Mark Hoover
email: mhoover@umich.edu

Imani Goffney
email: igoffney@umd.edu

The work of equitable mathematics teaching: Leading a discussion of student solutions

Keywords: Research on teaching, equity, work of teaching, mathematics teaching

Mathematics teaching produces and reproduces social injustice. It also has the potential to disrupt patterns of inequity and advance just communities of practice. Drawing from literature on equitable mathematics teaching, we analyze the work of leading a discussion of student solutions in ways that nurture healthy identities, relationships and societies. From a conceptual analysis of a Norwegian mathematics lesson, we first identify dynamics of race and gender at play, then identify three key aspects of mathematics teaching that can serve to disrupt these dynamics while creating opportunities for alternative identities, relationships and futures: (i) having regard for property and its use; (ii) taking up student thinking as participatory citizenship; and (iii) orchestrating collective mathematical work. We discuss nuances of this work and implications for research on teaching.

Inequities that play out in classrooms are well documented and a growing body of literature has proposed ways of teaching attentive to social differences (e.g., [START_REF] Ladson-Billings | Culturally relevant pedagogy 2.0: aka the remix[END_REF]NCTM Research Committee, 2018). Nevertheless, conceptualizations of equitable teaching are nascent. We need specifications that can undergird professional practice in a coherent educational system. At the same time, specific dynamics of oppression are intimately contextual. We need to more fully understand the dynamics of marginalization and privilege as they play out in classrooms and the work involved in altering those dynamics. In this study, we analyze a discussion of students' mathematical solutions to conceptualize further the work of equitable mathematics teaching. Video and transcript of this discussion was part of a collection of data shared with TWG19 participants at CERME11. Accordingly, descriptions given in this paper are abbreviated.

Conceptual and contextual background

We view teaching as responsible design and management of instructional interactions. We draw on features visible across developing theories of teachingthe didactical/instructional triangle, emphasis on interaction and joint action, and regard for milieu and broader environments [START_REF] Brousseau | Theory of didactical situations in mathematics: Didactiques des mathématiques[END_REF][START_REF] Cohen | Resources, instruction, and research[END_REF][START_REF] Jaworski | Investigating mathematics teaching: A constructivist enquiry[END_REF][START_REF] Wickman | A comparison between practical epistemology analysis and some schools in French didactics[END_REF]). We see teaching as professional practice identified through logical analysis in professionally useful domains and decomposable into constituent tasks for the purpose of examining, learning, and reconstituting new knowledge and skill into practice [START_REF] Ball | The work of teaching and the challenge for teacher education[END_REF][START_REF] Grossman | Back to the future: Directions for research in teaching and teacher education[END_REF][START_REF] Hoover | Common tasks of teaching as a resource for measuring professional content knowledge internationally[END_REF].

In this work, we focus on specifying teaching in ways that disrupt patterns of inequity and advance healthy communities of practice. We understand equity to mean being equal, fair, and even-handed, with "reasonableness and moderation in the exercise of one's rights, and the disposition to avoid insisting on them too rigorously" and with "recourse to general principles of justice (the naturalis aequitas of Roman jurists) to correct or supplement the provisions of the law" [START_REF] Equity | OxfordDictionaries.com[END_REF]. While recognizing no definitive authority for deciding what is equitable, we understand regard for equity as integral to teaching. In order to decide questions of equity, we draw on our sense of humanity and invite readers to do the same. Further, we want to be explicit that we understand education to be both a matter of development of the child and an invitation to children to reimagine the world. Equitable teaching is more than just not being inequitable. We understand teaching to be about engagement in and creation of a healthy learning communityone that both enriches without oppressing and is open and responsive to individuals, their experiences, and their spirit.

Design of the study

To understand the work of leading of a discussion of students' mathematical solutions, we analyzed a session from a national study in Norway. Intervention schools were provided an additional mathematics teacher, with the goal of evaluating whether increased teacher-student ratios, with increased small-group instruction, will increase student learning. For the session we analyzed, the additional teacher is experienced and locally recognized as skilled. At this school, rotating groups of 4-6 students leave their regular mathematics class to work in a small group with this teacher. The session is 21.5 minutes, with 5 year-four students, two boys (B1 and B2) and three girls (G1, G2, and G3). It occurred in 2017 on the 80 th birthday of the King of Norway. The teacher asked students to determine the year in which the king was born. Sitting around a large table, students work on their own for about 6 minutes, at which time the teacher has students share solutions.

Our analysis is empirical yet focused on concept formation [START_REF] Gerring | Social science methodology: A criterial framework[END_REF]. It is conceptualanalytic research in line with [START_REF] Sleep | The work of steering instruction toward the mathematical point: A decomposition of teaching practice[END_REF]. Drawing on our experiences, literature, and records of teaching practice (such as video, transcript, lesson plans, student work, and interviews), we generate and test ideas about recurrent tasks of teaching. We build, revise, and discard ideas based on logical coherence with purpose, enactment, and learning of teaching and on coordination of different perspectives, in particular of the discipline of mathematics and of pedagogy [START_REF] Thames | Coordinating mathematical and pedagogical perspectives in practice-based and discipline-grounded approaches to studying mathematical knowledge for teaching (K-8)[END_REF]. In this study of entailments of equitable teaching, we first examined potential patterns of structural oppression and marginalization. Then we analyzed opportunities for disrupting these patterns.

Potential patterns of structural oppression and marginalization

First we examine interactions in the classroom that might be indicative of the reproduction of societal patterns of marginalization and oppression. The teacher begins by encouraging students to cooperate, yet during the first six minutes of the video, G1 and B2 dominate the verbal bandwidth, primarily talking to themselves, or no one in particular, with occasional, mostly ignored responses. As the task is launched, B2 announces that the problem is "child's play" and that the answer is 1933, which he then amends to 1923. G1 says, "21 st of January," to which G3, the one identified black student, says, "It's not the 21 st of January, he has his birthday on the 21 st of February." G3's comment is participatory in the sense that she has listened to G1 and is responding substantively to her, yet G1 dismisses her concern, adjusts her thinking, and covers her statement with, "But, I'm writing when he was born, so the 21 st of February…January…." Perhaps her response is sincere and responsive, but its forceful expression seems to shut down and end the exchange. Sexism and racism are well-documented societal issues. Might they bear on these classroom interactions?

A similar dynamic plays out again two minutes later when G3 announces, "I know! It's 1937!" to which G1 puts up a hand to stop her and shouts, "You can't say it out loud! We're trying here…." The teacher also counters G3, "But remember, you remember what we've been saying. It's not the answer that's going to impress me. What's going to impress me the most is the way you got the answer." After a brief flurry then of students all wanting to explain and the teacher saying to wait, G3 reflects further, quietly stating, "It has to be 1937. I'm certain." This time, B2, sitting next to her says, "No, it isn't." This sequence is striking in several ways. G1 objects to G3 for announcing her answer, when B2 has announced his several times without objection. Likewise, each of the boys indicated that he has an answer, but the teacher did not counter them. G3, a black girl, is shut down by a white girl, white teacher, and white boy in succession. Given the documented marginalization of people of color internationally, this exchange warrants consideration. Might it be a case of the reproduction of patterns of marginalization and oppression that occur in the larger society? Whether it is or not, noticing it matters for teaching and can inform the development of disruptive practice.

