María María D Torres
email: mtorresg@ugr.es

María C Cañadas

Antonio Moreno

Structures identified by second graders in a teaching experiment in a functional approach to early algebra

Keywords: Functional thinking, structures, generalisation, teaching experiment

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Background and conceptual framework

Traditionally, algebra is not introduced until secondary school in most countries, Spain is one example (Ministerio de Educación Cultura y Deporte, 20141 ). That approach has induced failure in many students who react to the difficulties encountered by rejecting algebra and mathematics in general [START_REF] Kieran | Algebraic thinking in the early grades: What is it?[END_REF]. In early algebra the discipline is introduced in the first few years of schooling by working, among others, with the generalisation of functional relationships, known as the functional approach to early algebra. Children and adults both generalise by identifying regularity in inter-variable relationships. Notions such as generalisation and the structures in functional relationships are key elements in the study of students' functional thinking [START_REF] Twohill | Observations of structure within shape patterns[END_REF]. This paper addresses second-year primary school students' ability to identify structures and generalise in a functional approach to early algebra.

Functional thinking, as an approach to introduce algebra in the early grades, involves the relationship between two or more variables, and is based on the construction, description and representation about functions and their constituent elements [START_REF] Cañadas | Una aproximación al marco conceptual y principales antecedentes del pensamiento funcional en las primeras edades [Approach to the conceptual framework and background of functional thinking in early years[END_REF]. Patterns and regularities are of significant importance in the study of such thinking, for they interrelate the variables in a situation. Earlier studies reported that primary school students of different ages with no specific prior training were able to perform tasks involving functional thinking [START_REF] Blanton | A progression in first-grade children's thinking about variable and variable notation in functional relationships[END_REF]. The notion of structure acquires different meanings in early algebra [START_REF] Molina | La noción de estructura en early algebra [The notion of structure in early algebra[END_REF]. In the functional approach, it is associated with how inter-variable regularity is organised, which pupils may represent in different ways when working with specific values as well as when generalising the relationship between variables (Pinto & Cañadas, 2017a). Structures can be semantically equivalent. For example, in algebraic symbolism, x + 4, x + 2 + 2 and x + 1 + 1 + 1 + 1 are equivalent expressions (equivalent semantic structures) because these expressions are the same thing through arithmetic computations, but their external structure is different.

Limited studies have addressed these notions in primary education students. Pinto andCañadas (2017a, 2017b), in research on structures and generalisation in third and fifth-year students, described differences in the amount and variety of the structures identified, both of which were greater among the younger children. Third-year students tended to work with specific cases while those in fifth-year were able to establish generalizations; most of them generalised structures and used the same structure for both specific situations and the general case and their answers were more coherent. Three of the fifth-year students generalised in tasks involving specific cases. The authors also distinguished two types of generalisation: spontaneous (when students generalised in tasks describing specific situations) and prompted (when they generalised when asked to do so in connection with the general case). Tasks or worksheets were designed assuming that generalisation would arise in a later stage, so these students evidenced prompted generalisation (Pinto & Cañadas, 2017a). Generalisation and structure are not independent notions. Identifying structure, whether through specific or general situations, is requisite to generalising. We take generalising to mean seeing general patterns in specific situations [START_REF] Mason | Expressing generality and roots of algebra[END_REF]. The inductive model proposed by [START_REF] Cañadas | A proposal of categorisation for analysing inductive reasoning[END_REF] was implemented here to foster this kind of abstraction. At the outset, we used specific situations (variables with specific values), aiming to prompt generalisation (of the functional relationship) through the detection of regularities. Structures and generalisation may be represented in different ways by primary school pupils: natural language, algebraic symbolism, tables or graphs [START_REF] Radford | The emergence of symbolic algebraic thinking in primary school[END_REF]. The present study explored the identification of structures and generalisation in second year primary school students at the beginning and at the end of a teaching experiment involving functional thinking.

Method

In this paper, we focus on part of a broader teaching experiment [START_REF] Steffe | Teaching experiment methodology: Underlying principles and essential elements[END_REF], about functional thinking in second grades (7-8 years old). We present different problems during the five sessions of the teaching experiment and the two interviews. Three of the purposes of the teaching experiment were to: (a) explore how students relate the variables involved; b) identify structures on students' responses; (c) explore students' generalization process. This paper analyses the information gathered in two interviews: one held after the first session and the second after all the sessions were completed. The data were analysed to ascertain differences between the first and second interviews in terms of the structures identified and generalisation expressed by students.

