have a strong practical element (which is why this theoretical perspective is

attractive to us – RUM firmly advocates that maths at all levels is taught in collaborative groups

whenever possible).

4. A group may have a very short life span – just a few minutes in both an EY play scenario and in a

lesson on algebra in Y6.

5. Because whole-class membership is relatively unchanging over the school year in most schools,

successive regroupings of children throughout the year during lessons will tend, over time, to

contain similar membership and display similar characteristics that teachers and other classroom

professionals will recognise and want to discuss.

6. LPP provides educators with a framework for pedagogical discussion – a way of speaking about the

membership of a group and the relations between newcomers and experts.

“Communities of practice are groups of people who share a concern or a passion for something they do and

learn how to do it better as they interact regularly” (Wenger, 2000). This suggests that newcomers may

want to spend some time repeatedly joining a group in a peripheral role. This is what troubles teachers who

are under pressure to ensure that learning is individual rather than group based. Teachers worry that

children who are content to remain peripheral may be “missing out”. Individual children are tested

exhaustively in UK schools, beyond what is reasonably necessary. The focus is on the extent to which they

have absorbed the curriculum rather than what they can do with their knowledge and expertise. Group

knowledge has little value in today’s schooling when it comes to grading and assigning levels of


competence. In contrast we know that effective teamwork (including being a fully participating member of

the family group) is the key to successful living.

LPP locates learning in the relationships between people. Learning does not belong to individual persons,

but to the various groupings and discourses of which they are a part. In contrast, our education system

insists that the most important aspects of learning are in the possession of individuals – something that

exists in the head.

So LPP is useful in countering the arguments of the prevailing discourse of assessment, attainment targets

for individuals, grades and levels of attainment. The approach to learning fostered by advocates of LPP is

challenging and profoundly problematic to educational systems oriented to individual accreditation, and in

classrooms and schools which deny the relevance of relationships, collaboration and choice.

LPP and Lave and Wenger’s notions of communities of practice share some theoretical connections with

the work of Ivan Illich and his exploration of informal schooling and learning webs. Where Lave and

Wenger’s studies emerged from observations of groups, Illich began with a top down analysis of the

damaging effects of institutions on learning and individuals.

For those teaching mathematics, the challenge is to:

develop classroom practices that build on children’s interests;

create opportunities for children to share and develop know-how rather than know-what;

provide contexts for learning that support and draw benefit from collaboration;

promote choice for children in how they tackle maths;

create opportunities where the planning is done by children as well as by adults;

be open to the possibility of learning from the children as they work in collaborative groups.

References

Donaldson, M. (1978) Children’s Minds. London: Harper Collins.

Evans, Z. (n.d.) Booklets available from Hendre Craft, 'Old Barns', Newton St., Cyres, Exeter, EX5 5BY.


University Press.

Skemp, R. (1987) The Psychology of Learning Mathematics. London: Routledge.

Wenger, E. (1998) Communities of Practice: Learning, Meaning, and Identity. Cambridge: Cambridge

University Press.

Wenger, E. (2000) “Communities of practice: A brief introduction” Available at http://wenger-

trayner.com/theory/.

http://wenger-trayner.com/theory/

http://wenger-trayner.com/theory/


T11 The Learning Process – a review of dominant psychologies

Tony Brown (2013) This discussion summarises the viewpoints expressed in The Really Useful Maths Book. We take the view

that language is at the heart of learning per se, and therefore at the heart of learning mathematics. It

follows that our view of learning is constrained by certain limits; we are mainly interested in children’s

learning through the pre-school and school years 3–18, while the age range we focus on in terms of

activities and curriculum is 3–11.

Young children’s learning is itself dominated by language learning. As authors, we are interested in

classrooms, formal learning situations wherever they occur, informal situations where maths learning is the

focus – maybe in the home or a free school – and the self-taught learner, which we all are to some extent.

Our starting points are the critical consideration of those theories of language and learning that have a

strong connection with individual children who learn in groups, learning in formal classrooms and pre-

schools, and informal settings like environments in which children are home-taught.

Modified Piagetian theories are useful, because they emphasise the biological bases of learning, the role of

the physical environment and the fact that children learn by exploring the physical spaces they inhabit. Also

useful in this respect is Piaget’s ideas about accommodation, and the problems we face when our previous

experience doesn’t fit with a new setting, causing a conflict between earlier learning and immediate

experience, a “this doesn’t make sense” feeling which we overcome by accommodating the new with the

already learnt.

For us, the Piagetian stages of development are unhelpful, overturned by later researchers such as

Margaret Donaldson in the 1970s, and Valerie Walkerdine and Martin Hughes in the 1980s. Donaldson, and

Hughes who trained with her, reported the importance of children’s sense-making and their ability to

display development that Piaget thought impossible when they were given contexts that made sense to

them. Thus the results of Piaget’s mountain experiment (relevant to Swiss children perhaps) argued that

children of a certain age could not see a mountain feature in the distance from another person’s

perspective. In contrast, Donaldson and Hughes showed conclusively that, in settings that made social and

cultural sense to young children of the same age as Piaget studied, children could understand another

person’s viewpoint (i.e., “We need to hide teddy where he can’t be found. Can you put him where he can’t

be seen by X?”). Walkerdine’s research was more pertinent to everyday classroom practice because she

demonstrated how children are socially conditioned very soon after arriving in formal settings, to give

teachers what they imagine the teacher wants. Walkderine showed that children will say anything,

including what they know to be nonsense from their perspective, if the teacher appears to want that sort of

answer. The power of the classroom and the status of the teacher trumps commonsense and “what I

know”. This evidence raises lots of issues for teachers who want to develop and refine their practice.

Vygotsky was rediscovered when a large body of his work was translated into English in the 1960s. As a

Marxist in 1920s–1930s Russia, where it was lethal to be out of fashion, he worked on children’s

development from a perspective that was not only relevant to the times, but also highly relevant to

Western Anglophone classrooms of today where teachers are sensitive to the fact that children learn

together in groups, rather than independently. It was only in the 1970s that schools were building language

laboratories with individual booths that separated students from all interaction with their peers, on the

assumption that the decoded text and the disembodied voice from a tape machine were all that was

needed for language learning.


Vygotsky’s work emphasised the social aspects of learning, and the potential of collaborative learning to

provide support for less-experienced learners through the actions of more experienced peers and adults.

Ideas of “scaffolding” have emerged from his work, providing support that leaves the learner feeling

independent rather than submissive but which tries to ensure that effort pays learning dividends. To this

Marxist viewpoint we can add the research of Lave and Wenger, borrowed for use as it has been, with

children and formal classrooms. Their idea of communities of practice has been (somewhat forced) into use

in classroom settings. Their idea of legitimate peripheral participation we find politically useful because it

emphasises the role of choice in learning. Many classrooms in Anglophone countries still limit children’s

choice drastically. Ways of working, who to work with, how much work to do are not considered to be

appropriate decisions for children. We argue otherwise and suggest that only by organising classrooms so

that children can exercise choice will they learn how to do this in increasingly sophisticated ways. Denying

children access to decision-making undermines their ability to make choices.

