Much remains under-researched in how learners make use of domain-specific feedback. In this paper, we report on how learners’ can be supported to overcome logical circularity during their proof construction processes, and how feedback sup- ports the processes. We present an analysis of three selected episodes from five learners who were using a web-based proof learning support system. Through this analysis we illustrate the various errors they made, including using circular reasoning, which were related to their understanding of hypothetical syllogism as an element of the structure of mathematical proof. We found that, by using the computer-based feedback and, for some, teacher intervention, the learners started considering possible combinations of assumptions and conclusion, and began realising when their proof fell into logical circularity. Our findings raise important issues about the nature and role of computer-based feedback such as how feedback is used by learn- ers, and the importance of teacher intervention in computer-based learning environments. Keywords Computer-based feedback · Proving · Logical circularity · Geometry 1 Introduction or 6 years of age, children show a preference for non-circular explanations and that this appears to have become robust by In research on the teaching and learning of proof and prov- the time youngsters are about 10 years of age. ing, an under-researched issue is the extent to which students While learners’ preference for non-circular explanations are competent in identifying logical circularity in proofs and may be robust by the time they are 10 years old, within how such competency can be enhanced (Hanna and de Vil- mathematics education Heinze and Reiss (2004) report that liers 2008; Sinclair et al. 2016; Stylianides et al. 2016). Rips from grade 8 to 13 in Germany an unchanging two-thirds (2002) has argued that the psychological study of reasoning of pupils fail to recognise circular arguments in mathemati- should include a natural interest in patterns of thought such cal proofs. Such evidence illustrates that pupils are in need as circular reasoning, since such reasoning may indicate of considerable support in order to identify and overcome fundamental difficulties that people may have in construct- circular reasoning in mathematical proofs. As Freudenthal ing, and in interpreting, even everyday discourse. However, (1971) observed “you have to educate your mathematical Rips claims that up until his study in 2002 there appeared to sensitivity to feel, on any level, what is a circular argument” be no prior empirical research on circular reasoning. While (p. 427). All these studies, and the statement by Freudenthal, Rips reports on a study of young adults, Baum et al. (2008) suggest that there are still many aspects to be examined in report findings with younger students—indicating that by 5 order to have deeper understanding of students’ ways of thinking concerning deductive proofs, so that they can be provided with better learning support. Considering the current situation and gaps described * Taro Fujita t.fujita@exeter.ac.uk above, this paper explores issues of how learners’ can be supported to overcome logical circularity during their proof Graduate School of Education, University of Exeter, Exeter, construction processes, and how feedback supports the pro- UK cesses. We particularly focus on learners’ use of feedback Southampton Education School, University of Southampton, as the latter is a key aspect of assessment for learning and Southampton, UK something which is recognised as “one of the most powerful Institute of Education, Shinshu University, Nagano, Japan Vol.:(0123456789) 1 3 700 T. Fujita et al. Fig. 1 A ‘visual proof’ of influences on learning and teaching” (Hattie and Timper - Pythagoras’ theorem ley 2007, p. 81). Despite considerable research related to assessment and feedback there remain many open questions. In particular, much remains under-researched about how domain-specific formative feedback can improve learners’ learning processes. For example, Stylianides et al. (2016) state that it is necessary to investigate “productive ways for assessing students’ capacities to engage not only in produc- ing proof but also to engage in processes that are ‘on the road’ to proof” (p. 344). Fig. 2 A rectangle from Fig. 1 The aim of this paper is to consider an overarching ques- tion of how feedback can support learners who accept or construct a proof with errors. In order to achieve this pur- pose, in this paper we work with the following specific research questions (RQs), which we consider as useful to explore and expand our thinking on how to support students’ proving processes; 2 Logical circularity in geometrical proofs RQ1: What patterns of proof construction processes We begin by clarifying logical circularity in deductive can be identified as learners use the web-based learn- proving. In mathematics, Euclid’s Elements is one of the ing support system? oldest texts that organised various mathematical state- RQ2: How is the feedback from the online system ments logically. Each proposition is carefully ordered so used by learners to overcome logical circularity dur- that only already-proved propositions are used to prove ing proof construction? new propositions. Thus, for example, the proposition ‘the To address our research questions, we first clarify the base angles of an isosceles triangle are equal’ is not proved nature of logical circularity in geometrical proofs and why by using an angle bisector, as is common in current text- students might accept or construct a proof that contains a books, because this can fall into logical circularity if a logical circularity (Sect. 2). In particular we show that issues geometric construction of angle bisector is proved by with propositions and hypothetical syllogism are the under- using the proposition that the base angles of an isosceles pinning ideas that can inform feedback to support students’ triangle are equal. Such an approach entails assuming just proof learning. In Sects. 3 and 4, we review relevant litera- what it is that one is trying to prove (Weston 2000, p. 75). ture on learners’ use of feedback, including computer-based In logic, reasoning using circular arguments is considered feedback, and develop conceptual ideas for characterising a fallacy as the proposition to be proved is assumed (either the types of feedback provided by our web-based learning implicitly or explicitly) in one of the premises, and this support system for learning deductive proofs (hereafter, the results in logical circularity. system). In Sect. 5, after describing our system, we provide Such circular reasoning can happen within a proof. For our methodology for studying the use of computer-based example, Bardelle (2010) provides an example of some feedback during the proof construction process. We present, undergraduate mathematics students in Italy being pre- in Sect. 6, an analysis of selected episodes collected as stu- sented with the diagram in Fig. 1 as a ‘visual proof’ of dents worked on proof problems using the system. These Pythagoras’ theorem. The students were asked to use the episodes qualitatively illustrate how learners who have just figure to help them develop a written proof of the theorem. started learning to construct mathematical proofs made vari- Bardelle relates how one student focused on the rectan- ous mistakes, including using circular reasoning, and how gles that surround the central square. By defining a as the these relate to the use of their universal/singular proposi- short side and b the longer one (as in Fig. 2), the student tions and hypothetical syllogisms in their proof construction 2 2 used Pythagoras’ theorem to get c = a + b and thence, processes. Finally, in Sects. 