Nadim Kobeissi

Natalia Kulatova

Ledger Design Language: Designing and Deploying Formally Verified Public Ledgers

Cryptocurrencies have popularized public ledgers, known colloquially as "blockchains". While the Bitcoin blockchain is relatively simple to reason about as, effectively, a hash chain, more complex public ledgers are largely designed without any formalization of desired cryptographic properties such as authentication or integrity. These designs are then implemented without assurances against real-world bugs leading to little assurance with regards to practical, real-world security.

Ledger Design Language (LDL) is a modeling language for describing public ledgers. The LDL compiler produces two outputs. The first output is a an applied-pi calculus symbolic model representing the public ledger as a protocol. Using ProVerif, the protocol can be played against an active attacker, whereupon we can query for block integrity, authenticity and other properties. The second output is a formally verified read/write API for interacting with the public ledger in the real world, written in the F ⋆ programming language. F ⋆ features such as dependent types allow us to validate a block on the public ledger, for example, by type-checking it so that its signing public key be a point on a curve. Using LDL's outputs, public ledger designers obtain automated assurances on the theoretical coherence and the real-world security of their designs with a single framework based on a single modeling language.

Introduction

Blockchain technology, pioneered by Bitcoin [START_REF] Nakamoto | Bitcoin: A peer-to-peer electronic cash system[END_REF], provides a globally-consistent append-only ledger that does not rely on a central trusted authority. This underlying public ledger technology has led to the rise of more advanced cryptocurrencies with diverse and innovative use-cases. Ethereum [START_REF] Wood | Ethereum: A secure decentralised generalised transaction ledger[END_REF], for example, deploys an entire state machine on top of a public ledger, which can then run programs called "smart contracts" in a globally shared namespace. Zerocash [START_REF] Sasson | Zerocash: Decentralized anonymous payments from bitcoin[END_REF], on the other hand, focuses on making cryptocurrency trading and exchange more anonymous by employing zeroknowledge proofs. All of these technologies utilize the same underlying public ledger concept, where a series of append operations, each constituting a block, create a chain of blocks each cryptographically authenticating itself and the last block, hence the name "blockchain". All of these public ledgers are based on decadesold concepts in computer science, namely binary hash chains and Merkle trees [4]. The Bitcoin blockchain is, in essence, an extended binary hash chain capable of supporting more complex data types and with certain rules regarding append and read operations. More complex designs such as Ripple [5], for example, uses a Merkle tree-shaped public ledger.

A Language for Designing Public Ledgers

Given what we know about the concepts underlying all public ledgers, and the fact that they currently shoulder currencies of significant value, we propose Ledger Design Language (LDL), which combines two different aspects of formal verification in order to bolster the security of new public ledger designs in two distinct ways. LDL is composed of three components, illustrated in Figure 1: 1) Input: A Modeling Language for Public Ledgers. The LDL language describes public ledgers as a series of block types. As seen in Figure 3, each block type contains: LDL's architecture aims at simplifying the design process of public ledgers, allowing designers to immediately obtain a way in which to rationalize regarding the effective security goals of their public ledger and to test it in a practical setting. Modifying the resulting ProVerif model allows live prototyping on a protocol logic level, while the F ⋆ API allows for performancerelated and other kinds of practical testing.

Related Work

Despite the unified and well-understood nature of the concepts underlying public ledgers, there appears to be, to the best of our knowledge, little to no prior work on applying formal verification to public ledgers. The Solid Ether project [8] targets Solidity, the language in which Ethereum "smart contracts" are written, and includes translation to F ⋆ from both Solidity and its underlying bytecode which is what is stored on the Ethereum public ledger. Simplicity [9] follows another route by introducing an alternative to Solidity that lends itself more easily to writing correct programs. However, the structure of the ledger itself is not within the scope of these project's verification or correctness efforts.

The LDL Language

LDL prioritizes simplicity, aiming to construct a modeling language with the least amount of top-level components. In this section, we describe minimal LDL syntax and give a practical use-case example.

Language Semantics

LDL syntax (described in Figure 2) offers only three possible kinds of top-level declarations:

• Block Declarations. As demonstrated in Figure 3, block declarations allow the block types of a public ledger to be defined, first as a data structure and then with properties. The mandatory before and branch properties, specified for each block, are sufficient to obtain the full topology of the public ledger being described, be it a Merkle tree or a hash chain. • Actor Declarations. Actors, defined using a minimal syntax (as demonstrated in Figure 4) represent "entities" which are specified to "own" a particular chain or branch that is part of the ledger. This is useful when translating to a symbolic protocol model and when generating the F ⋆ API, as described in §3. • Structures. A simple syntax exists for defining data structures, which can then be used as shorthand within blocks (for example, a userInfo structure can contain three strings describing a user's name, email and phone number.) Block data types such as signature and nonce carry connotations of cryptographic security, which are translated in a special way by the LDL compiler.

LDL Features

LDL can be used to express some advanced features that do not usually appear in simpler public ledger designs (such as a ledger that tracks banking transactions). Nevertheless, these features do appear in more complex ledgers that are the basis for our case study in §4:

• Verifiable Random Functions. VRFs [START_REF] Micali | Verifiable random functions[END_REF] are essentially a type of hash function with public key elements. The owner of an asymmetric key pair (sk, pk) for V RF can compute V RF sk (x) = (p, y), where x is an input value and y is a pseudorandom output value with uniform distribution. Meanwhile, p is a "proof value" that can be used by any public observer that possesses (pk, x, p, y) to verify that y is indeed a correct pseudorandom mapping of x under a VRF keyed with secret key sk. VRFs are important for many advanced public ledger designs: they are used to encrypt all block contents by generating encryption keys that can be authenticated by the public ledger owner. In events where the public ledger is structured as a Merkle binary prefix tree, VRFs are used to randomize the indices to which blocks are allocated on the tree. The public ledger owner can then selectively reveal B i p , the proof value for any block B i , allowing the correct index to be confirmed. • Cross-Referencing. LDL also allows for crossreferencing, wherein blocks may include index references to other blocks on the chain. • Standard Cryptographic Functions. In LDL, all encryption operations are translated into ProVerif and F ⋆ as operating under the AES-GCM AEAD cipher. All public key operations are translated as using Curve25519, and all signing operations use ED25519. This design ensures that safe primitives are used always, lessening the degree to which LDL model designers can make low-level cryptographic mistakes. We accept the expense this decision has on cipher suite flexibility.

• Key Rotation. Using the signpub datatype, a public ledger user can append a block that determines their identity's future signing public key for all blocks coming after this block, achieving a form of key rotation that is essential for implementing some of the public ledger designs discussed in §4.

Current Security Queries

We are currently able to query for the following basic security properties:

• Block Integrity. Any internal, non-leaf block delivered to the reader process must have been written as-is by the writer process. • Block Authenticity. Any block delivered to the reader process must be cryptographically validated by a signature belonging to a certain actor if and only if it was sent by the writer process under that actor's identity. • Block Privacy. Specifying option vrf; in the LDL model enables "VRF mode", in which the usage of a VRF is assumed for encryption across the entire ledger. In the event that the ledger describe is shaped as a Merkle binary prefix tree, the LDL compiler output will also assume the randomized, privacy-preserving distribution of indices. This latter property is important for public ledgers such as CONIKS (as discussed in §4.)

• Reference Integrity. A block on the ledger can contain references to other block indices. We want LDL output to model and give assurances on whether these pointers will always point to the correct blocks as intended. In some structures, this is impossible to guarantee: we want the compiler output to reflect this and to give useful insight to the designer.

Additional Work on Security Queries

In addition to the scope of this paper, we plan to include additional security queries which will allow us to reason on the security of the following events:

• Chains with Shared Ownership. Examples given in this paper show chains owned by a single actor. We want to expand our formalisms to include scenarios where Alice and Bob both contribute to the same chain, but with a specific logic governing which signatures are expected for which block. At this stage, inferences can be made based on block address references and hierarchy which allow LDL to determine which signatures to expect at which time. • Block Ordering Reliability. Our current reasoning on the constructed ProVerif output assumes a perfectly ordered sequence of messages between the writer and accumulator processes.

We want to expand this model to include security queries with regards to unreliable communication between these to processes, so that we can ask whether this can lead to invalid or unintended public ledger architectures being constructed.

From LDL Model to Formal Verification

In this paper, we give a high-level description of the reasoning and strategy behind LDL's outputs. Future work will discuss details such as translation rules and the logic of underlying security goals.

Verifying Public Ledgers as Symbolic Protocols

ProVerif models are composed of a set of process descriptions, functions, reductions and constructors in the symbolic model. In this model, all cryptographic primitives are perfect "black boxes". Then, a top-level process execution is described, which could include a parallel conjunction of processes. These processes send and receive information over a set of channels. On public channels, an active Dolev-Yao attacker can monitor and tamper with messages. In this context, events and queries are defined with respect to the exposure (or lack thereof) of certain secret variables.