Participants

The subjects were four 7-to 8-year-old, second graders from a Spanish school. They had not been previously exposed to generalisation tasks involving linear functions. Their prior knowledge included the numbers from 0 to 399, addition with carrying and subtraction with borrowing. After the first session of the teaching experiment, in which a worksheet was administered to the entire class (24 pupils), the students were classified into two groups (intermediate and advanced). Students in advanced group generalized the functional relationship; students in intermediate group did not generalized. Four students, two from each group, were selected for the interviews. The subjects selected in each group were chosen by the classroom teacher considering that they should be participative.

Information gathering instrument

The five classroom sessions conducted were video-recorded. In each session we posed one or two tasks in different contexts involving different functions. Each task was described on a separate worksheet. Two semi-structured individual interviews were held with the four students selected, one after the first session and the other after the last.

Classroom sessions

A contextualized task involving a linear function was posed in each session. In all sessions, students first worked with specific cases of the function from which, after identifying the structure, they could pose new specific cases or generalise. Some of the tasks were original and others taken from the studies cited in section 1 above. The context and function addressed in each session are listed in Table 1. We first involved an additive structure (it is supposed to be easier following previous studies) and then we combined the additive and multiplicative structures. We observed that the students had more difficulties to reach the generalization in the later, so we involved only multiplicative structure. We selected contexts close to the students' experiences according to the students' teacher. The objective of each session was to explore students' generalization within different contexts. The dynamics were the same throughout the experiment. Three members of the research team were present in the classroom: the teacher-researcher (he was not the usual teacher), a support researcher and a researcher who video-recorded the session. Each session was divided into several parts: first we explained the task to the whole class, then we distributed the session-specific worksheets and lastly the task was discussed by the group as a whole. The students received no feedback on their replies. The worksheet used in the first session is described below as example and because the interviewees were selected on the grounds of their performance in that session.

Session 1 worksheet

The session 1 worksheet contained five questions about a ball dispenser that when fed balls returned a certain number further to the functional relationship f(x) = x+3. We introduced the context to the students as follows: "we have a machine in which we introduce and leave balls". Four questions involving specific cases were posed as shown in Figure 1. We organized the balls in such a way that did not facilitate the structures identification through visualization.

How many balls will come out from the machine if we feed it 8 balls? ____balls

Figure 1. Specific case question

The last question, involving generalisation, is shown in Figure 2. This worksheet helped to classify the students into two groups based on their answers: intermediate and advanced. The first group included students who had identified regularity in several of the questions and the other students who were able to generalise. The classroom teacher then chose two students from each category (intermediate, S2, S1; advanced, S4, S3). Hereafter, the students are identified by the letter S and a randomised number for anonymity purposes.

Interviews

We designed two semi-structured interviews, each with a different protocol. Two tasks involving additive linear functions were designed for each interview. In the first the context was the same as in session 1 ( ). In the second interview the context was the age of two superheroes, one 4 years older than the other (

). The structure was the same in the two interviews: We began with specific, non-consecutive (to avoid recursive patterning in students' answers) situations and progressed inductively toward generalisation. The design of the second interview, which built on the first, also involved an addition function to ensure inter-sessional comparability of the students' replies. The guides for the two interviews were analogous. The guide for the second interview is presented in Figure 3. We note that in the last question about generalization, letters were random values that served to express examples, the question did not involve or asked for symbolic letters.

the specific cases (spontaneous generalisation) and the general case (prompted generalisation). S2, S3 and S4 generalised only when prompted. The following extract exemplifies S1's spontaneous generalisation when answering questions about specific situations.

I (Interviewer):

when Iron Man was 5 years old Captain America was 9, when Iron Man was 7 Captain America was 11 and when Iron Man was 3 Captain America was 7.

Can you tell me how those numbers are related? S1 (Student 1): Captain America always wins: he's 4 years older.

Comparison of interviews

In the first and second interviews, S1 and S3 identified only one structure, correctly in both cases. S2 and S4 both identified more than one structure, likewise in both interviews. In the second interview S2 continued to identify x+x, although in the specific cases only, ultimately identifying the right structure as the interview proceeded. The contrary was observed in S4, who incorrectly identified the structure x+x in the second interview. Fewer differences were observed around the general case. S2 and S3 correctly identified the structure, whereas S1 initially identified a multiplicative structure in the second interview, 4x, despite having correctly identified the structure in the first. In the second interview S4 correctly identified the relationship between the variables, having failed to do so in the first. All the students correctly identified at least one structure in the specific or general cases and all expressed generalisation in at least one case. In the second interview, however, only one student, S3, identified the sole same structure in the specific and general cases. Two students did so in the first interview. While seeing no clear structure in the specific cases, S4 correctly identified it in the general case. Generalisation was expressed in various ways in both interviews. In the second interview, only S1 generalised spontaneously, whilst in the first both S1 and S3 did so.