7 and 8, we discuss our findings by squaring both sides, the student obtained Pythagoras and answer our research questions. Through answering our 2 2 2 theorem c = a + b . This is another, and rather local research questions and subjecting our findings to critical example, of a student using a circular argument or circulus discussion, we aim to provide insights into productive ways probandi (arguing in a circle). While we acknowledge it is of using assessment in proof learning, as well as into issues important to educate students to evaluate critically vari- related to the teaching and learning of mathematics with ous processes of circular reasoning between theorems or computer-based learning environments, and methodological within proofs, in this paper we focus on the latter because approaches to studying learning processes. 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 701 our focus is lower secondary school students who have just take feedback as “information with which a learner can started learning deductive proofs. confirm, add to, and overwrite, tune, or restructure infor - In the case of the teaching of proof in geometry, trian- mation in memory, whether that information is domain gle congruency is commonly used (Jones and Fujita 2013). knowledge, meta-cognitive knowledge, beliefs about self In this context at least two types of logical argument are and tasks, or cognitive tactics and strategies” (Winne and employed to structure deductive reasoning. One is universal Butler 1994, p. 5740). Shute (2008) identified two main instantiation, which takes a universal proposition (such as, functions of formative feedback: verification (simple in congruent triangles all corresponding interior angles are judgement of whether an answer is correct) and elabora- equal) and deduces a singular proposition (for example, if tion (providing relevant cues to guide the learner towards ∆ABD ≡ ∆ACD, then angle ABD = angle ACD). The other the correct answer). Clark (2012) states “The objective of type of logical argument is hypothetical syllogism, where the formative feedback is the deep involvement of students in conclusion necessarily results from the premises (Miyazaki meta-cognitive strategies such as personal goal-planning, et al. 2017a). monitoring, and reflection” (p. 210), and, as such, it is Appreciation of proof structure is recognised as an impor- related to self-regulated learning. tant component of learner competence with proof (Heinze In the teaching and learning of mathematics, feedback and Reiss 2004; McCrone and Martin 2009; Miyazaki et al. can be used by students to choose appropriate procedures or 2017a), and this inclusion might relate to why students improve problem-solving strategies. Rakoczy et al. (2013) accept or construct a proof with logical circularity. For found that written process-oriented feedback (i.e. “suggest- example, Kunimune et al. (2010) report on data from grade 8 ing how and when a particular strategy is appropriate” p. 64) and 9 students, showing that as many as a half of grade 9 and might foster grade 9 students’ mathematical learning. This two-thirds of grade 8 pupils were unable to determine why implies that certain types of feedback might be more effec- a particular geometric proof presented to them was invalid; tive than others. Hattie and Timperley (2007) claim that that is, they could not see circular reasoning in the proof in “Effective feedback must answer three major questions asked which the conclusion (‘the base angles of an isosceles trian- by a teacher and/or by a student” (p. 86), namely, ‘Where am gle are equal’) was used as one of the premises for deducing I going? (What are the goals?)’, ‘How am I going? (What the two triangles are congruent. We consider this oversight progress is being made toward the goal?)’, and ‘Where to as being due to a lack of understanding of the role of syllo- next? (What activities need to be undertaken to make better gism, which would lead to accepting or constructing a proof progress?)’. In order to realise ‘how learners are going’, they which includes a circular argument. A proof of the proposi- identify the following four elements (p. 90): tion ‘the base angles of an isosceles triangle are equal’, for example, can be done by connecting two deductions: (1) Task: “Feedback can be about a task or product, such as deducing two triangles are congruent; (2) deducing if two whether work is correct or incorrect.” triangles are congruent then corresponding angles are equal. Process: “Feedback information about the processes However, if a learner lacks an understanding of hypothetical underlying a task also can act as a cueing mechanism syllogism, he or she may use ‘the base angles of an isosceles and lead to more effective information search and use of triangle are equal’ as one of the premises in order to deduce task strategies.” that the base angles of an isosceles triangle are equal. In so Self-regulation: “Feedback to students can be focused at doing, he or she would be using a circular argument. the self-regulation level, including greater skill in self- Our interest in this paper is in how students who accept or evaluation or confidence to engage further on a task.” construct such proofs can be supported in their learning of Self: “Feedback can be personal in the sense that it is proof structure, and we focus on the use of computer-based directed to the “self,” which… is too often unrelated to feedback in this paper as an example of a way of providing performance on the task.” such support. They argue that while task-based feedback may be the least effective form, it can help when the task information 3 Feedback supporting learners’ proof is subsequently used for “improving strategy processing or construction processes enhancing self-regulation” (pp. 90–91). From these existing studies, and given that what makes 3.1 Feedback for learning feedback most effective for learners is complex, it remains uncertain whether, or how, a combination of task- and pro- Feedback is one of the strategies for assessment of learn- cess-based feedback might be effective when students are ing that is known to promote learning (Black and Wil- learning sophisticated mathematical topics such as deduc- iam 2009). Amongst many definitions of feedback, we tive proving. 