Claxton’s work on what makes and breaks good learning combines his Buddhist interests with his research

as a psychologist. The psychological components of secure learning resemble Buddhist ideas about right

mind, right practice, right view, right intention, right speech, right action, right livelihood, right effort and

right mindfulness. These qualities and dispositions have been adapted to be recognised more easily in

secular education; resilient, resourceful, reflective, robust, intuitive, immersed, intellectual and

imaginative. Claxton presents these as attributes of the individual, which we accept. However, the

acquisition and maintenance of these personal qualities in western societies, we would argue, are seen

very much as being influenced by others. We are vulnerable to believing ourselves to be the constructions

of others to such an extent that identity is much more a contested idea than even a few years ago. Thus the

climate of the classroom, the behaviour of teachers and the climate of repression or nurturing experienced

by pupils is central to their self-perception.

Reuven Feuerstein helps us to focus on the outliers, the atypical learners whose face difficulties with

learning that the majority of children do not experience on a regular or profound basis. Although

Feuerstein’s work is relevant to all learners, he worked for over half a century on studies of children with

learning difficulties. He looked at learning strategies, dispositions, organised clusters of thinking–behaving

which he terms modalities. In many children these modalities are underdeveloped. Feuerstein created

highly specific activities to compensate, improve and provide alternative ways of learning called mediated

learning experiences. One of the most challenging tasks for teachers is to develop strategies for ensuring

that children with learning difficulties feel they are integrated, full members of the classroom community.

There are specific ways of working that Feuerstein’s work provides, which make this easier for teachers to

accomplish.

For all learners, the particular structure and nature of mathematics and the school maths curriculum

presents challenges. This is due in part because of the abstract nature of maths (y = mx + c is powerful

because this single abstract statement represents every straight line function and an infinite number of

graphs that could be invented and drawn). Its very generality is what poses a challenge for young learners.

There is a danger that primary school pupils with good memories will try to memorise what they are

taught, rather than integrate new teaching with prior learning. Some memorization is inevitable and

necessary, but as a strategy it fails when maths problems become too complex and where maths factual

knowledge and skills have to be applied in new and very different situations.

Pillar learning is a danger. Learners need to be taught expressly how and why a new piece of maths should

be associated in the learner’s mind with specific knowledge and skills that have already been learned and


can be applied. By demonstrating the links or bridges between new facts and prior learning, the teacher or

more experienced peer can help the less experienced more readily integrate new knowledge. This bridging

process recognizes Piaget’s ideas of accommodation, Vygotsky’s recognition of the power of social

contexts, Claxton’s views of the need for a secure learning base and Feuerstein’s view that we need insight

into learners’ typical modalities or clusters of learning/thinking/behaving in order to predict as best we can

the likely problems that different learners can experience when learning something new. A new maths

topic will pose a range of challenges to learners with different psychological, emotional and learning

dispositions. Learners will react differently to learning challenges, depending on the nature of the new

knowledge and how it is taught. They will respond differently dependent on how they process topics that

are visual, highly abstract, procedural, easily represented kinaesthetically or explored through a computer

program, game or toy.

Feuerstein’s concept of “bridging” is anodyne to pillar learning. We look for the links, the associations, the

bridges between one area of mathematics and another, between different curriculum areas, outwards to

the hidden curriculum, the family, the community, wider society and the world.

Over the years we have been moved, inspired, excited and brought back to the ground (earthed) by many

other teachers and writers including Howard Gardner, David Kolb, Ivan Illich, John Holt, Paulo Friere, Maria

Montessori and Jerome Bruner.

See this superb Learning Theories Info-graphic, leading to infed and Wikipedia links. You might also like to

look up: Caleb Gattegno, Reggio Emilia, Bernice McCarthy, Fritz Schumacher, Seymour Papert, Zoe Evans,

Edward De Bono, Fridjov Capra, Sugata Mitra and Ellen McArthur.

References

Claxton, G. (1999) Wise Up: The Challenge of Lifelong Learning. London: Bloomsbury.

Claxton, G. (2002) Building Learning Power. Bristol: TLO Ltd.

Donaldson, M. (1978) Children’s Minds. London: Harper Collins.

Feuerstein, R. (1983) Instrumental Enrichment. Baltimore: University Park Press.

Hughes, M. (1986) Children and Number. Oxford: Wiley-Blackwell.


University Press.

Piaget, J. (1965) Child's Conception of Number. New York: W. W. Norton & Company, Inc.

Vygotsky, L. (1978) Mind and Society: The Development of Higher Psychological Processes. Harvard, MA:

Harvard University Press.

Walkerdine, V. (1988) The Mastery of Reason. London: Routledge.

http://cmapspublic3.ihmc.us/rid=1LGVGJY66-CCD5CZ-12G3/Learning%20Theory.cmap


T12 The Uncanny: Contexts for Learning, Surprises, Unusual and

Unfamiliar Activities

Tony Brown (2013) [Link to S4]

Children are highly sensitised to recognising what is familiar and what is unfamiliar in their lives. This is not

surprising. Children experience new thoughts, ideas and perspectives on a daily basis. This is partly because

of their youthfulness and relative inexperience. It is also partly because of the opportunities that schooling

offers. One of the purposes of schooling is to ensure that, whatever children’s origins and previous

opportunities, schools can offer something new socially, culturally, experientially, emotionally and as a key

part of what we understand as education.

Because children enjoy puzzlement when they are operating from a secure base, i.e., when they feel

comfortably safe, they are almost always eager to explore something new, even if it weird. We can create

puzzlement and make good use of children’s response to it in several ways:

by connecting different bits of maths that do not obviously relate, so that what children experience

does not easily “make sense” and needs to be untangled in some way;

by varying the ways in which we present ideas, activities and ways of working, “ringing the changes

in how we present maths”;

by inventing other worlds for them to inhabit (not only in their minds, but also in the classroom, by

creating objects that together we can invest with “otherness”).

The strategy that we refer to in The Really Useful Maths Book is a culmination of what we learned as

teachers:

1. We can connect maths to the world of lived experience and to imaginary worlds (as teachers we

can get inspiration from novels, stories, poetry, music, dance, film, TV, etc.).

2. We can push maths to the limit, how does maths work with the very big, the infinitesimal?

3. We can explore global issues and go beyond the world to ask questions about the universe, and we

can also invent a maths problem with a single apple, a knife and the wish to share the apple fairly

between three children.

