{'fraction_non_alphanumeric': 0.03153544120686092, 'fraction_numerical': 0.02037611076668733, 'mean_word_length': 5.091641490433031, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Teacher’s Actions to Promote Flexibility in Mental Arithmetic

In this article, we discuss teachers’ actions to develop flexibility in students’ mental arithmetic. We analyze the actions of a second-grade teacher during the whole-class discussion of a task, included in a sequence of tasks designed to promote calculation flexibility. Flexibility is a competence that students should develop at school, and teachers’ actions are crucial for fostering flexible mental calculation by which students establish number relations. Data were collected through observation by the authors along with videotaping the activity and transcribing recorded speech. In data analysis, we examined how the teachers’ actions and questions were critical for students to focus on the properties of numbers and operations and thereby improve their calculation flexibility. The teacher’s supporting/guiding and challenging actions were most frequently used during the whole class discussion, and they showed a greater influence in developing a conceptual understanding underlying the calculation flexibility. These specific actions, in conjunction with other actions, such as inviting and informing actions, appear fundamental in leading students forward in establishing number relationships.

This article is part of a project which aims to characterize the development of students' numerical thinking and flexibility in mental calculation from 6 to 12 years and to describe teachers' practices that facilitate that development. Calculation strategies can be used by students in a mechanized way, regardless of the context of the task or the numbers involved (Brocardo, 2014), and in this sense, students can show mastery of procedural skills without understanding the underlying mathematics. Hence, the ability to properly apply the knowledge that has been acquired in another situation to a new one is fundamental to the development of mathematical proficiency (NCTM, 2000). For Threlfall (2009), students use a flexible mental calculation when they meet the contextual variables of a given task or specific numbers, and establish relationships with their previous numerical knowledge.
How teachers promote and organize classroom discussions and the kind of questioning may be an important role in the development of students' flexible "reflects an attention to both solution methods as well as number patterns and relationships embedded in problem structures" (Rathgeb-Schnierer & Green, 2019, p. 8).
The study reported here takes the perspective of Threlfall (2009) as well as Rathgeb-Schnierer and Green (2013), considering strategic flexibility as the way an arithmetic problem is affected by circumstances to be mentally solved. These circumstances may be related to specific features of the tasks or contextual variables or personal characteristics. Threlfall (2009) argues that in a flexible calculation, "the calculation-strategy is not selected and applied, it is arrived to" (p. 548). For him, the emergence of strategies occurs through a process named zeroing-in, which is a not fully conscious mechanism, involving partial exploratory calculations arising from noticing specific features of the numbers involved and their respective relationships. This perspective diverges from several authors (Heinze et al., 2009;Heirdsfield & Cooper, 2004;Star & Newton, 2009) who define flexibility as knowing multiple solution pathways as well as the capacity to choose the most appropriate for a given problem. Like Threlfall (2009) and Rathgeb-Schnierer and Green (2013), Baroody and Rosu (2006) rejected this strategy-choice model, referring to flexibility in calculation as related to the fact that children discover number patterns, as they develop number sense, thereby building a network of numerical relationships. Rechtsteiner-Merz and Rathgeb-Schnierer (2015) introduce the term 'Zahlenblick' as "the competence to recognize problem characteristics, number patterns, and numerical relationships immediately, and to use them for solving a problem" (p. 355). For these authors, the approach of 'Zahlenblickschulung' is an important vehicle to develop a flexible mental calculation.
According to Baroody and Rosu (2006), what matters is the ability to produce new known facts from old ones, acting as an autonomous meaningful knowledge generator instead of the ability to efficiently produce meaningless answers from a memory network. Gray and Tall (1994) distinguish known facts from deduced facts, and they argue that known facts do not necessarily lead to the deduction of new facts, which depends on the flexibility to use known facts. One example of such flexibility can be found in arithmetic doubles, which we highlight in the study reported here. For example, starting with the double of 25: 25 + 25 (50)then 51 is a near double (50 + 1 or 25 + 25 + 1); or in reverse, 51 -25 = 26, because 26 = 25 + 1 and 25 is half of 50. Thus, the double/half relation can be used to deduce new facts.
Developing flexible calculation strategies involves developing networks of numerical relationships in which numbers become objects of thought (Gravemeijer & Bruin-Muurling, 2019;Gravemeijer et al., 2016). As reasoned by Gravemeijer and Bruin-Muurling (2019, p. 18), "numbers as objects allow for flexibly using number relations to solve tasks in the domain of number." Students can think about additions and subtractions in a relational way in order to form coherent sets of number relations (for instance, 34 + 20 = 54, 20 + 34 = 54, 54-20 = 34 and 54-34 = 20). In this way, sums and differences become mathematical objects with which students can act and reason. This contrasts to numbers as cogs in a series of algorithmic steps. Numbers must be structured by students so that they are composed and recomposed in a wide variety of ways that, when combined, constitute a rich conceptual structure (Gray & Tall, 1994). "Numbers, sums, and differences have to become mental objects that can be composed and decomposed in different ways, which in turn forms the basis for flexible arithmetic and an expansion to bigger numbers, and later for integers and algebra" (Gravemeijer et al., 2016, p. 37). As argued by Gravemeijer and Bruin-Muurling (2019), the flexible use of number relations may enhance a deeper understanding of the operations involved.