1 3 702 T. Fujita et al. 3.2 Learners’ use of computer‑based feedback 4 A web‑based system to support the learning of deductive proofs The use that learners make of computer-based feedback, in geometry defined as “assessment feedback to students created and delivered using a computer” (Marriott and Teoh 2013, p. 5), 4.1 Online feedback provided by the system continues to be a growing interest in educational research (Wang 2011; Bennett 2011; Attali and van der Kleij 2017). Given the various errors that learners can make in the pro- Based on their meta-analysis, Hattie and Timperley (2007) cess of learning to prove, they are likely to benefit from reported that computer-assisted instructional feedback is support and feedback not only in recognising errors but also one of the effective forms of feedback in that it can provide in ways to refine their proof in accordance with the type of cues or reinforcement for improving learning. Narciss and error they may be making. Our system is designed to support Huth (2006, p. 310) termed informative tutoring feedback such learning (the current system is online at http://www. as that providing “strategically useful information for task schoo lmath .jp/flowc hart_en/home.html). In particular, the completion, but [which] does not immediately present the system is designed to support overcoming of students’ dif- correct solution” and bug-related tutoring feedback as that ficulties in proofs that are particularly related to the use of “guiding students to detect and correct errors.” They found universal/singular propositions and hypothetical syllogisms both to be particularly effective because such feedback can (see Sect. 2). As we showed in an earlier study, adopting a provide useful strategies to correct errors as well as requir- flow-chart format and closed/open problems can enrich the ing learners to apply corrective ways to further attempts to learning experience of the use of universal/singular proposi- solve the problems. This is similar to process-based feedback tions and hypothetical syllogism (see Miyazaki et al. 2017b). described above. In our system, flow-chart proofs (see Ness 1962) are Nevertheless, learning with computer-based feedback is used and various proof problems in geometry are available, not clear-cut. For example, Attali and van der Kleij (2017) including ones that involve the properties of parallel lines report on their experimental research in which they exam- and congruent triangles. Learners tackle proof problems by ined the feedback effects of different question formats dragging sides, angles and triangles to cells of the flow-chart (multiple choice/constructed response), timing (immediate/ proof and the system automatically transfers figural to sym - delayed) and types (knowledge or results/correct responses/ bolic elements so that learners can concentrate on logical elaborated feedback). They found that the effects of different and structural aspects of proofs. Feedback is shown when types of feedback and timing can vary and that this might answers are checked. The geometry proof problems include be related to learners’ initial responses to the problems and both ordinary proof problems such as ‘prove the base angles their prior knowledge concerning the problem. As such, of an isosceles triangles are equal’ (an example of a ‘closed’ elaborated (or process-based) feedback is useful in general, problem) and problems by which learners construct different but when the learners’ prior knowledge to the problem is proofs by changing premises under the given limitation to low, it is not particularly effective. In contrast to this finding, draw a conclusion (these we categorise as ‘open’ problems). Fyfe et al. (2012) found that feedback can be more beneficial In the latter case, the correct answers can be reviewed so that for learners with little prior knowledge compared with those students may be encouraged to find other proofs. who have some knowledge. Perhaps, as Attali and van der For example, the problem in Fig. 3 is intentionally Kleij (2017) wrote, it is important that “Prior knowledge designed so that learners can freely choose which premises is considered to be the most important factor to consider they use to prove that AB = CD (note that information such for adapting instruction to an individual learner” (p. 167), as AB//CD is not stated explicitly at this level of problem something that might indicate the importance of human because this problem is for practicing how to use singu- interventions in the computer-learning environment. Pan- lar and universal propositions with two-step reasoning in ero and Aldon (2016) also reported that, with technology- later stages the problems are stated with more mathemati- based learning environments, both teachers and students cal rigor). A learner might decide, for instance, that a sin- might become more effective at using feedback by seeking gular proposition that ∆ABO and ∆CDO are congruent efficient ways of using it. may be used to show that AB = CD by using the universal Of these many complexities, one interesting area that proposition ‘If two figures are congruent, then correspond- needs further study is domain-specific computer-based ing sides are equal’. Based on OA = OC as an assumption, feedback in advanced mathematical topics, such as proving, ∆ABO ≡ ∆CDO can be shown by assuming BO = DO and as the existing studies have rather focused on “lower-level angle BOA = angle DOC using the SAS condition. However, learning outcomes such as rote memorisation” (Attali and other solutions are also possible. One approach might be to van der Kleij 2017, p. 155). use the fact that ∆ABO ≡ ∆CDO can be shown by assuming 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 703 Fig. 3 Feedback for circular arguments from the system OA = OC, angle BOA = angle DOC and angle OAB = angle designed to prompt the learner to think why they received such OCD, using the ASA condition for congruency. Two stars a message and to re-examine their proof. show this problem has two solutions, and each of them We take feedback from the system as ‘information given changes to yellow when found. As learners can construct by the computer to learners, which they can use to check their more than one suitable proof, we refer to this type of prob- answers, modify their answers and strategies for better proof lem situation as ‘open’. This open situation can be used to constructions, and seek different proofs’. For describing such scaffold students’ understanding of the structure of proofs, in feedback in detail, we use Hattie and Timperley’s framework particular the use of universal propositions and thinking for- ‘Where am I going?’, ‘How am I going?’ and ‘Where to next?’. wards/backwards to seek premises and conclusions in proofs Our system provides cues for ‘Where am I going?’ by (see Miyazaki et al. 2015, for the case without technology). clearly stating the goal of the problem, and for ‘Where to next?’ by giving a message such as ‘This is correct! But it is not the only answer. Find out more!’ or ‘You have found 4.2 Domain‑specific computer‑based feedback all answers’. The system also provides ‘How am I going?’ for supporting students’ learning of deductive feedback through task-, process- and self-regulation feedback. proofs ‘Self’ type feedback is outside the remit of our web-based sys- tem because it is more linked to the role of the teacher. Table 1 In the main, our system gives bug-related tutoring feedback summarises the overall features of the system’s feedback for (Narciss and Huth 2006); that is, once a learner clicks ‘Check ‘How am I going?’