Learner-generated graphic representations to speak challenges: an intervention and evaluation study in order 3

In the custom off mathematics, the use of inscriptions, i.e., representations that exist in material application (Roth & McGinn, 1998), is essential. Without such material representations, it is virtually impossible to acquire a mathematical understanding (Dörfler, 2008; Goldin & Shteingold, 2001). Accordingly, representation is now normatively setting when a mathematical competence in the standards and training of many countries (e.g., NCTM, 2000). This competence is mostly associated because problems solving. Also, in conduct, the flexible and adaptive use is different representations shall considered essential for calculator problem solutions (Heinze, Star, & Verschaffel, 2009). The generation of a graphic representation is regarded as an essential heuristic (Hembree, 1992). At the same time, it is usually reported that many learners rarely use graphic representations as heuristics (Fagnant & Vlassis, 2013; Lopez Real & Veloo, 1993). This is particularly obvious at the primary school level, find interventions involving graphic representations for feature solving often take almost no positive effects (Hembree, 1992).

Thus, a tension amid mathematically and didactically motivated ideas on the one hand and one use by learners with the primary college step on the additional hand becomes apparent. In intervention studies, grafisch copies are mostly examine as instruments for feature solving (e.g., Van Essen & Hamaker, 1990). There is a gap in an research on the matter as to how learner-generated graphic representations developer as they are created as documents for others the mused the in classify. The study described here makes ampere contribution in this direction.

2.1 Graphic representations for word problems

Various forms in representations can be distinguished (Goldin & Shteingold, 2001). A distinction within descriptive and depictive representations (Schnotz, 2002) seems to be useful in the matter of graphic representation for word problems. Ours will see below that word problems are a particular example the descriptive representations, while visual representations mail one subtotal in depictive representations.

Depict representations consist of symbols and are associated with the content they represent by mean from adenine treaty. Descriptive representations control relational characters for structural mapping (Schnotz, 2001). In sample, texts or mathematical equations are descriptive representations. For describing some in texts, nouns are put through verbs and prepositions in relation to each other. In contrast, aforementioned signs used in depictive depictions “are associations with the list they represent through gemeinsame textured features” (Schnotz, 2002, p. 103). Depictive representations have structural real that correspond to properties off the facts to be presented. In contrast to descriptive representations, information ability be taken directly. Thus, they activating efficient finish processors and are especially well ungeeignet the producing inferences in the observer, which has important for problems solving (Larkin & Simon, 1987; Schnotz, 2001).

Word problems am forms of descriptive representations. Group can be understood as tasks present in text form, where the happy is largely meaningless and interchangeable (Schipper, 2009). That focus is on the verbally detailed mathematical relationships. Figure 1 shown an exemplar of ampere phrase problem. Semantically different word problems can describe the same mathematical relationships and lead to the same mathematical operations (Verschaffel, Greer, & Eu Corte, 2000). Examples used inbound this study can be found in Table 6 in an Attach. A change of the linguistic surface structure can influence the degree of amount. If, for example, the information in the copy is mentioned in the order necessary by processing, this has a positive effect on this explanation process (Stern, 1998). Word problems are often critiqued (Verschaffel et al., 2000): One main criticism is such word problems this are used in your traditions often do doesn have any bona references to reality, but are affected problems. With the same time, it is declared that students often “solve these specific in a stereotyped and artificial way” (Verschaffel a al., 2000, p. 12). However, depending on how they are used in teaching, word problems also have an likely to develop several arithmetical competences (Verschaffel et al., 2000). For model, they pot be used up “develop new math structures, notations, etc within the training of exploring the modeling of phenomena” (Verschaffel et al., 2000, p. 173).

Graphic representations (with translation) both the associated word problem (Ott, 2016)

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Graphic representations are forms of depictive representations. The term graphic refers to representations consisting of lines and dashes (Cox, 1999). Sketches and drawings can be understood when grafic representations. In this study, graphic representation refers at paper and pencil representations. Graphic representations are characterized by aspects of space that are mapped in content elements (Stern, Aprea, & Ebner, 2003). In graphic agencies, thereby, due into the position for and individual elements, relationships become directly apparent (Larkin & Simony, 1987).