What is interesting about imaginary worlds is that many of the ones which have the greatest resonance for

children, in terms of excitement, curiosity and the willingness to suspend disbelief, are those which provide

an element of the uncanny: according to Freud, the uncanny occurs where something is familiar, yet

foreign at the same time, resulting in a feeling of it being uncomfortably strange whilst also being

uncomfortably familiar. Without wanting to make things too uncomfortable, we can make great use of this

state of mild discomfort.

The experience of something uncanny has the potential to create cognitive dissonance: we puzzle about

how to how to resolve the inherent conflicting information. Giants are useful in this regard because they

don’t exist – (do they?) – but if they did they’d be very dangerous, even if we met them in the supermarket,

or skateboarding. Would you lend your skateboard or your guitar to a baby giant? Is there enough food in

the supermarket for a giant to throw a retirement party for her friends and family? What would we eat if

our local supermarket was constantly frequented by hungry giants?


Children’s awareness is heightened and their curiosity sensitised when faced with the paradoxical tensions

of being attracted to and wary of an object at the same time. This is what we can capitalise on. We can find

ways to engage in maths by:

drawing on maths to answer problems of an unusual nature (what would be the shoe size that a

giant would look for in a shoe shop? How many XXXs before the L would there need to be on a T-

shirt label?

extending the maths to unusual limits (especially in terms of measures) creates a new space in

which to use and apply maths;

making things for non-typical purposes (a pair of braces to hold up her dungarees?).

Most of the time children expect mathematics to relate to the everyday world that they and we inhabit. In

school we have wonderful opportunities to disturb this everyday reference frame and we can do it in many

ways. By linking maths to English and other subjects, but most often through telling and reading stories and

poetry, we can reproduce a sense of the uncanny in one lesson and capture it for use in maths at another

time.


T13 Vygotsky and Language

Tony Brown (2013)

Psychological theory of classroom learning

Lev Semionovitch Vygotsky occupies an interesting pedagogical position in UK education. He was born in

Orcha in 1896, during the period of the Russian Empire, and died in Moscow in 1934.

He was the originator of a theory of human development that not only drew on ideas of race and heredity

but also insisted that human psychology develops as a result of being immersed in social and cultural

processes.

To survive in Soviet Russia in the Stalinist era meant managing expectations in ways that recognised and

promoted state ideals. It is no surprise, and indeed possibly a strength for us a hundred years later, given

the US-dominated twentieth century obsession with the cult of the individual, that Vygotsky’s theoretical

psychology emphasised social practices.

His theoretical position is that development of cognitive functions in children, such as higher order

reasoning, is possible only through practical activity in a social (read socialist?) environment. Teachers in

schools will easily recognise that practical activities in social settings are what typify classrooms. They are

social spaces where children learn, often in collaboration, through engagement with practical activity. That

is certainly the theoretical basis of The Really Useful Maths Book.

There is also a political dimension to this dependence on Vygotsky. His social and socialist perspective on

education is a valuable counter to the reductionist processes that have led children and parents in the UK

to be rebranded by politicians as customers to whom education can be sold. No longer viewed as members

of social groups who own the schools that they are paying for through taxation, ordinary people are now

customers who are forced to join the market that will sell them education, despite the fact that they must

pay twice for the privilege.

We are faced in the UK today with the excesses of the cult of the individual. Individual children are assessed

more often in UK schools than almost any other country, and not to their benefit. The purpose of Margaret

Thatcher’s Education Reform Act (1988) was to provide a basis for the sale of schools and the privatisation

of education, which is only now being foisted on us by the coalition government. From President Reagan,

Thatcher was taught the link between measuring children, measuring teachers (in terms of performance

and competence) and the measuring of schools in order to make the most attractive schools available for

sale.

The UK coalition government, via Tony Blair’s interest in faith schools, has found a slightly different way to

sell education: by modifying the definitions of “free” schools and academies, and by re-defining taxpayer’s

money as incentive contributions to be made available to businesses wishing to own and run schools. The

same process is now underway in universities through the removal of the block teaching grants which

provided tax payers’ money to the universities to educate for free the children of those paying taxes. Now

tax payers pay twice or three times: once in initial taxation to maintain universities, second when their

children pay for teaching in those universities and third when taxpayers’ money is given to private

businesses that create fee-paying colleges and universities (Collini, 2013).

Vygotsky’s theories were largely unknown beyond Russia until some 30 years after his death. They gained

prominence when translated and used by some Anglophone teachers and educationalists as a counter to


post-war teaching methods in the UK and elsewhere. At the time there were two or more powerful cultural

forces at work, competing for dominance. The totalitarian and fascist threats of the 1930s which were

popular with many British people (see Oswald Mosley, the British Union of Fascists and the Blackshirts)

were only just averted in 1945.

In educational settings, there followed a surge in individualist theories and practices that emphasised the

cult of the individual – “free to do whatever is desired” in terms of wider society and, in terms of education,

“learning is the result of individual changes in cognition brought about by activities with the material

world”. The approaches of the 1950s failed to, among other things, recognise the many experimental

educational and social practices in Britain that had continued from the 1920s, for example with schools run

by Anna Freud and others in London. In the event the theories of the individual, bolstered by evidence of

the very enormous gains in wealth made by the US from the 1950s onwards, propelled British schools

towards a schism: pre-schools and “infant” schools (4–7) focusing on play and exploratory learning, whilst

“junior” (7–11) and secondary (11–14) schools focused on teaching through a curriculum subject list.

Vygotsky’s theories and emphasis on the social nature of learning provided a valuable alternative

perspective, theorising the experience of children and the effective practices of teaching in terms of mutual

support, social interaction, learning from others and the centrality of language as a means of not only

communication and transmission of knowledge, but also the actual construction of meaning, by children, of

their world. Vygotsky is criticised by some for an over-emphasis on the role of language in development

and the relative lack of discussion of emotional factors in his theories of child development.

Vygotsky is recognised within the UK as having had a significant impact on pedagogy, as witnessed in an

Early Years publication by Teaching and Learning Scotland (2005), Let’s Talk About Pedagogy, which lists

Froebel, Montessori, Steiner, Piaget, Vygotsky, Bowlby, Isaacs and Bruner as important early influences on

EY practice.

Perhaps today in the UK, Vygotsky’s name is most strongly associated with “zpd” – described by some

commentators as one of the most misunderstood and misapplied, yet most referenced of psychological

theories. Some argue that current notions of zpd – zone of proximal development – are a long way from

Vygotsky’s original zona blizhaishego razvitiia (ZBR), but there are precious few who can untangle the

differences, linguistically and pedagogically. Another example, possibly, of pseudo-critical discussion of

Vygotsky that lacks actual substance. We know from observation that a child’s competence in applying

newly acquired skills varies with the context in which the child finds itself. We are frequently reminded that

children often realise this for themselves, “Can you help me – I can’t do this by myself.” By ZBR, Vygotsky

appears to have been theorising this variability in skill application. The lower limit of the zone is the level of

skill achieved when working independently while the upper limit is the level achieved with the skilled

assistance of another child or adult. For Vygotsky the notion of ZBR offered a useful way of explaining the

complex interplay between children’s learning and cognitive development.