An LDL model is translated into this setting by specifying three main processes:

• Writer Process. This process issues the blocks in the order mandated by the structure of the public ledger, using the identities defined by actor declarations.

• Accumulator Process. This process accumulates the blocks as received over the network from the writer process, producing a structure that acts as the record for the public ledger. • Reader Process. This process then reads blocks from the ledger. Read operations are sent to the accumulator process, which then delivers block information over the network to the reader process. This allows ProVerif a chance to expose this block information to the active attacker.

Generating a Verified Public Ledger API in F ⋆

In F ⋆ , we want to obtain an API implementing the public ledger in a way that allows for safe, highassurance read-write operations and with a formally verified cryptographic context. We expose an API with top-level access functions generated through LDL actor declarations:

Blocks as Dependent Types

Using dependent types in F ⋆ , we can describe blocks as a collection of simple datatypes and also algebraically validated types. This allows us to typecheck a block as valid within the context of a valid signature (which authenticates to a particular identity and is well-formed as two points on the curve, in the case of Ed25519) and a hash that is coherent within the hash chain of the branch in which it belongs.

Functional Correctness and Runtime Safety

The F ⋆ API will also be able to guarantee that all operations on the public ledger target a ledger that is well-formed, according to spec, with no room, for example, for misdirected block references.

Verified Primitives with HACL ⋆

HACL ⋆ [START_REF] Zinzindohoué | Hacl*: A verified modern cryptographic library[END_REF] is a formally verified cryptographic library in F ⋆ . Primitives within HACL ⋆ are verified for functional correctness, memory safety and against side-channel attacks. These guarantees carry when HACL ⋆ is translated to C code. The LDL compiler calls HACL ⋆ primitives for all cryptographic operations and thereby benefits from these security guarantees.

Case Study: an LDLChain Implementation

We present a case study that uses LDL to implement and verify "LDLChain", a full public ledger design that can be used for managing public key information. In order to understand how to best shape LDLChain, we based ourselves on an extensive literature review of existing public ledger systems. LDLChain is based on CONIKS [START_REF] Melara | Coniks: Bringing key transparency to end users[END_REF], a public ledger approach to certificate transparency by Melara, Blankstein, Bonneau, Felten and Freedman. LDLChain is also is inspired by ClaimChain [13], a recent design by Kulynych, Isaakidis, Troncoso and Danezis, which proposes a public ledger that lends itself towards storing claims made by users regarding their long-term identity keys for various cryptographic systems that employ public-key encryption.

Given the space constraints within this paper, we have found it impossible to describe our case study implementation in detail, including vital information such as the full LDL model, its translation into the applied-pi calculus and its F ⋆ API. A longer version of the paper will contain a complete version of the case study.

LDLChain Design and Security Goals

LDLChain blocks (described in Figure 3) can contain self-identifying information, such as usernames, and endorsements of claims made by other users (by referencing the block index on which they were made). LDLChain blocks can also contain key information: LDL allows for key rotation using the signpub datatype.

LDLChain is largely based on ClaimChain, and its security properties are exactly that of ClaimChain. These properties can be described in LDL, with the exception of non-equivocation, which is currently slated for future work.

The LDLChain API in F ⋆

We briefly describe LDLChain's F ⋆ API. 2 LDL's F ⋆ library comes with implementations for standard data structures such as Merkle trees and Skip lists. Depending on the LDL model, these libraries are used in the generation of an F ⋆ Spec. In it, blocks are represented as complex types with dependent subtypes depending on the structures described in LDL, as seen in Figure 5. For that F ⋆ type, an API is generated, including, for example, a function addClaim, which uses lemmas and logic contained in the LDL Merkle tree F ⋆ library in order to manage the secure insertion of the claim into the public ledger.

Conclusion

In this paper, we introduce LDL as a domainspecific modeling language that can be used to describe public ledgers. The LDL compiler validates these descriptions and generates two distinct outputs that allow users to reason about the security of public ledgers, obtain meaningful conclusions and deploy them with practical, verified implementations.

1. The main reason why we implement LDLChain instead of simply ClaimChain is that, to the best of our knowledge, ClaimChain's public ledger data structure is not currently fully defined by its authors.

2. Due to space constraints, we cannot include a similar description of the ProVerif output in this version of the paper.

Future Work

Future work includes extending the LDL syntax to capture more public ledger use-cases and continuously improving the language's functionality and expressiveness in accordance with modern public ledger design norms.

Currently, there exists no formal proof guaranteeing that the LDL translation rules fully preserve all security goals that are targeted by an LDL model as it is translated to an applied pi calculus model or to F ⋆ code. Future work aims to remedy this.

Furthermore, different avenues could be explored for LDL output. The CryptoVerif [14] verifier accepts a similar syntax to ProVerif, but can be used to describe computational models with algebraic properties. LDL's F ⋆ translations themselves could also be expanded to capture more of the protocol logic which we currently verify using ProVerif, creating a formal relationship between the two representations.

Figure 1 .

 1 Figure 1. LDL Architecture Outline

Figure 4 .

 4 Figure 4. LDL actors that are part of LDLChain, discussed in §4.

Figure 5 .

 5 Figure5. Target F ⋆ code representing the datatypes generated from Figure3by the LDL compiler. Signatures are dependently typed so as to be verified as valid and authentic.

Output: A Symbolic Model Representation of the Public Ledger. A

	ad-
	mitted directly preceding blocks, and whether
	it is followed by multiple branches (like a
	branching hash chain) or whether it is followed
	by a single branch (like a hash chain resembling
	the main Bitcoin blockchain). If the binary
	qualifier is used to describe the branchType of
	all block types on the ledger, then the LDL
	compiler types the ledger as a Merkle binary
	prefix tree. Once these two properties are de-
	scribed for each block, we can obtain a full map
	of the public ledger's structure.
	2) model, written in the
	applied-pi calculus represents the public ledger as
	a protocol being played across a network with
	parallel processes. This model is adapted for anal-
	ysis using the ProVerif [6] protocol verifier, which
	then analyzes it against an active attacker in the
	Dolev-Yao modeland attempts the validation of
	whether specific cryptographic properties, such as
	authenticity, are obtained.
	3)

• Block Data Structure. A listing of the elements this block contains. Each element can have a simple type, such as string or integer, or a type that carries security connotations, such as signature (a cryptographic signature which must be validated according to certain rules) or nonce (an identifier which must never be reused across the public ledger's existence.) ⟨ldl⟩ ::= ⟨ledger⟩ | ⟨option⟩ ⟨ledger⟩ ::= 'ledger ' @identifier '{' (⟨st⟩)*'}' ⟨option⟩ ::= 'option ' ⟨opt⟩ ⟨opt⟩ ::= 'vrf' ⟨st⟩ ::= ⟨block⟩ | ⟨actor⟩ | ⟨struct⟩

⟨block⟩ ::= 'block ' @identifier '[' (⟨type⟩ @identifier ';')* ']' '(' ('before' @identifier ';' 'branch' ⟨branchType⟩) ')'

⟨actor⟩ ::= 'actor ' @identifier '['']' '(' ('ownsChain' @identifier ';' 'ownership' ⟨ownershipType⟩) ')'

⟨struct⟩ ::= 'struct ' @identifier '[' (⟨type⟩ @identifier ';')* ']' '()' ⟨type⟩ ::= 'string' | 'address' | 'nonce' | 'integer' | 'signature' | 'date' | 'reference' | 'struct' | 'signpub' ⟨branchType⟩ ::= 'single' | 'multiple' | 'binary' ⟨ownershipType⟩ ::= 'single' | 'multiple' Figure 2. LDL syntax.

• Block Properties. Two mandatory properties, before and branchType, describe the

Output: An API Implementing the Public Ledger

	. F ⋆ [7] is an ML-like language aimed at
	program verification. Using its advanced features,
	such as dependent types and refinements, we can
	obtain a real-world API implementing the public
	ledger in a way that allows immediate use in
	production systems, while maintaining runtime
	safety, type safety and employing formally verified
	cryptographic primitives.

  Figure 3. A partial LDL model showing two block declarations. A UserChain describes the root block of an append-only Merkle branch and indicates that it can occur only before the Root block for the entire public ledger. Meanwhile, the Claim describes a check which can only have a single, non-branching chain of Claim blocks as children. The full model describes the ledger discussed in §4.

	block Root[
	date currentDate;
	nonce merkleRoot;
	string ledgerName;
	](
	before UserChain;
	branch multiple;
	)
	block UserChain[
	string name;
	signpub initialKey;
	](
	before Claim;
	branch single;
	)
	block Claim[
	date timestap;
	struct claimData;
	signpub nextKey;
	](
	before Claim;
	branch single;
	)
	actor ChainOwner[] (
	ownsChain Root;
	ownership single;
	)
	actor Alice[](
	ownsChain UserChain;
	ownership single;
	)

  1 Only readers authorized by the Claim-Chain owner to access a claim at a particular point in time can read the content of that claim, at that point in time. At a particular point in time, a ClaimChain owner cannot provide two authorized readers of a claim with different content for that claim.

