Hyman Bass
email: hybass@umich.edu

Reidar Mosvold
email: reidar.mosvold@uis.no

Teacher responses to public apparent student error: A critical confluence of mathematics and equitable teaching practice

Keywords: Mathematical knowledge for teaching, teaching practices, positioning, agency, authority

This paper uses analysis of shared video data to discuss teacher responses to public student apparent error. Drawing upon conceptions of student thinking and theories of agency, authority, and mathematical identities, we argue that certain ways of responding to public student apparent error have the potential to elevate student thinking and empower students with mathematical agency, authority, and standing. We also suggest that such teaching practices may be a highleverage site to develop equitable mathematics teaching.

Introduction

Maria's 7 th grade class is beginning to learn arithmetic with integers. She is called to the board to show her solution to the subtraction problem, 38 -52. Maria writes and says the following: 3 8 "8 take away 2 is 6" -5 2 "3 take away 5 is -2" -2 6 Is Maria right? What is there to understand here about place value, with negative numbers? How might the teacher respond to Maria? And what effect will this have on her?

These questions, about this and about countless other similar scenarios, are the object of our study. We are concerned here with the interactions of the mathematics, the student's thinking and standing, and the impact of the teacher response on all three. We are not proposing that there is a "right" teacher response, but we do argue that the response is consequential for the individual student's learning and standing, as well as for that of the collective. Moreover, much knowledge and insight goes into understanding the range of possible responses and their consequences. In particular, what mathematical knowledge for teaching, what knowledge of student thinking, and what cultural and psychological sensitivities might be involved? So, our research question is:

What kinds of teacher responses to public student apparent error can help cultivate students' mathematical agency, authority, and standing? Student mathematical productions are rarely flawless. Overt error may distract teacher attention from otherwise robust reasoning. A student's lack of technical vocabulary may cloud insightful conceptual expression. Such apparent flaws happen with all students, but they are understandably common among students who are educationally disadvantaged, because of race, ethnicity, poverty, language, etc. For this reason, we feel that our research question matters for equitable instruction.

The title refers to a core task of teaching. Three terms there -"public," "apparent error," and "equitable" -deserve comment. "Public" indicates that this is about collective learning and norm setting, more than bilateral teacher-student transactions. The student work and teacher response communicate to the whole class, not just to Maria. "Apparent error" draws attention to the teacher's perception without evaluating the student's thinking. [START_REF] Rougée | How do mathematics teachers manage students' responses in-themoment?[END_REF] has studied teacher responses to "apparent student error," but her study explored the teacher stress and anxiety this can precipitate. As indicated above, we believe that our focus is a high leverage site for "equitable teaching practices." The latter are often described in terms of inclusive classroom culture and participation structures. Our interest focuses further on how equitable practices intersect with content.

Our method is to first examine records of practice (video and other artifacts), to generate hypotheses about teacher responses to public student apparent error that can either cultivate or degrade students' mathematical agency, authority, and standing. Robust conclusions of this kind will ultimately require more longitudinal data.

We will here explore the above task of teaching in two cases, one involving 4 th graders identifying fractions on the number line, the other showing another group of 4 th graders figuring out a difficult subtraction problem before having learned a general subtraction algorithm. These examples will help to open the analytical space behind our question.

Conceptual frameworks

Teacher responses to public student apparent error, the focal phenomenon of our study, is a site of confluence of three research traditions -1) mathematical knowledge for teaching, 2) student mathematical thinking, and 3) agency, authority, and identityeach with its own theoretical frame and perspective. It is our task to coordinate, if not integrate, these traditions for our work. To understand the "work of teaching" we use the "Instructional Triangle" (IΔ) of [START_REF] Cohen | Resources, instruction, and research[END_REF], as later elaborated by [START_REF] Ball | Just dreams and imperatives: The power of teaching in the struggle for public education[END_REF]. The "stuff" in our case is "mathematics." The inner circle represents instruction, where mathematical knowledge for teaching primarily resides. Student mathematical thinking spills into the environment. Agency, authority, and standing constitute an arena in which the instruction may attempt to disrupt inequitable environmental influences. Much mathematics education research is situated at a vertex, or along an edge of the IΔ. Our question spreads across the entire diagram.