Discussion and conclusions

This paper presents evidence of functional thinking in second-year primary school students. Like the fifth-year primary students who participated in the Pinto and Cañadas (2017a) study, most of the second-year pupils in our survey generalised structures, identifying the same structure in the specific and the general cases in the first and second interviews. Their answers were found to be coherent. The number and variety of the structures identified were greater in the first interview.

Although the students might have identified 2x instead of x+x for the specific cases in both interviews, none did. Students evidenced the x + x structure more easily than the 2x because of the kind of structure (additive versus multiplicative structure). All the students expressed generalisation verbally, a finding consistent with the Cañadas and Fuentes (2015) results. All the students generalised in both the first and the second interview. That finding differed from the results reported by Pinto and Cañadas (2017b) for third-year students, only one of whom generalised. In the second interview generalisation was consistently and simply phrased as 'adding plus 4'. In our study generalisation was normally prompted. In one case, S4 in the first interview, expressed the general relationship as y>x. This student was able to see that 'y is always greater', without quantifying to what extent is greater.

Whilst differences between the intermediate and advanced groups was not explored in this study, the findings showed that students' classification in one or the other had no impact on their ability to correctly identify structures. In other words, the advanced students did not correctly identify more structures than the others. In fact, the contrary was found: S1, an intermediate group student, was the only one to correctly identify the structure involved in the specific and general cases in both interviews. Work to enlarge on this study is ongoing to determine how the working sessions might have affected the differences between the two interviews.

Figure 2 .

 2 Figure 2. General case question

Table 1 : Characteristics of session tasks

 1

Ministry of Culture, Education and Sports of Spain.

Acknowledgement

This study was funded by Spain's Ministry of the Economy and Competitiveness under National R&D Plan project EDU2016-75771-P.

Exploration of specific cases

 Identify the functional relationship  Apply the functional rule in different specific cases 1. Specific cases When Iron Man was 5 years old, Captain America was 9. When Iron Man was 7 years old, Captain America was 11. When Iron Man was 3 years old, Captain America was 7. 2. Specific cases proposed by the student Give me an age for Iron Man (__). If he's that old, how old is Captain America?

Generalisation

Data analysis

The data gathered during the two interviews were analysed with a system of categories based on the structures identified by the students in both specific and general cases. Students were considered to have identified a structure when they answered two or more questions referred to the same task with the same regularity or generalised a regularity.

Results

Table 2 summarises the results for the specific and general cases posed in the first interview. The analogous information for the second interview is given in Table 3. The structures identified by the students as they worked are listed in tables 2 and 3 in the order in which they were observed and expressed using algebraic symbolism, although the students did not use that system.

Student Specific cases

General case All four students generalised the functional relationship verbally (Table 2, last column). Three identified the structure as y = x+3, which they expressed verbally in different ways. S1 and S3 generalised both the specific cases (spontaneous generalisation) and the general case (prompted generalisation). S2 and S4 generalised only when they were prompted. S1 first gave examples of specific cases and then generalised by adding that the answer was found by adding 3: '1 ball we get 3, 2 balls we get 5, 3 balls we get 6, 4 balls we get 7. You always have to add 3. One million, you get 1 million and three.' That answer was classified as prompted generalisation because the student formulated it when asked the general case question (Figure 2). S4, in turn, noted that 'more balls come out than went in', generalising without quantifying: 'I know the machine spits out identical balls and what I know is that inside it has lots more balls' (prompted generalisation). The other two students' answers showed that they were thinking in terms of x+3. Three students identified the structure correctly (x+3) in both the general and the specific cases. As in the first interview, students identified one or two structures for the specific cases. Three types of structures, including the ones given for the general case, were found: x+4, 4x and x+x. S2 and S4 exhibited the same type of structures as in the first interview, identifying the relationship in the specific cases to consist in doubling the value given. Generalisation was expressed verbally by all four students. The structures observed in their generalisations were: 4x and x+4. The second was identified by all the interviewees. S1's initial assertion that the age of the older superhero was found by 'multiplying by four' was indicative of their difficulty in distinguishing between addition and multiplication. Student S4 generalised the relationship to be x+4 saying: 'you have to add plus 4'. S2 and S3 both instantly answered that what they had to do was 'add plus 4'. S1 generalised both