Turning to the teacher's engagement with student solutions, several other observations can be made. Following the explanation of G1's approach (reducing 80 by 17, then subtracting 63 from 2000), the teacher praises her ability to figure out the calculation, when "you haven't really learned this." He again repeats that he is "extremely impressed" and "should almost write an article" about her solution. Then, following B1's explanation (removing 17 from 2017, subtracting 80, then adding 17 back in), the teacher characterizes it as a "clever way". In contrast, following G2's explanation (conventionally subtracting 80 from 2017), he acknowledges that she did it "elegantly and nice", but then makes the point that B1's approach is easier and "really smart". Given the lavish praise given to G1, the teacher's qualified praise for G2's work seems inconsistent, as G2 performed the "difficult calculation" and did so flawlessly. G2 began the session expressively, but with the dominance of the talk of G1 and B2, her voice remains muted for the remainder of the lesson.

The last student to present was G3. She says, "What I did was first: Nineteen… [Writes 1900.] And then I did... Plus... 20 since the King is 80 years old. Then I added 17. [Writes 1937.]" On the board, she now has, "1900 + 20 = 1920 + 1937". The teacher says she started with the year 1900, "But now you're challenging me, I'm actually not entirely sure how you thought. Why did you start at 1900 and why did you add 20?" The teacher has an exchange with G3 and ends by saying he sees and that it is pretty clever, but a full explanation is not elicited, and the teacher does not offer a recapitulation to the other students as he does with each of the approaches.

These exchanges suggest societal dynamics of power and privilege that might be forming and playing out. Both boys in the class speak with authority and a sense of entitlement, without hesitancy and with an expectation that others care and will listen. One talks extensively, with limited regard for others around him and little apparent understanding of the problem or its solution. The other works independent of the group, quickly generating a correct answer and presenting his solution with conviction and little attention to audience. These ways of being are consistent with what [START_REF] Mcintosh | White privilege and male privilege: A personal account of coming to see correspondence through work in women's studies[END_REF] identifies as white male privilegewhere privilege is unearned benefits resulting from societal patterns of discrimination and oppression. Dynamics of privilege are central to sexism and racism [START_REF] Collins | What is white privilege, really? Teaching Tolerance Magazine[END_REF][START_REF] Keith | Masculinities in contemporary American culture: An intersectional approach to the complexities and challenges of male identity[END_REF][START_REF] Lipsitz | The possessive investment in whiteness: How white people profit from identity politics[END_REF] and deserve thoughtful consideration in teaching and learning.

Turning to the girls, the first also talks extensively, forcefully inserting herself into exchanges about the problem, in contrast to the second girl, who seems to retreat after a few early comments. These two ways of being in the classroom are suggestive of the overbearing woman and the deferential woman sub-stereotypes respectively [START_REF] Kite | Gender stereotypes[END_REF]. They exist in relation to one another within two core dimensions of gender stereotypes: agentic being more male (active, confident, competent, and independent) and communal being more female (emotional, expressive, understanding, and concerned with the welfare of others) and remain relatively consistent across age and nationality (see [START_REF] Kite | Gender stereotypes[END_REF] for a review). How might these dynamics of privilege and enacted stereotypes be symptomatic of larger societal patterns? How might they be creating such patterns? How do they shape these students' identities and what sort of patterns do they establish for relationships these students have with others in the future? Also noteworthy are patterns related to racial discrimination. In the wake of the WWII, reference to race became taboo in European politics and academia [START_REF] Wodak | Discourse and racism: European perspectives[END_REF]. Rising immigration, though, has resurfaced its visibility and prompted policies aimed at monitoring racism and advancing integration and cultural diversity. As Maeso and Araújo (2017, p. 29) argue, though, these anxieties and policies "fail to address its [racism's] embeddedness in political culture, and therefore in institutional structures and practices." Or, as De Genova (2018, p. 1765) contends, the "migrant crisis" is an "unresolved racial crisis that derives fundamentally from the postcolonial condition of 'Europe' as a whole." Norway is implicated in this history. In their analysis of the 2011 attack in Oslo, Mulinari and Neergaard (2012, p. 15) point out that Norway has parliamentary representation for culturally racist parties and that mainstream discourses, policies, and practices make this permissible, again implying that racism is alive and well in Norway and in its institutions.

Might dynamics of sexism and racism be playing out in these classroom interactions? The boys act with entitlement and are treated in kind. The girls navigate in relation to the boys. G3 is negatively critiqued and her approach devalued. It is unclear whether her solution has been understood. It is not sufficiently supported to be useful to others. How is this experience likely to shape her sense of self? What does it suggest about how these children are likely to engage across gender and race in the future? Blacks are often seen as less capable; Black women as invisible. The enactment of privilege, stereotypes, and discrimination is hard to dismiss in this session.

Opportunities for equitable teaching

Next we examine teaching's possibilities for disrupting patterns of inequity and advancing healthy communities of practice. Teaching is a powerful force. It shapes learning and lives. The children's experiences shape their sense of themselves as doers of mathematics, as capable or not. It shapes their sense of their relations to others, with regard to both connection and power. How do they work together? How do they contribute their ideas, take up the ideas of others, and develop new thinking? In many ways, participation in this class is practice for participation in the world. How might teaching shape that practice and the future world? We start by observing that the potential power of teaching in this session is extensive in large part because of the choice of the problem for these students, the orchestration of the session as problem solving, and the focus on explanation. These could be used to create rich opportunities for forming identities, relationships, and ways of working together in the world. Within teaching, though, are discretionary spaces, often filled with habitual talk and actions, unconsciously accumulated from being in the world as it is [START_REF] Ball | Just dreams and imperatives: The power of teaching in the struggle for public education[END_REF].

Teaching is a complex space, with much to consider. Our focus on disrupting patterns of inequity and advancing healthy communities led us to identify three key aspects of teaching as it plays out in this session: (i) having regard for property and its use; (ii) taking up student thinking as participatory citizenship; and (iii) orchestrating collective mathematical work. These are related in significant ways, but we distinguish them as offering different lenses on the dynamics of the work of teaching related to this session.

First is an issue of property. Who has say over what? At the start of the session, the teacher tells students to "close the book for now." G1s book is open and she is looking between her book and a worksheet. While talking, the teacher nonchalantly reaches down, closes her book, and slides both it and the worksheet away from her and into table space in front of him. Whose property is this? Respect for personal space and personal property are important, as are agency and skills of selfmonitoring, transition, and collaboration. What message does this action send? This may seem like a small matter, but it continues throughout the lesson. For instance, to whom does the whiteboard belong? The teacher? Or is it the class' whiteboard and a resource for collective work? Extending this to intellectual property, who has authority when it comes to student explanations? The teacher maintains extensive control over G1's explanation. He stands at the whiteboard, creates a representation for her approach, and makes claims about her thinking. It is as if he uses her work as a medium for presenting his own thinking, in a way that might be experienced and seen as co-opting her thinking for his purposes. Who owns ideas and how do we know?