Vygotsky offers something significantly different from alternative contemporary theories. Constructivists

would argue that development must precede learning. Development being associated with the

accommodation between already internalised concepts and what we might call experiential challenges to a

current way of thinking (i.e., a child knows that heavy things sink, but finds when playing with a heavy bowl

that it can be made to float, forcing a need for accommodation between the original conceptual

understanding of what sinks and an experience that appears to contradict this mode of thought). Maturity

precedes understanding in the constructivist world. Behaviourism presents a different view: learning

develops simultaneously with experience. Behaviour is modified primarily by further adjustments to


previous behaviour and this is what leads to development. Gestalt psychology argues that learning and

development are separate but interdependent and interactive with each effectively preparing the mind for

the influence of the other. For Vygotsky, learning precedes development.

There are connections to be made between the thinking of Vygotsky and Bruner (see RUM p. 212–3), much

of whose writing extends across the second half of the 20th Century. Bruner has made extensive use of the

notion of scaffolding, a term that appears to have been first used in the literature in Wood, Bruner and

Ross (1976).

Most teachers in the UK will be familiar with the term as applied to the differential help offered to children

to support learning, based on the child’s perceived level of performance: more support and perhaps a

different quality of support for children struggling with their studies; less support and perhaps qualitatively

more complex support offered to children clearly coping with new ideas.

Scaffolding is intended to maintain the child’s engagement with the task in hand so that the child remains

within the ZPD, an essential element of which is the child’s ability to function linguistically in ways that

promote thinking and learning. According to Vygotsky, speech allows children to communicate with and

learn from others through dialogue in ways that bring the child’s under-developed concepts under the

influence of the teacher’s more systematic and logical thinking. Later psychologists may argue that

language, both external and internalised, structures thought and is inseparable from higher-order thinking.

Mathematics presents a number of challenges to learners including:

1. Its hierarchical nature (if you don’t understand multiplication and have a good grasp of

multiplication facts then understanding division isn’t going to come easy. If you are not very aware

spatially, then aspects of geometry can be hard to grasp, etc.)

2. The special language – pentagon, octagon, fraction, multiple, angle, division, etc.

3. The special use of language – a child in the classroom could hear the following utterances from the

teacher in a five minutes period: Don’t go into the hall until I tell you. I haven’t got time to go into

that now. Six goes into 18 three times.

Language, it seems, not only supports thinking but also structures thinking. Maths understanding depends

on language fluency as much as on any other single factor. There are two dimensions to mathematics that

distinguish it from many other areas of learning:

1. Maths and maths relationships can be represented by objects: stones, money, number lines, tables

and lists, equations, plastic cubes, wooden rods and paper shapes. Using them supports thinking

and also structures new thoughts.

2. Maths can be represented by concrete objects but the purpose of much of the maths we do is to

forego reliance on physical objects in favour of symbolic reasoning. Somewhere in the middle of

this process of increasing abstraction are written objects like x + y = z which are generalised

statements (in this case containing only 3 letters and 2 other symbols whose power is in its ability

to represent an infinite number of possible situations. Moving to an abstract understanding of

mathematical relationships is difficult for most learners, who need lots of experiences with single

instances of x + y = z

(5 + 7 = 12) (0.5 + 0.7 = 1.2) (-5 + -7 = -12) – minus 5 and minus 7 equal minus 12

(5 + 7 = 10) – in base 12.

before they can use the generalised formula.


Scaffolding in maths is often provided by physical objects: pencil and paper, calculators and computers,

measuring tapes, etc. There is no inherent maths in these objects, although there may be relationships

between the different cubes, as for example in a set of Dienes blocks or Cuisenaire rods.

Relationships are part of a conceptual structure in the brain. The cubes and rods jog the mind, help us to

pose questions and remember the steps needed in solving a problem. The physical objects help us break

thing s down into a logical sequence that we can understand. They help us record. They challenge us to

name the patterns we perceive and to use (and invent) language to think something new – something that

we haven’t thought before.

Vygotsky’s theorising is helpful in understanding this process. He makes a different argument from that of

the constructivists, because Vygotsky suggests that we can provoke development rather than wait for the

child to mature sufficiently before more complex thought is possible. By structuring the challenges that we

present to children, we can invite them to think beyond their current knowledge and then, by scaffolding

imaginatively, we can provide sufficient support to allow thinking to proceed, without overdependence on

memory (which is a weak faculty in most humans), and without stealing the intellectual kudos from the

child, so that overcoming the challenge, solving the problem, is pleasurable for the child because it is

achieved with a degree of independence and intellectual creativity.

References

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

Bruner, J. (1974) Going Beyond the Information Given. New York: Norton.

Bruner, J. (1986) Actual Minds, Possible Words. Harvard, MA: Harvard University Press.

Collini, S. (2013) Sold Out. London Review of Books, 35(20), 3–12.

Teaching and Learning Scotland (2005) Let’s Talk About Pedagogy.

Vygotsky, L. (1978) Mind and Society: The development of higher psychological processes. Harvard, MA:

Harvard University Press.

Wood, D. J., Bruner, J. S., & Ross, G. (1976) The role of tutoring in problem solving. Journal of Child

Psychiatry and Psychology, 17(2), 89–100.


T14 Using an Urdhu number grid

Henry Liebling (2013) Display an Urdhu counting grid from 0–99 or 1–100 on an OHP or computer screen, projector, IWB or

visualiser (see p. 30 of The Really Useful Maths Book). You as the teacher will need to be able to point to it

and cover and uncover it in some way. Use it with the whole class (see N3 Arabic and other number

Systems pp. 26–31 and N4 Working with Grids pp 32–39 for a full description).

Study it in silence on your own for a minute.

Say what you see. What is important? What do you think it is? What do you notice? Any clues? Look

carefully.

Collect the responses and build on them. Ask someone to act as a scribe if you wish.

I’m turning the image around clockwise through 90 degrees. Now what do you think? Just tell me what

comes into your head. That way we can build up ideas together. Can anyone add to what X has just said?

Who agrees? Who doesn’t? Who couldn’t care either way? Why not? Does it matter which way round it is?

I’ll turn it clockwise through another 90 degrees. Any comments? And again. What now? What do you

notice? How has it changed?

Reinforce what has been observed. Try to highlight main points, strategies and insights.

Where is the origin, where it starts? In which direction do the symbols go?

An aside Tifinar is a script used by the Tuareg which can be written and viewed from any side.