Claire Wladis
email: cwladis@bmcc.cuny.edu

Jay Verkuilen
email: jverkuilen@gc.cuny.edu

Sydne Mccluskey
email: smccluskey@gc.cuny.edu

Kathleen Offenholley
email: koffenholley@bmcc.cuny.edu

Dale Dawes
email: ddawes@bmcc.cuny.edu

Susan Licwinko
email: slicwinko@bmcc.cuny.edu

Jae Ki Lee
email: jklee@bmcc.cuny.edu

Relationships between procedural fluency and conceptual understanding in algebra for postsecondary students

Keywords: Elementary algebra in college, conceptual understanding, concept inventory

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In the US, elementary algebra and other developmental courses have consistently been identified as barriers to student degree progress and completion in college, with only as few as one fifth of students ever successfully completing a credit-bearing math course (e.g., [START_REF] Bailey | Referral, enrollment, and completion in developmental education sequences in community colleges[END_REF]. There is evidence that students struggle in these courses because they do not understand fundamental algebraic concepts (e.g., [START_REF] Givvin | What community college developmental mathematics students understand about mathematics, Part II: The interviews[END_REF][START_REF] Stigler | What community college developmental mathematics students understand about mathematics[END_REF], and many research studies have documented the negative consequences of learning algebraic procedures without any connection to the underlying concepts (e.g., [START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF]. However, developmental mathematics classes currently focus heavily on recall and procedural skills without integrating reasoning and sense-making [START_REF] Goldrick-Rab | What Higher Education Has to Say about the Transition to College[END_REF][START_REF] Hammerman | Strategies for Developmental Mathematics at the College Level[END_REF]. This focus on procedural skills in isolation may actually be counter-productive, in that students may often attempt to use procedures inappropriately because they lack understanding of when and why the procedures work (e.g., [START_REF] Givvin | What community college developmental mathematics students understand about mathematics, Part II: The interviews[END_REF][START_REF] Stigler | What community college developmental mathematics students understand about mathematics[END_REF].

In this paper we explore student responses to conceptual questions at the end of an elementary algebra course in college. We combine quantitative analysis of responses (using latent class analysis and distractor analysis) with qualitative analysis of cognitive interviews to better understand different typologies of student reasoning around some basic concepts in algebra, and to better understand how conceptual understanding and procedural fluency may relate to one another in this context.

Research questions

Is there a latent class structure that adequately represents the heterogeneity of item responses on an algebra concept inventory among college developmental algebra students? If so, are patterns observed with distractor analysis, procedural assessment scores, and qualitative analysis of cognitive interviews consistent with the class structure, and can they contribute to class interpretation?

Theoretical framework

In this paper we use Fishbein's (1994) typology of mathematics as a human activity as a framework for analyzing student responses and thinking about how conceptual understanding and procedural fluency might inform one another. Fishbein outlines three basic components of mathematics as a human activity: 1) the formal component (which we call conceptual understanding), which consists of axioms, definitions, theorems and proofs, which need to be "invented or learned, organized, checked and used actively" (p. 232) by students; 2) the algorithmic component (which we call procedural fluency), which consists of skills used to solve mathematical problems in specific contexts and stems from algorithmic practice; and 3) the intuition component, which is an "apparently" self-evident mathematical statement that is accepted directly with the feeling that no justification is necessary.

In this study we use the term conceptual understanding not only to denote a formal understanding of abstract concepts (e.g. axioms), but also of how, when, and why procedures can be used. This is in contrast to procedural fluency in standard problem contexts, in which a student may be able to quickly solve particular types of standard problems correctly but may not understand of how, why, or when these methods work. We say that an item tests conceptual understanding if a student must use logical reasoning grounded in mathematical definitions to answer correctly, and it is not possible to arrive at a correct response solely by carrying out a procedure or restating memorized facts. We define a procedure as a sequence of algebraic actions and/or criteria for implementing those actions that could be memorized and applied correctly with or without a deeper understanding of the mathematical justification. Using these definitions, no question is wholly conceptual or procedural, but falls on a spectrum. We recognize that the definition of conceptual understanding and its relationship to procedural fluency has been much debated in the research community with no clear consensus (e.g., [START_REF] Baroody | An alternative reconceptualization of procedural and conceptual knowledge[END_REF][START_REF] Star | Reconceptualizing procedural knowledge[END_REF], and that the two forms of knowledge are interrelated (e.g., [START_REF] Hiebert | Conceptual and procedural knowledge in mathematics: An introductory analysis[END_REF]National Research Council, 2001).