Mathematical knowledge for teaching

For decades, a core problem in research on mathematics teaching has been what knowledge is needed to teach well. Researchers have approached this problem in different ways. Some have produced lists of what teachers should know. Others have investigated what knowledge teachers have, and yet others have studied knowledge teachers use. [START_REF] Ball | Uncovering the special mathematical work of teaching[END_REF] suggested flipping the question by focusing instead on what mathematics the work of teaching entails (see also [START_REF] Ball | Content knowledge for teaching: What makes it special[END_REF]. This shift was not new. Already two decades ago, it was elaborated in several publications (e.g., [START_REF] Ball | Crossing boundaries to examine the mathematics entailed in elementary teaching[END_REF][START_REF] Ball | Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics[END_REF][START_REF] Ball | Research on teaching mathematics: The unsolved problem of teachers' mathematics knowledge[END_REF]. The present paper follows this line of research and targets the challenges that are embedded in a particular mathematical task of teaching. Eliciting, interpreting, and responding productively to student thinking, all entail deep and flexible mathematical knowledge.

(Interpreting and responding to) student thinking

As we indicated above, mathematical thinking, by experts as well as students, is rarely flawless. It is typically a mix of ideas, some intuitive or sketchy, some more formalized, some of it sound, some of it not yet fully formed, some of it incorrect. Simply calling this complex package "error" is to ignore the substantial mathematical value there to be discerned. The phrase "apparent error" is our way of keeping open this broader range of possibilities.

When a child publicly presents a problem solution, or a mathematical explanation or comment, what tasks of teaching does this present? One is to understand the child's thinking. This may draw on several knowledge resources: Of the mathematical terrain and the diverse ways it can be represented; of typical student (mis)conceptions and representations of the ideas; and of the child's identity and background, and the culturally diverse forms of children's communication (language, gesture, etc.). Once the child's thinking is understood, what can the teacher do to help make this understanding shared by the other children? And if the teacher does not at first understand, what can she do to probe the child's thinking to gain such understanding? The above are all challenging tasks of teaching.

Once the teacher possesses an adequate understanding of the child's thinking, what is the next move? In some common practice, the teacher will first notice (perhaps even before fully understanding the child's thinking) the mathematical flaws, and either announce them, or simply pass on, without comment, to another student in search of a more "acceptable" response. This can be characterized as a deficit teacher response. The opposite kind of response, sometimes called assigning competence (e.g., [START_REF] Cohen | Producing equal-status interaction in the heterogeneous classroom[END_REF], or asset oriented response, is to offer some detailed public appreciation of the positive mathematical qualities of the child's thinking, without prematurely announcing that the mathematical work is done. Sometimes the teacher might even invite other students to comment on "what was good" about the child's contribution. Again, enactment of this kind of move calls upon a fine-tuned understanding of the mathematics as well as of the student thinking.

Agency, authority, identity

Teaching is relational work, challenging in its complexity. We highlight three aspects of this complexity: agency, authority and identity.

Agency is the capacity, freedom and autonomy of an actor to act in a given environment. The environment of concern here is the mathematics classroom. Agency may either be encouraged and supported, or constrained and suppressed, for example through the influence of such factors as race, social class, religion, gender, ethnicity, perceived ability, customs, etc.

Authority is characterized by Benne (1970, p. 393) as a social relationship between the bearer (of the authority, a person or group), the subject(s) (of the authority), and the field (context). The bearer receives willing obedience from the subjects while helping provide for their need of advice, leadership, guidance, or direction in the field. This is an example of "reasoned power," but distinct from more general forms of power, which may be based on domination and coercion. Benne distinguishes three kinds of authority: Expert (as in the doctor-patient relation); Rule-based (as in say the playing of baseball); and "Anthropogogical" authority (partly exemplified by the doctormedical student relationship, an apprenticeship into practice). In rule-based authority, the rules that govern behavior represent an implicit consensual authority of the community of players, who are thus individually subjects, and collectively bearers.

Authority in a mathematics classroom has at least two aspects. One, common to all classrooms, is about behavioral norms, based on socio-cultural norms. Another, more pertinent here, is about disciplinary authority, about how to certify the authenticity of a piece of mathematical knowledge. Education reforms over the past half-century have moved in the direction of giving students more disciplinary agency and authority. How can that be interpreted in the above framework? For Benne, the 'bearer' of authority is a human agent, a person or group of persons; so how can we speak of "the authority of the discipline?" [START_REF] Amit | Authority and authority relations in mathematics education: A view from an 8 th grade classroom[END_REF] suggest that mathematics, with its "fixed set of rules," supports a kind of rule-based authority. We think instead of the discipline as a growing (inanimate) body of knowledge and methods, but that it is the cumulative product of (centuries of) human effort. How do new concepts or methods gain entry into (the current form of) this corpus? There are disciplinary norms for such membership, but human judgment (peer review) is needed to decide if these norms have been adequately met. Thus, in our view, disciplinary authority is mediated through those (human) agents empowered to certify proposed disciplinary claims.