The writing was usually from the bottom to the top, although right-to-left, and even other orders, were also

found.

http://en.wikipedia.org/wiki/Tifinagh


Imagine a group of children sitting around a teacher who is writing/drawing symbols in the sand. There is

no front or back. Do you know any languages written from top to bottom or right to left? Why might that

be? Do you know that there are some 23 scripts which have been found but not yet deciphered? (Here, you

can talk about the Rosetta Stone).


On the Urdhu Grid, can you see 1, 2, 3? Imagine making these marks on paper. Try skywriting now with me.

Make 1 then 2, then 3 downward strokes from right to left. Can you see how the marks for 2 join up and the

marks for 3 join up? When we look at them in this way we can see where the versions of 1, 2 and 3 that we

use today have come from. The writing/mark making goes from right to left.

What other numbers do you recognise? Are they false friends? One looks like a seven but isn’t. What of the

number five?

Try to find anchor points on the grid, symbols you feel sure of, then work out the symbols either side.

I’ll point and you all say what you think each numeral is.

(In Silent Way mode, go from 0 to 9 then down the tens, then wherever they direct you with the accuracy

of their responses. Don’t be afraid to encourage any one individual who gets it right to carry on and the rest

join in when they think they understand. Go back over some sequences if you wish (see N2 Saying and

making numbers pp. 16–25).

Cloud Put a paper cloud over one or more numerals and ask what is covered.

Mask Cover the grid with a sheet of paper with a small hole in it which reveals only one number. Can they

work out what it is? Slide the paper up and or down a number to give a clue or, easier, left and right. You

are trying to get them to reveal the algebra within the grid. Finding X when you reveal X-10, X+10, X+1, X-1,

etc. (see N4 pp. 34–35).

Plenary Try to collect strategies used to work out where the origin is, what direction the numbers on the

grid take, the value of the numerals, what helps and what doesn’t, etc. Will these strategies work with all

the grids? Are they useful anywhere else? In other areas of maths, other subjects, outside school, at home?

Any part of the above activity could be used as a starter activity and developed over a number of weeks

with different number grids.

Opportunities for further work/bridging Repeat this lesson with a different number grid, and try two contrasting systems such as Chinese and

Mayan. Try similar systems such as Urdhu and revise the following week with Arabic.

Work on origins (the starting point) and frames of reference (which way round, orientation and direction),

as concepts developed for pragmatic reasons, but in some sense arbitrary and changeable. Show grids with

different origins such as the bottom left hand corner, or even ones that spiral from centre. Consider the

nature of the four quadrants for an X, Y axis where values can be positive or negative. Explain and explore

this notation for +ve and -ve values of X and Y.

Extend this to grids other than counting grids, such as multiplication grids, addition grids, etc. (see pp. 37–

39).

Jigsaws Print out any grid onto card and cut it up to make a jigsaw. Try to make them of different

complexity from 4–10 pieces. Better still, get the pupils to make them. Try to do them yourself first. Put the

pieces into a sturdy envelope, label them and use them as a short revision activity to use with pairs of

pupils at the end or start of a session. Watch what happens and listen to the dialogue to assess pupils’

strategies and understanding (see p. 36).


Try making sets of dominoes with different number systems such as Mayan, Chinese or Korean (see pp. 28–

29).

Create your own grids using a spreadsheet and different font sets for counting and multiplication.

Discuss or ask them to research Indian number systems and how they evolved; the importance of zero; the

acceptance and stability for 100 years of the Arabic number system which has now become international;

or the burdensome nature of Roman numerals (using letters for numbers).

Develop work on saying numbers in different languages (see p. 24).

Develop target boards using different, combined number systems (see N5 Working with target boards pp.

40–45).


T15 “What would happen if we used the principles of living systems as

tools for learning?”

Henry Liebling (2011) Using the headings from F. Capra http://www.ecoliteracy.org/nature-our-teacher/ecological-principles

Can this become an ecology of teaching and learning?

Networks of relationships

How can we encourage networks of relationships? With learning sets, critical friends, circle-time style

activities, an environment where sharing is encouraged, and support groups. See also the school/college as

a community.

Diversity

Encourage diversity in topics covered and in how tutors and students respond to tasks and assignments,

and celebrate diversity and difference rather than trying to flatten everything to make grading assignments

and evaluating student performance easier. Allowing students to set their own agendas and choose their

own assignment titles gives a greater range and diversity of material to feedback to their peers and future

groups.

Nested systems

Acknowledge the nature of learning as complex with systems and procedures each with their own rules. Try

to help students come to terms with and question these systems, including referencing, structuring,

presentation, participation and planning. Consider schema theory: each schema is a sort of mini system and

schema are built up or nested within one another, e.g. getting up, washed, dressed, fed and to

work/school! Or gathering, preparing, cooking and eating food. This appears linear, and indeed some

actions have to be sequenced in a particular way for good reason, but in other ways the schema are nested.

Cycles

Resources are exchanged and moved around in cycles, one organism’s waste is another organism’s food.

How can we relate this to teaching and learning? By an analysis of misconceptions and making good use of

our errors and misunderstandings, and by considering seasons, a sense of place over time, life cycles, rise

and fall in populations, etc.

Flow of energy

This reminds me of Sylvia Ashton-Warner in Teacher (1963) describing organic teaching and learning. The

teacher breathes out and the learner breathes in, then the teacher breathes in and the learners breathe

out. There is a rhythm and balance to the day, and within each session, as well as within each dialogue. Is

the continuous flow of new students part of the energy flow? The leavers and freshers. What of the flow of

new ideas and innovation? Where can/does/should the learner and the teacher each get their energy

from?

Development and change over time

This is learning itself I think. Adaptation, evolution, Piagetian stages, schema, NLP, etc.

Dynamic balance

The resilience of the combined community, the ethos of the school/learning community where each is

valued and useful and all are still learning.

http://www.google.com/url?q=http%3A%2F%2Fwww.ecoliteracy.org%2Fnature-our-teacher%2Fecological-principles&sa=D&sntz=1&usg=AFQjCNHpY-kkWGBPq0F4Y3DhHARYCcZRtg


















Reference

Ashton-Warner, S. (1963) Teacher. New York: Simon & Schuster.


T16 Fritjof Capra’s ‘Speaking Nature's Language: Principles for

Sustainability’

Henry Liebling (2011) The book was Ecological Literacy: Educating Our Children for a Sustainable World, developed by the Center

for Ecoliteracy. One essay in particular, “Speaking Nature's Language: Principles for Sustainability” by Fritjof

Capra, cofounder of the Center for Ecoliteracy and systems thinker, provided a framework that was crystal

clear.

In this essay Capra explains that, in order to design sustainable societies, we must first embrace a new way

of seeing the world that, in many ways, runs counter to traditional Western science and education. This

way of thinking, known as systems thinking, emphasizes the qualities of relationships, connectedness, and

context in any system, whether an ecosystem or a school system.