. your answers’, something which can be done at any time, the Our research interest is in how the above computer-based system checks for any error via a database. These errors are feedback, in particular information for ‘How am I going?’ recognised in terms of the use of singular/universal proposi- (task/process/self-regulation based), is used in the context of tions and hypothetical syllogism. For example, Fig. 3 shows learners construction of deductive proofs in geometry. feedback for a proof of an ‘open’ problem where the proof falls into logical circularity. In this case, the conclusion AB = CD is used as the one of three conditions to deduce the congruence of triangle ∆ABO ≡ ∆CDO. As a result, the system shows a message ‘You cannot use the condition to prove your conclu- sion!’. This message does not provide a correct answer but is 1 3 704 T. Fujita et al. Table 1 Types of feedback provided by the system Type of feedback Example Task Indicating simple errors by suggesting the alternative choice of universal/singular proposition is the correct answer “Be careful of the order of comparisons” “Is this a correct reason to draw the conclusion?” “You may select again a condition of congruence of two triangles” Process Not only indicating errors related to the use of universal/singular propositions and hypothetical syllogism but also cue- ing to search for a better solutions or relationships without directly telling correct answers “Let’s find the included angles of these two sides” “Let’s find two angles at the end of this side” “You cannot use the conclusion to prove the conclusion!” Hint for reminding conditions of congruent triangles Self-regulation For encouraging them to find alternative answers, learners can review already completed proofs by clicking yellow stars This was used for the interviewers to assess their initial 5 Methodology understanding of the structure of proofs. Finally, they were asked to solve problems that include 5.1 Study design, data collection and participants two steps in the proof, and, if they were very success- ful, then more difficult problems. We initially piloted the English-language version of the system in 2010–13 in the UK with a range of individual Participants’ activities were observed and recorded by or grouped learners (with groups of up to 4). These learn- a video camera. As stated above, we particularly used the ers had previously learned about congruent triangles, but problems with one or two steps in the proof, as our par- none had much prior experience of deductive proof based ticipants had relatively little experience in constructing on properties of lines and angles and congruent triangles. geometrical proofs. The interventions from the interview- They used our web-based system to tackle one or more of ers were kept to a minimum, because we wanted to see the problems, either with or without explicit instructions how the feedback from the system would be used by the from researchers. During this pilot study, it was gradually learners. However, we sometimes had to give ad hoc inter- noticed that students often misused universal propositions ventions when they totally lost the notion of what they had in order to justify their reasoning, and produced proofs to do, or needed clarification on what to do, etc. This is with logical circularity when they undertook open proof something we learnt from our early trialling; that learners problems with two steps of reasoning. can spend too long a time on just one proof problem and As stated above, the web-based proof learning system develop some frustrations towards learning proofs—which was primarily developed to support learners’ learning of was not our primary research interest. deductive proofs in geometry. During the pilot studies, For this paper, we have selected three cases from our it was evident that the system provided a research tool data; one case was a pair of high-attaining secondary not only to reveal students’ lack of understanding of syl- school students aged 14 years (WS1 and WS2), a second logism, but also to study the learning processes by exam- case was an individual undergraduate primary trainee ining how learners respond to feedback messages from teacher (R), while the third case was a pair of undergradu- the system when they make various errors. Therefore, we ate primary trainee teachers (J1 and J2). We chose these decided to collect data systematically from a total of 15 three cases because we found interesting reactions to the learners’ experiences using the system, focusing on their system, and the feedback received, during the proving errors and how they used feedback during sessions that processes as well as their experiencing of correct/incor- took 30–60 min. The typical session comprised the fol- rect reasoning. Table 2 summarises their activities and the lowing structure: durations of the video data. First learners were introduced to the system by interview- ers, during which it was explained how to use it with an 5.2 Data analysis procedure introductory open problem. One computer was shared within small groups in order to encourage their collabo- After initial examinations of the video data, we selected the rative learning and dialogue. If necessary, learners were following problem-solving episodes from each case, and reminded of the conditions for congruent triangles. then extracted in total 432 utterances, and then numbered Following this introduction, they were asked to under- them for data analysis. take one or two more relatively easy open problems. 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 705 Table 2 Participants’ proof construction experience with the system we first identified in total 36 proof construction ‘phases’. Each phase commenced with learners’ attempts to construct Participants Activities a proof and ended when they managed to complete a cor- WS1 and WS2 Lesson II-2 (one step open proof, 8 min) rect proof or they completely lost their directions despite Lesson III-1 (two steps open proof, 8 mi) receiving feedback of various kinds from the system. By Lesson III-2 (two steps open proof, 7 min) identifying these phases we were able to examine the learn- Lesson IV-4 (one step open proof, 12 min) ers’ proof construction processes more closely. For identi- J1 and J2 Lesson II-1 (one step open proof, 5 min) Lesson IV-3 (one step open proof, 13 min) fied phases, we undertook a detailed qualitative analysis to Lesson III-1 (two steps open proof, 6 min) ascertain patterns of proof construction processes in terms Lesson III-2 (two steps open proof, 6 min) of errors which learners made (informed by Sect. 2) and the Lesson V-1 (closed two steps open proof, 10 min) types of feedback they received (informed by Sect. 3) and, R Lesson II-1 (one step open proof, 7 min) where necessary, interventions by the interviewers. We use Lesson II-2 (one step open proof, 4 min) Lesson IV-3 (one step open proof, 11 min) this analysis as evidence to answer our research questions. Lesson III-1 (two steps open proof, 6 min) For example, phase III-2/Ph2 in Table 3 is the second phase