We do not mean to imply that any one of these actions is good or bad. Our point is that issues of property in classrooms are important sites of power and that without thoughtful consideration will likely be places where patterns of marginalization play out. On the flip side, they are sites for disruption. Providing a sense of property in a classroom, where respect for property is accorded, can give students, who may otherwise feel marginalized, a sense of belonging, of being valued, of having a legitimate place at the table of knowledge. While implicit and explicit messages tell girls and children of color they do not belong in mathematics, being a property owner in a mathematics classroom can create a strong counter narrative. Teaching can help this happen. Equitable mathematics teaching reflects on the multiple forms of property that exist in a mathematics classroom, ways property might be distributed, with what effect, and how to ensure respect for it. This leads us to a second important aspect of teaching: taking up student thinking. Whether student thinking is taken up, and how it is taken up, matter. As already mentioned, the teacher dominates the presentation of G1's work. It is then striking to watch B1 stride to the whiteboard, take the pen from the teacher's hand, and explain his approach. With white male privilege, B1 claims the space and his ideas. He begins, "I thought like this. We remove 17 so there's only 2000 left." Then, midway in B1's presentation, the teacher steps in to "repeat" what B1 said for the benefit of others. The teacher says, B1 "didn't bother about the 17 at first," but thought, "What if it we're in the year 2000 now?" This differs from what B1 actually said and is likely confusing for other students. B1 began by "bothering" about the 17. He began by removing 17 to simplify the subtraction, with the idea of compensating later. This is left implicit in B1's explanation but is consistent with what he said. Greater focus on B1's thinking, perhaps with questions to ask why he removed 17, or at least checking in with him about an interpretation, might push back on the low demand placed on his explanations as a privileged white male. It might also create for other students, more equitable access to his thinking. In contrast, for a marginalized student, focus on their thinking might serve to include them more fully and communicate value for their thinking. Either way, the work of hearing students and interpreting and treating their thinking with integrity is crucial to equitable teaching and opportunities for disrupting patterns of privilege and marginalization. Our point here is that everyone suffers from oppression, in different ways and to different extents. If teaching acquiesces to white male privilege and skips over marginalized students, it may routinely fail to provide learning opportunitiesprematurely accepting explanations of white males and dismissing the explanations of others.

For the first three students who explain their approach, the teacher steps in to explain the student's thinking, often couched as necessary for other students' understanding. We suspect that this is a pedagogical impulse that many teachers feel. Teachers often feel responsible for students' understanding and know they are, as the teacher, the one who is supposed to know the content. As they struggle to understand unconventional, emergent thinking, they may feel compelled to explain each student's thinking to the other students. Unfortunately, the demands of knowing intimately what others think are great and such a teaching practice often serves to reenact dynamics of privilege for some and marginalization for others. It runs a risk of causing students to feel unheard, silenced, and invisible. And it can undermine students' opportunities to learn from and with their peers. It is also a missed opportunity in the sense that a focus on student thinking can be used to counter patterns of oppression. Honoring each student's thinking as offered, with support for its expression and its respectful, authentic reception by other students, can nurture the healthy communities of practice that can rebuild our world. This relates to the third key aspect of teaching implicated: the nature of the collective mathematical work. Collective work is an obvious avenue for addressing identities, relationships, and future worlds. This session again suggests challenges and opportunities. The teacher starts by characterizing the task as challenging and encouraging students to "cooperate". What, though, does it mean to cooperate on this problem? What might students need to be taught about how to cooperate for this work? A minute later, as a way to encourage students to focus on thinking about the problem and not jumping quickly to an answer, the teacher says, "A good tip right now is not to trust that one sitting beside you … only trust yourself … think for yourself." At this point, students are working relatively independently, and this comment reinforces independent work.

As the class shifts from problem solving to presenting their approaches, the nature of the collective work changes, as does the work of teaching. To prepare productive citizens, schools need to provide students with skills and experience crucial to public engagement, even in mathematics classes. Students need to learn how to present their mathematical ideas to an audience, be aware of who has and has not had airtime, and respond substantively to other's mathematical ideas. Mathematical work has a specific form and function. Students need to learn what counts as a mathematical explanation and how mathematical claims are decided, but many of its features extend broadly to public discourseclear communication, attention to audience, space for others to express themselves, thoughtful listening, respect for differences, and so forth. Equitable mathematics teaching creates opportunities for collective mathematical work and accountable discourse. In addition, it supports students' contributions, making sure students are ready to contribute, instructing them on how to present, teaching others how to listen, and supporting productive response. The teaching in this session creates opportunities for collective work and public discourse, but it provides inadequate student support (e.g., not teaching students how they might cooperate), inserts itself in ways that close off opportunities (e.g., explaining students' thinking for them), and truncates exchanges in ways that undermine authentic public discourse (e.g., providing a teacher summation in lieu of responses from others and collective resolution).

Our analysis of leading a discussion of student solutions in this brief session reveals three key aspects of teaching that can serve to disrupt inequitable dynamics. Each involves nuanced work, operating in big and small ways in classrooms. Together, they provide an image of sensibilities and skills that would serve teachers well. They also suggest the depth that would be required to disrupt patterns of privilege and marginalization and avoid reproducing them.

Acknowledgment

This material is based upon work supported by the National Science Foundation.


Lina Kayali
email: l.kayali@uea.ac.uk

Irene Biza
email: i.biza@uea.ac.uk

'Balancing' the 'live' use of resources towards the introduction of the Iterative Numerical method

Keywords: Knowledge Quartet, foundation, transformation, connection, contingency, resource

 to analyse an introductory to the Iterative Numerical Method Year 13 lesson of a secondary mathematics teacher who uses a range of paper based and electronic resources including Autograph, a mathematicseducation software. Data were collected during one lesson observation and a follow up interview with the teacher. Analysis identifies the different aspects of the Knowledge Quartet dimensions: foundation, transformation, connection and contingency, in relation to the introduction to the Iterative method and to the teaching of Year 13 students. Findings demonstrate how the teacher used students' contributions as resource for his teaching; how he balanced his use of resources; and how he created connections between these resources while he remained attentive to exam requirements.

