ATM Vision Group

Visualisation, Imagery and Spatial Understanding Working Group

This working group is intended to enhance the use of visual, diagrammatic, and spatial approaches across the mathematics curriculum. It will look to replace talk and text with more use of visualization and diagrams by creating resources and designing tasks drawing on research, current good practice and creative thinking. The group will meet regularly, to discuss and develop tasks, it will invite speakers with expertise in visualisation, and discuss research and other publications in the area. The Working Group will potentially have a website/blog. We would expect to produce publications for teachers, and to contribute to ongoing research.


Mid Conference Plenary

Mid-Plenary - Simon Peyton Jones & Ian Benson

Title: Computing and maths: 1+1 = 3?

In 2014 the new national curriculum re-launched the ICT curriculum as “Computing”. For the first time anywhere in the world, the new programme of study establishes computer science as a subject discipline that all children should learn, alongside maths and natural science, from primary school onwards.

But computer science is mathematics incarnated (made flesh) in tangible form. The pioneers of computer science, from Babbage to Turning, were prominent mathematicians. Computer scientists, just like mathematicians, abstract the key properties of a real-world problem (the velocity of the ball, the force of gravity), compute or reason about this abstract model, and then re-interpret the results into the real world. And so on.

So, instead of having “maths lessons” and “computer science lessons” in distinct silos, what if we had “computational lessons” – pick your own title – instead? Perhaps we could illuminate, motivate, and embody mathematical concepts in the form of executable programs? In this talk, and subsequent workshop, we will leave our silos behind, and explore (with your help) how we could make 1+1= 3.

Professor Peyton Jones has been working on K-12 computing education for over 10 years. He was motivated by the fact that there didn’t seem to be any connection between the subject that his children were learning at school and computer science, the subject to which he has devoted his professional life. He started with colleagues a guerrilla movement, Computing at School (CAS), to try and reform the UK computing curriculum. Rather to their astonishment, they have been very successful. CAS now has a membership of over 30,000 computing teachers and academics. The English National Curriculum now explicitly says that all children should learn computer science from primary school onwards as a foundational discipline in the way that they do maths or physics. And for the same reasons: that is, not because they're going to become mathematicians and physicists, but because an elementary understanding of these foundational concepts enables you to be an empowered citizen in a complicated world.

Professor Peyton Jones is a Fellow of the Royal Society. He is a Distinguished Fellow of the British Computer Society, granted for his work to advance the development of computer science education in the UK. He is an Honorary Professor of the Computing Science Department at Glasgow University, where he was a professor in the 1990s, and he is currently a Principal Researcher at Microsoft Research. He has recently been appointed Chair of the UK National Centre for Computing Education.

Dr Ian Benson convenes the ATM working group on functional programming and computer algebra. The working group is updating ATM guidance on the implications of computer systems for teaching algebra at A level. He is Principal Investigator for the Tizard programme which is reintroducing early algebra as pioneered by Georges Cuisenaire and Caleb Gattegno in their Numbers in Colour text-books. Tizard applies early algebra to teach programming at Key Stage 1. Such computational modeling shows promise for mathematics and computer science education and for professional development.

Ian is a Trustee of ATM, and is working with Simon to develop these sessions. Further details will be available at the conference or from


Creativity, compensating dynamics and retention

Event: ATM Conference
Location: Chesford Grange, Warwick
Date: Tuesday 16 April 2019
Session: B03
Time: 0900-1030
Facilitators: Anne Haworth, Ian Benson
Booking (Day Delegate Rates available)

Swift recall of multiplication tables and number bonds is taken as a measure of primary school success in mathematics. Traditional teaching aims to achieve success through repetition and memorizing, perhaps aided by structured drawings. But memory is a poor mechanism on which to build mathematical understanding. It is often accompanied, at a later time, with forgetting. In Cuisenaire-Gattegno mathematics retention is based on awareness. In this talk we will look at how free algebraic writing, compensating dynamics and computation path analysis help learners to navigate tables and number bonds, and consolidate a firm foundation for arithmetic

Aimed towards: Initial Teacher Education, Key Stage 1, Teacher Professional Development


ATM Joint Primary Expert Group

Download file "maths.pdf"


iOS apps show and tell

Event: ATM Joint Mathematics Festival
Location: National STEM Center, University of York
Date: Saturday 15 September 2018
Session: A03 (Classroom 3)
Time: 1100-1155

Ian Benson works with teachers revitalising Cuisenaire-Gattegno mathematics. In this interactive session he will demonstrate the top 12 games he works with to share with young learners an experience of the algebraic foundation of arithmetic. Bring along your favorite iOS maths app and be prepared to show us and tell us why you like it



Caleb Gattegno's contribution to mathematics education ranged from innovation in the school curriculum ("Algebra First") to a theory of learning - the Science of Education.