of proof construction lesson III-2. In this phase, WS1 and WS2 were undertaking a proof requiring two steps in an WS1 and WS2: Lesson II-2 (64 utterances), III-1 (76 open problem context with the interviewer T. utterances), III-2 (63 utterances) In this example, pair WS1 and WS2 started their proof J1 and J2: II-1 (29 utterances), III-1 (36 utterances), III-2 construction with errors. As they did not notice that they (65 utterances), V-1 (37 utterances) were using the conclusion to prove the conclusion (utter- R: II-2 II-2 (16 utterances), III-2 (46 utterances) ances 24–28), we categorised them as lacking an under- standing of syllogism. Subsequently, following the process- We chose these cases because these episodes were par- based feedback related to logical circularity (e.g., utterance ticularly related to learners’ use of universal propositions 28) and task-based feedback related to the use of a universal and errors of logical circularity, as well as their reactions proposition (e.g., utterance 32), they noticed that they used to feedback including overcoming difficulties in their proof the conclusion (as well as a wrong universal proposition) in construction processes. The methodological challenge was, their proof and corrected these by themselves (utterances as Stylianides et al. (2016) pointed out, how to assess and 29–32). In utterance 33 they received feedback on a correct analyse students’ processes on the road to proof. To do so answer. The interviewer T encouraged them to find another Table 3 Analysis example of proof construction by WS1 and WS2 in III-2/Ph2 Utterance Subjects Transcript Description/analysis 24 WS2 Right. Two pairs of sides… included angles are equal. So it Proof construction with errors as they were using the conclu- would be that one and that one, that one and that one, and sion ‘angles ABO = ACO’ as one of the premises of their that one and that one (angles ABO and ACO) proof 25 WS1 No, we cannot use those angles [ABO and ACO] again, so WS1 now suggested ABO and ACO cannot be used as they are the conclusion 26 WS2 Yes but ABO and ACO, if you do that, here… So, that one WS2 still used ABO and ACO. WS1 did not notice this time. and that one, BO and CO Also, the process-based feedback was ignored 27 WS1 Change [indicating a tab from ‘angles’ to ‘sides’] WS1 suggested ‘if congruent then sides are equal’ 28 WS2 [Click, feedback] Process-based feedback (circular argument) was given 29 WS1 No? Ah, look at that, you can’t, we used those two [ABO WS1 noticed again ABO and ACO were used thanks to the and ACO] process-based feedback 30 WS2 Oh. [changing angles, click. Feedback, as the angle is not Process-based feedback (cuing to seek correct pairs of included anymore] Ah, that is wrong now, because if angles) was given we use these two we need to change these two [sides to AO = AO] [click, feedback] Uhmm 31 WS1 Maybe… 32 WS2 Oh, yes that one, wrong one, that is why [indicating a tab WS2 noticed the wrong universal proposition was used, but from ‘sides’ to ‘angles’] after this task based-feedback, they could complete a cor- rect proof 33 T Well done. Try once more, then you can delete all red Move to the next problem indications. Click that one, then all go. Yes 1 3 706 T. Fujita et al. proof. We summarise this proof construction process as a proof constructions, often without any formative feedback pattern ‘Proof construction with errors → Process/task based from the system. R often started proof constructions with feedback → Proof construction without errors’. Our approach errors, but after receiving feedback from the system and ad to the analysis of our qualitative data is to see what patterns hoc interventions by the interviewer, could complete two- can be identified in each case. step proofs. Meanwhile, on one occasion, R did attempt We are aware that the sample size is small and therefore to construct a proof that included a circular argument we do not intend to propose generalised findings. Also, we (III-1/Ph3). WS1 and WS2 received much feedback from do not claim effectiveness of the web-based system based the system, plus ad hoc interventions by the interviewer, on a few sessions; that is, we do not intend to claim that by e.g., explaining the goals of tasks, two-step proofs, etc. using our system learners can completely overcome their dif- Sometimes they lost their directions (e.g., II-2/Ph1 and 3 ficulties in their learning of proof. Yet, as proving consists of or III-1/Ph2 and 4). Towards the end of the session, they rather complex mathematical processes, we consider qualita- started completing proofs with feedback only from the sys- tive analysis is necessary prior to undertaking a larger-scale tem without ad hoc support by the interviewer, and in fact study. As the students’ proof construction processes that we received less feedback. However, they still constructed a identified contain a rich source of information, we use these proof with a circular argument. data as a step towards more in-depth research. 6.2 Students’ use of feedback 6 Findings While J1 and J2 could use both process- and task-based 6.1 Patterns in the proof construction process feedback well to improve their proofs, both R, and WS1 and WS2, in contrast, needed interventions by the inter- By following the above procedure of analysis, we identified viewer (as captured in Fig. 4) in addition to feedback from a total of 36 proof construction phases. Based on an analysis the system. For example, the interviewer had to clarify of these 36 proof construction phases, we could identify 12 the goal of a two-step proof for R. In a first attempt, R patterns of proof construction (see Table 4) grouped into needed a further clarification of how to complete a proof four broad categories: in a flow-chart form (III-1/Ph2). While R could use task- based feedback well to correct the proof format in gen- Proof construction without errors (PC without Es) eral, further clarification from the interviewer was needed Starting proof construction with errors, reacting to feed- when R received the first process-based feedback in III-1/ back, finishing without errors (PC with Es → FB → PC Ph3. This further clarification related to why a circular without Es) argument is not allowed in a proof, as well as how to use Starting proof construction with errors, reacting to feed- singular propositions properly. back with interventions, finishing without errors (PC For the case of WS1 and WS2 in their II-2 and III-1 with Es → FB with interventions → PC without Es) phase, their proof construction started from ‘PC with Starting proof construction with errors, reacting to feed- Es → Process FB → PC with Es’ or ‘PC with Es → Pro- back with interventions, finishing with errors (PC with cess/task FB → PC with Es’, meaning they could not use Es → FB → PC with Es). feedback from the system by themselves and their learning was a rather ineffective trial–error approach. Ad hoc inter - Necessary interventions were forms of ad hoc support by ventions by the interviewer were necessary. In particular, the interviewers, which were given when the participants they really struggled to construct a correct proof with two were completely lost after they received feedback from the steps (III-1/Ph2 and III-1/Ph4), showing confusion about system. The 12 patterns of proof construction, grouped into using singular propositions. the four broad categories, are summarised in Table 4. Figure 4 shows the process of proof construction by each 36 WSs [trying several proofs and then received feedback as they used BO = CO twice] of the three sets of learners, WS1 and WS2, J1 and J2, and 37 WS2 Let’s use these ones, because they are bigger (AB = CD R (with the shaded boxes being identified patterns in each again) phase of the proof construction). 