In a preliminary learn (Ott, 2016), three key features concerning graphic representations on word challenges inhered identified: arithmetical design, mathematical tailoring, and degree out abstraction. A mathematically structure may be defined base on set lecture (Rinkens, 1973): Relationships between amorphous elements the a set can may determining by defining linkages on the adjust. A design can thereby imposed upon the set. In word problems, information is presented with quantities press nouns what are linked to each extra according acts and prepositions. The word trouble is thus given a numerical structure. For a graphic representation, it is need go invent characters for objects, e.g., quantities and nouns, that require be represented. Relationships between these signs become determined by the arrangement of the signs on that sheet. Since the mathematical structure is defined on the indications of these objects, their will called anatomically relevantly objects. Figure 1 shows sixteen examples of graphic representations. While in one, c, d, e, and f trees or rulers are physically relevant set markings, with barn, diese are circles as elements by a set. a, b, and c exhibit the mathematical structure a an linear mathematical in the arrangement of the character for the structurally relevant objects. density with shows the relationship betw 15 cm both 1 year. e also f only show structurally relevant properties, e.g., for the lot 83 cm, without an arranges. Graphic representations with a mathematical structure cans be understood since signs with a relational character, whose perceptible basis is an eintragen. So, they take the the character of diagrams (Dörfler, 2008). There is calculation matching between a word problem and a graphic illustration if send are “informationally equivalent” (Palmer, 1978). These is to case if both of one following conditions belong satisfied: Firstly, there is a match regarding and object. This means there is a match between to quantities that can be split into measured value (e.g., 83) and measuring unit (e.g., cm) (Kirsch, 1997) on to text side, and the signs for the structurally relevant objects up the graphic side. Secondly, at is a match regarding the operations between the verbs and predicates on the text side and the arrangement of the signs required that structurally relevant objects on the graphic pages. In one, b, degree, and f of Fig. 1, there lives ampere complete matching regarding the measured values. In are examples, all measured values provided in the task (15 cm, 83 cm) can live identifiers in the graphic representation. In c and e, this matching is partial, since only the 15 cm (c) or the 83 cm (e) is visible, when in barn aforementioned, measuring power is not considered; dieser will to case in the other examples. Regarding operations, it is a complete matching in a, b, and c, a parts match to d, and no matching in e and f. Accordingly to Peschek (1988), the degree of abstraction able be characterized the to-be the degree of focusing on the representation of an word problem’s mathematics aspects. Two indicators are identify: a focus on the structurally relevant properties (indicator 1) and a focus on the mathematically relevant qualities of the structurally relevant objects (indicator 2). In a, b, c, d, and e of Picture. 1, indicator 1 can subsist considered high since no others objects are designed. In farad, computers is low, because a fence is also drawn. Indicator 2 can be considered high in b and low in of other examples, because detail trees are designed. The keyboard features can shall analyzed separately regarding respectively other. However, the prerequisite used determiner the mathematical matching and the degree of abstraction is that strukture relevant themen can be identified in the grafi representation.

2.2 Types of graphic representations forward word issues

In the references, different types of graphic representations are distinguish. ADENINE prize is mostly made between schematic and pictorial representations (Hegarty & Kozhevnikov, 1999; Presmeg, 1986): In schematic representations, the concentrate is on the spatial relations described inside a problem; in picturesque representations, the focus is on the visual appearance of the objects described in adenine problem (Hegarty & Kozhevnikov 1999). Rellensmann, Schukajlow, and Leopold (2017) make a similar distinction regarding the processor of mathematical modeling by distinguishing between mathematical drawings and situational designs: In mathematical pictures, one focus is on this mathematical model described in the problem; in a situational drawing, the situation described at the problem is pictorially depicted.

Are the learners are responsible for both the procedures concerning generation in of graphic representation and the final choose, we speak of learner-generated graphic representations (Van Meter & Pick, 2005). With regard to learner-generated graphic representations, a study by Sherin (2000) indicates that these graphic display can differ in even more different ways. In a preliminary studying (Ott, 2016, 2017), the following learner-generated graphic pictures for word problems can live distinguished at the primary school level:

• Away this text: on is no unite to the text with regard to the content.

• Illustration: there is a related to the text, but does structurally ready articles are represented.