So conceived, it now makes sense to speak of the distribution of disciplinary agency and authority in a mathematics classroom. In the traditional transmission model of instruction, the teacher alone bears expert authority to which the student subjects are expected to conform. In contrast, distributing agency among students means that they can participate in the construction of new knowledge and methods, finding new solution strategies, noticing patterns, speculating, conjecturing, proposing evidence. Distributing authority means that students themselves will judge the adequacy of claims of their peers, and find ways of resolving disagreement. This is somewhat captured in the third mathematical practice standard of the Common Core (2010): Construct viable arguments (agency) and critique the reasoning of others (authority). This broad distribution of mathematical agency and authority among students, as co-producers (with the teacher) of knowledge, resembles what [START_REF] Benne | Authority in education[END_REF] calls anthropogogical authority.

(Mathematical) identity refers to how students consider themselves as mathematical doers, thinkers, and learners. [START_REF] Aguirre | The impact of identity in K-8 mathematics: Rethinking equity-based practices[END_REF] define mathematical identity as "the dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives" (p. 14). Positive mathematical identity involves willingness to take risks to engage in discourse and to see one's self as capable and worthy of being heard [START_REF] Berry | Positioning students as mathematically competent[END_REF]. This is related to productive disposition, the habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy (NRC, 2001). A premise of our work is that positive mathematical identity can be cultivated and supported by equitable teaching.

Case 1: Fractions on the number line

This diverse class of 9 to 10-year old children has learned how to name what fraction of (the area of) a rectangle is shaded. The method is to first assure that the (whole) rectangle is divided into a number (d) of equal size parts, to then count the number (n) of those equal parts that are shaded, and finally to name the fraction as n/d. The lesson segment in the video shows the start of their learning how fractions correspond to points on the number line. This is based on linear measure, in contrast with the area measure of shaded rectangles; on the number line one is counting lengths of intervals, and the unit of measure (the whole) is taken to be the interval from 0 to 1. We see the students presented with the following task (Figure 2). The correct answer is 1/3, but the student notebooks show a variety of answers. The teacher asks who would like to come up and show their work; she calls on Aniyah. The teacher reminds the class that they should "listen closely and see what you think about her reasoning and her answer." Aniyah puts "1/7" next to the orange arrow (an "evident error"), at which point Toni asks, in surprise, "Did she say one-seventh?!" Aniyah responds, "Yeah, because there are seven equal parts, . . ." as she counts, with her spread thumb and index finger, the seven intervals (not hash marks) visible on the displayed (portion of the) number line. Here we see the "public student apparent error" to which the teacher responds, not to Aniyah, but to the class: "Before you agree or disagree, I want you to ask questions if there's something you don't understand about what she did." Toni asks, "Why did you pick one-seventh?!" The teacher, affirming "That's a very good question," also says, "Let's listen to her answer." Aniyah says, ". . . because there's seven equal parts." Lakeya asks, "If you start at the zero, how did you get one-seventh?" She may have been thinking about counting hashmarks instead of intervals. Dante then struggles to formulate a more speculative, but difficult to understand question, perhaps to the effect that, "If you moved the orange arrow to where the one is, would you still know it was one-seventh to put it where the orange line is now?" Several things are noteworthy about the teacher response to Aniyah's "apparent error." Though Aniyah's "one-seventh" is wrong, it is so far not challenged or corrected either by the teacher or the students, though, at the teacher's invitation, the children pose questions to understand Aniyah's thinking, which is made central to the instruction. The mathematical issue is resolved by the end of the lesson (beyond the video segment), but we here see the teacher publicly sheltering Aniyah's standing, as author of well articulated mathematical reasoning, which is flawed only in not using the standard unit (interval) as the whole. Further the teacher confers on Aniyah's peers the authority to question, and eventually evaluate, Aniyah's solution, free of teacher judgment. In these ways, the students are given/assigned remarkable collective agency and authority to develop the mathematics.

Case 2: Subtracting before you know how

In this case, the teacher presents a group of five 4 th graders with the following problem: "The king celebrates his 80 th birthday today" (this was in 2017). "In what year was he born?" This amounts to calculating 2017 -80. The teacher has not yet taught these students how to subtract large numbers by using a standard algorithm, though some may know this already. The students show four different approaches. One starts by taking away 17 from 80 to get 63, and then subtracts 63 from 2000 by using a standard algorithm. Another student uses the standard algorithm directly to subtract 80 from 2017. In both these cases, the teacher explains their thinking to the other students, and provides a lot of praise. Two other students come up with more unexpected approaches to solving the problem. Below, we take a closer look at one of these student's method.