	• Authenticity. The information stored in a
	ClaimChain has been input by the owner of the
	chain.
	• Integrity. The information stored in a Claim-
	Chain has not been modified since it was added
	to the chain.

• Privacy. • Non-equivocation.

Acknowledgments

We thank Karthikeyan Bhargavan, Bruno Blanchet and Antoine Delignat-Lavaud for discussions regarding translation to ProVerif and F ⋆ . We also thank Hadi Kabalan and Russell O'Connor for discussions regarding use-cases. We also acknowledge Diffie the poodle for his profound insight and moral support.


domain: math shs

text:
Margareta Engvall
email: margareta.engvall@liu.se

Teachers learning through participatory action researchdeveloping instructional tools in mathematics primary classrooms

Keywords: Mathematics education, action research, teacher professional development, multilingual classrooms, instructional tools

Introduction

In many classrooms, from first grade through the whole school system, there are many students whose mother tongue is not the same as the teachers' language used for instruction [START_REF] Khisty | Effective teachers of second language learners in mathematics[END_REF]. In Sweden, newly arrived immigrant children with limited Swedish language knowledge are learning Mathematics together with children who have spent their entire life in a Swedish context. Considering the large number of students with limited knowledge in the language of instruction one of the most important tasks for teachers in Swedish primary education is to create conditions to support the development of mathematical knowledge in these students.

Lately, the role of language in mathematics education has received a profound interest in educational research. Researchers have emphasized the importance of teachers using specific strategies to facilitate the classroom communication to and support students' mathematical thinking (O'Connor & Michaels, 1993). Specifically, in the syllabuses [START_REF] Skolverket | Läroplan för grundskolan, förskoleklassen och fritidshemmet 2011[END_REF] it is particularly prominent that mathematics is dominated by discourse-intensive approaches, and the use of instructional tools such as talk moves, give ample opportunities for student learning [START_REF] Chapin | Academically productive talk: Supporting students' learning in mathematics[END_REF]. Similar strategies for supporting students' learning in mathematics have received attention among effective teachers of second language learners in mathematics [START_REF] Khisty | Effective teachers of second language learners in mathematics[END_REF]. This study uses action research which is characterized by ongoing processes of self-reflection, which can be thought of as a spiral of self-reflective cycles on planning a change, followed by acting and observing the process and reflecting on the process and then re-planning and so forth (Kemmis & Wilkinsson, 1998). Using PAR gives an attempt "to help people investigate and change their social and educational realities by changing some of the practices which constitute their lived realities" (Kemmis & Wilkinsson, 1998, p.22).

Method

The poster gives a brief presentation of a one-year research project where four primary teachers at the same school (year 2, 4 and 5) have been working together with a researcher, using participatory action research (PAR) (Kemmis & Wilkinsson, 1998) to develop their instructional tools in order to support students' mathematical development in multilingual classrooms. Data collection has continued throughout the whole action research process during the academic year. The empirical data includes teachers' logs, teacher questionnaires with open answers, researcher's notes, audio-Thematic Working Group 18 Proceedings of CERME10 recorded discussions from the meetings twice a month in the project group and 3-4 video-taped mathematics lessons in each classes, 14 lessons altogether

Results

Although the focus in this project has been on instructional tools for supporting students' talk in order to enhance their development in communicating and reasoning mathematically, it is noteworthy that the teachers express their development, not only in terms of (1) instructional tools but also regarding other aspects such as (2) classroom organization and (3) focus on mathematical content.

Methods structured in these three themes above constitute a teacher tool kit to support students' learning mathematics in multicultural classrooms.

Conclusions

By using PAR, the teachers had the opportunity to reflect critically, analyze and act as coparticipants in the challenge to change the practices in which they interact, which also challenged their approach to teaching.

When teachers act and reflect on their use of specific strategies of classroom talk they also start reflecting and acting on other aspects of teaching, such as classroom organization and how to keep attention to the taught content. Thereby, the change in practice became more than just temporary changes.

Proceedings of CERME10


text:
Pessia Tsamir
email: pessia@post.tau.ac.il

Dina Tirosh

Ruthi Barkai
email: ruthibar@post.tau.ac.il

Esther Levenson
email: levenso@post.tau.ac.il

Noticing aspects of example use in the classroom: Analysis of a case

Keywords: Examples, case analysis, teacher education, theoretical tools

This paper investigates promoting knowledge of example use in mathematics education by way of analyzing a case using theoretical tools. Participants were both prospective and practicing teachers attending a university course. An event taken from a tenth grade geometry class was analyzed in terms of example use, and then discussed. Participants related to the type of example given, the timing of the example, agency, what the example was an example of, and the aim of giving the example.

Introduction

Examples and non-examples are an integral part of learning and teaching mathematics. They are used in concept formation, when seeking relations between elements, abstraction, and generalization (e.g., Smith & Medin, 1981;[START_REF] Watson | Qualities of examples in learning and teaching[END_REF]. Acknowledging the importance of example use in mathematics education, [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] included knowledge of examples within the category of pedagogical content knowledge (PCK). This knowledge, he claimed, is essential for representing the subject, so that it will be comprehensible to others. [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] also included teachers' use of examples in their description of 'the knowledge quartet', a framework for thinking about the ways subject matter knowledge comes into play in the classroom. In their framework, the ways teachers use examples and the types of examples they use, are manifestations of teachers' own content knowledge, their meanings and descriptions, being transformed and presented in order for students to learn the mathematics. [START_REF] Ball | Content knowledge for teaching: What makes it special?[END_REF] also noted the importance of teachers' knowing how to sequence examples. Although example use is a complex matter and promoting teachers' knowledge of example use could be quite challenging, few studies specifically investigated this aspect of teacher education. This study proposes fostering prospective and practicing teachers' knowledge and awareness of example use by applying research and theoretical tools to analyze cases, helping to bridge theory and practice for mathematics teachers. The material, which is the focus of this paper, is a case based on a classroom event, and used as an exemplar. The case material consisted of a classroom transcript, along with guiding questions to be answered by each participant. This was followed by a group discussion with the teacher educator. Specifically, we ask: What aspects of example use do individual participants notice when studying an authentic case? Can those aspects be traced back to theories learned during the course? What additional aspects arise during group discussion?

Example use in mathematics education

There are many aspects of example use which have been investigated. To begin with, there are different types of examples. [START_REF] Arcavi | The role of visual representations in the learning of mathematics[END_REF] [START_REF] Tsamir | Intuitive nonexamples: The case of triangles[END_REF]. Intuitive examples are those examples which students immediately identify as such and are often derived from practical experience. Likewise, intuitive non-examples may encourage visual, rather than analytical thinking. On the other hand, it seems that non-intuitive examples and non-examples can encourage students to use reasoning based on critical attributes. [START_REF] Tsamir | Intuitive nonexamples: The case of triangles[END_REF] also discussed the sequencing of examples and non-examples and its effects on students' learning.

The difference between examples and non-examples is dependent on the mathematical lens one is looking through. When looking for polygons, a triangle is an example; when looking for quadrilaterals, a triangle is a non-example. In other words, another aspect of example use to consider is the way an example is being used. [START_REF] Zazkis | What makes a counterexample exemplary?[END_REF] introduced the idea of pivotal examples which can cause the learner to change his or her cognitive perception or way of approaching a problem. [START_REF] Rowland | The purpose, design and use of examples in the teaching of elementary mathematics[END_REF] differentiated between two uses of examples in teaching. The first is when examples are used to motivate generality. In this case, the examples are examples of something where the aim is to teach a general procedure or to support abstraction and concept formation. The second type of example use is for students to practice what was taught. That is, students are given many examples to practice some procedure. This type of example use allows students to experience variation and can lead to additional awareness and understanding. [START_REF] Watson | Qualities of examples in learning and teaching[END_REF] found that examples can be used as templates for dealing with other class members, indicating a relation between classes or to express equivalence.

If we take into account that the way an example (or non-example) is used by a learner is determined by the focus of that learner, then it becomes the teacher's task to choose examples and set them up in such a way that students will view those examples through the intended lens and focus on the intended pedagogical aim. [START_REF] Zodik | Characteristics of teachers' choice of examples in and for the mathematics classroom[END_REF] identified six types of considerations employed by teachers when selecting or generating examples: starting with a simple or familiar case, attending to students' errors, drawing attention to relevant features, conveying generality, including uncommon cases, and keeping unnecessary work to a minimum. They found that the most frequent consideration was choosing to begin with the simple or familiar example. They also found that on the spot, teachers often choose an example that will attend to an error which arose in class, whereas pre-planned examples tend more to consider uncommon cases.