In fact, each of Fishbein's three components may interact. Intuition may facilitate learning when it is consistent with logically justifiable truths (i.e. the formal component) but may also be an obstacle to learning in cases where it is inconsistent. In this paper we explore how students who develop algorithmic/procedural skills in isolation from formal/conceptual understanding may be prone to developing intuitions that are inconsistent with logically justifiable truths, and as a result, may exhibit simultaneously increased procedural fluency in standard problem contexts and decreased conceptual understanding. We consider how student justifications of answer choices may exhibit intuition components (which may be either correct or incorrect), and we consider how these intuitions may relate to both the processes of developing procedural fluency as well as conceptual understanding.

Methods

This study focuses on student responses to the multiple choice questions on the Elementary Algebra Concept Inventory (EACI). For details on the development and validation of the EACI, see [START_REF] Wladis | Development of the Elementary Algebra Concept Inventory for the College Context[END_REF]). Here we focus on 698 students who took the inventory at the end of their elementary algebra class in 2016-2017. In order to supplement quantitative data, 10 cognitive interviews were conducted towards the end of the semester with students who were enrolled in an elementary algebra class and were analyzed using grounded theory [START_REF] Glaser | The Discovery of Grounded Theory: Strategies for Qualitative Research[END_REF], although a full qualitative analysis is not presented here due to space constraints. All students currently enrolled in the course were recruited via email to complete the EACI and then to be interviewed about their thinking immediately afterwards. The distribution of interviewees among the three classes was not significantly different from the whole quantitative sample.

In this paper we used latent class analysis (LCA) of the nine binary scored (right/wrong) multiplechoice items on the inventory. LCA is a latent variable model that presumes that items are locally independent conditional on a discrete nominal latent variable (e.g., [START_REF] Collins | Latent Class and Latent Transition Analysis: With Applications in the Social, Behavioral and Health Sciences[END_REF].

Description of the classes

Latent class analysis on binary (correct/incorrect) scoring of the items was used to look for common response patterns among the students in the quantitative sample, and this analysis revealed three distinct classes of students. While latent class analysis was used to generate the three classes, more fine-grained analyses was needed to generate potential interpretations of each class. After the initial classes were generated by latent class analysis, item response patterns, distractor analysis, and qualitative coding of cognitive interviews were used to generate characterizations of each class, and evidence was found among these different complementary approaches for the following potential characterizations of each class. The remainder of this paper describes in more detail the different approaches that were used to develop the class interpretations presented here:

• C1 (27%): Answers to most items are indistinguishable from random guessing, likely due to low procedural/conceptual knowledge, low self-efficacy, and/or low motivation. • C2 (28%): Likely have some procedural skills but limited conceptual understanding.

• C3 (45%): Likely have procedural skills and emergent conceptual understanding.

These descriptions emerged from different analyses of the data. Firstly, we consider the response patterns of students from each of the three classes, and we see some clear trends (see Figure 1). Student responses in class 1 do not vary much from what would be expected for random guessing on four-option multiple choice items. While all of the items on the test were designed to test conceptual understanding, some of them (items 2, 3, 5, and 6) use more abstract or non-standard formulations of algebraic ideas. Class 2 answers significantly worse than chance on questions 2 and 6 because of the presence of attractive distractors that likely tap into misconceptions related to the misuse of procedures. Classes 2 and 3 are distinguished by improved performance on the items = 0.000).

overall as well as different proportions of key misconceptions. Students who passed the class were most likely to be in class 3, then class 2, and least likely to be in class 1. An end of course standardized assessment that measures procedural fluency in standard problem contexts showed a similar outcome. To illustrate how different response patterns distinguish these three classes, we performed a distractor analysis and analyzed cognitive interviews for three exemplars: items 2, 4, and 6. We used the Bayes modal assignment to determine the class membership of interviewees.