When Brian (pseudonym) is called to the board to present his solution, the teacher comments that he noticed how Brian finished the problem quickly. He continues, "So I'm very excited to hear what you were thinking!" Brian explains that he just got rid of the 17, so that you only have 2000. He writes down 2000 -80. Brian then states, "So, then we have 1920, which is pretty easy!" When Brian has written down this partial answer, the teacher interrupts, "So, now I would just like to repeat what you did. I just want to repeat what you did, so that I understand your thinking." The teacher asks Brian to move a little bit to the side, before he continues, "I understood from what you were saying, Brian, what you did that I understood, was that, to begin with, you didn't want to deal with the 17. You just jumped back to (points to 2000), and thought: What if we are in 2000 now? And then, when we go 80 years back, we get to 1920. And, now I'm eager to know, what did you do next?" Brian jumps up to continue his explanation, "And then I just added 17, which is pretty easy, simply 20 plus 17." The teacher proclaims that this is "simply fantastic," before Brian moves on to write down the answer on the board. When Brian has finished explaining, the teacher asks, "Did anyone understand this clever way?" Anna responds that she understood it, and that she thought it looked like a good way of thinking about the problem. The teacher then continues to rephrase what Brian did when he decided to "jump back 17" and start on 2000, and how "he already then knew that he had jumped back 17 years too long (…) So, when he got to 1920, he had to jump 17 years forward again, to correct it."

One of the students early displayed an incorrect answer and seemed otherwise distracted at times. He was the only student not called to present his work. His was the only appearance of apparent error. For each of the others, at issue was the uncertainty of their solution strategies, for which they were each praised in varying degrees.

Figure 1 .

 1 Figure 1. Revised instructional triangle[START_REF] Ball | Just dreams and imperatives: The power of teaching in the struggle for public education[END_REF]

Figure 2 .

 2 Figure 2. Naming fractions on the number line.

Acknowledgment

Our thinking about these ideas owes much to discussions with Deborah Ball and Mark Hoover.

Concluding discussion

The choices made by the teachers in these two cases differ in several respects. We will here highlight and discuss two particular differences in the teachers' responses. The first relates to how the teachers assign competence to students. In case 1, we notice only one instance of explicit teacher affirmation or praise, whereas the teacher in the second case offers a lot of praise to the students. We argue that both teacher responses can be described as asset oriented, and we posit that assigning competence does not have to include direct praise or public teacher statements. The teacher in case 1 assigned competence to Aniyah by inviting her to be the teacher. She positioned Aniyah as a recognized contributor, with standing, in the mathematical discussion, and she conferred on Aniyah's peers the authority to analyze and critique Aniyah's work. In the case 2, the teacher assigned competence to Brian by re-voicing his thinking and praising his contribution. This leads to the second highlighted difference, relating to the distribution of mathematical agency and authority, which we next discuss.

The work of mathematics teaching demands a mathematical knowledge that exceeds ability to solve a problem by using the standard algorithm. The teacher in case 2 chose not to confine attention to the two solutions using variants of the standard algorithm, thus disrupting the rule-based authority that is common in many mathematics classrooms. Instead the teacher highlights an unexpected student solution strategy, which he has to interpret and evaluate on the fly and assure that all of the students understand. By bringing Brian to the board, and expressing excitement and eagerness to understand Brian's thinking, the teacher publicly affirms Brian's agency and standing. It is interesting to notice how the teacher, in the middle of Brian's explanation, decides to interrupt to repeat what Brian did and make sure he understands Brian's thinking. It is likely that the teacher in fact understood well what Brian had done, but, by saying "so that I understand your thinking," the teacher was nominally putting himself on the same level as the other students, thus modeling the kind of inquiry needed to understand another student's thinking. In case 1 above, the teacher assigns this work to the students, whereas the teacher in case 2 does not do so. He himself is modeling, by example, how such inquiry is done. This effort is further reinforced when, after Brian has finished his explanation, the teacher again decides to re-voice Brian's explanation, to emphasize the crux of his thinking. Throughout this episode, the teacher demonstrates, and so models, attentive listening, and a pressing need to understand Brian's mathematical thinking. Brian is positioned as a mathematical authority, but one whose thinking needs to be publicly analyzed and sanctioned by the teacher, so that Brian's knowledge is not only validated and praised, but becomes collective, thus hoping to support all of the students' understanding. In case 1, Aniyah also gains mathematical authority, but only after public resolution of the integrity of her thinking by her peers, who are assigned this responsibility.