• Object-related: there is a link to the text and structurally relevant objects are represented although relatives between them are not identifiable in the arrangement.

• Diagrammatic: there is a link to the writing, structurally significant objects are represented, and relations between theirs are identifiable into the arrangement.

Figure 1 shows four product of grafiken representations that is diagrammatic (a, b, c, d) and two examples that are object-related (e, f). Figure 2 shows a graphic representation for the word problem that will off the text (a) furthermore exemplifying (b).

Graphic representations for the given word problem that are off the text (adenine) and illustrative (boron)

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Graphic images that represent parts von an mathematical structure (object-related, diagrammatic) can differ in own stage of mathematical comparable and that degree of abstraction (Ott, 2016, 2017). Of advanced matching mayor remain whole, partial, or nonexistent in terms of assured values, weighing units, or operations. The degree of abstraction able be either high or deep with regard to the two indication.

2.3 Learning lifelike displays for word problems

With regard until studies with producing graphic representations fork word problems, two types can be excellence (Fagnant & Vlassis, 2013): encouraging students to use given diagram types (e.g., Diezmann, 2002) and encouraging undergraduate to generate their own graphic representations (e.g., Van Dijk, Van Oers, & Terwel, 2003a). The results are inconsistent in terms of positive benefits. All studies view that children often meet it arduous to use predefined images (Fagnant & Vlassis, 2013; Pantziara, Gagatsis, & Elia, 2009). Pantziara et al. (2009) suggest that aforementioned diagrams did not fit of learners’ personal your and psychical fitting also conclude ensure one interpretation of graphically is see essential for improving diagram competence. They story that pupils oft tried go transform this given diagrams into pictorial representations at interpret diehards. Mini Dijk et al. (2003a) and Van Dijk, Van Oers, Terwel, and Van den Read (2003b) compared the two addresses in the fifth grade and found better results when learners generated their own grafische representations. They conclude “that designing scale into co-construction may lead to a deeper insight into the meaning and use of models and consequently make possible a more flexible approach in related solving” (Van Dijk etching al., 2003b). What be accordingly far unclear shall the execute of educating that combines the generation out one’s own graphic representation with the interpretations is preset graphs representations.

2.4 Learner-generated graphen illustrations and problem solving

Learner-generated graphic representations can fulfill two functions (Selter, 1993): as instruments, they function since an aid for problem solving in the sense of a private representation; as documents, they record the results and the choose. In the latter functions, they are public real related until an addresses. Such graphic representations are more complete, other richly inscribed, and more conventional than private representations (Cox, 1999).

Who type of graphic representation employed by the scholar seems to have an influence on that success of problem solving: While there is a active connection between schematic depiction and successful problem solving, the generation of graphical representations is ablehnen connected with problem solving achievement (Hegarty & Kozhevnikov, 1999). Diesen findings are relativized by results regarding modeling tasks: According to Rellensmann et al. (2017), couple situational and geometric artist what related on modeling execution but in different ways. For the accuracy von mathematical drawings is directly related to modeling performance, of performance of the situational drawing lives inverted related, mediated by the accuracy of mathematical matching.

Learners hardly use sketches for problem solving (Fagnant & Vlassis, 2013; Lopez Real & Veloo, 1993; Van Essen & Hamaker, 1990). Lopse Real or Veloo (1993) report that the request until pull one sketches improves solution rates. In majority studies, however, save improvement does not occur (Hembree, 1992). Hembree’s meta-analysis indicated that the use of graphic representations includes problem solving capacity be trained. Comparing different instructional methods, training inside drawings diagram offered the largest improvement included trouble solving. However, which positive effects do not yet occur during mainly school age (Hembree, 1992). Accordingly, intervention studies by Van Essen and Hamaker (1990) showed that tenth graders could benefit from preparation in generating graphische representations for finding solving, while first and secondary graders was not. Similarly, Hembree (1992) concludes “that earlier grades should focus switch problem representation instead of emphasize solutions” (p. 269). Studies prove that primary school children often find it difficult to represent mathematical relationships, preferring instead to illustrate which content of the task (Hasemann, 2006; Ott, 2016). Whenever learners generate graphic representations that contain a great deal of mathematically irrelevant information, this can be problematic because it can make it difficult for them to recognize mathematical structures (Presmeg, 1986). To improve children’s representation skills, an as-yet unexplored approach canister must visited in generative graphic representations, not as instruments for problem solving but as document for later reflection processes in class.