Where a "bit" might be said to be a unit of study in information theory, a gene in biology, or an atom in physics -- Gattegno's scientists of education study an "awareness."

In these two articles developed as part of booklets with the ATM Science of Education Working Group, I look at how exercises with the rods can stimulate various mathematical awarenesses, whose "temporal hierarchy" is recorded in the pathways in the Curriculum Graph.

The reader can try the exercises here.

Download file "IaBeBlackRod.pdf" from On teaching and learning mathematics with awareness. ATM, 2018
Download file "IaBePatterns.pdf" from Mathematical imagery, ATM, 2016

The full publications are available at the ATM shop.


Curriculum Chart April 4 2018 BCME

Location: University of Warwick (British Congress of Mathematics Educators Session D34)

Dates; Wednesday 4 April, Time: 14:00-15:30

Facilitators: Anne Crosby, Ian Benson

With his Curriculum chart Gattegno demonstrated how a study of permutations and combinations of Cuisenaire rods can animate an enriched mathematics curriculum, extending from KS1 to KS5. The chart is organised as a directed acylic graph: its nodes labelled by concepts, and its edges representing a temporal hierarchy. Each root corresponds to the study of a restriction on the pattern of all permutations equivalent in length to a rod. This is a practical session using both Cuisenaire rods and a related software system. We will explore the mathematics and computer science that teachers need to know to use the chart.

The chart is reproduced on page 6 of Working with the rods and why. An ATM open resource.

Visit BCME to reserve your day delegate place.


Cuisenaire-Gattegno Course 2017

Location: Fox Primary School, London
Dates; Tuesday 25 May, Thursday 8 June; Thursday 29 June, Time: 13:30-17:30
Facilitators: Jenny Cane, Lorraine Hartley, Ian Benson

The use of concrete manipulatives is essential in developing conceptual understanding in mathematics. One common, but at times underused, resource is Cuisenaire.This series of sessions aims to support Year 2 teachers and/or KS1 Maths Co-ordinators in using Cuisenaire to explore mathematical structures and relationships. In addition to the face to face sessions which will include demonstration sessions with a group of learners, we encourage participants to work with other teachers in between the sessions, observing each other at their schools. All resources and activities created during the sessions will be shared with other teachers within the hub.


ATM Event Tuesday April 11 2017

The ATM have posted a new public resource which discusses the impact that computer algebra languages and systems
(CALS) are having on learning and teaching mathematics.

ATM have convened a working group of members and other experts drawn from mathematics education both university and pre-university, technology vendors and mathematics education research to update this 1995 booklet..

A special session is planned for ATM Conference on Tuesday 11 April at 14:00-15:30

Visit ATM to reserve a day delegate place.

(the illustration shows a learner using a CALS system to compose a successor and predecessor function for a user defined datatype that models the Cuisenaire-Gattegno "staircase" (c) Sociality Mathematics CIC, MMXVII. Learner aged 6 after 4 terms of The Primary Mathematics)


Early Algebra Event Nov 15/16

Facilitator: Rachael Rudge
Location: Kesgrave High School (15/11) Tudor C of E Primary Academy (Sudbury) (16/11)
Time: 13:30-16:00

The Norfolk & Suffolk Maths Hub are running a FREE 2 day course looking at introducing and developing algebra in Key Stage 1 classrooms. The course is being run by Rachel Rudge who started on this workgroup with the Maths Hub in 2015/2016 and is continuing this further this academic year. The course will be based on introducing Cuisenaire rods and the concepts of equivalence, addition and difference and how to write equations using the letters to represent the rods and the plus, minus and equals sign. There will be a gap task followed by a second event in February. This will feature feedback from teachers on the gap task and will look at the concepts of multiplication, division and fractions as operators using the rods. The rods will be assigned numbers for the second session.

Download file "CPD sessions.pdf"


Early Algebra Event December 5

Facilitators: Jenny Cane, Ian Benson
Location: Bursted Wood Primary School
Time: 16:00-18:00

We are pleased to announce a twilight event on Early Algebra for KS1 teachers and mathematics co-ordinators. Places are available on a first-come, first-served basis here..