Three example questions: illustrating different class response patterns

First we consider Item 4: 4. Which of the following is a result of correctly substituting 𝑥 -4 for y in the equation 3𝑦 -2 = 𝑦 2 + 1

a. 3𝑥 -4 -2 = 𝑥 -4 2 + 1 b. 3𝑥 -4 -2 = 𝑥 2 -4 2 + 1 c. 3(𝑥 -4) -2 = (𝑥 -4) 2 + 1 d. 3𝑥 -3 ⋅ 4 -2 = 𝑥 2 (-4) 2 + 1
The correct answer is c. C1's responses are scattered in a pattern consistent with random guessing (see Figure 2). By contrast, C2 and C3 have a high probability of choosing the correct response, with C3's probability significantly higher than C2. This item is behaving the way we would expect if C1, C2, and C3 were ordered on conceptual ability. Selecting c is also highly correlated with scores on the procedural exam, corresponding to a score that is higher by 10.8 percentage points. (𝑝=0.000)

C2 (chose C):

So usually when a math question says, "substituting" that's basically putting the numbers that they give you into 𝑥 or 𝑦 that they say to put it. I automatically substituted it in, and my correct answer was 3(𝑥 -4) -2… I didn't pick any other answer, because I didn't see the parentheses.

C3 (chose C):

I didn't choose A because when trying to multiply the 𝑦, which is 𝑥 -4, you have to put the parenthesis behind 3, unless you already multiplied 3 times 𝑥 -4… [choice D] does have parenthesis on -4, but then, it will be missing the complete equation for 𝑦 because -4 is not the only equation that equals to 𝑦𝑦 is 𝑥 -4.

In these examples, the C1 student demonstrates an intuition about substitution that the 𝑦 should be replaced with 𝑥 -4 in the equation, but this does not include an awareness of the equation structure (e.g., that the 𝑥 -4 must be treated as a single unit). The C2 student also demonstrates a partiallycorrect intuition that parentheses must be used around whatever is being substituted, but doesn't execute this procedure completely correctly on both sides, and doesn't demonstrate any awareness of why the parentheses are necessary. In contrast, the C3 student shows both an awareness of the need for the parentheses and an understanding of why the parentheses are logically necessarybecause without them, the structure of the equation will be altered. Now we consider item 6, which shows a different pattern of responses: The correct answer is b. C2 and C3 were strongly attracted to option a (see Figure 3), likely because they have intuitions stemming from their experiences with procedures associated with the distributive properties, but they do not recognize the critical differences between distributing multiplication versus exponents-likely because they have no deeper conceptual understanding of how the distributive property works. Unlike item 4, selecting the correct answer is negatively correlated (and selecting the attractive distractor was positively correlated) with scores on the procedural exam-students who selected the incorrect option a scored on average 7.1 percentage points higher on the procedural exam (𝑝 < 0.000) than others. This suggests that in this context (where procedures are typically taught in isolation from concepts) procedural fluency in standard problem contexts can be inversely related to conceptual understanding of the distributive properties.

Looking at student interview responses reinforces our interpretation of the three classes, and sheds light on how intuitions developed from procedural practice may impede conceptual understanding. C2 (chose A): I feel like that's correct because in order to solve 𝑥² and 𝑦, you have to distribute…. Because I've seen problems like this before and it's like you have to solve it, there is no not solving it because there is no … there is no solution.

C1 (chose

C3 (chose A): That's how you kind of get rid of the parenthesis and get rid of the outer exponents by distributing it in the inside. Whether it's with another exponent or with a number… You want to add or multiply that exponent [outside the parentheses] to the ones inside the parentheses but I can't remember whether you add or multiply…

In these examples, the C1 student notices that there is a difference between the two equations and has an intuition that it is important, but doesn't actually know how to perform the distribution correctly. For the C2 and C3 students, we see a number of ways in which students are citing incorrect intuitions that stem from algorithmic experience-we have included only a few samples here. None of the students interviewed was able to describe when and why it is possible to distribute-they all cited incorrect intuitions related to procedural methods that they had learned in class. Next we consider item 2, which reveals another interesting pattern of responses: For this question, the correct answer is b, (the most popular choice for students in classes 1 and 3) but no examinee in class 2 chose it (see Figure 4). They were strongly attracted to option d, which was also the second most popular choice for students in both of the other classes, although at a much lower rate. Option d is a common response from students asked to solve a system of linear equations for x and y, which may explain its popularity. Both C2 and C3 were attracted to option d, again suggesting they are attempting to apply inappropriate procedural reasoning. This item is responsible for the notable separation between C2 and C3 because no examinee in C2 chose the option b, the correct response. Looking at student interview responses reinforces our interpretation of the three classes, and sheds light on students' reasons for choosing both correct and incorrect options. C2 (chose D): What I assumed was the x term and the y term, you would have to substitute. And I know there are certain numbers that will add up to ten, so there could be two solutions, since there's only a x term and a y term… Like x could equal 5, y could equal 5… since it is two terms, so you could say two different solutions.