Introduction

For some decades, it was thought that access to more teaching resources (e.g. textbooks, mathematics-education software) meant better teaching practice [START_REF] Cohen | Resources, instruction, and research[END_REF]. However, "more resources do not necessarily lead to better practice" (Adler, 2000, p. 206), and priority should be given to how such resources are used [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF][START_REF] Cohen | Resources, instruction, and research[END_REF]. This is especially important when digital resources are used as these are thought to add to the complexity of teaching [START_REF] Clark-Wilson | Hiccups within technology mediated lessons: A catalyst for mathematics teachers' epistemological development[END_REF], also due to the overwhelming number of resources available online nowadays. Teachers interact with these resources, choose from them and manage them; and "these interactions play a central role in the teacher's professional activity" (Gueudet, Buteau, Mesa, & Misfeldt, 2014, p. 141) and have great impact on students' learning [START_REF] Carrillo | Building mathematical knowledge in teaching by means of theorised tools[END_REF]. Resources influence and are influenced by teachers' professional knowledge, and a close look at the interactions between the two help identify opportunities to develop both [START_REF] Rowland | The knowledge quartet: The genesis and application of a framework for analysing mathematics teaching and deepening teachers' mathematics knowledge[END_REF]. The study presented here explores secondary mathematics teachers' ways of managing and balancing the different resources that influence their work, especially when they use a range of resources including mathematics-education software [START_REF] Kayali | One of the beauties of Autograph is … that you don't really have to think": Integration of resources in mathematics teaching[END_REF]. We are interested in how teachers use the available resources in relation to their teaching aims and how they materialise and manage their lessons in the light of the available resources. To reflect on such complex teaching situations, we draw on the Knowledge Quartet [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] to analyse lesson observations and post observation interviews. Here, we present analysis of an introductory to the Iterative Numerical Method Year 13 lesson observation and a follow-up interview from one participant. With this analysis, we aim to investigate the resources used by this participant and the characteristics of his work with these resources by using the Knowledge Quartet. In the following two sections we offer an overview of what we consider as resource in this paper, and of the Knowledge Quartet. Then, we present a detailed analysis of selected parts of the lesson and teacher's reflections. We conclude the paper with a discussion on our findings.

Resources

Resources include artefacts, teaching materials, as well as social and cultural interactions that can be used or affect a teacher's teaching and teaching preparation work [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF]). An artefact is an object that is designed and used for a specific purpose [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF]. It could be a mathematical technique for solving a specific problem, a mathematics-education software or a tool like a pen. Teaching materials are textbooks, calculators, and any other materials designed for teaching mathematics [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF]. Social and cultural interactions are the interactions with the environment, students and colleagues, for example a student's feedback on an activity [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF]. The term "resource" can also be "the verb re-source, to source again or differently" (Adler, 2000, p. 207). This implies that teachers interact with resources, manage them and reuse them to achieve their teaching aims [START_REF] Gueudet | University teachers' resources systems and documents[END_REF]. And "the effectiveness of resources for mathematical learning lies in their use, that is, in the classroom teaching and learning context" (Adler, 2000, p. 205). In this paper, we closely examine one teacher's use and interaction with resources, in the wider range defined above, using the Knowledge Quartet lens.

Knowledge Quartet

The Knowledge Quartet [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] was developed as a tool for analysing teachers' knowledge and beliefs. It can be used to trigger teachers', teacher educators' and researchers' reflections towards the development of mathematics teaching [START_REF] Rowland | The knowledge quartet: The genesis and application of a framework for analysing mathematics teaching and deepening teachers' mathematics knowledge[END_REF]. It is defined by four dimensions: foundation, transformation, connection and contingency [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]. The first dimension, foundation, represents teachers' knowledge and beliefs. This includes their knowledge about mathematics as well as about the teaching and learning of mathematics, and their beliefs about mathematics and its teaching and learning [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]. The codes listed under this dimension look at teachers': adherence to textbook; awareness of purpose; concentration on procedures; identification of errors; overt subject knowledge; use of theoretical underpinning; and use of terminology. The transformation dimension "concerns the ways that teachers make what they know accessible to learners and focuses in particular on their choice and use of representations and examples" (Thwaites, Jared, & Rowland, 2011, p. 227). Its codes cover aspects such as choice of examples; choice of representation; demonstration; and use of instructional materials. The connection dimension looks at teacher's choices in terms of their lesson plans, sequence of activities, and their connections between procedures and concepts where possible. Its codes concern teachers' anticipation of complexity; decisions about sequencing; making connections between procedures; making connections between concepts; and recognition of conceptual appropriateness. Finally, the contingency dimension examines how teachers respond to "unanticipated and unplanned events" (ibid, p. 227). This involves their responses to unexpected student contributions; deviation from agenda; use of opportunities; and responses to the (un)availability of tools and resources (ibid). The four dimensions of the quartet afford a focused look at the details of teachers' work in class which has great impact on students learning and achieving learning outcomes [START_REF] Carrillo | Building mathematical knowledge in teaching by means of theorised tools[END_REF]. In this paper we use these dimensions to investigate the characteristics of one teacher's use of resources.

Methodology

The study is conducted in East Anglia region of the UK and looks at upper secondary mathematics teachers' work when they employ mathematics-education software along with other resources. It provides qualitative findings established on an interpretative research methodology that values the participant's views and reflections and looks for meanings within the participant's environment [START_REF] Merriam | Qualitative research. a guide to design and implementation[END_REF][START_REF] Stake | Qualitative research: Studying how things work[END_REF]. Here, we discuss one video-recorded lesson observation (50-minute long) and the audio-recorded follow-up interview of one participant, George, with 15 years of teaching experience mostly in upper secondary education. George was teaching a mixed gender group of Year 13 students (17-18 years old), who were preparing for their school leaving examination. The lesson was on Numerical Methods. George started the lesson by introducing the chapter "Numerical Methods" from Wiseman and Searle (2005, pp. 118-135) including four methods: Change of Sign method, Iterative method, Mid-ordinate rule, and Simpson's rule by justifying why teaching this chapter is important. Then, he briefly introduced the four methods, and started explaining the Iterative method in detail using Autograph (a dynamic environment with visualising graphs affordances, see http://www.autograph-maths.com). George then showed his students some past exam questions about the Iterative method and went over their mark schemes. Finally, he asked his students to solve some textbook questions of his choice in the remaining time of the lesson. Later, George was interviewed (by the first author), and he reflected on his teaching of Iterative method and especially on points the first analysis had identified, such as his choices of resources and actions in the lesson (e.g. his aims, justification of his choices). An analysis of George's actions during the lesson and his responses in the interview was performed using the Knowledge Quartet. In the next section, we offer a detailed account of selected parts of the lesson observation, where his work with resources in the class was more visible through his communication with students, and relevant parts from the interview. These parts regard the introduction to Numerical methods and the demonstration of the Iterative method to the students also with the use of Autograph.

Introducing the Iterative method-A look through the Knowledge Quartet lens

Introducing "Numerical methods"

In his introduction to the chapter of "Numerical Methods" and its four methods: Change of Sign method, Iterative method, Mid-ordinate rule, and Simpson's rule; George justified its importance:

The whole chapter is about things that you can't solve by doing the algebra to them, or just by integrating them, okay. And, it turns out that there are way more functions and things that you can't solve algebraically than there are ones that you can, ok. There are loads of horrible things to integrate that are too difficult to integrate, that it's just easier to do a numerical thing and kind of approximately, a bit like a trapezium rule George reminded the students that they saw trapezium rule and Simpson's rule in a previous lesson. He explained that Simpson's rule and the mid-ordinate rule were about finding the area and extension of the trapezium rule, while the change of sign method and the Iterative method were about solving equations. Then, he briefly introduced the four methods, and mentioned that he was going to focus on the "Iterative method" and when this lesson's methods are good to use. This introduction reflected George's awareness of purpose and how he was addressing connections (e.g. numerical methods and trapezium rule). He tried to connect concepts/procedures, for example, to connect Simpson's rule and integration used in previous lessons to the methods he was planning to teach in this lesson.