To notice something means to make a distinction, to stress some perceived feature and ignore others [START_REF] Mason | Epistemological foundations for frameworks which stimulate noticing[END_REF]. As shown above, various aspects of example use can be considered when analyzing a classroom event. In this study we investigate which of these aspects, and perhaps other aspects of example use, practicing and prospective teachers notice when analyzing a case.

Methodology

The participants in this study were 13 practicing teachers (denoted as T) with between 1 and 11 years of experience (mean years of experience was four) and 10 prospective teachers (PTs) who had completed their first degree in mathematics or a mathematically rich field of study, such as engineering. The specific course which is the context of this study, aimed to promote participants' knowledge of explanations and examples, and the relationship between them in mathematics education. The course consisted of a total of 13 lessons which took place once a week for a period of 90 minutes each. The first six lessons were devoted to explanations in mathematics education. Different types of explanations were reviewed, such as conceptual, procedural, mathematicallybased and practically-based explanations (e.g., Levenson, Tsamir, & Tirosh, 2010), and theories concerning the roles of explanations in mathematics education were discussed (Levenson & Barkai, 2008). The last seven lessons were devoted to examples in mathematics educations. Theoretical perspectives of examples, like those described in the background of this paper, were read and discussed.

During the fourth course lesson on examples (lesson #10), the teacher educator handed out a transcript of a geometry lesson which took place in a tenth grade classroom. The geometry lesson had taken place within a few weeks of the course lesson and was observed by three of the prospective teachers attending the course, lending the case authenticity and relatedness. The overall aim of the geometry lesson was to introduce students to Thales' theorem and show its connection to similar triangles. The students had previously learned about similar triangles in ninth grade. The case transcript began with the classroom teacher stating Thales' theorem. This was accompanied by a drawing on the board, made by the teacher, of two similar triangles, under the headline: Geometry -Thales' theorem. Next to the drawing it says that if the given is BC║DE then from Thales' theorem we conclude proportional line segments (see Figure 1). The case transcript was handed out in the beginning of the course lesson with instructions to read it from beginning to end, without interruption, in order to understand the context and get a feel for the classroom. After reading through the transcript, participants were asked to reread the transcript and fill out a worksheet with the following questions: What did you learn about the use of examples during mathematics instruction from the examples given in the case presented? What would you do the same as the teacher did with regard to examples? What would you do differently from the teacher with regard to examples? After the participants wrote their answers and handed them in, a discussion followed. This discussion was audio-recorded and transcribed.

Data analysis

When analyzing participants' responses to the worksheet, our guiding question was, what aspects of using examples did the participants relate to when studying the case. We then used the literature background to help form a categorization scheme of those aspects. as responding to a student's error. Because participants wrote freely, it happened that one sentence could encompass more than one aspect of example use. For instance, one teacher (T12) wrote "the teacher gave a numerical example, which in my opinion served the purpose of making it easier for the students to understand and generalize the idea." T12 refers to the type of example (numerical) as well as the purpose of the example (to help students generalize). Participants' comments that were thought not to be related to the giving and use of examples were categorized as 'unrelated'. Table 1 lists the categories along with examples from the data of each category.

Aspect of example use Sample data

Type: What type of example is given?

The example uses simple numbers; it is a visual example of Thales' theorem.

Agency: Who is giving the example? Only the teacher gives examples.

Aim: what seems to be the aim of giving the example?

The example is given to explain it again; the example shows the students how the theorem works.

Timing: When is the example given? The example is good for the beginning of a lesson.

Example of: What is the example an example of?

The teacher gives examples of proportions.

Unrelated

The teacher presents a dry definition of Thales' theorem.

Findings

This section analyzes participants' comments from the worksheet on two segments of the case along with excerpts from the discussion which focused on those case segments. Thus, we review what participants noticed individually and what they discussed collectively. Analysis of participants' comments is carried out according to the aspects of examples listed in Table 1.

Segment one-Introducing Thales' theorem

The case transcript began a picture of the teacher at the board presenting to his students an example (see Figure 1). Accompanying the picture were the following lines from the case transcript: There are several ways to look at the examples in the above segment. One way is that there are two explicit examples. First, there is the drawing on the board (see Figure 1). Second, there is the numerical example given by the teacher in Line 1. However, the drawing on the board can be an example of Thales' theorem or an example of similar triangles, and indeed, that is what the teacher is trying to convey. In addition, depending on one's point of view, the example on the board and the numerical example may be considered one complete example, with the example on the board written in a general matter, using parameters (a, b, c, and d), and the oral example, a specific example given with numbers.

1 T: In Thales' theorem, it
The most frequent aspect of examples mentioned by participants was example type. Participants used such descriptions as: a simple example (PT2), an unfinished example (T8), a visual example (T12), a numerical example (T12, T13), and an intuitive example (PT 22). Although most participants wrote that the teacher gave the example, we do not take this as commenting on the agency, but rather as a description of what is going on. On the other hand, for a different segment, one participant wrote, "Only the teacher gives examples… but he should have requested examples from the students." This participant is not merely describing the situation, but commenting on who is and who should be giving the examples. The aim of giving the examples was mentioned by T8 who wrote, "In Line 1, the teacher gives an example that students have to finish. He is checking to see if the students are listening and if they understand." T13 wrote, "The teacher gave a numerical example so that the students could understand the example and draw on their previous knowledge of similarity and proportional triangle sides." Timing of the example, that the example was appropriate for the beginning of the lesson, was noted by PT2. Two participants commented on what the example was an example of -T12 wrote that it was an example of the theorem, meaning that the example showed how the theorem could be applied. PT22, referring to the numerical example in Line 1, wrote that it was an example of equivalent ratios.

Segment two -Proportional segments

The following case segment is a direct continuation of the first one:

4-6.

(The teacher and students review the concept of similar triangles.) 7.

T: So, what is the ratio of their corresponding sides (referring to the example given in Line 1)? 8-10. Students:

1 to 3. 3 to 1. It depends on how you look at it.

T:

The ratio is … 3 to 4 because BD and CE are not sides of the triangle. So 6 is to 8 like 12 is to 16. Now, …, what came first in Euclidian geometry? First, there was Thales' theorem and only after that came the similar triangles theorems. So, let's say we are in ancient Greece and we don't know yet about similar triangles, but we do know Thales' theorem. With that theorem, we can prove proportional sides in similar triangles.

In lines 4-11, no new examples are given. Instead, the teacher and students still refer to the first examples given on the board. Like the comments on the first segment, here too, most comments related to the types of examples being given: simple numbers (PT7), a non-intuitive example (T11), a general example with parameters (T17), and a numerical example with familiar numbers (T19). None of the participants mentioned aspects of agency, aim, or the timing of examples. T11 noted that the example was an example of ratios. Two PTs wrote remarks connected with the story of ancient Greece. PT23 wrote, "There is an example from real life that I like -in ancient Greece." PT16 wrote, "In Line 11, the teacher made the material come alive when she told the story about Thales' theorem from ancient Greece ... This example raises the question of why the students first learn about similarity and then about Thales." T23 calls it an example from real life. Yet, it is not a mathematical problem related to real life. The term 'example' when describing the ancient Greece story is not in line with the notion of examples discussed in the course.

Discussing the case

After the worksheets were handed in, the teacher educator (TE) opened up the discussion by asking who wished to comment on the case. After nine minutes of discussing general ideas, the discussion turned to the example given in Line 1 of the case transcript. Note that PT21 was present during the geometry lesson as part of their field work.

TE:

Is it an explanation or an example? Can an example also be an explanation?

PT21:

No. An explanation can be accompanied by an example. There can be an example and then the explanation generalizes it.

T8: I felt that way about Line 1. It feels like an explanation, and also like an example. On the one hand, there are numbers. On the other hand, it's not complete.

The above excerpt gives us a glimpse into PT21's and T8's concept image of an example. Both infer that an example has numbers, but it must be complete, without any missing parts. There is also the question of the different roles an example may play in the classroom. Can an example be an explanation? Must all examples be specific and only the explanation can generalize it? These are some of the questions that the participants are grappling with. In the following excerpt, PT22, who was also present as an observer of the geometry lesson, tells what he observed.

PT22:

He was trying to show how Thales' theorem is really intuitive. That is, he gave an intuitive example.

TE: An intuitive example of what?

Many voices: Equivalent fractions.

PT22:

And most answered correctly. The teacher was trying to show how easy it is.

TE:

But what was it an intuitive example of?