Once these perceptual shifts are made, Capra explains, one can begin to study sustainability in the language

of nature through eight particularly important concepts that describe the patterns and processes by which

nature sustains life: networks, nested systems, interdependence, diversity, cycles, flows, development and

dynamic balance.

“These concepts, the starting point for designing sustainable communities, may be called principles of

ecology, principles of sustainability, principles of community, or even the basic facts of life,” writes Capra.

“We need curricula that teach our children these fundamental facts of life” (see Bennett, 2009)

“Creating communities that are compatible with nature's processes for sustaining life requires basic

ecological knowledge.

We need, says Center for Ecoliteracy cofounder Fritjof Capra, to teach our children — and our political and

corporate leaders — fundamental facts of life:

Matter cycles continually through the web of life.

Most of the energy driving the ecological cycles flows from the sun.

Diversity assures resilience.

One species’ waste is another species’ food.

Life did not take over the planet by combat but by networking” (see Stone, 2006).

Nature's patterns and processes Understanding these facts arises from understanding the patterns and processes by which nature sustains

life. In its work with teachers and schools, the Center for Ecoliteracy has identified several of the most

important of these. It has helped teachers identify places in the curriculum where students can learn about

them. They include networks, nested systems, cycles, flows, development and dynamic balance.

Networks

All living things in an ecosystem are interconnected through networks of relationship. They depend on this

web of life to survive. For example, in a garden, a network of pollinators promotes genetic diversity. Plants,

in turn, provide nectar and pollen to the pollinators.

Nested systems Nature is made up of systems that are nested within systems. Each individual system is an integrated whole and, at the same time, part of larger systems. Changes within a system can affect the sustainability of the


systems that are nested within it as well as the larger systems in which it exists. For example, cells are nested within organs within organisms within ecosystems.

Cycles Members of an ecological community depend on the exchange of resources in continual cycles. Cycles within an ecosystem intersect with larger regional and global cycles. For example, water cycles through a garden and is also part of the global water cycle.

Flows Each organism needs a continual flow of energy to stay alive. The constant flow of energy from the sun to Earth sustains life and drives most ecological cycles. For example, energy flows through a food web when a plant converts the sun’s energy through photosynthesis, a mouse eats the plant, a snake eats the mouse, and a hawk eats the snake. In each transfer, some energy is lost as heat, requiring an ongoing energy flow into the system.

Development All life, from individual organisms to species to ecosystems, changes over time. Individuals develop and learn, species adapt and evolve and organisms in ecosystems co-evolve. For example, Hummingbirds and honeysuckle flowers have developed in ways that benefit each other; the hummingbird’s colour vision and slender bill coincide with the colours and shapes of the flowers.

Dynamic balance Ecological communities act as feedback loops, so that the community maintains a relatively steady state that also has continual fluctuations. This dynamic balance provides resiliency in the face of ecosystem change. For example, Ladybugs in a garden eat aphids. When the aphid population falls, some ladybugs die off, which permits the aphid population to rise again, which supports more ladybugs. The populations of the individual species rise and fall, but balance within the system allows them to thrive together.

References

Bennett, L. (2009) “Greening a K-12 curriculum”. Available at www.ecoliteracy.org/essays/greening-k-12-

curriculum

Capra, F. (2005) “Speaking Nature's Language: Principles for Sustainability”. In M. K. Stone and Z. Barlow

(Eds.) Ecological Literacy: Educating Our Children for a Sustainable World (pp. 18–29). San Francisco, CA:

Sierra Club Books.

Stone, M. K. (2006) “Ecological Principles”. Available at www.ecoliteracy.org/nature-our-

teacher/ecological-principles

Stone, M. K. & Barlow, Z. (2005) Ecological Literacy: Educating Our Children for a Sustainable World. San

Francisco, CA: Sierra Club Books.

http://www.ecoliteracy.org/essays/greening-k-12-curriculum

http://www.ecoliteracy.org/essays/greening-k-12-curriculum

http://www.ecoliteracy.org/nature-our-teacher/ecological-principles



T17 “Sense & Sustainability” (Extracts)

Ken Webster and Craig Johnson (2008) Systems approaches based on feedback are the key to understanding participatory learning, to

understanding design processes, to understanding how change happens; to the circular or closed loop

economy, to understanding ecology, living systems and other feedback led processes such as climate and

indeed democracy.

...

Our intention is to contrast the mechanical industrial worldview with a living systems worldview. The

former uses metaphors around machines and parts and proportionate linear relationships – where, for

example, society is seen as just a collection of individuals and Nature is seen as an unlimited Resource and

Waste Depository.

By contrast people with a living systems world view see the whole as greater than the sum of the parts and

where connections and relationships matter more than parts – they recognise that these relationships are

based on loops and feedback. Here Nature is Teacher and Capital.

...

Education for sustainability becomes the debate around how the insights of Nature can be best applied

explicitly to a modern world in transition and to the processes of learning itself, which is also based on

participation and feedback.

Link

The article is available as a downloadable PDF here.

http://www.ellenmacarthurfoundation.org/education/resources


T18 Pedagogical Approaches: Pedagogic elements and aims and for EfS It is crucial to EfS that the learning process fosters key skills that help people to engage effectively with

sustainability. Certain elements are central to EfS pedagogy and these elements can serve both as strategic

principles for institutional plans as well as specific indicators for curriculum innovation.

Futures thinking

Futures thinking engages people in imagining preferred visions for the future. It involves the exploration of

assumptions and of meaningful understandings and interpretations of sustainable development. This

process of envisioning futures leads people to take ownership and responsibility for more sustainable

futures.

Critical and creative thinking

Critical and creative thinking enables people to explore new ways of thinking and acting, make informed

decisions and create alternatives to present choices. It involves reflecting on how people interrelate with

one other, understanding cultural differences and creating alternative ways to live together.

Participation and participatory learning

The engagement of people is needed to build sustainable futures collectively. Engaging diverse

stakeholders and communities is essential as they value and include differing knowledge systems and

perspectives. The process of participation is also important for creating ownership and empowerment.

Systemic thinking

Thinking systemically is essential to sustainable development, as piecemeal approaches have proved not to

work, instead resolving one issue while creating other problems. Sustainable development requires

approaches which go beyond analysis in terms of “problem-solving” and/or “cause–effect”.

Partnerships

Partnerships are a motivating force towards change. They empower people and groups to take action, to

take part in decision-making processes and to build capacity for sustainable development. Intercultural and

multi-sectoral partnerships in particular are often highlighted as critical in EfS approaches.

Key points raised in Engaging People in Sustainability include:

• There is no one definition of “sustainability” – instead EfS prioritises learning processes,

collaboration and debate, critical thinking and strategic change to promote constructive

educational responses to questions of sustainability.