Rearranging an equation -Why ?

After the introduction to the chapter, George started explaining the method, he mentioned that he needed to use an equation that cannot be factorised but can be solved using the quadratic formula.

He spontaneously chose the quadratic equation , and asked his students to suggest rearrangements of the equation to the form ⋯ (e.g.

). After that, he asked them to find the equation roots using what they previously learned on how to solve quadratic equations or using calculators. Students used calculators and gave the approximate answers " " & " ". George was spontaneous in his choice of example. He chose a quadratic equation with the aim that students would be able to solve it with the formula but not with factorisation; which would show them that the Iterative method worked. His choice was based on awareness of purpose which is to show the students that the Iterative method gives them a good approximation of the solutions. George also tried to connect procedures when he asked his students to create different rearrangements of the equation. Later, after finding the different rearrangements, George again tried to create a connection between the equation's solutions students found on their calculators and the Iterative method would "zoom in on the answers". He was always responding to students' ideas, commenting on when two rearrangements were the same and just written differently and on when a student completed the square instead of rearranging the equation. George was demonstrating about iteration to students while at the same time inviting them to contribute. These contributions were essential parts in the flow of his lesson, for example, when students suggested the rearrangements and calculated the quadratic solutions. We consider these students' contributions as a resource [START_REF] Adler | Conceptualising resources as a theme for teacher education[END_REF][START_REF] Gueudet | University teachers' resources systems and documents[END_REF] that George triggered and used during the lesson, along with other resources.

The use of Autograph

After working with the students on finding the equation's different rearrangements and solutions, George's next step was to use Autograph (instructional material) in order to further explain about the Iterative method and show how the different rearrangements would work. He used Autograph to graph the functions and the other side of the rearrangement (e.g. for he graphed and ). The first rearrangement he graphed on Autograph (Figure 1) was and it did not "work" as George said while pointing at the graphs on Autograph:

Now, I can see that the blue one [ ] does not quite, I think it almost touches, but not quite touches the red one [ , okay. So, they in fact don't cross. So, for that particular rearrangement sadly it's not going to work out. So, let's not do that one, let's pick a different one. Which one do you want?

We noticed here one contingency incident (use of opportunity: (un)anticipated outcome on the software) that happened when George entered the equations from one rearrangement on Autograph and noticed that the graphs of these equations did not cross, in this case he commented that some rearrangements did not work and asked the students to pick a different one. Again, a student's contribution here is a resource that George used to demonstrate ideas. In this case, one student picked . George graphed and on Autograph (Figure 2 left) and commented that the two graphs crossed this time at two points. Having noted that the two points are close to and , George explained that the Iterative method is based on replacing the in the denominator of this rearrangement by and replacing the isolated by . Thus, he suggested the formula to be used to complete a table with a starting value . This shows how George used different representations such as tables and graphs; and tried to create connections between procedures and also between different resources (student's contribution + Autograph). You know like when you know your tan graph at 90 degrees, that 90 degrees on a tan graph is effectively saying we've got an infinitely tall triangle which wouldn't be a triangle at all, because it would be at 90 degrees. You're talking two parallel lines that's why tan of 90 doesn't work. This is a contingency moment (responding to students' ideas) where George used Autograph as instructional material and connected concepts students had met previously (dividing by zero, tan of 90 degrees, parallel lines, asymptotes of the graph of the tangent line) to respond to student questions.

How to zoom on to a specific root?

Afterwards, George worked on different formula rearrangements and different starting points to show how some rearrangements "work" (when the two functions cross at two points) and some do not (when the two functions do not cross). He also wanted to show how a rearrangement that "works" can lead to one root or to different roots of the equation, when different starting values ( ) are used. But, until this point, his work on different rearrangements with different starting points was leading to the same root. This made the students question how they could zoom to a specific root and how they could tell if they were going to zoom in on a certain root in the exam. George said "I don't know the answer to this" but he promised to comment on this question later during the lesson (not deviating from the lesson's agenda). He returned to this point later when he invited students to practice on past-exam questions by saying that exam questions would provide both rearrangement and initial point. His response reflected his attention to exam requirements and their importance to his teaching. After that conversation about how to zoom to specific roots, a student suggested the use of calculators to do the Iterative method. The teacher approved the student's use and method of use of calculators by commenting that it was a "really quick way". This is an example of him responding to the availability of resources (student's contribution + calculator). The use of calculators facilitated the zooming in on the other root, as this way was quicker students tried different starting points and found out that x=3 would lead to the second root (-3.732). George used x=3 on Autograph and showed the students how this number worked for the purpose of zooming in on to the other root. This is an example of connections between resources (student contribution +calculator +Autograph+ staircase diagram).

George's reflections

In the interview, George was invited to reflect on three aspects. First, about his spontaneous choice of equation, he said he preferred working "live" in the class over showing the students "a specific and special example". He added that he was happy to work slowly with different examples to show that not all examples work. George's spontaneous choice of equation reflected his confidence to deviate from his agenda which could create more contingent moments during the lesson. His purpose was to "show" the students that not all rearrangements "work out". Second, about what the "overflow" means on Autograph, he said it indicated that the Iterative method "instead of converging, it will diverge" at that point. During the lesson, George did not use the opportunity to explain the meaning of this outcome to the students. His focus was on getting a number and comparing this number to one of the equation roots students had found. Third, he was asked about the use of calculators. He said:

That's what they need to do in the exam. So, the Autograph is just to explain, show, demonstrate it in a lesson. When they then move onto the calculators, that's them saying yeah they are happy with it, they understand it, and this is what they then need to do. This is how they do it in the exam.

George's comment on the use of calculators reflected again his considerations of exam requirements while at the same time his attention to connections between resources.