Discussion

During teacher preparation and professional development, participants are introduced to various theories. While field work is important, theories can help prospective and practicing teachers make the most of their field work by focusing their attention on different elements of practice. Findings showed that analyzing the case gave participants a chance to apply their knowledge of example theory when examining a classroom situation. Findings also showed that participants did not necessarily draw on the same theories when analyzing the same event. The example given in Line 1 of the case transcript was described alternatively as simple, numerical, visual, and intuitive. Each of these types can be traced back to different theories discussed in the course, but they focus on different issues. An intuitive example may also be numerical, but if the prospective teacher specifically comments on its intuitiveness, then that participant is remarking that a student will easily recognize it as an example [START_REF] Tsamir | Intuitive nonexamples: The case of triangles[END_REF]. In other words, that participant is integrating knowledge of students [START_REF] Ball | Content knowledge for teaching: What makes it special?[END_REF] when analyzing example use and has appropriated a specific theory to accompany this integration. As teacher educators, we wish to encourage such integration of knowledge.

During the discussion, additional aspects of example use arose. Participants grappled with the nature of examples, and whether or not an example can be an explanation. This link between examples and explanations could have stemmed from the first half of the course which dealt with the topic of explanations in mathematics education. In any event, this question can help teachers and prospective teachers focus on the roles examples may play in the greater picture of teaching and learning mathematics. Finally, we also note how the integration of an authentic case, one that at least some participants actually observed, can help bridge theory and practice. In the discussion, participants held different views regarding what the example in Line 11 was an example of. Those who had actually observed the lesson had a chance to review the lesson again, focusing now on example use. They were also able to share with others some background of the lesson, perhaps adding to everyone's sense of 'being there'. This excerpt illustrates how fieldwork may be integrated into course work. It also reminds us, as teachers and as teacher educators, that it is not enough to offer examples. As [START_REF] Goldenberg | Shedding light on and with example spaces[END_REF] said, exemplification is dependent on one's point of view. Analyzing a case using theories, and then discussing these analyses with participants, can raise awareness of how students may view examples and encourage planning example use in mathematics classrooms.

Figure 1 :

 1 Figure 1: Presenting Thales' theorem on the board

  For example, we examined participant's responses for comments related to various types of examples that were discussed in class and were present in the case, such as intuitive examples, non-examples, familiar examples, and uncommon examples. Participants also commented on didactical aims of giving examples, such

  discussed visual representations, while Tabach et al. (2010) focused on numerical examples. In addition to the form of the example, studies have categorized examples by how they are identified. For instance, there are intuitive and non-intuitive examples and non-examples

Table 1 : Categorizing participants' comments

 1

  He wanted to give an example of a ratio, but… Instead, he asked the students... He told the students to guess. And they did. He didn't really give an example, in my opinion. But, in the next example...

	TE:	Let's look at Line 1.
	PT21: But it's not complete… TE: Where is the next example, in your opinion?
	PT21:	Line 11… In my opinion, it's an explanation with a few numbers and that's so you
		can see… I can't decide. It's, like, a numerical example. Here (in Line 11) is what
		was missing beforehand (in Line 1).

  As stated in the background, what an example is an example of, depends on your focus and point of view. PT22 claims that the examples in Line 1 and Line 11 are examples of Thales' theorem and that the teacher used intuitive examples to simplify the concept for his students. However, other participants, those not necessarily present during the geometry lesson, see those examples as intuitive examples of equivalent fractions or of a proportion.

	T12:	Of proportions. I know that it's proportional because I know, I recognize it. It
		doesn't really have anything to do with Thales' theorem. Simply, 6 is to 2 as 12 is
		to 4. That's it. That's the example.

Proceedings of CERME10

Acknowledgment

This research was supported by The Trump Foundation (grant No. 145).


text:
Amanjot Toor

Joyce Mgombelo
email: jmgombelo@brocku.ca

Math centers: A pedagogical tool for student engagement in intermediate math class

Keywords: Professional learning, flipped classroom, student-centered, engagement, intermediate mathematics. Thematic Working Group 18

This paper draws from a qualitative exploratory case study that aimed at exploring the learning experiences of teachers as they engage in professional learning project. The case study involved three elementary school teachers' professional learning experiences as they engaged in developing a practical, research-based approach to differentiated instruction using a flipped classroom and student-centered pedagogical approaches that would result in enabling students to be engaged with mathematics..

Introduction

Research indicates that professional learning, which is job-embedded [START_REF] Joyce | Student achievement through staff development[END_REF], collaborative (Garmston & Wellman, 2003), occurs over time, and is driven by the needs of the teachers involved [START_REF] Fullan | The limits and the potential of professional development[END_REF][START_REF] Lawler | Planning for effective faculty development: Using adult learning strategies[END_REF]Little, 2002), is effective. Furthermore, effective professional learning is focused on student outcomes, integrated into the teacher's day-today culture, and often tied to the school's improvement process [START_REF] Way | Teacher development: A strategy for school improvement[END_REF]. The paper draws from a study that aimed at exploring the learning experiences of three teachers engaged in a professional development project in Ontario at an Intermediate level (grade 6, grade 7 and grade 8). The professional development project is part of an initiative of The Elementary Teachers Federation of Ontario (ETFO). ETFO invited and provided support for teams of teachers from the same school or in similar roles at different schools to come together and conduct professional learning projects relevant to their specific professional needs, circumstances and interests. In addition to the three teachers a university researcher was invited to collaborate with the group and conduct a case study of the professional development project. The three teachers in the professional development project collaborated in developing a practical, research-based approach to differentiated instruction using a flipped classroom approach and student-centered pedagogical approach that would result in enabling students to be engaged with mathematics. This flipped classroom approach and studentcentered approach involved the use of grade-appropriate math centers where students would engage in a variety of math problems and/or topics; have opportunities to practice and consolidate basic facts and operational skills; use technology and manipulative as learning tools; become efficient communicators in math; and develop a sense of self-awareness toward their own math skills. Students would also grow in their ability to work independently and cooperatively as they work through various math centers, allowing the teacher(s) to conference with individuals and small groups of students. In this paper we describe some of the findings from the case study. The case study is guided by two questions: How did the professional development project facilitate teachers' understanding of the use of math centers in a flipped classroom and student-centered approach for teaching and learning mathematics? How did the teachers negotiate constraints and possibilities as they engaged in their professional development project? .

Research has demonstrated that engaging students in the learning process increases their attention and focus [START_REF] Jonathan | Flip Your Classroom, Reach Every Student in Every Class Everyday[END_REF]. Further it motivates students to practice in higher-level of critical thinking while promoting meaningful learning experiences. Chickering & Gamson (1987.), states that "learning is not a spectator sport. Students do not learn much just by sitting in classes listening to teachers, memorizing pre-packaged assignments, and spitting out answers. They must talk about what they are learning, write about it, related it to past experiences and apply it to their daily lives. They must make what they learn part of themselves".

In this sense, a flipped classroom and student-centered learning is essential. Educators who adopt to a flipped classroom and student-centered approach as a pedagogical method find that it increases student engagement, which allows for learners to successfully achieve the learning objectives [START_REF] Jonathan | Flip Your Classroom, Reach Every Student in Every Class Everyday[END_REF]. For a flipped classroom and student-centered mathematics class to be effective, a shift in the role of the teacher and students in the classroom must be adapted. On the one hand, teacher's role is viewed as a guide for students' constructive processes towards mathematical meanings and mathematical ways of knowing. On the other hand, learning is viewed as an active, constructive activity in which students wrestle through problems that arise as they participate in the mathematical practices in the classrooms [START_REF] Cobb | A constructivist alternative to the representational view of mind in mathematics education[END_REF]. Recently, there has been an upsurge of interest in instruction that focuses on flipped classroom approach for teaching mathematics.

Type of Math Centers

There were various types of math centers that each participating teacher used in their classroom. The choice of a math center was based on teachers' professional judgment of the students and students' needs.

Inquiry based Center:

-A group of 4-5 students rotated from one station to another to learn about various topics. Examples of these topics included: explore and connect station, "what happens when…", various word problems, "Reflection on this", "Test your knowledge." Each station had the option of students either working by themselves and/or in their respective groups. Teachers used as one or more of inquiry based centers to develop curiosity of a given topic among the students.

Resource Center: -This center was available for students all the time. This station consisted of graph papers, blank papers, mathematics dictionaries, mathematics textbook (e.g. Math on call, Math on hand, and Math makes sense 8). This center also included measuring tools such as meter -sticks, rulers, weight measuring scale and measuring tape. This center gave student the opportunity to select an appropriate tool for themselves in order to learn a topic/concept at hand.

Online Research Center:

-Students had the opportunity to use their own technology or/and the computer station located in class to deepen their understanding of topic/concept at hand. Students also had the option of exploring a given topic at home via online research.