• The span of EfS includes all subject areas in the formal curriculum, as well as a range of research

and informal learning activities underway across the university.

• EfS can and should serve as a vehicle for critical academic development, across all disciplines and

professional subject areas. The focus of EfS is not merely the environment but includes the widest

range of topics relating to quality of life.

Source: Tilbury, D. & Wortman, D. (2004) Engaging People in Sustainability. Cambridge, UK: Commission

on Education and Communication. Available at

http://insight.glos.ac.uk/sustainability/education/hefcelgmquality/documents/engaging%20people%20in%

20sustainability.pdf.

http://insight.glos.ac.uk/sustainability/education/hefcelgmquality/documents/engaging%20people%20in%20sustainability.pdf

http://insight.glos.ac.uk/sustainability/education/hefcelgmquality/documents/engaging%20people%20in%20sustainability.pdf


T19 Some Pedagogical Approaches being Used in Schools

Henry Liebling (2011)

Type Description and examples

entrepreneur Social, fair-trade cafe

set-aside time Carving out specific time for activity

cross-curriculum department

Working across department’s courses, levels, cross-fertilization. Peer tutoring

cross-curriculum interdisciplinary

Using tasks, tools and methods from other curriculum areas

re-search Looking for/gathering material, thinking about, analysis, thesis, synthesis…new ideas/thinking. Can split up task and re-combine at end

stimulus Use a wide variety/range of stimulating material, artefacts, surprises, visitors, video, DVDs, images, data, graphics, music, mp3, art , poetry, story/narrative. (VTSD from Sahara project)

presentation Students/pupils present research findings/information/work/results/artefacts to peers or wider audience. Maybe into public domain.

collaboration Learning sets, group work, peer learning, team work, joint projects, shared assignments, team teaching, joint presentations

participation Student voice, encourage meaningful participation

problem solving Enquiry based, problem solving approaches

role-play Through other eyes (toe), hot seat, expert mantle. Develop empathy, compassion, equanimity. Other viewpoints à informed world view.

creation Using arts and design to innovate, improvise, create, express feelings, report action, discussion, thinking or solve problems.

reflection Encouraging learners & teachers to examine and question their thoughts, feelings & actions in the light of what they have learned/done. Critical friend, buddy, circle time, journal, critical incidents/significant events file

action Exploring individual or collective action to address issues of “esdgc”…

events/projects Organising special events, projects, days in/out, displays or exhibitions

campus Using environment created by buildings & grounds as a tool for teaching & learning

gardening Growing food, fruit trees/bushes, herbs and other plants. East feast

community Involving parents &/or the community in learning for sustainable acts.

local2global Using links along local to global dimension. Face to face, virtual, simulated.

experts Inviting experts in. Using expertise from within the group. (languages, hobbies, interests, expertise!!) Offers/Needs and Requests/Strengths.

Adapted from: Gayford, C. (2010) “Learning for Sustainability in Schools: Effective Pedagogy”. WWF.

Pedagogical principles for ESDGC (short version) Envisioning the future, then back-casting.

Dealing with complexity. Critical/creative thinking (challenge). Ethics and conflict resolution.

Systems thinking, integrated, interdisciplinary.

Participation and participatory learning.

Partnership working (no one pathway to sustainability. CONTESTED).

Asking questions about traditions and innovations.

Source: Based on notes from a conversation with Dr Alex Ryan.


T20 Some Background to Exploring Non-linear Pedagogical Approaches

to ESDGC

Henry Liebling and Tony Brown (2011) There is some agreement that we need to think differently in trying to find a way to live with the planet

rather than destroy it. We need a change of heart that helps us to see the planet’s resources as finite and

that there is no “away” for our waste.

In the natural world there is a cyclical flow of resources, with many feedback loops. The waste from one

living thing becomes the food of another and so on in networks of micro-organisms, plants and animals

living on the varied surfaces and environments of the planet, from the tropical rainforest to the bottom of

the oceans, in a dynamic dance of life where diversity assures resilience.

Much remains to be discovered, understood and learned from:

the cyclical nature of life;

the way each molecule of sugar, oxygen, water and carbon dioxide is re-used and even reformed,

created and broken down into constituent atoms;

the balance between photosynthesis and respiration, with the sun’s energy as the ultimate driving

source of energy;

the self-similarity shown in patterns of clouds, sand dunes, sea shores, coastlines, rivers, valleys

and mountains, where systems function in ways that are non-linear, dynamic and ever changing.

We live in a world that cannot be described adequately using Euclidean geometry and a mechanistic,

reductionist worldview. Many mechanistic models are intra-systemic – they can be used only within a single

system. They are not inter-systemic; they don’t always work across systems. They do not provide accurate

representations of the world when used in super-complex systems. Yes, current models have value, but

they also have serious limitations. They are inadequate for tackling the “wicked” problems we face. For

example, current models that “explain” how humans can “benefit” from “exploiting” the world are mostly

derived from business and economic models that exclude costs associated with maintaining biodiversity

and the environment. Instead these models limit measures to intra-systemic factors like profitability, sales

performance, advertising penetration and market share. These models assume endless growth measured

intra-systemically, whilst excluding most inter-systemic measures. Thus they exclude calculations of the

carbon footprint of resource acquisition and industrial exploitation, and the biological, health,

environmental and social costs of activity. Intra-system models are the ones most familiar to us. These are

within-system models that limit measurement to carefully selected indicators – often ones that are geared

to “proving” the success of activity described by the models. Because they refer to closed systems (such as

economic exploitation, growth of economic markets and measures of operating costs compared to

investment in new technology), they can make claims for predictability (within their chosen models) in an

increasingly unpredictable world: a world which demands better understanding of super-complexity and

which needs models that show how activity in one system can have far-reaching effects in others.

Additionally, there is often a lack of understanding about the relationship between prediction and control.

Many economic models imply that greater prediction will improve opportunities for control. We are

encouraged to believe that greater measurability of market forces will help businesses outperform their

competitors. However, many systems are too complex to understand sufficiently well and to be controlled.


For example, we know the weather is difficult to predict, and measuring it with greater accuracy can help

us prepare for extreme weather events but will not automatically help us to control them.

Survival appears to depend on diversity. Diversity in all forms of life benefits survival and adaption to new

challenges. Survival depends on the continued existence of diverse environment and habitat, culture,

language, and so on. Human intervention in the natural world is too often at odds with this natural system.

Instead of maintaining diversity, human activity too often involves a convergent approach that leads to

decreased diversity and fewer species, cultures and languages.

Industrialised Europe and the US have promoted a view of humans as separate individuals, fiercely

independent, with rights that protect the individual’s life choices. Equating individual with separate is a

fundamental mistake, but this western view, driven by an expanding global consumerist economic

perspective, is increasingly influential.