Teachers' Actions
How teachers question students during the whole-class discussion is a fundamental aspect of classroom communication (Boaler, 2003;Ponte & Quaresma, 2016). Different points of view should be considered, and students should be encouraged to explain and justify their reasoning and solutions, thus developing, with the teachers' support, their understanding (Ruthven et al., 2011). One important challenge is that the ideas students develop are aligned with the mathematical concepts and procedures that we want them to know. In any case, they should be able to establish numerical relationships by thinking of numbers as mental objects. For that, teachers should plan lessons that include questions that support and extend students' thinking. So, a fruitful whole class discussion requires a strong commitment from the teacher, beginning with lesson planning, anticipating student resolutions, thinking about which questions should be made, and supporting students when they work in pairs trying to solve the task. In that context, teachers need to identify student works to be discussed and in which order, and culminate with wholeclass discussion, where a teacher's actions are decisive. For analyzing a discussion, Ponte et al. (2013) distinguished four different teachers' actions related to mathematical aspects: (a) Inviting, seeking to initiate a discussion; (b) Supporting/Guiding to lead students to present information or to focus on a specific mathematical idea, making questions or observations; (c) Informing/Suggesting, introducing information, giving suggestions, validating students' responses; and (d) Challenging to encourage students to produce new representations, interpret a statement, establish connections, or formulate reasoning or an evaluation.
With regard to arithmetic teaching, teachers should support students to expand the variety of number relations and to transcend the "splitting /jumping dichotomy" (Gravemeijer & Bruin-Muurling, 2019, p. 26)this dichotomy refers to the two mechanized procedures of calculating, either by splitting or jumping. For these authors, it is important that, under the guidance of the teacher, students reason with number relations when they solve problems of addition and subtraction; that is, "they will come to think of adding and subtracting as composing and decomposing mathematical objects" (p. 26) in contrast to traditional memorization that emphasizes speed and accuracy over meaning. For example, tasks such as 85-49, might be solved by comparing it with 85-50, which is easier and then, 85-50=35 and 85-49=36. According to the authors, from this task, the teacher can also work other relations with students, such as 85-36=49 and 49+36=85. So, the teacher must be able to maintain the cognitive demands of the task, fostering "the transition from jumping and splitting as procedures to construing sums and differences as objects" (Gravemeijer & Bruin-Muurling, 2019, p. 22), focusing students' work on the goal of understanding the underlying mathematics and not so much on mastery of procedural skills.

Methodology
Our study focused on examining the teacher's actions during a planned whole-class discussion of a task designed to foster flexibility in mental arithmetic. That strategy provided data about both cognitive flexibility and teacher actions summarized above.
This study follows a qualitative approach framed in an interpretative paradigm (Bogdan & Biklen, 1994). It is focused on the educational processes and the meanings of the study participants (Erickson, 1986). The study is part of a large project whose main goal is mentioned before. The project adopts the modality of design research conceived to produce local teaching theories and teaching sequences that can contribute to informing teachers' practices (Gravemeijer & Cobb, 2006). In particular, these local teaching theories will involve teaching practices that promote flexibility in mental arithmetic.
This article focuses on the whole class discussion of the task "Cards with numbers" fulfilled in October 2015, in the 2nd grade with 26 students, and whose goal is to use known facts to establish number relations and to calculate flexibly. The task was the 3rd of a sequence of 6 tasks which were previously discussed with the teacher and adapted by her. This teacher was a primary teacher with more than 10 years of teaching experience and who had active participation in projects and in-service education courses related to mathematics education; she was selected for her availability to collaborate with our project. For the task, 18 cards shown in Figure 1 were distributed to the students. The cards included number values the teacher believed her students knew as number facts as well as expressions with numbers close to the previous ones so that the students could deduce them by relating them to the known facts.
Then, each student received a sheet to record the expressions on the cards, separating them into two columns: those that students knew quickly and those that students did not know quickly. A similar task was used in Rathgeb-Schnierer and Green (2015) where the students were asked to sort 12 two-digit addition and subtraction expressions displayed on cards as either 'easy' or 'hard' and to explain their sorting.