Debriefing Center: -This is usually available at the end of a lesson, where students come back into their respective groups. Here students are given the opportunity to consolidate their learning as a group, clarify any misunderstandings and learn from one another.

An example of inquiry based math center

Students were given the opportunity to solve a real life problem, using a task called, "how big is a trillion?". In this problem, students were asked two open-ended questions: 1) Is it possible for trillion rice to fit into this room? Explain your solution 2) How much distance around the Earth can the rice cover if each rice is lined up in a line? Compare this with another non-metric unit. Students were asked to solve this problem in group of 4-5. Each group was given 6-7 classes (40 minutes each) to solve the problem in class. Students were allowed to do any background inquiry that they thought was necessary at home. The purpose of working on this problem in class was to have a common working place as a group, where they negotiated their learning and solution. At the end of the 7th class, groups were asked to submit their solution in form of poster which each group presented in the following class. Mathematically, this problem required students to learn about how to measure a unit in real life, length, metric versus non-metric units, volume, and capacity. Other than the mathematical knowledge, students had to discuss their ways of solving the problem, which means selecting appropriate tool to solve the problem in most efficient way, while self-regulating their learning and progress as a group. By the time this particular problem was given, students often became competitive. Teacher noticed that while students helped members of other groups with background mathematical knowledge, groups often tried to keep their solution a secret. This was because groups often wanted their solutions to be a unique solution.

Preparing the students for flipped classroom with math-center included teacher presenting and discussing math center code of conducts, where the purpose and the importance of self-regulation for one's own learning was discussed in depth as a class. In order to utilize various centers, students were divided in groups of 4-5 students, these groups were often changed and redesigned either by the teacher or the students. Students were informed that they may seek help support from their teacher at any point, however they are encouraged to first discuss it with the members of their respective groups.

Flipped classroom

Flipped classroom approach for teaching mathematics is considered as an effective way for engaging students in active learning as well as in meaningful peer-to-peer and peer-to-teacher interactions during the in-class learning process [START_REF] Forsey | Flipping the sociology classroom: Towards a practice of online pedagogy[END_REF][START_REF] Pluta | PBL and beyond: trends in collaborative learning[END_REF]. Moreover, Bergmann and Sams (2012) indicated that flipped classrooms enable teachers to take individual students' needs into account as well as to facilitate more interactions among peers and teachers in the classroom. The teaching and learning context of flipped classroom approach consists of two kinds of activities: in-class and out-class. In-class time is utilized for inquiry, application and assessment in order to better meet the needs of the individual learners. Technologyassisted out-of-class time involves personal instruction, where students acquire responsibility for their own learning, through studying course material on their own, using various sources (selfdiscovered and/or provided by teacher). The main goal in flipping a class is to cultivate deeper, richer, and active learning experiences for students where the instructor is present to coach and guide them. Further, emphasis is on higher-order thinking skills and application to complex problems, and which might include collaborative learning, case-based learning, peer instruction and problem set. In this sense the role of the teacher is to facilitate the learning process by helping students individually and in groups. According to Bergmann & Sams (2012), there are many ways of implementing a flipped classroom approach. For this study, the participating teachers utilized various math-centers (discussed previously in the article) where students self-regulated their own learning in a math class.

Methodology

A qualitative research methodology was used to conduct this case study. According to Yin (2003) a case study design should be considered when: (a) the focus of the study is to answer "how" and "why" questions; (b) the behavior of those involved in the study cannot be manipulated; (c) you want to cover contextual conditions because you believe they are relevant to the phenomenon under study; or (d) the boundaries are not clear between the phenomenon and context. A case study was chosen because the study could not be considered without the context of flipped classroom approach, and more specifically the math centers classroom settings. This case study involved three mathematics teachers at a Canadian middle school, who taught intermediate level (grade 6, 7 and8) students. Two of the teachers were intermediate level mathematics teachers with their own classroom. One teacher was the resource teacher assigned by the district school board. The role of the resource teacher was to support the two classroom teachers by finding necessary resources needed to run the project. The two classroom teachers applied the flipped classroom approach in their mathematics classes. In preparation for the project, all three teachers sought opportunities to enhance their professional knowledge of using technology in mathematics teaching and learning. Further, each teacher read a number of monographs provided by the Ontario Ministry of Education in order to develop efficient knowledge of the Ontario elementary school mathematics curriculum, in particular knowledge of the mathematical processes such as problem solving, communicating, reasoning and proving.

Data was collected from: 1) Participating teachers' observation of their teaching and learning experiences. Each teacher recorded field notes based on their own reflections as well as observation of their students in-class events related to math-centers (e.g., counting the number of students being engaged per center, how teachers guided the off track students to get back to work, etc.). In addition to these data was collected from teachers' notes of their conferencing with students as individuals and in groups.

2) Transcriptions and field notes of group meetings (selectively audio-recorded). Each teacher's observations were shared, discussed and reflected upon by teachers as a group for professional growth while focusing on their own professional growth and their students' engagement in respective mathematics class.

3) Teachers' artifacts such as lesson plans and assessment rubrics, and the teacher team's final project report.

Findings and discussion

All transcribed data, field notes and teachers' artifacts were analyzed by the university researcher and one teacher independently to identify major themes related to the guiding questions-How did the professional development project facilitate teachers' understanding of the use of math centers in a flipped classroom and student-centered approach for teaching and learning mathematics? How did the teachers negotiate constraints and possibilities as they engaged in their professional development project? The findings will be discussed in the following themes which emerged from the analysis: time for teachers to meet; awareness of initial resistance from parents and students; enhancement of student learning; and challenges. Findings from the study suggest that in order to sustain a collaborative professional development project teachers need time and need to meet. Teachers in this study were able to plan collaboratively and develop a practical, differentiated math program based on flipped classroom and student-centered pedagogical approach using math centers. The flipped classroom approach using math centers allowed students to engage in purposeful practice while freeing up the teacher to meet with individual and/or small groups of students for teaching and/or learning. However the two participating teachers taught different grades in different buildings of the same school, which became somewhat challenging to coordinate schedules for sit-down meetings. Often, the conversation between the three teachers would occur either in between classes (as the teachers would pass by each other's classroom) or through email (keeping each other informed on their status with the project). Although the teachers planned collaboratively, they had fewer time to compare what they had initially planned. This however, had an unexpected positive outcome which ended up by providing teachers with two different ideas of math-centers and student engagement. Both classroom teachers did utilize flipped classroom, student-centered pedagogical approach, however during their final group meeting, teachers discovered that they had each taken a different approach to the math center idea. This allowed for each teacher to talk about their thought process behind their choice for developing the math centers the way they did. Further, this provided both teachers to learn from one another's professional learning experience with their respective math centers. As different as each participating teachers' math centers were, the participating teachers observed that there were common themed categories to the math centers (these categories were presented earlier in this paper).

Another theme that emerged from the case study is that the professional development project provided opportunity for participating teachers to be aware of and understand about the initial resistance from the students and their parents toward flipped classroom and student-center pedagogical approach. Teachers developed awareness of the fact that both the students and their parents perceived mathematics teaching and learning in a traditional manner. And that for both the students and their parents, mathematics was a subject where the teacher taught a lesson, the students completed assigned tasks like doing practice questions from a mathematics textbook related to the lesson, followed by an assessment in form of a test. The students in participating teachers' classes and their parents' perception of how mathematics should be taught presented with complex challenges. This resulted in the professional development project allowing teachers to learn about strategies for alleviating these challenges including having to do a lot of community building exercise in class, while also having conversation with parents through emails, phone calls and/or one-on-one meeting, about the importance of math centers for their child's learning. These conversations with parents often revolved around the topic of how math centers not only helped students to become more engaged with mathematics but also helped to develop importance skill set of becoming more self-regulated toward their own learning.

Another theme that emerged is how the professional development provided opportunity for teachers to enhance students' learning and development through using math centers. Teachers noticed that after the initial resistance, the students began to be engaged with math centers and by the end of school year, they began to self-regulate their learning. One teacher noted about a grade 8 student who reflected on his journey with math centers and stated that it helped him to become more resilient to mathematics learning. First, I did not know why we were doing math centers. I felt that the teacher did not want to teach anything…. but now, when I go through different centers in class I know that I am able to do things on my own…I feel happy…. I have done these many [math centers] today…which means I can do math…I just have to take my time with each center and not worry about how much time my group members are taking with centers.

In relation to the same theme, another teacher expressed how her grade 8 student commented, I used to think that my teacher should know everything…you know, like all the answers…but now I know that I can find all the answers…and if I am stuck, I can take help from my friends…which is okay, because we are learning together Both of the participating teachers experienced a sense of fulfillment in terms of their professional development experiences in relation to their impact on growth in their students in terms of both mathematical understanding and self-regulation toward their own learning. Teachers noted that many of their students grew stronger in their ability to self-regulate, as they had to make choices toward their learning in terms of what to work on, how long to work for and with whom. Teachers expressed satisfaction on how the one-on-one time with the teacher allowed the struggling learners to take risk and seek clarification without feeling restricted by the classroom environment.