Discussion and conclusion

In this study we are interested in how teachers, in our case George, use the available resources in relation to their teaching aims and how they materialise and manage their lessons in the light of the available resources and we used the Knowledge Quartet lens to analyse teacher actions and justification of these actions. In our analysis, we noticed aspects of George's foundation knowledge including his awareness of purpose, use of terminology, identifying errors, concentration on procedures, his subject knowledge as well as his knowledge about the mathematics-education software (Autograph here) and exam requirements. When a student asked him a question he could not answer, he responded that students were not required to know the answer for the exam, so he could not answer the question because it was "beyond exam requirement". This indicates how important the exam requirements are in his teaching priorities. In his transformation of ideas, he was spontaneous in his choice of "live" examples and representations, and he seemed confident in these spontaneous choices. His use of instructional materials included the use of Autograph, textbook, past-exam papers and calculators. He used the textbook as source of exercises for classroom practice. He also asked for students' contributions and used these contributions as an essential resource in the flow of his teaching also in connection to other resources and instructional materials including Autograph. For him, students' contributions were predictable, as they were within a range of possible answers (like choosing one rearrangement of equation). He made decisions about sequencing the different parts of the lesson and frequently connected concepts and procedures. He was responding to students' ideas and sometimes using these as opportunities to create connections between concepts. In terms of contingency, George was responding to students' ideas and to the availability of tools and resources, and sometimes deviated slightly from his lesson's agenda without losing his overall aim. We distinguish the students' ideas under the contingency dimension from the students' contributions under the transformation dimension, as these are unpredictable students' ideas that George did not directly invite. While students' contributions are a deliberate result of George questioning students and engaging them in his demonstration of mathematical ideas, thus, we see them as a characteristic part of his teaching. As we consider students' contributions as resources used frequently by George, we suggest an extension of the code "use of instructional materials" that includes the use of resources. We recommend the code to be rephrased to "use of resources" to involve both instructional materials and students' contributions under the transformation dimension. We also saw George connecting between different resources he used (e.g. Autograph, textbook and student contributions). We recommend the code "connection between resources" to be added to the connections dimension. These suggestions are established on the episode included in this paper as well as other episodes with George. Overall, our standpoint is that the use of resources is intertwined with teachers' aims and actions [START_REF] Gueudet | Towards new documentation systems for mathematics teachers[END_REF]. The analysis of George's lessons, as well as of other lessons in this study, indicate that this use is a 'balancing' act on resources that become 'live' in their lessons. These resources go beyond textbooks or other digital (or not) materials and include students' contributions as well, which teachers use, balance and connect in their work.

  Figure 1: and on Autograph

Acknowledgment

This research is funded by a studentship from the Faculty of Social Sciences, UEA.


Elisabet Mellroth
email: elisabet.mellroth@kau.se

Jesper Boesen
email: jesper.boesen@ju.se

Noticing mathematical potential -A proposal for guiding teachers

Keywords: Mathematical ability, highly able pupils, teachers

It has been shown to be problematic for teachers to use Krutetskii´s definition of mathematical abilities to recognize mathematically highly able pupils (MHAPs). Aiming to concretize what teachers can notice in pupils'

problem-solving processes, we connect a 10-year-old boy's problemsolving process to some of the abilities defined by Krutetskii. The results give clear descriptions of what teachers can observe in pupils' mathematical activities to notice their mathematical potential. We concretize, for example, how a pupil's abilities to grasp a problem's formal structure and to generalize can be observed. To be able to notice MHAPs, teachers need research-based support on how and what to observe in their pupils. Our proposed guide needs to be tested and validated to explore if it will help teachers to notice MHAPs s and subsequently support their learning.

Background

 [START_REF] Schnell | Mathematics enrichment for all: Noticing and enhancing mathematical potentials of underprivileged students as an issue of equity[END_REF]have shown that teachers need the capacity to notice pupils' mathematical potential in order to support all pupils in their development. [START_REF] Mellroth | Elementary teachers on orchestrating teaching for mathematically highly able pupils[END_REF] have shown that this is a challenging task for teachers when it concerns highly able pupils. It is therefore important to give teachers research-based guidance on how to discover pupils who need more challenges than others, for example MHAPs. If a teacher does not notice these pupils and provide greater challenges for those who, for example, improve their skills quickly, the pupils "may slip into a stage of boredom" [START_REF] Liljedahl | On the edges of flow: Student engagement in problem solving[END_REF](Liljedahl, , p. 1147). [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] work on the mathematical abilities is, for many, still the seminal work on the nature of mathematical ability. However, using his ideas to notice pupils' mathematical potential poses significant difficulties for many teachers. This is shown by [START_REF] Mellroth | Elementary teachers on orchestrating teaching for mathematically highly able pupils[END_REF] in a study where teachers working in groups analysed pupils' potential to show, in their problem-solving processes, the abilities defined by Krutetskii. The following transcript of a discussion among three teachers (A, B, and C), shows some of the teachers' frustration:

1 B: So, we are supposed to use those and apply them to a problem we choose.

A:

The words are pretty difficult.

3 C: This is harder, oh! Teacher A reads aloud the Swedish translation of one of the abilities from Krutetskii's work.

B:

Oh my God! Primary teachers are the ones who first meet children in formal learning situations, and they should therefore be given the best support to notice pupils' mathematical potential. It is also these teachers' responsibility and duty to support and challenge all children. Teachers in primary school do not usually have deep knowledge in mathematics. It is, however, reasonable to argue that all teachers of young children want the best for their pupils and want to give them the challenges necessary for learning. To do this for pupils with high ability in mathematics, teachers need to be able to notice them [START_REF] Shayshon | Teaching mathematically talented students: A cross-cultural study about their teachers' views[END_REF].

The aim of this paper is to propose a structure to help primary teachers notice their pupils' mathematical potential. More precisely, through richer descriptions and commented examples from a pupil's problem-solving process, we concretize what teachers can notice as characteristic of mathematical high ability. In this paper we will present and describe the development of such an emerging structure, based on the seminal work of [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF]. How well the structure works to improve teachers' observational skills remains for further studies to explore.

Krutetskii's way of describing mathematical ability

There are newer frameworks than [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] describing mathematical ability, such as those used by [START_REF] Schnell | Mathematics enrichment for all: Noticing and enhancing mathematical potentials of underprivileged students as an issue of equity[END_REF]. It is, however, easy to ascertain that these frameworks are descendants of [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], so we have chosen to use the original framework, since it is unquestionably the most used in the research field of education for MHAPs.

Several researchers suggest that pupils should be active in problem-solving activities that are meaningful for them, in order for their teachers to be able to notice their mathematical potential [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF][START_REF] Schnell | Mathematics enrichment for all: Noticing and enhancing mathematical potentials of underprivileged students as an issue of equity[END_REF][START_REF] Singer | Problem posing: Students between driven creativity and mathematical failure[END_REF]. In particular, socalled rich problems fulfil many criteria known to be important for supporting and stimulating MHAPs by, for example, demanding higher order thinking [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. Therefore, when MHAPs work with rich problems, it can be assumed that they are participating in an activity that is meaningful for them.

In his work, [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] describes mathematical abilities and categorises them into four groups. His four groups divide the components of mathematical ability in the following way (pp. 350-351): 1. Obtaining mathematical information; 2. Processing mathematical information; 3. Retaining mathematical information; and 4. General synthetic component. The first three groups proceed from basic stages in problem solving. The fourth, specified as a mathematical cast of mind, is a more general component that Krutetskii identified during his experiments. To fulfil the aim of this article, we here present and explain some of the mathematical abilities described by Krutetskii.