Figure 1. Cards Distributed to Students.
The teacher introduced the task and answered students' questions about the task. Then students began to solve the cards in teacher-assigned pairs. During that time the teacher observed the work of her students and answered some occasional doubts. When all the pairs had solved the task (with all 18 problems completed), the teacher put all the cards on the blackboard and started the whole class discussion asking one pair to go to the blackboard. The discussion began by focusing on one of the pair's most difficult cards. The discussion reported here lasted 30 minutes.

Promote Flexibility in Mental Arithmetic
The observation data collection was done by the authors of this article, complemented with field notes and videotaping of the lessons. The names of the students have been changed to ensure confidentiality. The analysis focused on field notes, video recordings, and transcriptions. The analytic categories shown in Table 1 are the teachers' actions related to the whole-class discussion, as considered by Ponte et al. (2013).
These categories were used because they are closely linked to wholeclass discussion management and are also suitable to analyze discussions focused on fostering flexibility in mental arithmetic. The data were coded separately by the two authors of this article and then put together. The level of agreement was 90%. The situations with disagreement were discussed and consensually agreed upon.

Results
Results are reported sequentially for selected pairs of students. Adequate excerpts from the whole-class discussion are incorporated where the teacher's actions are more visible.
With all the cards posted on the board, the teacher invited the pair Mónica and Gil to go to the blackboard and choose the most difficult card for them (Code: Inviting Action). Gil picked up the card 100-48 as being the most difficult and put it to the side on the board.  With her actions, the teacher invited Mónica and Gil to present a difficult card and then guided and challenged them through her questioning. Other students tried to justify the relations between the cards. Maria did not know the answer but decided to point out the fact that 52 is an even number. Gabriel used the double/half relation to get 100-50. Then the teacher guided him to relate this number fact to deduce 100-48. She asked the students to sit down, and the discussion continued. The teacher asked again why Gil and Mónica had chosen the card 100-52, looking to have a justification, challenging them. Other students contributed to the discussion. Ilda seemed to relate 50 with both 52 and 48. Gabriel seemed to have understood the property of compensation, and he was able to apply it appropriately. Renato showed some facility with calculation. Analyzing the sequence of interactions, we infer the teacher's actions sought to keep students focused on establishing number relations to deduce new facts from known ones and The teacher tried to guide the students to relate the expressions recorded on cards (100-48; 100-52) by challenging them to interpret the situation. As the students remained silent, the teacher rephrased what Armando had said: Teacher: This 52 appears as the result of 100-48=52, and this 48 appears as the result of 100-52=48. Why does this happen? (Code: Challenging Action).
The teacher asked again for a justification from the students, but they continued in silence. She did not give up and continued challenging them to obtain a justification for the situation. From the conversations we had before with this teacher, we infer that for her it was important to lead students to go beyond the calculations and understand what happened beyond the concrete numbers. So, she decided to overcome the situation by asking them to present another similar example: Teacher: So, give me another example where this also happens (…) One other example where the numbers also appear exchanged? (Code: Supporting Action) Gabriel went to the board and wrote 100-51=47, but quickly rectified: Gabriel: Gives 49. Paulo (talking sitting in his place): 100-49=51 [Gabriel's board work is shown in Figure 3]

Figure 3. Gabriel's Example Illustrative of the Part-whole Relation.
The teacher supported the students' efforts to achieve the part-whole relation by suggesting they find a new example. Then she continued pressing them to justify the exemplified relationship: With the teacherś challenging action, Luí s seems to have understood that sums and differences become numbers, and he appears to work with them as both mental objects and number relations. Luis understood the relationship between addition and subtraction and the relations between different numbers in both operations. When the teacher went to the board and rephrased what Luí s had said, she tried to ensure that all the students understood the situation. Probably, Luís's generalization derives from understanding number relations as well as the inverse between addition and subtraction.
After that, the teacher invited another pair to go to the blackboard to write the most difficult expression for them (Code: Inviting Action). The pair Joã o and Paulo went to the blackboard and chose the cards 52-29 and 50-29 as being difficult, relating them to 50-30.  Figure 4 shows the records done by Paulo. With actions of guiding, supporting, and informing, the teacher tried to help the students understand the relations between different numbers and the subtraction proprieties. She used the informing action to help students remember the property of invariance, and she used the guiding action to lead students to understand the compensation resulting from changing the additive (which troubled Paulo).
Continuing the whole class discussion, the teacher knew that Nadia and Santiago had used the empty number line to solve the same problems (100-52 and 100-48; 52-29 and 50-29), so she invited: Teacher: Someone did it differently? (Code: Inviting Action) Nadia and Santiago went to the blackboard and drew their scheme on the board, for 100-52 and 100-48. Then the pair drew their scheme relative to 52-29 and 50-29, shown in The pair Nadia and Santiago used the empty number line in a different way than the teacher, putting the subtractive on the line instead of being placed above the arrow (the jump). In 52-29, they decided to subtract from 52 other number with the same units, 22, to obtain exact tenths, 30, and then plus 1 (more than 22) to obtain 23 in the jump, insofar as 29 is 1 less than 30, as observed by Nadia ("29 is before 30"). Therefore, the pair determined the difference, 23, in the jump, and not on the line. Given this use of the line, the teacher asked the pair why 52-23 helps to calculate 52-29 if there was not that card, but they could not answer. So, the teacher challenged all students to think about it. It was Luí s who went to the blackboard and said: With these new expressions, Luí s replicated the reason he had invoked in the previous situation with 100-48=52 and 100-52=48. The teacher validated Luís's answer, trying to have students involved in the same reasoning.
Among all the teacherś actions observed and reported here, the supporting/guiding and challenging ones were the most used. In terms of frequencies, the action codes in descending order of use were:

Discussion
Relative to the first empirical research question, this teacher used all four coded actions during the whole class discussion. The guiding action was used to focus students' attention on the relations between the expressions represented on the cards, and the challenging action was used mainly to elicit justifications from students. Inviting and informing actions were less used regarding all the teacher's actions. With respect to inviting action, it was used to ask students to present their resolutions and reasoning. The informing action was used mainly to rephrase students' statements to make their meaning explicit, thereby producing clearer mathematical language. Rephrasing was used by the teacher to validate students' responses and to focus their attention on mathematical ideas. The informing action was also used to remind students of operation properties already learned, as happened with the property of invariance ("if we want to keep the result, what I do in the additive, I have to do it in the subtractive"). The way the teacher managed the discussion might correspond to her teaching style of showing a great concern with deepening the arithmetic aspects in the discussion.
Concerning the second empirical research question, we infer that the teacher's actions are closely related to students' flexibility in mental arithmetic, especially the more frequent guiding and challenging actions. Given the limitation of a single lesson segment, the study reported here has insufficient data to draw unequivocal conclusions about the teacher's impact on her students' mental flexibility. The teacher's guiding and challenging actions (Ponte et al., 2013) were relevant in leading the students to further development in establishing numerical relationships (Threlfall, 2009). With these actions, the teacher favored the conceptual understanding of her students. As argued by Gravemeijer and Bruin-Muurling (2019), conceptual understanding plays a fundamental role in developing flexibility in arithmetic.
The guiding action is associated with the teacher's clear awareness of the goal of the task. In this vein, she keeps in mind the concern of maintaining the focus of the whole class discussion on achieving new results through the connections between the several cards' expressions. To lead the students to establish these connections, the teacher guides them, making questions centered on the similarities of number expressions and on known facts (as "How much is 100-50?"). In this way, she hoped that her students would be able to relate the known results with the new ones (Gray & Tall, 1994) (for instance, "How can this help to make 100 minus 48? Let's think."). The results of this study illustrate the importance of guiding actions for leading the students to relate numbers and operations, thereby contributing to the development of their flexibility in mental calculation.
This teacher seems to have high expectations towards her students' and uses the challenging action to encourage her students to reason about number relations ("How do you know? [100-52=48]"), asking them to justify what they say ("Why?") and to interpret the relationships established ("What does it mean?"). The teacher is also concerned with maintaining the cognitive demand of the task (Gravemeijer et al., 2016;Stein & Smith, 2009), not giving the answer and making the students' progress in their thinking.
It appears that the challenging action played a crucial role in students' development of flexibility in arithmetic because it is based on a deeper conceptual understanding as a foundation for a flexible calculation. In contrast to what happens when students show mastery of procedural skills without understanding the underlying mathematics (Brocardo, 2014), the second graders' answers reported here show a flexible use of numbers, grounded in building number relations which are established comprehensively. This type of calculation requires numbers to become objects of thought (Gravenjeijer & Bruin-Muurling, 2019), and the networks of number relations are promoted substantially by the teacher's challenging questioning. Moreover, this action owes its effectiveness to its conjunction with other actions, namely inviting students to participate, supporting their ideas, guiding them to the intended goal in each moment, and clarifying what has been discussed. So, in the context of the lesson segment, it seems that the teacher's actions were fundamental in generating meaningful arithmetic knowledge (Baroody & Rosu, 2006;Rathgeb-Schnierer & Green, 2013. The teacher's questioning, in particular, appears, on one hand, to increase the students' flexibility in mental calculation, leading them to move forward in establishing relationships between numbers and operations, and on the other hand, to facilitate their communication of those relationships.
This study suggests several areas of future research. For example, we need to understand much better how students work with mental objects and create networks of number relations. We also need more empirical evidence about how teachers' actions may foster the development of mental flexibility, which would ideally come from a longitudinal study.