Given the many positive outcomes of the professional development project that focused on flipped classroom and student-centered pedagogical approach through math centers, there were some challenges. These were mainly due to the fact that teachers became aware that some students needed more time with this approach, which was not possible given that there was a limited time that these students were with their mathematics teachers and that the teachers were expected to cover the curriculum expectations. Also teachers in the professional development project realized that for a small number of students, it was extremely difficult to adapt to this approach, even if they loved mathematics. This was because these students had only experienced learning only from a textbook teaching approach in mathematics, and flipped classroom and math centers approaches were a significant departure from their past mathematics learning experiences.

Implications

This project utilized a case study research design and was conducted at one Canadian middle school. Hence, the findings of the study should not be read in terms of generalizability, but of transferability to other cases [START_REF] Creswell | Educationl research: Planning, conducting, and evaluating qualitative and quantitative research[END_REF]. Recently, mathematics educators have realized the potential for a flipped classroom and student centered pedagogical approach for enhancing student engagement and learning. However, very little is known in terms of the implementation of this approach in elementary schools. This study explored the mathematics professional learning experiences of elementary school teachers as they implement the flipped classroom and student centered pedagogical approach. The professional development project provided opportunity for teachers to enhance their understanding of flipped classroom and its impact on students' learning. Teachers noted that their students became engaged with mathematics and self-regulated toward their own math learning. The findings suggest that given opportunity to learn in a professional development setting that ensures autonomy, teachers learn and are capable of teaching through flipped classrooms and student centered pedagogical approach. The study also suggests that professional development project provides opportunity for teachers to be aware of the need for communication and collaboration among teachers, parents and students regarding the benefits and implementation of flipped classroom. As a result, further research is needed on how professional development can facilitate teachers' learning about how to communicate and collaborate with parents and students in flipped classrooms.

Proceedings of CERME10


text:
Ute Sproesser
email: sproesser@ph-heidelberg.de

Markus Vogel
email: vogel@ph-heidelberg.de

Tobias Dörfler
email: doerfler@ph-heidelberg.de

Andreas Eichler
email: eichler@mathematik.uni-kassel.de

How to deal with learning difficulties related to functionsassessing teachers' knowledge and introducing a coaching

Keywords: Mathematics education, teachers' professional development, functions, learning difficulties, teacher survey

Students often have difficulties with the content area of functions. If their teachers are not aware of these problems and lack of adequate teaching methods, they cannot counteract pointedly in their classrooms. This paper presents a project developing and evaluating a coaching to foster teachers' pedagogical content knowledge about several learning difficulties with functions and about how to respond to them. As this work is still in progress, we here focus on the project description as well as on the development of the survey to measure teachers' corresponding knowledge.

Introduction

Being able to adequately reason with functions is considered to be a central goal of mathematics education (e.g. [START_REF] Eisenberg | On the development of a sense for functions[END_REF]KMK, 2003;NCTM, 2000). More precisely, reasoning with functions characterizes a specific way of thinking in interdependencies, relationships or changes [START_REF] Vollrath | Funktionales Denken[END_REF]) that is especially required when working on inner-and extra-mathematical problems [START_REF] Hinrichs | Modellierung im Mathematikunterricht[END_REF]NCTM, 2000).

However, several studies show that learners have particular difficulties in this domain (see for an overview [START_REF] Nitsch | Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge[END_REF][START_REF] Vogel | Mathematisieren funktionaler Zusammenhänge mit multimediabasierter Supplantation[END_REF]. For instance, they may experience problems with the meaning of the parameters (e.g. [START_REF] Schoenfeld | Learning: The microgenetic analysis of one student's evolving understanding of a complex subject matter domain[END_REF], conceive graphs as pictures (e.g. [START_REF] Monk | Students' understanding of a function given by a physical model[END_REF], confound the slope and the height of graphs (e.g. [START_REF] Hadjidemetriou | Children's graphical conceptions: assessment of learning for teaching[END_REF] or have difficulties with word problems in the sense of the word-order-matching-process (e.g. [START_REF] Clement | Algebra word problem solutions: Thought processes underlying a common misconception[END_REF]. Often, their teachers are not aware of these difficulties [START_REF] Hadjidemetriou | Teachers' pedagogical content knowledge: Graphs, from a cognitivist to a situated perspective[END_REF]Sproesser et al., in press) and therefore cannot counteract explicitly. Moreover, the study of Nitsch (ibid.) revealed systematic differences between school classes referring to learning difficulties with functions. She concludes from this finding that some teachers are more successful than others in responding to such difficulties.

Theoretical background

The findings mentioned above raise the assumption that teachers' professional development (TPD) focusing on such typical learning difficulties may enhance teachers' pedagogical content knowledge (PCK, see e.g. [START_REF] Shulman | Knowledge and teaching: Foundations of the new reform[END_REF], their instruction and mediate also students' learning in this field. This is also in line with the general understanding that teachers need TPD in order to meet the challenges that they encounter in their professional lives as university studies cannot satisfy all of demands from practice (cf. [START_REF] Mayr | Lehrer/innen als zentrale Ressource im Bildungssystem: Rekrutierung und Qualifizierung[END_REF]. To our best knowledge, there is no empirical evidence about the effects of a TPD related to learning difficulties with functions, especially taking into account the interplay between the teacher and student level, yet.

Particular TPD-characteristics that have already proven to be effective in general can be implemented in a TPD referring to dealing with learning difficulties in the domain of functions. In this context, [START_REF] Lipowsky | Theoretische und empirische Perspektiven zur Wirksamkeit von Lehrerfortbildung[END_REF], for instance, found that TPD should be related to one specific domain instead of focusing on different domains. Furthermore, long-term TPD courses enable to integrate input, practice and reflection phases. The study of Lipowsky (ibid.) additionally confirmed that giving feedback (e.g. [START_REF] Shute | Focus on formative feedback[END_REF] supports learning also in the context of TPD. Teacher coaching represents a specific form of TPD that can also implement the mentioned characteristics. In adaptive (teacher) coaching [START_REF] Leutner | Instructional design principles for adaptivity in open learning environments[END_REF], the coach refers back to the teachers' statements and activities. If teacher coaching focuses on a concrete classroom situation, it is, for example, possible to encourage teachers to reflect on this situation [START_REF] West | Content-Focused Coaching: Transforming mathematics lessons[END_REF] or to train them in giving supportive feedback to students showing a particular learning difficulty.

In several studies, such a focus on responding to students' difficulties or errors (e.g. through giving feedback) has already shown to be useful in order to measure or promote teachers' PCK concerning different mathematical content areas: For instance [START_REF] Biza | Using tasks to explore teacher knowledge in situation-specific contexts[END_REF] propose to measure (preservice) teachers' PCK by requesting them to analyze wrong student solutions and to formulate supportive feedback. The study of [START_REF] An | Enhancing mathematics teachers' knowledge of students thinking from assessing and analyzing misconceptions in homework[END_REF] revealed that teachers' PCK can be fostered through asking them to analyze students' errors and to develop approaches how to correct them.

Research goals

The teacher coaching developed in this project aims at building up teachers' pedagogical content knowledge related to learning difficulties with elementary functions and hence to support also their instruction and students' achievement in reasoning with functions. As a narrow content focus has proven to be a characteristic of effective TPD [START_REF] Lipowsky | Theoretische und empirische Perspektiven zur Wirksamkeit von Lehrerfortbildung[END_REF], we here refer to specific PCK components as defined by [START_REF] Ball | Content knowledge for teaching. What makes it special[END_REF]: In the case of 1) knowledge of content and students (KCS), we focus at fostering teachers' knowledge about typical learning difficulties and about students' thinking related to functions; in the case of 2) their knowledge of content and teaching (KCT), we train them in "adequately" responding to such specific learning difficulties. Within the content area of functions, we concentrate on linear functions and on the understanding of the concept of a bivariate functional relationship in order to assure a narrow content focus. The emphasis on this subdomain also takes into account that viable concepts about elementary functions appear to be crucial for understanding higher-order functional classes later on.

As the majority of existing TPD courses is not carried out in an experimental design, it cannot be clearly identified which of their characteristics would be responsible for a certain effect (e.g. [START_REF] Yoon | Reviewing the evidence on how teacher professional development affects student achievement[END_REF]. Therefore, this teacher coaching is brought out via two variations, namely with and without focus on feedback. This procedure takes into account findings from other studies showing positive effects of giving feedback (see above) but additionally evaluates the effectiveness of this characteristic (explicitly training to give feedback to students showing concrete learning difficulties). In this sense, the main goal of the project described in this paper is to prove what effective aspects of an adaptive teacher coaching are.