38 WS1 We just used this one! Table 4 and Fig. 4 capture several points related to 39 WS2 Hang on. [feedback, you cannot use the conclusion …] learners’ proof construction processes, and feedback use 40 WS1 We just used them, just used them (this is, of course, still limited to the context of proof con- 41 WS2 That is wrong! struction with the system). First, it seems J1 and J2 have sound understanding as they have successfully completed 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 707 Table 4 Patterns in proof construction processes with the web-based system Category Pattern Description PC without Es PC without Es Learners construct a correct proof without errors, receiving no process/task based feedback from the system Self-regulation FB → PC without Es Learners construct a correct proof without errors, receiving self-regulation based feedback Self-regulation FB + Intv → PC without Es Learners construct a correct proof without errors, receiving self-regulation based feedback and inter- vention Intv-PC without Es Learners construct a correct proof without errors with short interventions PC with Es → FB → PC without Es PC with Es → task FB → PC without Es Learners start constructing a proof with errors, but after receiving only task based feedback, learners construct a correct proof PC with Es → process FB → PC without Es Learners start constructing a proof with errors, but after receiving only process based feedback, learn- ers construct a correct proof PC with Es → process/task FB → PC without Es Learners start constructing a proof with errors, but after receiving both process and task based feed- back, learners construct a correct proof PC with Es → FB with Interven- PC with Es → task FB + intv → PC without Es Learners start constructing a proof with errors, but tions → PC without Es after receiving task based feedback, learners con- struct a correct proof with intervention clarifying feedback PC with Es → process FB + intv → PC without Es Learners start constructing a proof with errors, but after receiving process based feedback, learners construct a correct proof with intervention clarify- ing feedback PC with Es → process/TASK FB + Intv PC without Learners start constructing a proof with errors, Es but after receiving both process and task based feedback, learners construct a correct proof with intervention clarifying feedback PC with Es → FB → PC with Es PC with Es → process FB → PC with Es Learners start constructing a proof with errors and receive process based feedback but they are not able to correct their errors, resulting losing completely their directions PC with Es → process/task FB-PC with Es Learners start constructing a proof with errors and receive both process and task based feedback but they are not able to correct their errors, resulting losing completely their directions PC proof construction, FB feedback, Es errors, Intv intervention by an interviewer Ph6). These cases indicate that the feedback from the system 42 WS1 Leave that one. Maybe that one and that one… how about that one (AB) and that one (BO), the line that one (AB) might not be enough and it might be necessary to intervene and that one (BO) [WS1 is highly confused now] in students’ early proving processes, if they repeatedly make 43 WS2 You cannot re-use it mistakes in both singular and universal propositions (e.g., 44 WS1 Why not? the pattern ‘PC with Es → Process/task FB → PC with Es’). 45 WS2 No [feedback for using the same sides of the triangle twice] BO and BO, you cannot use the same one twice 6.3 Learners’ experience with circular reasoning with the system After this failure, the interviewer suggested that they start again, and not use the SSS condition. With these ad In the lesson III-1, J1 and J2 completed the two proofs hoc interventions they finally managed to complete cor - with little difficulty. On being prompted by the interviewer, rect proofs with process-based feedback from the system they then tried to find a different proof, and, in the phase as well as a clarification by the interviewer (III-1/Ph5 and III-1/Ph3 (categorised as ‘PC without Es’), they not only 1 3 708 T. Fujita et al. 1 3 Learners J1&J2 Pattern / Lesson/PC phase II1/Ph1 II1/Ph2 II1/Ph3 II1/Ph4 III1/Ph1 III1/Ph2 III1/Ph3 III2/Ph1 III2/Ph2 III2/Ph3 III2/Ph4 V1/Ph1 The goal of the problem was clarified by the interviewer PC without Es 1 1 1 1 1 1 1 1 Self-regulation FB →1 PC without Es Self-regulation FB + Intv → PC without Es Intv → PC without Es PC with Es → Task FB → PC without Es 1 1 PC with Es → Process FB → PC without Es 1 PC with Es → Process/task FB → PC without Es PC with Es →Task FB + Intv → PC without Es PC with Es → Process FB + Intv → PC without Es PC with Es → Process/task FB + Intv → PC without Es PC with Es → Process FB → PC with Es PC with Es → Process/task FB → PC with Es Learners R Pattern / Lesson/PC phase II2/Ph1 II2/Ph2 II2/Ph3 III1/Ph1 III1/Ph2 III1/Ph3 III1/Ph4 The goal of the problem was clarified by the interviewer PC without Es 1 Self-regulation FB → PC without Es Self-regulation FB + Intv → PC without Es Intv → PC without Es 1 PC with Es → Task FB → PC without Es 1 PC with Es → Process FB → PC without Es PC with Es → Process/task FB → PC without Es 1 PC with Es →Task FB + Intv → PC without Es 1 PC with Es → Process FB + Intv →1 PC without Es PC with Es → Process/task FB + Intv → PC without Es PC with Es → Process FB → PC with Es PC with Es → Process/task FB → PC with Es Learners WS1&2 Pattern / Lesson/PC phase II2/Ph1 II2/Ph2 II2/Ph3 II2/Ph4 II2/Ph5 II2/Ph6 III1/Ph1 III1/Ph2 III1/Ph3 III1/Ph4 III1/Ph5 III1/Ph6 III1/Ph7 III2/Ph1III2/Ph2 III2/Ph3 III2/Ph4 The goal of the problem was clarified by the interviewer PC without Es 1 Self-regulation FB → PC without Es Self-regulation FB + Intv → PC without Es 1 Intv → PC without Es 1 PC with Es → Task FB → PC without Es 1 PC with Es → Process FB → PC without Es PC with Es → Process/task FB → PC without Es 1 PC with Es →Task FB + Intv → PC without Es 1 PC with Es → Process FB + Intv → PC without Es PC with Es → Process/task FB + Intv → PC without Es 1 1 1 PC with Es → Process FB → PC with Es 1 PC with Es → Process/task FB → PC with Es 1 1 1 Fig. 4 Patterns of proof construction process (the shaded boxes are identified patterns in each phase of the proof construction) Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 709 refuted using the SSS condition for the problem (utter- 33 R: Am I not using the same lines? ances 27–28), but also eliminated other possibilities for 34 T: You are using the same lines answers (utterances 31–36). This illustrates their capacity 35 R: … but angles are not on the same lines… to identify logical circularity, grasping the relationship 36 T: That is right between premises and conclusion; that is, as a combination of universal instantiations and syllogism. This was a short but important moment for R, and finally, after some thought in silence (2 s), R noticed the 27 J2 You could do all the … correct proof was the one already completed. R reasoned 28 J1 All the sides? that it was not possible (utterances 37–44 below) to create 29 J2 Yes… actually no, because… different proofs using SSS because in this case R realised 30 J1 and J2 You are trying to prove [AB = CD] … that AB = CD had to be used as one of premises, which 31 J1 And if you can’t use this line [AB] had already been rejected by the system. This shows that then we can’t use the other angle… R might have started developing an understanding of this because it is not included… aspect of syllogism. 