Due to space limitations, we have chosen to focus on the abilities for which we can show clear evidence in one empirical case. This paper focuses on primary teachers. As the third component has been shown to be easier to observe in older students (Szabo & Andrews, 2017), we exclude it from this paper. The fourth component requires observation over a longer period of time [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], so we also exclude also it. We describe here how the abilities chosen are interpreted in this paper.

Obtaining mathematical information

"1a. The ability for formalized perception of mathematical material, for grasping the formal structure of a problem" (Krutetskii, 1976, p. 350) This is the ability to grasp the formal mathematical structure. Children with this ability want to understand the mathematical structure of the problem. They are not only interested in individual variables, but also in the relationships between different variables. Children with this ability have a need or desire to understand the mathematical structure of the problem and to discover relationships available to connect these relationships. Children who lack this ability often work unsystematically. They mainly utilize the mathematical connections they already know, even if the connection to the actual problem may be weak.

Processing mathematical information

The second group contains six subdomains, a-f; of these, a, b, d and f are accounted for below.

"2a. The ability for logical thought in the sphere of quantitative and spatial relationships, number and letter symbols; the ability to think in mathematical symbols" (Krutetskii, 1976, p. 350) This is the ability to think logically and to understand mathematical symbols. Children with this ability think logically-for example, they easily identify a common principle in a series of numbers or pictures (Vilkomir & O'Donoghue, 2009)-and they have no problem understanding and working with mathematical symbols.

"2b. The ability for rapid and broad generalization of mathematical objects, relations, and operations" (Krutetskii, 1976, p 350) The ability to generalize mathematically materials can be viewed in two ways: 1. The pupil can apply a known general concept to a specific case, that is, based on a known general concept, the pupil can propose a concrete case that applies in the given situation; 2. Based on a concrete case, the pupil can rapidly extrapolate to a general formula. Krutetskii makes a distinction between the pupil using an already-known generalization and deducing a new generalization. When using the term rapid, he is not primarily referring to time, but rather to the number of concrete examples a pupil needs before he or she can deduce or apply a general relationship [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF].

"2d. Flexibility of mental process in mathematical activity" (Krutetskii, 1976, p. 350) The is the ability to have a flexible mind-set. Children with this ability are not limited by only using known methods. They do not necessarily try to apply a known method to the problem. The important thing for these children is the search for solutions, and they easily change strategies.

"2f. The ability for rapid and free reconstruction of the direction of a mental process, switching from a direct to a reverse train of thought (reversibility of the mental process in mathematical reasoning)" (Krutetskii, 1976, p. 350) The is the ability to recognize and work with reverse problems. Children with this ability easily solve a reverse problem without any special instructions. They quickly identify it as the opposite of what they had just solved, and it is not difficult for them to switch between a direct and a reverse way of thinking. Given the equation 12/3=4, an example of a reverse problem would be 4×3 = 12.

Method

For this paper, we use empirical data from a previous study [START_REF] Mellroth | Hur man kan identifiera och stimulera barns matematiska förmågor [How to identify and challenge children's mathematical abilities[END_REF]. Data consists of audio recordings, field notes, and the pupil's written solutions from six 50-minute-long problem-solving sessions. During these meetings, the pupil worked with nine tasks, five of which were rich problems [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. Field notes and audio recordings were taken by one of the authors over a period of one semester. The pupil was a 10-year-old boy, "Marcus", in his fourth year at a public primary school. The sessions were carried out during school hours, at the school. Each session comprised either one or two tasks. Sessions were led by one of the authors, and only the pupil and the researcher were present during the sessions. The sessions had the following format: The task was introduced both orally and on paper, and the pupil was instructed to think aloud. In general, no specific feedback was given to the pupil during the sessions, but in some instances feedback of the following nature was given: "This isn't completely correct, can you try to explain it in a different way?" Sometimes encouraging statements, such as "you can continue in this way", were made. The pupil was chosen because his teacher had identified him as potentially highly able in mathematics.

In addition, Marcus's problem-solving process clearly fulfils the criteria of MHAPs given by [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], demonstrated below. Therefore, the case of Marcus is a good example that concretizes how teachers can notice mathematical potential. In advance of his participation, Marcus and his parents were informed about the study, the aim, and the methods, and the pseudonym Marcus was chosen by the boy. The study was performed following ethical recommendations for research.

Here we show results of the analysis for one problem, the flowerbed. Analysis was carried out in the following way: A) a holistic overview of the whole solution process was written down, B) each of Krutetskii's components (1, 2a-f, 3, and 4) was analysed separately, meaning that we sought examples of evidence for that specific component in the whole solution. This was repeated and discussed between the authors until we reached agreement. We first present the task, then Marcus's solution process, and finally how we interpret the abilities to be concretized in Marcus's solution process.

The Flowerbed

Dilan has a flowerbed. He wants to put paving stones around it. The flowerbed has four sides of the same length, each paving stone is the same size as the flowerbed (see left picture in Figure 1).

1) How many paving stones does Dilan need to be able to encircle the whole flowerbed?

2) Dilan decides that he want to have two flowerbeds next to each other and put paving stones around them in the same way (see right picture on in Figure 1).

How many paving stones are needed if he wants to have the following number of flowerbeds:

3) 10? 4) 30? 5) 90? 6) If Dilan has 50 paving stones, how many flowerbeds can they be used to encircle? 7) If he has 79 paving stones, how many flowerbeds can they be used to encircle? In this problem, hands-on material can be used, but eventually it will become impossible, or very time consuming, to build or draw a solution. To be able to completely solve the problem, the pupil must work in a structured way and find a connection between the number of paving stones and flowerbeds.

Marcus's problem-solving process

Marcus very quickly grasped the structure of the flowerbed. After introduction of the problem, he immediately started to work on the solution; it was a rare quiet moment. However, he seldom did any writing, but he was thinking aloud and was encouraged to do so. His way of approaching the problems was similar in all the problem-solving sessions. The flowerbed consists of subtasks 1-7, and as soon as Marcus had finished one subtask, he was eager to continue with the next.

Holistic description of the solution process

When working with the problem, Marcus quickly gave correct answers on the first three subtasks.

After answering the third subtask, he was asked to explain his thinking.

5 Marcus: Well, it is 10 flowerbeds, and 2 on each side, that equals 12, times 2, equals 24.

Then there are also 2 on the edges, so 26.

The researcher continued to give Marcus subtasks 4 and 5, and he answered correctly and quickly, without making notes. Thereafter, subtask 6 was given. Marcus thought a little bit more, but less than three minutes. His answer, 24 flowerbeds, was incorrect. The researcher asked him to explain his thinking. At this point, there was an illustrating picture on the table (see left image in Figure 2). The researcher explained her thought process to explore if Marcus was able to change his thinking. Marcus was convinced his solution was correct, but he followed willingly and without difficulty.

8 Researcher: I am thinking: Double the number of paving stones going out from the number of flowerbeds, then add 6, since it will always be 6 on the sides. Then we can check.

If we have 24 flowerbeds, how many paving stones are needed? First we will have 2 rows of paving stones… Marcus interrupts the researcher. 9 Marcus: It must be 22 flowerbeds.