C1 (originally chose

C3 (chose B):

Ten could equal to many things. Like five plus five could equal ten. Nine plus one could equal ten. Seven plus three. That's why I chose that, because it could be any number that will equal to ten. It's not just one certain number.

The C1 student initially chose "no solution" because they didn't know what x and y could be, but as they discussed their answer, they started to relate this to the idea that x and y could be "anything". While their reasoning is not strictly correct, they are beginning to explore the idea that x and y may have many possible values, and they show no evidence of faulty intuitions stemming from procedural practice. The C2 student exhibits an intuition about what the equation means (perhaps from standard substitution problems) to find a single solution, but they do not explore whether there might be others. They also confuse the number of solutions with the number of variables in the solution set, suggesting that their intuitions about the definition of a solution set are likely incorrect. The student from C3 describes how this equation could have multiple solutions, demonstrating some conceptual understanding of solution sets, including the fact that they describe the relationship between the two variables.

Discussion and Limitations

This study revealed several trends. Firstly, roughly one quarter of students at the end of the course appeared to guess somewhat randomly on conceptual questions, likely because of low knowledge, self-efficacy, or motivation. However, cognitive interviews suggest that these students may be able to make some progress towards conceptual understanding by relying initially on more naïve reasoning and that they are not typically hindered by incorrect intuitions stemming from misuse of procedures. About one quarter of students demonstrated some mastery of procedures in standard problem contexts, but demonstrated many misconceptions related to misuse of procedures. In contrast, roughly half the class showed evidence of emergent conceptual understanding, with lower frequency of misconceptions related to misuse of procedures. For a number of conceptual questions, particularly those that were more abstract or non-standard, conceptual understanding and procedural fluency were significantly strongly inversely related. Cognitive interviews revealed that this may happen when students develop incorrect intuitions stemming from the use of procedures.

Both local observation of courses at the research site, and national research literature on developmental mathematics classes in college (e.g., [START_REF] Goldrick-Rab | What Higher Education Has to Say about the Transition to College[END_REF][START_REF] Hammerman | Strategies for Developmental Mathematics at the College Level[END_REF] provides evidence that almost all instruction in this context teaches procedures isolated from underlying concepts, so this research may be revealing patterns specific to this type of instruction-it may be that repeated procedural practice in isolation may worsen certain kinds of conceptual understanding. We note that this study did not attempt to link student response patterns to specific types of instruction-there is a pressing need for future research to examine this relationship in order to determine which types of instruction impact student growth in conceptual understanding.

Figure 1 :

 1 Figure 1: LCA profiles of student responses in each class

Figure 2 :

 2 Figure 2: Item 4 Distractor Analysis Looking at student interview responses reinforces our interpretation of the three classes, and reveals how higher classes appear to have more robust and correct intuitions about substitution. C1 (chose B): It says 𝑥 -4 for 𝑦, this is what I think like because 𝑦 2 . It could be like changed to a 4 2 . I put together like 3𝑥 -4 -2 = 𝑥 2 -4 2 + 1. [I didn't pick c or d because] they [pointing to the 𝑥 -4 in the item stem] didn't have no bracket around them. [I picked B with the 𝑥 2 in it instead of A, which doesn't have the 𝑥 2 ] because 𝑥 equals 𝑦 2 so it has to have an 𝑥 2 in it because the 𝑦 is squared there.

6.

  A student is trying to simplify two different expressions: i. (𝑥 2 𝑦 3 ) 2 ii. (𝑥 2 + 𝑦 3 ) 2 Which one of the following steps could the student perform to correctly simplify each expression? a. For both expressions, the student can distribute the exponent. b. The student can distribute the exponent in the first expression, but not in the second expression. c. The student can distribute the exponent in the second expression, but not in the first expression. d. The student cannot distribute the exponent in either expression.

Figure 3 :

 3 Figure 3: Item 6 distractor analysis

  B): [The difference between the first and second equation] is that there's a plus right there [pointing to the second equation]. I think for this one [pointing to the second equation], you have to add and for this one [pointing to the first equation] you don't…. Actually, I think like over here [pointing to the second equation] you add a 3. 3 plus 2. [For the first one] you do 𝑥 2 times 𝑥 2 and 𝑦 3 times 𝑦 3 .