The reality is that we are completely dependent on other people, animals, vegetation, fungi, microbes and

viruses. No single person can create a mobile phone, a life-saving drug, fuel for cooking, or a shop that

supplies food to the community. We are completely dependent on others and the planet as a whole for the

air we breathe, the water we drink, the food we eat and the soil in which we grow most of our food. Where

are the rights of the earth and all non-human life?

As Capra says, “Interdependence – the mutual dependence of all life processes on one another – is the

nature of all ecological relationships” (1997, p. 20).

The consumerist model that is shaping our relationship with today’s world is a linear one. It can be

illustrated as follows.

1. Think of something that people will buy.

2. Persuade people to want to buy it using a range of advertising.

3. Make it cheaper and more desirable than its competitors.

4. Minimise costs and maximise financial profit.

Another way of representing this linear way of thinking and acting is to think about the use of resources.

1. Find, extract or create materials and energy in the cheapest way possible.

2. Use them to produce goods and services that can be sold at a profit.

3. Make it as easy as possible for people to acquire things.

4. Make it as easy as possible for people to throw things away and buy something else.

The problem with this linear approach is that it doesn’t tell the whole story. It’s not a complete picture. It

simplifies certain aspects of the human relationship with the natural world, but it hides others. It promotes

a certain view of the world and our role in it, while it discourages other views. It favours simplicity whilst

discouraging complexity. In reality linear models of the world are a distortion: natural systems are circular

not linear, they are dynamic with emergent properties such as self-organisation. Ecological systems like

rainforests have evolved over hundreds of thousands of years to become super-complex interdependent

systems. At a macro level they survive the variability of the climate. At a micro level they stimulate

diversity. They change and adapt in cycles. They follow circular, not linear, functions.

There are three major tasks with which to engage.


1. To understand the dangers of trying to predict, manage and control our relationship with the

natural world in terms of linear models.

2. To move away from behaviour that relies on linear models.

3. To educate people to see the importance of adopting and adapting non-linear alternatives.

For educators this means asking some new questions about the impact of taking a non-linear approach:

Can we describe teaching and learning in terms of non-linear approaches?

What kind of pedagogy might result?

What might happen to our worldview?

Will non-linear approaches to teaching and learning reveal the non-linear paradigm to us?

Moving to a non-linear approach to pedagogy will influence our responses to teaching and learning in many

areas:

feedback;

collaboration, negotiation and partnership;

participation and empowerment;

reflection on learning;

self-regulation;

learning to learn and the restructuring of knowledge.

A non-linear approach to pedagogy opens up possibilities for creative work in a range of more interactive

activities such as role-play, simulation, concept mapping, braided learning, problem-based learning,

unconferences, building a wiki, bridging, cradle to cradle, circle time, cycles, critical incidents, searching for

nested systems and fractals.

Source: http://esd.escalate.ac.uk/2508

Reference

Capra, F. (1997) The Web of Life: A New Scientific Understanding of Living Systems. New York: Anchor

Books.

http://esd.escalate.ac.uk/2508


T21 Some Current and Some Older References and Links to Pedagogical Approaches to Sustainability and

Global Learning Author Title Year Pub/Format Notes/Websites

Tilbury, D. Pedagogical Approaches 2010 From 2004

pub’n


Gayford, C. Learning for Sustainability in Schools:

Effective pedagogy

2010 WWF http://esd.escalate.ac.uk/2520

Sterling, S. et al. LinkingThinking: New perspectives on

thinking and learning for sustainability

2004 WWF http://esd.escalate.ac.uk/2526

Sterling, S. The Future Fit Framework 2012 HEA http://www.eauc.org.uk/the_future_fit_framework_an_intro

ductory_guide_1

Horton, P. Sustainable Development and Gaia

Theory

2010 Online

ESCalate

http://esd.escalate.ac.uk/juggins

Mitra, S. The Child-Driven Education 2010 TED talk http://www.ted.com/talks/sugata_mitra_the_child_driven_e

ducation.html

Stibbe, A. (Ed.) The Handbook of Sustainability

Literacy: Skills for a changing world

2009 Green books

and online


http://www.sustainability-literacy.org/

Capra, F. The Web of Life 1997 Harper

Capra, F. Speaking Nature’s Language: Principles

for sustainability

Ecoliteracy

website

http://www.ecoliteracy.org/nature-our-teacher/ecological-

principles




http://www.eauc.org.uk/the_future_fit_framework_an_introductory_guide_1

http://www.eauc.org.uk/the_future_fit_framework_an_introductory_guide_1

http://esd.escalate.ac.uk/juggins

http://www.ted.com/talks/sugata_mitra_the_child_driven_education.html

http://www.ted.com/talks/sugata_mitra_the_child_driven_education.html


http://www.sustainability-literacy.org/




Webster, K. &

Johnson, C.

Sense and Sustainability 2008 Book & online http://www.ellenmacarthurfoundation.org/education/public

ations

Preston, C. Unconferences: From practice to praxis

in informal professional learning

contexts

2010 Online paper http://www.mirandanet.ac.uk/casestudies/254

We are also reminded of the following work, which we feel needs refreshing, rewinding and rebooting!

Author Title Year Format Notes

The Ecologist Blueprint for Survival 1972 Penguin Radical proposals for immediate action. Ah well!

De Bono, E. Teaching your Child to Think 1992 Viking Encourages divergent non-linear thinking and balances use of

more logical approaches e.g. “6 hat technique”

Ben-Hur, M. On Feuerstein’s Instrumental

Enrichment

1994 IRI Skylight Really made me stop and think about education.

Gardner, H. Intelligence Reframed 1999 Basic Natural intelligence a more recent addition

Gattegno, C. The Science of Education 1987

ff

Educational

Solutions,

Inc.

Inspiration to many a maths teacher and beyond.


Novak, J. & Gowin,

D.

Learning How to Learn 1984 CUP Seminal book on concept mapping and the little known Gowin’s

V. Mainly applied to Science Education.

Taylor, J. V. Enough is Enough 1975 SCM A bishop cautions our ruthless, unbridled, unthinking excess.

http://www.ellenmacarthurfoundation.org/education/publications

http://www.ellenmacarthurfoundation.org/education/publications

http://www.mirandanet.ac.uk/casestudies/254



Illich, I. Deschooling Society 1973 Penguin Maybe we need this now even more than then!

Meadows, D. H. et

al.

Limits to Growth 1972 Universe

Books

Recently re-visited

Papert, S. Mindstorms: Children, computers and

powerful ideas

1980 Harvester I found learning LOGO alongside children a liberating experience.

This book guided me.

Schumacher, E. F. Small is Beautiful 1973 Abacus A study of economics as if people mattered.

Pike, G. & Selby,

D.

Global Teacher, Global Learner 1988 Hodder Seminal, influential and still valid

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