We have presented two examples of teacher responses to "public student apparent errors." We described, in each case, teacher moves that might have potential to cultivate the mathematical agency and authority of the students. In both cases, student agency is well supported, but in case 2, more of the mathematical authority appears to still reside with the teacher


Sharon Calor
email: s.m.calor@uva.nl

Rijkje Dekker
email: r.dekker@uva.nl

Jannet Van Drie
email: j.p.vandrie@uva.nl

Bonne Zijlstra
email: b.j.h.zijlstra@uva.nl

Monique Volman
email: m.l.l.volman@uva.nl

Effects of a Scaffolding Model for small groups in mathematics

Keywords: scaffolding, mathematics, collaborative learning, mathematical level raising, teacherstudent interaction

We investigated whether teachers' use of the Small-Group Scaffolding model (SGS model), when supporting heterogeneous student-groups in a collaborative setting in math lessons, raised students' mathematical level more than teacher support without use of the model. Participants were eight teachers and 266 students of two schools working collaboratively in 73 small heterogeneous student-groups on the topic of Early Algebra. The five teachers in the experimental condition were taught to scaffold the groups in their classes according to the SGS model (give support adapted to student-groups' needs). The three teachers in the control condition were not given prior instruction of how to support the students. Analysis of a pre-and post-test on mathematical level raising showed that more mathematical level raising occurred in the Scaffolding condition than in the control condition. We concluded that using the SGS model can lead to more mathematical level raising.

Background

According to [START_REF] Freudenthal | Revisiting Mathematics Education -China Lectures[END_REF] and [START_REF] Van Hiele | Structure and insight[END_REF], the learning of mathematics occurs in discrete steps, implying the existence of levels of mathematical thinking. [START_REF] Van Hiele | Structure and insight[END_REF] described four levels: 1) Visual level: the forms of mathematical objects are the object of study, 2) Descriptive level: the properties of mathematical objects are the object of study, 3) Theoretical level: relations between the properties of mathematical objects are the object of study, and 4) Formal logical level: relations between theorems are the object of study. [START_REF] Freudenthal | Revisiting Mathematics Education -China Lectures[END_REF] built upon this theory and stated that the levels are more relative than discrete, meaning that level raising occurs every time a mathematical activity (performed at a lower level) consciously becomes the object of reflection (at a higher level). [START_REF] Freudenthal | Geometry between the devil and the deep sea[END_REF] defines a mathematical activity as an activity of organizing subject matter, which can be matter from reality or mathematical matter, according to mathematical patterns or new ideas. For example, at a lower level, patterns of the number of tables and chairs in a table setting can be studied. At a higher level, these patterns themselves become the object of reflection when one tries to create a formula with which the number of chairs in a particular table setting can be calculated when the number of corresponding tables is given.

According to [START_REF] Freudenthal | Weeding and Sowing: Preface to a Science of Mathematics Education[END_REF], when students collaborate in a heterogeneous small group (a small group of students of mixed ability) on one task, there will often be at least one student who experiences an "Aha moment" (jumping to a higher level) when understanding the subject matter. A typical higher-level activity will follow for that student, such as reflecting on how he/she mastered the subject matter and explaining to other students in the group what he/she just learned. In other words, discussions between students in a heterogeneous group while performing mathematical activities (mathematical discussions) enhance mathematical level raising.

A reoccurring problem with attaining mathematical level raising when learning Early Algebra in secondary education is that students find it difficult to make a shift from studying patterns at a lower level to understanding formulae at a higher level [START_REF] Kieran | The learning and teaching of school algebra[END_REF][START_REF] Sfard | The gains and pitfalls of reificationthe case of algebra[END_REF][START_REF] Van Stiphout | The development of students' algebraic proficiency[END_REF]. [START_REF] Janvier | Translation process in mathematics education[END_REF] addressed this problem by defining Algebra representations, situation, graph, table and formulae as four ways to describe the relation between two variables in a formula. He calls the ability to switch between the representations a translation skill. According to [START_REF] Janvier | Translation process in mathematics education[END_REF], translation skills are best learned when they are taught in a pair-wise manner. For example, the translation skill modeling is learned best when students first learn how to construct a situation from a formula, followed by learning how to construct a formula given a situation, or vice versa. In this study, all translation skills involving the representation formulae are considered to contribute to mathematical level raising. These translation skills are: Parameter recognition (formulae to situation), Computing (formulae to table), Sketching (formulae to graph), Modeling (situation to formulae), Fitting (table to formulae) and Curve fitting (graph to formulae).

Collaborative learning is considered an important means to improve students' learning of mathematics [START_REF] Webb | Student interaction and learning in small groups[END_REF]. Several studies have shown that mathematical discussions in small heterogeneous student-groups contribute significantly to mathematical reasoning and level raising [START_REF] Freudenthal | Revisiting Mathematics Education -China Lectures[END_REF][START_REF] Dekker | A process model for interaction and mathematical level raising[END_REF][START_REF] Dekker | Teacher interventions aimed at mathematical level raising during collaborative learning[END_REF]. [START_REF] Dekker | A process model for interaction and mathematical level raising[END_REF] developed a 'Process model' (PM), which promotes and operationalizes mathematical discussions. This model distinguishes three types of learning activities: key activities, which help mathematical level raising; regulating activities, which regulate key activities and mental activities that occur in students' minds. They found that stimulating mathematical discussions by the teacher resulted in more mathematical level raising compared to providing content support [START_REF] Dekker | Teacher interventions aimed at mathematical level raising during collaborative learning[END_REF]. However, teachers find it difficult to support mathematical discussions in small-group learning [START_REF] Webb | The teacher's role in promoting collaborative dialogue in the classroom[END_REF].

