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.

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.

(1) C. Gattegno, Thinking Afresh About Arithmetic, Arithmetic Teacher, v6, 1, 1959, p30-32 http://www.jstor.org/stable/41184118

(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 http://www.jstor.org/stable/30037729

(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

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.

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

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.

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

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).

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)

Title: Co-operative Maths Reform

Speaker: Ian Benson

Event Date: 13/11/2014 (10:00-14:00)

Venue: Birmingham, UK

Abstract:

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

The new English mathematics curriculum continues to rely first on counting, not algebra. In an opinion piece, from the September edition of the Ring, the magazine of the graduate association of the Cambridge University Computer Laboratory, I argue that computer science in schools can deliver this essential entitlement.

Sociality Mathematics CIC has published the early results of its work developing an approach to formative assessment in Year 7 based on Gattegno's early algebra. A short video can be viewed here. Although there are many reports of the Cuisenaire-Gattegno approach being trialled in primary school, Fortran is the first reported study to reproduce Gattegno's findings at secondary level. The study was part funded by the Sutton Trust. The result is a successful proof of concept and a step change in the understanding of teachers and learners.

The attached presentation was discussed at the annual conference of the UK Association of Teachers of Mathematics (ATM) in Sheffield on Wednesday 3rd April 2013. The video links in the presentation are as follows:

- Dr G and Algebra
- St Michael's School (1961) Grade 1
- Pattern for the dark green (Year 1) [d]
- Checking and Explaining Writing (Year 1) oo|rBB
- Substitution and Transformation (Year 1) r + r + p = t
- Substitution and Transformation (Year 1) p + r + r = o - r
- Substitution and Transformation (Year 1) p + r + w + w = o - (w + w)
- Substitution and Transformation (Year 1) 4r = (o + r) - p
- Fractional arithmetic by renaming (Year 2/3) 1/5 = 1/2 * 2/5
- Equivalent fractions (Year 3/4) 1/2 ~ 2/4 ~ 3/6 ~ 9/18 etc

For more on the colour codes see Trivett, The coloured sticks. Professor Trivett served as Chairman of the ATM during its first decade. My slide 7, Cui unfolding, refers to page 193 (web 201/208) of the ebook Gattegno's Common Sense of Teaching Mathematics. Click here for the background to Sociality Mathematics CIC and the Tizard project.

Recent results in a paper by John Jerrim and Alvaro Choi suggest that, although average math test scores are higher in East Asian countries, this achievement gap does not increase between ages 10 and 16. They conclude that reforming the secondary school system may not be the most effective way for England to ‘catch up’ with the East Asian nations in the PISA math rankings. Rather earlier intervention, during pre-school and primary school, may be needed instead.

The role of Cuisenaire rods with Gattegno's textbooks is the subject of a recent blog exchange. Readers can experience this themselves by visiting the tizard widget.

''The growing importance of maths shows that we need to do more to make sure that children speak that language too ...... the only way to create the next generation of Turings and Lovelaces is to make fluency in the universal language of maths our top priority.'' So says Education Minister, Elizabeth Truss in her speech at the North of England Education Conference. In defining what fluency and arithmetic proficiency mean the NCETM cite the work of the Tizard network as their sole example of best practice. More on the Tizard cohort 6 can be found at the NCETM website, where a case study of four years exposure to the Cui curriculum is described by Caroline Ainsworth.

What can self-organising biological systems tell us about learning and teaching? A good deal, according to Enrico Coen, a plant geneticist. In ``Cells to Civilizations'' he presents a unified account of the emergence of living organisms, and highlights common principles of development across levels -- from evolution and cell development, to learning and human culture.

In her review of the recent TIMMS and PISA results and findings from a recent study that shows little change in mathematics proficiency since the 1970's Elizabeth Truss, Education Minister, wrote, ``At primary level,
(the new primary curriculum) will mean increased focus on arithmetic and taking it off data; requiring not
only that pupils learn things like their tables earlier – at Year 4 instead of Year 6 –
but also that they develop structured arithmetic, developing the foundations for
algebra.''

The Cui approach is the only proven way of learning and teaching early algebra.

Ian Benson and Anne Haworth are offering a 3 hour workshop for primary teachers and educationists at the ATM conference in Sheffield, UK on Wednesday 3rd April

Part 1: Early Algebra

Gattegno maintained that we can exhaustively identify the awarenesses needed in any domain and redefine teaching as the activity which leads students to cover this ground for themselves without missing any essential steps and without wasting time. To this end he developed the Cui curriculum and related textbooks. Like the proposed UK primary curriculum, Gattegno covers all four arithmetic operations, fractions and product tables at KS1. He did this by introducing algebra as a formal language first, before number. What does algebra look like to infants? We will cover Cuisenaire code, trains, staircases, patterns, decimal fractions and percentages in practical exercises.

Part 2 Metamathematics and Formative Assessment

Few primary teachers are familiar with mathematics as a language. What do teachers look for when they observe students working with rods and algebraic writing? What does Gattegno mean by equivalence and how does he harness the idea to create rich opportunities for students to learn? Reasoning with equivalence: colour, length, difference, parity, fractions as magnitudes, products. Reasoning about equivalence: sets, functions, domains, objects, arrows, permutations and combinations. Exercises with Complete Patterns. Formative assessment of student work in Years 1-6. Sessions based on eight years experience of re-introducing Cui in the Tizard network of primary schools.

Anne Haworth has now collated the evidence submitted to the UK Advisory Council for Mathematics Education and ACME have presented it to Government. In it ACME recommend developing a coherent pre-algebra strand in the primary mathematics curriculum. They write,

''We had asked specifically about algebra.... Many respondents said that the Programme of Study (PoS) does not provide a coherent progression towards formal algebra, one claiming that it provided a weaker foundation than the current curriculum because the procedural approach discouraged thinking before acting.

``Many respondents suggested ways in which preparation for algebra could start in Year 1, and that expectations of algebraic thinking could be even more challenging if they were based on reasoning about relations between quantities, such as patterns, structure, equivalence, commutativity, distributivity, and associativity, and models and representations of these.