2.

  Consider the equation x + y = 10. Which of the following statements must be true? a. There is only one possible solution to this equation, a single point on the line 𝑥 + 𝑦 = 10. b. There are an infinite number of possible solutions, all points on the line x + y =10 c. This equation has no solution. d. There are exactly two possible solutions to this equation: one for 𝑥 and one for 𝑦.

Figure 4 :

 4 Figure 4: Item 2 distractor analysis

C, but drifted towards B in the interview):

  𝑥 + 𝑦 equals nothing so it can't be 10. Right?... [Maybe infinite means] what could be like possible? I don't know. Like equal number maybe? 𝑥 + 𝑦 = 10. It could be possible like it equals 10. [D isn't correct] maybe because x and y could be equal to anything?


Alexander Wolff
email: wolff@imai.uni-hildesheim.de

Forming basic conceptions in dealing with quadratic equations

Keywords: quadratic equations, solving, transforming, understanding, basic conceptions

Introduction

Transforming and solving quadratic equations, such as , is part of the curriculum taught in school. To support the learning process of equations and their equivalent transformations, models such as the balance scale-, the box or the strip model, can be and are usually used in class (see Holzäpfel/ Barzel, 2010). The work and the support thereby reach its limits when dealing with negative coefficients in quadratic equations or by multiplying/dividing with negative numbers (see [START_REF] Vlassis | The balance model: hindrance or support for the solving of linear equations with one unknown[END_REF]. On the one hand its whole variety of formal expressions of quadratic equations and its equivalent transformations cannot be supported and visualized with the above mentioned models. On the other hand the work with the suggested models can lead to inadequate conceptions (misconceptions) of the mathematical concepts: E. g. facing problems to be able to interpret formal expressions such as or as equations as well (with an empty set of solutions). E.g. regarding the division of both sides of the equation by or the extraction of the square root as an equivalent transformation. But working with models and forming content adequate conceptions for the learning process is indispensable. According to vom Hofe/ Blum (see 2016) the formation of basic conceptions (Grundvorstellungen) belongs to a competent and sensible dealing with mathematical concepts. But which basic conceptions, that students are about to form in the learning process, can be cited up? Two basic conceptions are cited normatively and visualized in a model further down. They allow to interpret a quadratic equation and its equivalent transformations illustratively in a graphical (or tabular) way and consequently grasp the core of understanding

Model Figure 1: Graphical visualization of basic conception 1 (GV1)

Basic conception 1:

A quadratic equation as an intersection of two functions, whereby at least one has to be quadratic. To determine the set of solutions the x-coordinate of the intersections and has to be considered.

Example:

Study design

To evaluate if these normatively cited basic conceptions can be formed by students and hence can be proved empirically, special learning assignments have to be designed. Aim of the presented study is thus to develop a learning environment and subsequently to investigate if those basic conceptions can be formed by students. For this purpose, a teaching sequence of 2x45 minutes was designed. Both sets of 45 minutes dealt with one basic conception. Each set consists of a specially designed common starter with a subsequent individual working phase of 4 learning assignments. The utilized exercises in the working phase were supplementary designed with regard to a high potential of diagnosis implying a written response in addition to the actual task.

Preliminary results and outlook

A first postprocessing of students' solutions of the learning assignments revealed to evaluate the collected data quantitatively through a deductive procedure. This leads with regard to the formation process of the first basic conceptions to the following preliminary results: The solution quote over the first 4 learning assignments amounts up to 81%. In summary, it could be said, that the dealing with the exercises for the formation of the first basic conception (GV1) worked out well. Missing percentages are due to a lack of time, inaccurate drawing of graphs and mistakes in reading off exactly the relevant coordinates and distinguishing both from each other. The evaluation of the learning assignments for the formation of the second basic conception (GV2) showed up to be more difficult. The used tasks were created too open and required for its processing the correct and precise use of mathematical language. These circumstances made the students' solutions difficult to evaluate. The tasks are now going to be revised more closely. The following study is going to be carried out once again, under the revisions of the exercises, this year.

Basic conception 2:

An equivalent transformation keeps the x-coordinate (the set of solutions) constant and simultaneous leads to a change of the functional value of the intersections.

Example:

Figure 2 :

 2 Figure 2: Graphical visualization of basic conception 2 (GV2)