Download file "Invitation to Dec5 event.pdf"


Computing at School June 18

Title: Bridging Mathematics and Computer Science with Haskell
Venue: Computing at School Conference, Birmingham, UK

Workshop session summary: The statutory entitlements for mathematics and computer science require learners to be able to move fluently between representations of mathematical ideas and to master computational thinking. At university level conceptual mathematics unifies the study of algebra, geometry and logic. We have developed a set of (6) exercises that apply conceptual mathematics to meet these new entitlements.

In this session we will show how to apply this approach to Caleb Gattegno's early algebra. Gattegno was a pure mathematician and educationist who was founding secretary of the Association of Teachers of Mathematics (ATM). He uses Cuisenaire resources to teach all four arithmetic operations and fractions as operators in Year 1 as required in the new curriculum. His is the only proven way to do this. In Gattegno's concept graph algebra is introduced before arithmetic through the study of constructions made with colour coded rods. We have been exploring the use of Haskell as a language to model these elements and operations across the transition from primary to secondary school. We have found that it provides a useful bridge between the statutory requirements for mathematics and computer science as well as an invaluable vehicle for teacher professional development.

Speaker biography: Ian Benson is acting CEO of Sociality Mathematics CIC, a community interest company that supports a network of schools following the Cuisenaire-Gattegno approach to mathematics. He is a member of the General Council of the ATM. Ian has a PhD in Computer Science (Cantab), a Masters in Symbolic Computation (Stanford) and a Masters degree (Cantab) in Mathematics. This workshop describes the results of a pilot project with Year 6 and Year 7 mathematics and computer science students.


Seminar June 2

Gattegno’s cryptomorphisms: Modeling algebraic understanding in the early years
Room 102, Jordan Building, Stanford University

Caleb Gattegno was a pure mathematician and educationalist. A sometime associate of Jean Piaget and Jean Dieudonne, Gattegno pioneered a radical reconceptualization of pre-university mathematics, including advocating the study of algebra before arithmetic. Gattegno approached teaching the integer and rational number systems through student investigations using sets of ideograms of color coded cuboids (“Cuisenaire rods”) in a variety of configurations together with the equivalence relations between them. He wrote, "All mathematical discoveries of importance can be traced to a dynamic alteration within our mind of existing organized images, or ideas."(1) He elaborated a theory of learning to account for this process. A central idea was that, in learning to listen and speak, or to see and move, a baby has already developed the capacity to reason algebraically. Educationists have recently rediscovered the merits of "early algebraization."(2,3) Similarly his learning model forshadows "dual process" psychological theories of higher cognition.(4) To Gattegno, all mathematical reasoning was grounded in mental imagery that required a conscious “awareness” of algebraic structure for its manipulation. His proposal for the new (Cui) curriculum challenged the learner with a series of progressively more complex “rod worlds.” Elements and actions in the rod world are mirrored in virtual actions on imagery suggested by these actions. Color codes and notation are used to give names to these elements and actions. Learners are encouraged from Year 1 (aged 5) to read and write expressions and equations in simple formal languages. They learn to move fluently between these representations of mathematical ideas so that the elements and operations or actions of one structure can be substituted for the elements and operations or actions of the other: a “cryptomorphism.” In this seminar we will explore exercises drawn from Gattegno's text-books, and illustrate these activities with examples of students’ work.

Participants may wish to download virtual rod resources to their laptop, iOS or android devices: links to follow can be found here.

Download file "IntroToCuisenaire.pdf"

(1) C. Gattegno, Thinking Afresh About Arithmetic, Arithmetic Teacher, v6, 1, 1959, p30-32
(2) For an overview see Chapter 1: Treating the Operations of Arithmetic as Functions, David Carraher, Analúcia D. Schliemann and Bárbara Brizuela
Source: Journal for Research in Mathematics Education. Monograph, Vol. 13, 2005, NCTM

(3) Early algebraization : a global dialogue from multiple perspectives, Jinfa Cai and Eric Knuth, 2011

(4) Dual process theories of higher cognition: Advancing the debate. Johnathan St. B. T. Evans, Keith E. Stanovich, Perspectives on Psychological Science, 3, 223--241


Newham school cluster

The initial meeting of the London NE Maths Hub Early Algebra Workgroup was held at Stratford School Academy on January 7th. Delegates attended from 1, 2 and 4 form entry primary schools. You can learn more about the work here.