32 J2 You mean those [ABO and CDO]? 33 J1 Yes, it is not included [as AB cannot be used]… and we’ve already got 37 R: [2 s silence] I don’t think there is any more answer others… 38 T: No. So you are confident, just two [answers]? 34 J1 How about AO-OAB-AB? 39 R: Yes 35 J2 You cannot use these, because… 40 T: Yes that is right. Because in this problem these two 36 J1 Because these ones [AB and CD] (AO = CO) are assumed already. So you need to use, like which we are trying to prove… you did, if you choose angles AOB and COD, and angles ABO and CDO, then 41 R: … they are not on that line In particular, they exchanged their thoughts about a pos- 42 T: That is right. That is right. And we have discussed, we can- sible proof before they actually constructed a proof with not use that one SSS, and without support from the system during their 43 R: Because attempts. They also explored very similar reasoning during 44 T: Yes, because this is… Lesson III-2 on why angles ABO = ACO (the conclusion) 45 R: What you are trying to find! [laugh] cannot be used as one of premises. 46 T: That is right! [laugh] In the case of J1 and J2, they could argue why a proof cannot contain a circular argument without any feedback In the case of R, the process-based feedback worked from the system. However, the other two cases are quite well, but for WS1 and WS2 it was a lot more difficult than different from J1 and J2 in terms of dealing with a cir- expected for them to understand why the conclusion can- cular argument. For example, R firstly constructed cor- not be used as one of the premises in deductive proofs. For rect answers by using SAS and ASA individually during example, in the proof construction process of WS1 and phase III-1/Ph2 and 3, recognised as ‘PC with Es → Pro- WS2 (III-1/Ph2), without any hesitation, their first attempt cess/task based FB → PC without Es’. This suggests that involved using the SSS condition (Fig. 6). R could understand universal instantiation. However, R They could identify pairs of sides and angles in their considered that it would be possible to use the SSS condi- proof (Fig. 6), but they made a mistake as they put ‘angles tion as one of the answers of the open problem to prove AOB = COD are congruent’, rather than ‘triangles OAB AB = CD, which indicated that R was lacking understand- and OCD’; regardless, they chose SSS as a condition of ing of hypothetical syllogism. R then tried to change the triangle congruence. This suggests that they also did not condition (ASA) and then complete a proof (Fig. 5, left). have a good understanding of universal instantiation. More This failed because R used angles ABO = CDO instead of importantly, they failed to notice that they should not use angles OAB = OCD. the conclusion AB = CD in their proof. The system first The system gave the message “Let’s find two angles at highlighted the mistake of logical circularity. After receiv- the end of this side” (Fig. 5, right), encouraging R to find ing the message that “You cannot use the conclusion to a different angle or side of the triangle (utterance 27), as prove your conclusion!”, and with additional ad hoc inter- well as prompting R to explain why the proof was wrong. ventions from the interviewer, they started considering As shown below, R still had difficulty identifying the error that AB = CD should not be used in their proof in III-2/ (utterance 33), but managed to correct the mistake (utter- Ph5. With this consideration they began to understand why ance 35). AB = CD should not be used. 1 3 710 T. Fujita et al. Fig. 5 A proof constructed by R (above) and feedback (below) However, when they started a new problem in III-2, they shows that they did not see the whole structural relationship still used the conclusion as one of the premises in their between premises and conclusion. In order to identify the proof. Thus, the cases from WS1 and W2 illustrate that the circular argument as a serious error, learners need to under- understanding of the meanings and roles of premises and stand at least the role of syllogism which connects premises conclusions might be very difficult for learners who have with conclusions. just started learning mathematical proof. Moreover, from On the positive side, WS1 and WS2 gradually received the point of view of structure of proof, their experience less and less feedback from the system, and towards the end 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 711 Fig. 6 WS1 and WS2’s proof in III-1/Ph2 of the activity (Lesson III-2), at least WS1 started grasping 7 Discussion why the conclusion cannot be used in their proofs (III-2/Ph4, ‘Self-regulation FB-PC without Es’). In this paper our focus is the use of domain-specific com- puter-based feedback by students who are learning the struc- 46 WS1 [Reviewing the already answered proofs] ture of proof but accept or construct a proof with logical Three angles we can use, we have used circularity. In order to study this issue, we conceptualised two, so it must be the other one students’ difficulties in terms of the use of universal/singular 47 WS2 [Suggesting ABO again] propositions and hypothetical syllogism (Sect. 2). The ways 48 WS1 No, which could be. Go back that feedback provided by the system is related to students’ 49 WS2 So we use these ones [BAO and CAO] difficulties is shown in Sect. 4. In Sects. 6.1–6.3 we provide 50 WS1 Yes our analysis of learners’ proof processes using feedback 51 WS2 So then that one [ABO] from the system and, in some cases, ad hoc interventions. 