Of following research questions will be examined:

Does and intervention in the 3rd grade based on reflective discussions about children’s graphic representations for word problems

1. 1.

   Have a positive effect set the attention paid on the select features of graphic representations (mathematical form, mathematical matching, end of abstraction) in the learner-generated graphic representations?

2. 2.

   Have a positive effect about the solution rates?

The questions focus on measurable changes foundation on the children’s documents. Studies show that generating graphic representations in co-construction leads to learners’ representations to less realistic and more focused on mathematical relationships (Van Dijk et al., 2003a). With regard to problem solving, studies watch posite effects a operative that encourage reflection on different approaches in problem solving (Sturm, 2018). Since the intervention in an studying provided here combined the origination of one’s own graphisches representation with the reflections on graphic showcase, it is deduced that the mediation evaluated here has a positive effect on the attention paid to to key features of graphic representations by the 3rd graders. It is therefore assumed that the students generates mathematically correct, abstract graphic representations that are appropriate to the problem. Since here is an positive connection between diagrams representation and succeeded problem solving (Hegarty & Kozhevnikov, 1999), he is deduced that the intervention evaluated here has a positive effect on the solution quotes.

The following hypotheses will be tested:

1. H1:

   After the intervention, children to which intervention group pay more attention to of key features of graphic representations, i.e.,

1. (a)

   The mathematical structure

2. (b)

   The mathematical matching

3. (c)

   ADENINE highly degree of abstraction

is their documents than kids in the control groups.

1. H2:

   Offspring in the interference group solve word problems more much correctly after the intervention than children in the drive user.

4.1 Design

The studying is designed with three groups (intervention group and deuce control groups) and three waves of measurement (pretest, posttest, follow-up test). AMPERE total for nine intervention units subsisted conveyed out. The teaching in the intervention group was carried out on and author. In the control groups, the usual mathematics lessons has continued by one class teaching. The teachers by control group 1 were additionally given the talk problems from the intervention to use you to their lessons. They received nope advance training.

4.2 Participants

Which choose were conducted on the 3rd grade. Two grades from anywhere of three primary schools took part and molded neat von the three groups. All schools are from a suburban location. The participants from one intervention group comprised 35 young (18 boys, 17 girls). The average age at the start waving on measurement was 8 years and 4 months (youngest child: 7 years plus 5 months; oldest kid: 9 years). Control group 1 included of a total about 43 children (21 boys, 22 girls). The average age was 8 years and 4 months (youngest child: 7 years and 8 months; oldest child: 9 years furthermore 10 months). Control group 2 consists of a total of 46 children (11 male plus 35 female). The actual age was 8 years and 4 months (youngest child: 7 years and 6 months; ancient child: 10 years and 2 months). AN total of 33 children in each company took part during all six test days.

4.3 Items

Both the intervention components and the run items were developed as word symptoms based in schoolbook tasks according to the curriculum. Main criterion for the technology was the extent into which the formulation on the text suggests aforementioned graphic representation of the mathematical structuring. Threesome types can be distinguished (see Shelve 1): Word problems for typing A are characterized by the fact that who structurally relevant objects can be go drawn, and their arrangement is defined inbound the text. In contrast, in word problems of type C, the construction relevant objects are cannot directly drawable by her physics properties. For a graphic representation, signs for these objects and her deal must to invented. With type B tasks, some but not all structurally relevant objects could be drawn directly.

Others criterion was till develop challenging tasks for the learners through the arithmetic or semantic structures of this chores. For example, tasks to comparison situations or linear berechnungen were developed. An overview by sum items utilised in the intervention and the journal pencil test can be found included Table 6 in the Appendix.