The researcher and Marcus together checked the answer, based on his way of thinking, as shown in the middle image in Figure 2. The length of the rows with paving stones equals 22 plus 2, and two of these is 2•24=48. Finally, add the 2 paving stones on the sides to get 50 in total. Correct. The allotted time for the problem-solving session was running out; only 5 minutes remained, and Marcus looked tired, so the researcher suggested ending the session. But Marcus noticed that one more subtask remained, number 7, and he wanted to also do it before he joined his regular class, which was on break. The researcher told him that only 5 minutes remained; Marcus replied, 10 Marcus: I can do it! 11 Marcus: 36 flowerbeds and 1 paving stone is left over.

12 Researcher: How did you get that?

13 Marcus: First I searched for something that was reasonable.

Marcus gave his answer after thinking for 2 minutes. In the discussion, he explained that he tried 31 and 34 flowerbeds and then concluded that it must be 36 flowerbeds and that one paving stone would be left over.

Interpretations of how the abilities are concretized

With the goal of concretizing for teachers how MHAPs may show some of the abilities defined by [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF], we here describe the process and criteria we used to tie Marcus's problemsolving process to specific abilities, as describes in the section about Krutetskii's abilities.

1a. The ability for formalized perception of mathematical …

Marcus seemed to quickly grasp the problem statement, as is shown by his rapid and correct responses to the first three subtasks, where he did not explain how he came to the correct answer. However, when he explained his thinking-" Well, it is 10 flowerbeds, and 2 on each side, that equals 12, times 2, equals 24. Then there are also 2 on the edges, so 26"he showed that he had grasped the mathematical structure of the problem.

What further strengthens the conclusion that he has an ability to see the mathematical structure is that he showed no signs of unsystematically searching for solutions, that is, he did not randomly guess.

2a. The ability for logical thought …

In his solution there is no evidence of the use of mathematical symbols other than numbers and the word addition. What evidence for logical thought can be seen? We claim that his rapid answers in subtasks 1-3, together with the explanation in subtask 3, transcript row 5, shows evidence of logical deduction, as he identified a common principle from the two given figures. In addition, when the problem was turned around in subtask 6, where the number of paving stones is given instead of the number of flowerbeds, he had no difficulty understanding the problem. Furthermore, he also deduced, without hesitation, that there is a paving stone left over in subtask 7 (see transcript row 11).

2b. The ability for rapid and broad generalization …

Marcus showed proof of this ability when he explained his thinking on subtask 3 (transcript row 5). On his own, he deduced a general way to solve the problem. Thereafter, through his work on subtasks 6 and 7, it can be deduced that he used the general solution proposed by the researcher (transcript row 8), that is, after a single case, he adapted the given generalisation to new cases (transcript rows 9, 11, and 13).

2d. Flexibility of mental process in mathematical activity

When Marcus rapidly adopted the alternative strategy the researcher suggested for arriving at the solution (transcript rows 8 and 9), he showed that he can easily change strategies.

2f. The ability for rapid and free reconstruction … Subtask 6 reverses the problem. First Marcus gave the wrong answer. However, when he described his thinking (transcript row 6), it became clear that he did understand the question and that he had a strategy to solve it. Therefore, we claim that he quickly identified the reverse problem.

Discussion and implications

We strongly believe that teachers can notice pupils' mathematical potential when they are supported by clear explanations in conjunction with examples of how MHAPs can show their mathematical abilities. We also argue that this is a matter of schools´ responsibilities since MHAPs are present in most classes, pupils from homes with high socio-economic statuses will manage well, but MHAPs can origin from all homes, and hence it´s important that the school as organization have knowledge on how to support these pupils. Using previous research as a basis, we have proposed rich descriptions and provided and commented on examples of a MHAP's problem-solving process.". We have thereby concretized what teachers can pay attention to in their pupils' problem-solving processes in order to identify those who may be highly able in mathematics.

To continue the work begun in this paper, the next step is to conduct a study to explore if our descriptions-or ones similar to them-give teachers support in noticing MHAPs. We have designed and piloted a workshop with this aim. In it, teachers are asked to evaluate pupils' oral and/or written solutions, to note if they find indicators of a certain ability (based on those proposed by Krutetskii), and if they do find them, to rate how strong the indicators are. In addition, they are asked to grade, on a scale of 0 to 2, how easy it is to detect the ability based on the pupils' solutions. However, to show its value the workshop must be validated with a larger number of teachers.

To be able to notice MHAPs, teachers need support based on research. This support can be orchestrated, for example, through professional development programmes [START_REF] Shayshon | Teaching mathematically talented students: A cross-cultural study about their teachers' views[END_REF].

Observing mathematical abilities by using [START_REF] Krutetskii | The psychology of mathematical abilities in schoolchildren[END_REF] framework is likely not the only way teachers can notice MHAPs. For example, as [START_REF] Singer | Problem posing: Students between driven creativity and mathematical failure[END_REF] wrote, allowing pupils to work with problem modification is also a suitable way to notice MHAPs, who generally work in a structured manner, changing one variable at a time to control how changes may affect the problem. The results reported by [START_REF] Singer | Problem posing: Students between driven creativity and mathematical failure[END_REF] also provide important knowledge for teachers to use in their endeavours to notice MHAPs.

Naturally, the support teachers give to pupils after they have noticed their mathematical potential is important, for example to prevent boredom, which can be a result of too little challenge [START_REF] Liljedahl | On the edges of flow: Student engagement in problem solving[END_REF]. Of special importance is teachers' ability to notice and support mathematical potential in underprivileged pupils, for example those from disadvantaged socioeconomic backgrounds [START_REF] Schnell | Mathematics enrichment for all: Noticing and enhancing mathematical potentials of underprivileged students as an issue of equity[END_REF]. One way for teachers to support the learning of all pupils, including MHAPs, is to use rich problems [START_REF] Nolte | Challenging problems in a regular classroom setting and in a special foster programme[END_REF][START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. Pupils have different learning needs; some require more help to understand a concept or problem, while others need complex tasks to develop their learning [START_REF] Sheffield | Extending the challenge in mathematics: Developing mathematical promise in K-8 students[END_REF]. It is therefore important that research is conducted with the aim of giving teachers support on how to differentiate teaching in the regular classroom. To develop pupils' learning according to their potential is, of course, the main goal of teaching. To do this for MHAPs, however, the first step is to help teachers notice them [START_REF] Shayshon | Teaching mathematically talented students: A cross-cultural study about their teachers' views[END_REF].

Figure 1 :

 1 Figure 1: The flowerbed with surrounding paving stones

Figure 2 :

 2 Figure 2: Illustration of the problem, produced during the solution process 6 Marcus: First I took away those 2 [he points to the 2 dark grey side pieces in the middle image of Figure 2], then it is 48, then divide by 2, equals 24. So, 24 flowerbeds. The correct answer is 22 flowerbeds. The researcher drew the image on the right in Figure 2 to show her thought process. 7 Researcher: It is possible to count backwards to check if the calculation is correct. Let's try.