52 WS1 No, middle one [AOB] Now we discuss our findings in relation to our research ques- 53 WS2 But that one was used? tions, although, as we have already noted, these are neces- 54 WS1 No wasn’t sarily tentative because of the sample size. 55 WS2 No wasn’t [laugh]. So if we use that one In terms of the patterns of proof construction processes [AOB] [completing a proof] (RQ1), as we demonstrated in Table 4 and Fig. 4, we iden- tified various patterns of use of the feedback by learners As can be seen from the dialogue above, reviewing during their proof construction processes. These patterns already-answered proofs (self-regulation) was useful in guid- include proof constructions started without errors (‘PC ing WS1 and WS2 to construct a different proof by them- without Es’), ones that started with errors but by using selves (utterance 46). While WS2 still suggested using angle feedback from the system, learners could manage to correct ABO, which is the conclusion, WS2 was very confident that their errors and construct proofs without errors (‘PC with they should not use it. This resulted in their constructing a Es → FB → PC without Es’), ones that started with errors correct proof without formative feedback from the system. but, following interventions, finished without errors (‘PC At this point they had received both process- and task-based with Es → FB with Interventions → PC without Es’), and feedback from the system, and the ways they performed with ones that started with errors and then finished with errors the system indicate they had started internalising feedback (‘PC with Es → FB → PC with Es’). For example, in the case from the system and could correct errors by themselves. of J1 and J2 their patterns were mostly ‘PC without Es’ or ‘PC with Es → FB → PC without Es’. Other learners (e.g., 1 3 712 T. Fujita et al. WS1 and WS2) appeared to have difficulties with why a road’ to proofs (Stylianides et al. 2016). Our methodological circular argument cannot be used in a proof (e.g., ‘PC with approach was to segment the proof construction processes Es → FB → PC with Es’). into ‘phases’, and record what errors the learners made and We found, as did Panero and Aldon (2016) and Attali their reactions to the feedback given by the system. We and van der Kleij (2017), that in computer-based learning found that through learners receiving both task- and process- environments the teacher’s role continues to be important. based feedback supplied by our online learning system, this Here we found that feedback from the system was useful helped them overcome logical circularity in their proving for the interviewers to give specific ad hoc interventions for and to construct correct proofs. some learners who were relying on trial–error based learn- Our findings raise important issues about the nature and ing or ‘PC with Es → FB → PC with Es’ loops (e.g., WS1 role of computer-based feedback. For example, while both and WS2, II-2/Ph3–4 or III-1/Ph4–5). Thus, while we found Rakoczy et al. (2013) and Hattie and Timperley (2007) that computer-based feedback can be effective to improve state that feedback for effective strategies (process-based) learning (as did Narciss and Huth 2006 and; Wang 2011), is more effective than just stating right or wrong answers human intervention cannot be under-estimated. This conclu- (task-based), it may be that for advanced topics, such as sion applies, in particular, for those who have limited under- proofs, the combinations of both types might be necessary standing. In our study context, if patterns of attempted proof including human interventions. Also, the use of feedback construction such as ‘PC with Es → feedback → PC with Es’ might be related to learners’ understanding of proofs, and are repeatedly observed, then the feedback from the system this suggestion again echoes the claims of other studies on might not be enough and interventions might be necessary. learners’ prior knowledge and understanding and feedback In terms of how feedback was used to overcome logical use (e.g., Fyfe et al. 2012; Attali and van der Kleij 2017). circularity in proofs (RQ2), both task- and process-based In order to investigate these points in more depth, a larger feedback (Hattie and Timperley 2007) supplied by the sys- data set is needed set in order to evaluate if the learning with tem provided guidance on what might help learners con- open proof problems and the system, including the feedback struct correct proofs. Our cases showed that learners started format and timing, can effectively improve students’ under - bridging the gap in their logic in syllogism (e.g., R, and WS1 standing of deductive proof with computer-based learning. and WS2 towards the end of their proving) after receiving Our study is limited to congruency-based geometrical both task- and process-based feedback. Self-regulated feed- proof using the flow-chart format and, in particular, ‘open’ back rarely occurred in our cases, although learners were proof tasks. Even so, we obtained rich data from our sample directed to what different proofs could be produced (e.g., and found that our methodological approach worked well. WS1 and WS2, III-2/Ph4). Nevertheless, the challenge still exists to examine, for exam- Overall, in order to support students’ learning, this con- ple, how to assess proof construction processes in wider clusion suggests, for all cases, that considering possible contexts such as progressing from ‘open’ to ‘closed’ (i.e., combinations of premises and conclusion, and checking single solution) problem formats with more mathematically- whether or not a proof falls into logical circularity, some- rigorous conditions. In addition to this, insights are needed times prompted by the system’s feedback in the open prob- into how computer-based feedback can be used to support lem contexts, are useful to overcome logical circularity. proving processes in other proof formats (such as two col- This result echoes what Freudenthal (1973) suggested, that umn proofs), or in topics such as algebra. On top of this, in order to make learning mathematics meaningful “the first insights are needed into what pedagogical approaches would step is to doubt the rigour one believes in at this moment” be necessary in the classroom. A related issue that Sinclair (p. 151). et al. (2016, p. 706) identify in relation to overcoming logi- cal circularity in deductive proving is learners “understand- ing the need for accepting some statements as definitions 8 Conclusion to avoid circularity”. Further research into all these matters should enrich understanding of the teaching and learning of With feedback recognised as one of the effective ways to mathematical proofs and proving, not only logical circularity improve learning, our aim was to explore how domain- within proofs but also logical relationships between theo- specific, computer-based, ‘bug-related’ tutoring feedback rems, which we did not explore much in this paper. (Narciss and Huth 2006) is used by learners in order to Acknowledgements This research is supported by the Daiwa Anglo- overcome their difficulties in their proof construction pro- Japanese foundation (No. 7599/8141) and the Grant-in-Aid for Scien- cesses, especially when there was logical circularity in their tific Research (Nos. 15K12375, 16H03057), Ministry of Education, deductive proving. We took three cases with five learners Culture, Sports, Science, and Technology, Japan. We thank the three anonymous reviewers and the editor for their valuable suggestions. and examined their proof construction processes. It was still challenging for us to assess how the students were ‘on the 1 3 Learners’ use of domain-specific computer-based feedback to overcome logical circularity… 713 Open Access This article is distributed under the terms of the Crea- Marriott, P., & Teoh, L. (2013). 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Published: Jun 5, 2018
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