Content support adapted to the level of the student's understanding (contingency) is considered one of the key features of scaffolding [START_REF] Van De Pol | Scaffolding in teacher-student interaction: a decade of research[END_REF]. The other two key features of scaffolding are diminishing content support over time (phasing out) and returning responsibility for learning processes to the learner. Scaffolding in the context of group work seems to enhance learning outcomes in general ( [START_REF] Van De Pol | Scaffolding in teacher-student interaction: a decade of research[END_REF][START_REF] Webb | The teacher's role in promoting collaborative dialogue in the classroom[END_REF]. [START_REF] Van De Pol | Promoting teacher scaffolding in small-group work: A contingency perspective[END_REF] developed a model to describe scaffolding interactions, the Contingent Teaching Model (CTM), in which teacher support is adapted to the level of individual students. It consists of four steps: applying diagnostic strategies, checking the diagnosis, applying intervention strategies, and checking students' learning. In the CTM individual students are scaffolded in a small-group setting.

In the context of collaborative learning teachers often scaffold individual students within small groups. How teachers can scaffold small groups at group level and what the effect is on students' mathematical learning, has not yet been investigated. A combination of stimulating discussions and providing contingent content support adapted to the level of the group might result in more mathematical level raising. To help teachers support small heterogeneous student-groups we integrated the CTM ( [START_REF] Van De Pol | Promoting teacher scaffolding in small-group work: A contingency perspective[END_REF] and the PM [START_REF] Dekker | Teacher interventions aimed at mathematical level raising during collaborative learning[END_REF]) into a scaffolding model for small student groups [START_REF] Calor | Scaffolding small groups at group-level: Improving the scaffolding behavior of mathematics teachers during mathematical discussions[END_REF]. We call this model the Small-Group Scaffolding model (SGS model).

In Figure 1 a flowchart representation of the SGS model is shown. The model consists of six teaching steps: evaluating if the question is a question of the whole group, applying diagnostic strategies, checking the diagnosis, giving contingent content support, checking students' learning, and giving process support (stimulating mathematical discussions). Following the Small-Group Scaffolding model, teachers should first determine if the question is a question of an individual in the group or if it is a question of the whole group. In case the question is a question of the whole group, the teachers should diagnose the group's current level of understanding in the second step, by for example asking students what they have done so far and asking the students to show or tell their work to the teacher. It is possible that one or more students come up with an answer to the question during this step. In that case, the teacher should encourage the students to explain the answer to each other. If none of the students come up with an answer during step two, teachers can check if their diagnosis is correct in the third step. In the fourth step, the teachers should give contingent support to the group (increase support when the students show poor understanding and decrease support when students show high understanding) until one or more students understands the question. In step 5, the teacher should check if at least one of the students understood the content support. If this is the case, teachers should stop giving content support (phasing out, the second key feature of scaffolding) and return responsibility for learning back to the group (third key feature of scaffolding) ( [START_REF] Van De Pol | Scaffolding in teacher-student interaction: a decade of research[END_REF]. Returning responsibility for learning to the group can be done by encouraging students to explain and show their work to each other and to criticize each other's work, i.e. give process support, in the last step. In case the outcome of the determination of the question in the first step was that the question was not a question from the whole group, but rather a question of one or more individuals in the group, teachers should return responsibility for learning back to the group by giving process support.

In this study, we combined Fruedenthal's, Janvier's, and sociocultural perspectives as theoretical lenses to study mathematical lessons.

Research question

Does the use of the Small-Group Scaffolding model raise seventh grade students' mathematical level in Early Algebra?

Hypothesis

We hypothesize that the mathematical level of students will be raised more in the Scaffolding condition than in the control condition.

Method

In a quasi-experimental design, we compared the mathematical level raising of students who were taught by teachers who were trained to give help to students according to the SGS model with those of students who were taught by teachers who gave help without prior instruction. In both conditions students worked on an Early Algebra unit that was especially designed to invoke mathematical discussions and raise the level of mathematical knowledge [START_REF] Calor | Let's discuss math: Effects of shift-problem lessons on mathematical discussions and level raising in early algebra[END_REF].