More precisely, we evaluate the following research questions:

 What do teachers know about typical learning difficulties in the domain of functions and what ideas do they have how to react to them (pretest)?

 To what extent can teachers' KCS and KCT related to functions be fostered through two variations of teacher coaching (pre-and posttest)?

 Which impact do the coaching treatments have on students' domain-specific competence?

Methods

Pilot study

The content of the coaching was identified via a pilot study in the academic year 2014/15 (see Figure 1 for an overview of the project's structure): Part I of the pilot study revealed that all of the learning difficulties derived from the literature (see Introduction) occurred among students within our learning settings (paper-and-pencil-tests in 4 classes of grade 7 and 8). Moreover, we found that their teachers only knew some of these learning difficulties and that their knowledge about them and about how to respond to them was very heterogeneous (interviews with 4 teachers). Therefore, TPD in this domain appears to be useful. A summary of these results can be found in Sproesser et al. (in press). As to our knowledge there is no consensus about how to "accurately" respond to such learning difficulties or how to largely prevent them, teacher trainers and university educators were interviewed about these issues within part II of the pilot study. Via these expert interviews, we collected and further developed teaching ideas, methods and material for the coaching.

Main study

Within the main study, the teacher coaching (3 modules) accompanies the instructional unit of linear functions in grade 7 or 8, respectively: Module 1 is held before, module 2 during and module 3 after this unit. This structure enables to implement the content of the coaching in the teachers' classroom as well as to reflect on the teachers' experiences within the TPD. About 60 teachers of grade 7 or 8 are assigned to one of two treatment groups or to a control group. Both treatments contain input, reflection and activity phases in order to foster teachers' KCS and KCT related to learning difficulties concerning elementary functions. Only in one of the two treatment groups, teachers are specifically trained in giving supportive feedback to students facing a particular learning difficulty.

In order to gain empirical evidence about effective characteristics of the coaching, the teachers' PCK as well as their students' knowledge related to elementary functions are assessed before and after the coaching / teaching unit. This data structure allows using analysis tools such as multilevel analyses and hence to evaluate the interplay between the two levels. The student survey (pre-, postand follow-up-test) contains large parts of the test instruments developed by [START_REF] Nitsch | Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge[END_REF]: Via a number of tasks referring to elementary functions, several learning difficulties (see above) can be identified. Moreover, covariates such as students' cognitive abilities [START_REF] Heller | KFT 4-12+R[END_REF] or motivational variables [START_REF] Pekrun | Skalenhandbuch PALMA: 1. Messzeitpunkt (5. Klassenstufe)[END_REF] are gathered.

In order to measure KCT and KCS of the participating teachers, we developed a survey that particularly refers to several tasks of the student test. The development and the structure of the teacher survey will be presented in more detail in the next section. , on the basis of a wrong student solution they have to put theirselves in a student's position in order to make transparent his or her thinking process when working on the task. Hence, these tasks require knowledge of typical student (mis)conceptions and errors as well as about students' thinking. The third PCK item (question c) in Figure 2) corresponds to the knowledge component KCT (Ball et al., ibid.): Teachers need to know about mathematics and about teaching in order to sequence their instruction and hence to promote students' understanding. For instance, they need to know different methods and procedures and choose appropriate ones for their instruction. This means that KCT is particularly relevant when teachers respond to students' mistakes and difficulties or when they aim at building up viable concepts through their instruction. Ball et al. (ibid.) propose to measure KCT e.g. by asking for examples for simplifying particular content or how learning of a specific content can be facilitated. As displayed in Figure 2, such KCT items are also included within our test instrument: In question c), teachers are asked to outline how they would react to a concrete student mistake.

Within the whole survey, the sequence of PCK items is always as displayed in Figure 2: The first question a) is open-ended in order to collect the teachers' ideas and experiences without being influenced by specifications of the survey. Afterwards (question b)), teachers are confronted with a concrete students' mistake referring to this task and they are requested which (mis)conception could cause the mistake. As in real classroom situations, responding to a student mistake (cf. KCT) happens after its noticing (cf. KCS), the question sequences are always ended up by the KCT item (question c) in Figure 2).

This sequence of questions (a) open-ended, b) referring to a concrete mistake) was chosen to gather data about teachers' knowledge and experience concerning several student problems in general and related to specific mistakes. Within the teacher interviews of the pilot study, this sequence was also used and proved to provide essential findings. However, one particular limitation of this sequencing should not be disregarded: Teachers could add the mistakes and learning difficulties displayed in b) to the open-ended question in a) even if they had not thought of them without the indication of the survey. We decided to accept this possible drawback that may occur in field studies as the our rather than in laboratory studies because interviews instead of the paper-pencil-survey would be extremely time-consuming for the numerous participants of the main study and could irritate them; furthermore, a digital survey with time markers could hardly be implemented as the coaching is brought out in different schools where we cannot count on a safe internet-connection. In the teachers writings it can mostly be identified if they have come back to a previous item or not.

We consider the relevance and the validity of these PCK items as relatively high because of several issues: First, empirical studies show that the presented learning difficulties are common among students and hence they are relevant for teachers. In their research [START_REF] Ball | Content knowledge for teaching. What makes it special[END_REF] similarly have drawn typical student mistakes and learning difficulties from the literature. Furthermore and as pointed out above, the kind of questions that we use are also proposed by these authors. Hence, our approach is not arbitrary but systematical and can also be applied in other content areas.

Student task

Draw the graph according to the functional equation

y = 5x -2
in the given coordinate system.

Explain briefly how you proceeded. In addition to the mentioned PCK items, the teacher survey contains covariates for instance about their professional background (e.g. university degree, teaching experience), beliefs related to mathematics education (e.g. their constructivist conviction [START_REF] Staub | The nature of teachers pedagogical content beliefs matters for students achievement gains: Quasi-experimental evidence from elementary mathematics[END_REF], assumed determinants for mathematical ability [START_REF] Stipek | Teachers' beliefs and practices related to mathematics instruction[END_REF] or their experience with and motivation for TPD (see several scales in [START_REF] Jerusalem | Förderung von Selbstwirksamkeit und Selbstbestimmung im Unterricht. Skalen zur Erfassung von Lehrer-und Schülermerkmalen[END_REF].

Current status and future steps of the project

As mentioned above, the pilot study has already been carried out in the academic year 2014/15 and its evaluation is almost concluded. Student assessment and teacher interviews revealed that a TPD referring to dealing with learning difficulties related to elementary functions would be useful for teachers within our learning settings. Moreover, the expert interviews were helpful to gather "bestpractice-methods" and material for the coaching.

Concerning the content of the coaching, both treatments focus on the same learning difficulties (problems with the parameters, graphs-as-picture-mistake, slope-height-confusion, emphasis on the word-order in word problems). Teachers get information about their prevalence in empirical studies. Moreover, we illustrate the best-practice-methods and material how to prevent or overcome them that we have gathered through the pilot study. There are also active phases for the teachers: On the basis of the presented methods / material, they are asked to further develop tasks and material for their own classroom. Moreover, based on described classroom-situations showing concrete student mistakes, they are requested to think up a reaction to support the student to overcome his problem. In these tasks, the variation with / without focus on giving feedback comes into play: In the treatment with focus on feedback, teachers are asked to concretely formulate the feedback and explicitly explain the hints that they would use when being confronted with the corresponding student difficulty (e.g. "How would you respond to this learning difficulty in your mathematics classroom? Please be explicit: Verbalize your feedback and illustrate other ways to support the student."). In the treatment without focus on feedback, teachers are simply requested to mention adequate ways of responding to these learning difficulties in a more general way (e.g. "How would you respond to this learning difficulty in your mathematics classroom?"). Furthermore, in both treatments teachers' experiences in the course of the learning unit are discussed and reflected as well as their classes' results -if the teachers agree with students testing before and after the unit.

The coaching has already been carried out in the academic year 2015/16 and it is still offered in the year 2016/17. Hence, the main study is in progress at the moment and data will be gathered at the student and teacher level. Results are expected from the end of the academic year 2016/17 onwards.

Figure 1 :

 1 Figure 1: Outline and content of the project

  a) Which typical mistakes or learning difficulties would you expect from your experience in this student task? b) A student solved the task as displayed on the right. Which concept could underlie this solution? Please justify your answer. c) Imagine you would be confronted with this learning difficulty. How would you respond to it in your mathematics classroom?

Figure 2 :

 2 Figure 2: Sample item of the teacher survey

Proceedings of CERME10

Acknowledgment

This study is supported by the Ministry of Science, Research and the Arts in Baden-Wuerttemberg (Germany) and by research funds from Heidelberg University of Education.