Early Algebra Project

In AY 2014/15 in an action research project we piloted a scheme of work based on Book 1 of Gattegno's "Mathematics with numbers in colour" with Bentley and Copdock Schools. The project was funded by the NCETM Norfolk/Suffolk Maths Hub. The local authority have posted a report of our work here


Recent Events

ATM 2016 Conference
Mathematics as a Human Endeavour 
Chesford Grange

Session H3: Thursday 31st March 1600
Title: Graphs, Codes, Number Systems and Gattegno
Who: Ian Benson (ATM General Council), Anne Haworth (ATM Chair)

The statutory entitlements for mathematics and computer science require learners to be able to move fluently between representations of mathematical ideas and to master computational thinking. At university level conceptual mathematics unifies the study of algebra, geometry and logic. We have developed a set of exercises that apply conceptual mathematics to meet these new entitlements. In this session we will show how to apply this approach to Gattegno's early algebra. Gattegno uses Cuisenaire resources to teach all four arithmetic operations and fractions as operators in Year 1. We compare our approach to other ways of meeting the 2014 curriculum aims.


Windmill Cluster of Schools
City Heights e-Act Academy
Workshop Session: Monday 4 January
Title: Getting Started with Early Algebra
Who: Ian Benson, Suzanne Spencer

The Cuisenaire resources are increasingly popular in primary schools as a means of introducing learners to the relationship between numbers, both whole numbers and fractions as operators. Caleb Gattegno popularised Cuisenaire's invention and wrote a series of influential textbooks that taught algebra before arithmetic, using colour code names for the rods, and naming virtual actions, and patterns made with the rods with algebraic writing. We have integrated the Cuisenaire-Gattegno approach with the 2014 National Curriculum to create algebraFirst™ software tools and a curriculum unfolding. In this workshop you will learn about how our approach meets the key aim of the new NC - that pupils and teachers can "move fluently between representations of mathematical ideas."

For information on booking please email pporter-mill at sudbourne dot com and follow @windmillcluster on twitter


Workshops on algebraFirst™ July 9

We will be presenting our work at workshops at the More than the sum of the parts conference in London on July 9.

The 2014 national curriculum requires learners “to move fluently between representations of mathematical ideas.” Gattegno’s Cuisenaire resources are a proven way to meet this aim. In contrast to the traditional counting first approach Gattegno explores the algebraic structure of number before arithmetic. Mathematical ideas are experienced as concrete actions with Cuisenaire rods, virtual actions on mental images and written symbol systems. In these workshops we will discuss how to implement the cycle “concrete – virtual – symbolic” to meet the Year 1 aims. Delegates are encouraged to bring Cuisenaire rods, or tablet computers with the NumBlox app installed (iPad and Android).


Representing Functional Relationships

The study of permutations and combinations of Cuisenaire rods has proved to be a rich source of mathematical tasks motivating abstraction through algebraic symbol systems as well as mathematical generalisation. Cuisenaire-Gattegno is a unique approach that can meet the aim set by the 2014 English national curriculum, that learners “need to be able to move fluently between representations of mathematical ideas.”

In this article I take a rod permutation problem and show how teachers can use this task to support generalisation by employing a new formalism and diagramming convention that records the functional relationships between patterns of Cuisenaire rods. This convention was suggested by William Lawvere and Stephen Schanuel as an “external and internal diagram” for mappings between typed sets. (Conceptual Mathematics,1997,2004)

Download file "Introduction to Conceptual Maths.pdf"


Getting Started With Early Algebra

In two articles, we describe how we are meeting the ambitious Key Stage 1 objectives of the 2014 mathematics curriculum through mathematical writing. The 2013 curriculum requires learners ‘to move fluently between representations of mathematical ideas.’ Children now have to study all four arithmetic operations and fractions as operators for small numbers from Year 1.

Download file " PM March15 Getting Started.pdf"

Download file "PM May 2015 Experiences.pdf"



Title: Co-operative Maths Reform
Speaker: Ian Benson
Event Date: 13/11/2014 (10:00-14:00)
Venue: Birmingham, UK

The 2014 mathematics curriculum requires learners to move fluently between concrete, symbolic and numerical representations of mathematical ideas. Sociality Mathematics CIC is a network of primary and secondary schools who are working together to create high quality resources to meet this end. In this session you will learn how colour-coded Cuisenaire rods are being used to teach all four arithmetic operations and fractions as operators from Year 1, and how Gattegno's "Mathematics with Numbers in Colour" serves as an introduction to computational thinking.

Title: Using the Gattegno/Goutard approach to Cuisenaire rods to enable children to discover the structure of maths for themselves
Speaker: Caroline Ainsworth
Event date: 29/11/2014 (10:00-13:15)
Venue: Manchester, UK