Participants were eight teachers (7 males,1 female) and their 266 seventh grade students of two schools in Amsterdam, the Netherlands (5 teachers and 169 students of two schools in the Scaffolding condition and 3 teachers and 97 students of one school in the control condition). Two of the eight teachers (one in the Scaffolding condition and one in the control condition) worked with an Early Algebra unit in a collaborative setting in a previous study [START_REF] Calor | Let's discuss math: Effects of shift-problem lessons on mathematical discussions and level raising in early algebra[END_REF]. Other than that, the teachers had no experience with collaborative learning. Teachers were matched into pairs according to age and teaching experience. Of every pair one teacher was randomly assigned to a condition. Every teacher taught their own seventh grade class. Teachers in the Scaffoldingcondition were trained on the job so that they could be engaged in inquiry into the SGS model and reflection on its use and to provide an opportunity for them to make connections between their learning and classroom practice (Borko, Jacobs, & Koellner, 2010). Prior to the lesson series, teachers in the Scaffolding condition were trained in a two-hour training to give help to students according to the SGS model. In addition, before and after every lesson the use of the model was discussed individually with the teachers and questions were answered. The aim of the research was only known to the teachers in the Scaffolding condition. The teachers in the control condition were instructed to let the students work together on the assignments and give their usual support. All students worked in small heterogeneous groups (46 Scaffolding and 27 control condition). Small student-groups consisted of students of mixed ability according to track advice related to the students by their primary school administration. Ideally groups consisted of four students, but due to class size restrictions 24 groups of three students (13 Scaffolding and 11 control condition) and one group of two students were formed (Scaffolding condition). Students' age varied between 12 and 15 years. Groups in the Scaffolding condition were instructed by the teacher to explain and show/tell their work to each other, ask each other for explanations, and to criticize each other's work. Teachers in both conditions implemented the lessons as intended (teachers in the Scaffolding condition used the SGS model to support students, whereas teachers in the control condition gave support as they normally would). Implementation check was performed by the first author during and after lessons (check with teacher and watch videotape of lessons).

In both conditions, students worked on a lesson series of 12 lessons of 60 minutes on the topic of Early Algebra for five weeks. We replaced five of these lessons, with lessons that explicitly aim to invoke mathematical discussions and raise the level of mathematical knowledge [START_REF] Calor | Let's discuss math: Effects of shift-problem lessons on mathematical discussions and level raising in early algebra[END_REF]. During these five lessons, students worked collaboratively on the same assignment. Every lesson started with 10 minutes introduction by the teacher, followed by students working collaboratively for 50 minutes. During the remaining lessons students sat together in the same small heterogeneous groups and worked on regular assignments from the textbook.

Students' mathematical levels in Early Algebra were measured by means of a test. A pre-test was administered in the lesson prior to the intervention, and the same test was administered as a posttest in the lesson after the intervention. The test aims to measure the mathematical levels of students based on the translation skills from [START_REF] Janvier | Translation process in mathematics education[END_REF]. The highest level implies being able to translate from representation formulae to representations situation, tables and graphs and vice versa. The lowest level implies not being able to translate from and to the representation formulae at all.

The test consists of six questions and 17 sub-questions, of which 10 were used to measure mathematical levels. They consist of questions involving all translation skills of the representation formulae [START_REF] Janvier | Translation process in mathematics education[END_REF]. The other seven questions were discarded (not included in the scores) during determination of level raising. They consist of basic primary school arithmetic questions so that students would be able to answer at least some of the questions when the test was administered as a pre-test. Scores of 0, 1 and 2 were assigned; 0 meaning low mathematical level, 1 meaning medium mathematical level, and 2 meaning high mathematical level. The sub-questions of the test corresponded to the switches between the Janvier representations involving formulae [START_REF] Janvier | Translation process in mathematics education[END_REF]. The first sub-question only involved a less abstract representation of a formula; therefore, the score for this question was limited to 1 (medium mathematical level). The sum of the scores amounts to a maximal number of 19 points in total that students could score for the test. The levels that were assigned to the sub-questions were discussed with a second coder (the second author of this article). Students worked individually on the test for 60 minutes. An example of a basic primary school arithmetic question is: "Out of one package pancake mix you can bake six pancakes. How many pancakes can you bake with 12 packages of pancake mix?" An example of a question that measures mathematical level is: "Shane wants to get in shape, therefore he goes swimming. With the following (word) formula you can calculate Shane's costs. Number of times swimming ×3+30= costs. (Here, 3 stands for €3, the cost for swimming once at a swimming pool for a member and 30 stands for €30, the cost of a yearly membership to a swimming pool.) Draw a graph for this formula." A score of 2 (high mathematical level) was assigned when the graph was drawn correctly (linear relationship in the graph was depicted correctly); a score of 1 (medium mathematical level) was assigned when small errors occurred in the graph (e.g. graph starts at (0,0) rather than (0,30)); a score of 0 (low mathematical level) was assigned when the graph made no sense (e.g. when only names of axes were written down). Another example of a question that measures mathematical level raising is: "Eric saves money to buy a smartphone. Every month he saves €27,-. He has currently no money in his savings account. Give the formula with which the amount of money on Eric's savings account can be calculated". A score of 2 (high mathematical level) was assigned when the formula was correct; a score of 1 (medium mathematical level) was assigned when the formula contained small errors or when the linear relationship was explained in words; a score of 0 (low mathematical level) was assigned when the formula made no sense (e.g. when only 27 was written down).

Interrater reliability for the pre-and post-test between two coders, determined in a prior study, was excellent (ICC 0.95) [START_REF] Calor | Let's discuss math: Effects of shift-problem lessons on mathematical discussions and level raising in early algebra[END_REF].

A multilevel model for repeated observations on fixed occasions with an unrestricted covariance matrix [START_REF] Snijders | Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling[END_REF] was used to test for differential growth between the Scaffolding condition and the control condition. The fixed occasions are the measurements (pre-and post-test) nested in individual students. The measurements (pre-and post-test) are the first level, the individual students are the second level, the small groups are the third level, and the classes are the fourth level. This multilevel model (Snijder & Bosker, 2012) enables us to test the growth in the Scaffolding condition against the growth in the control condition.

Results

We present the means of the pre-and post-test on mathematical level for the Scaffolding condition and control condition in Table 1 Table 1 shows that the students' mathematical level improved a fair amount in both conditions. For the Scaffolding condition (SC), the mean score increased from 2.04 to 11.93, and for the Control condition (CC), it increased from 1.43 to 9.22.

A multilevel model for repeated observations on fixed occasions with an unrestricted covariance matrix [START_REF] Snijders | Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling[END_REF] was used to test for differential growth between the SC and the CC.

We report on the model for three levels (with measurements (pre-test post-test) nested in individual students and classes) since there was no significant improvement in model fit for four levels (measurements, individuals, groups, classes).

The expected outcome for the CC was pre-test and post-test . For the SC the expected outcome was pre-test and post-test . Thus, compared to CC the growth in the SPC is larger. This differential growth is significant .

Discussion/conclusion

In this study, we investigated whether the mathematical level of students who were taught by teachers who were trained to give help according to the Small-Group Scaffolding model was raised more than the mathematical level of students in a control condition.

With respect to our hypothesis, multilevel analyses showed that the mathematical level of students in the Scaffolding condition was raised significantly more than the mathematical level of students in the control condition.

This study was conducted at two schools, which limits the generalizability of our findings. In addition, the conditions were observed at different schools. Consequently, results might be affected by qualitative differences between the schools. For example, the average pre-test score for the Scaffolding condition was higher than that of the control condition. This could be indicative of a higher aptitude, which in turn might be favorable for the growth in the Scaffolding condition. However, this is not very likely since there are no large descriptive differences between secondary schools in the Dutch education system (e.g. track advice given in primary schools determines the track students follow in seventh grade).

In a follow-up study, we will report elaborately on the scaffolding behavior of the teachers in both conditions. We will also focus on the quality of mathematical discussions in the groups, since reflection during mathematical discussions may have a positive effect on mathematical level raising [START_REF] Freudenthal | Weeding and Sowing: Preface to a Science of Mathematics Education[END_REF].

We conclude that the Small-Group Scaffolding model is a useful extension of the Contingent Teaching Model (Van de [START_REF] Van De Pol | Promoting teacher scaffolding in small-group work: A contingency perspective[END_REF]. In addition, we conclude that using the SGS model for Early Algebra can raise the mathematical level of students (ability to switch back and forth between representation formulae and other algebra representations [START_REF] Janvier | Translation process in mathematics education[END_REF]). The SGS model can therefore be useful to help teachers support small heterogeneous student-groups.

Future research should focus on the investigation of what kind of help the teachers in the conditions gave to the students and whether or not the quality of the mathematical discussions mediates mathematical level raising. In addition, this study should be replicated on a larger scale to find more robust findings.

Figure 1 :

 1 Figure 1: A Flowchart of the Small-Group Scaffolding model (Calor et al., 2019)

  .

		Scaffolding condition	Control condition
		M (SD)	M (SD)
	Pre-test	2.04 (3.32)	1.43 (2.66)
	Post-test	11.93 (5.88)	9.22 (6.44)

Table 1 : Means and Standard Deviations Pre-and Post-test on Mathematical Level, Scaffolding condition (N=169) and Control condition (N=97), Maximum Score of 19

 1