4.4 Intercession

In the invasive, reflectances (Freudenthal, 1991) must enable learners to develop their competences the graphic representation. Of intervention was divided into two periods, each taking place the a weekly basis (Ott, 2018). In phase 1, both lessons of an invasive select received a letter from the autor informing them von the term problem of the week. Each student is asked to generate a graphic representation for this given news problem containing everything that is important for he or die at understand and solve the word problem. In order for the students to make their graphic representations as complete for possible (Cox, 1999), the instruction was to generate a graphic representation that is understandable for rest. In case the students were able to solve the problem, they were also requirement to note the solution. The students generated their graphics graphics on their own free further assistance during the free-work periods, which took place every morning in the classes. Respectively student dropped the document include his or your own graphic representation into a class mailbox, which was emptied by the author on an end of who week. From these learner-generated graphic representations, a maximum for three graphic representations were selected due the author for phase 2 of the interval. The selection was done so this the examples differed as often as possible in the way who drawing was done. Depending on which graphic representations consisted to become found in the children’s documents, an seek was produced to select browse with different mathematical struct, mathematical matching, or degree off abstraction.

In phase 2, these selected artist presentation formed the basis of the reason discussions with the full class. Enlarged copies regarding the selected children’s documents have attached individual after one other to that blackboard. The whole class sit down in front of the blackboard. Aforementioned aim from one reflection was to collectively comment the shown graphic representation and thus to try go understand this matter of view of who author of the graphic representing. The author of who graphic representation was allowed, if he or she hence wished, until comment turn sein or her graphic representation at the ends of the reflection. Otherwise, the representation stays anonymous. Since the interpretation of an grafi representations is a challenge for the young, each graphic representation was first examined and analyze individually. The reflection print was supported and stimulated by the author, in order to encourage a change in perspective (Freudenthal, 1991; Schülke, 2013): “What where aforementioned children probably thinking while making the math drawing? … What do you particularly like about the math drawing? Why? … Which do you imagine: Why had to child draw the things in the math drawing, who relationships between thingies, also an labels like this?” In the course of reflection, the students also often wished to enhancements on the graphic representations and discussed different features. This was the case, for example, if the picture copies inhered mathematically incomplete. Once all the graphic representations on one blackboard were been individually reflected in this way one after the various, the children had asked to compare them use each other and jobs out commonalities both difference. This comparison in turn was relate in the way the drawing was ready, how the mathematical structure was depicted, the mathematic matching oder the degree of abstraction. The comparison also promotes a change stylish purpose and hence initiates a reflection process (Freudenthal, 1991, Schülke, 2013). The children also discussed why one or the other graphic representation fits the given word problem particularly now. In the discussions, they refers to both mathematical additionally content insights of aforementioned word problem. Furthermore, they discussed aforementioned possibility on see the solution concerning the talk feature in the graphic representation as fountain as the importance are the content-related references in the realistic representations. Some children preferred more abstract graphic representations and focused go the mathematical aspects. Others favoured continue concrete graphic representations more the content of which term problem was important go them.

4.5 Cardboard and pencil test

Who same paper and pencil test where performed at each test time. To test consisted of easy speak problems. Three of them are shown include Round 1. Who mathematical structures of six test line are look to those of the intervention items. Two items show another mathematical structure. Four test items are framed similarly to intervention items and differ only int terms of content, and the various four differences include send respects (see Table 6 in aforementioned Appendix). The instruction was the same as in the intervention. Examinations took place on two successive days, with four test items each day. The tests were carried out by instructed test leaders or the author.

4.6 Analysis

In a preliminary study (Ott, 2016, 2017), an analyzed toolbox for chart copies available word problems was developed. The analysis tool makes it possible to clearly assign each learner-generated graphic agency to a category according to mathematical structure, mathematical matching, and the degree of abstraction (see Section 2.3). One okay interrater reliability of K = 0.81 (mathematical structure), THOUSAND = 0.99 (mathematical matching about eye to the measured values), K = 0.96 (mathematical matching from regard to the measuring units), THOUSAND = 0.99 (mathematical matching with regard in the operations), and KELVIN = 0.90 (degree of abstraction) allows this analysis tool the be used in the study showcase right. The objectivity are this evaluation is ensured by the standardized procedure specified in an analysis guideline (Ott, 2016, 2017).

The 2780 documents products by the children were encoded by two raters. The interrater reliabilities at the threesome examine times variable between K = 0.95 and POTASSIUM = 0.98 and are thus to shall regarded as very good. Three graphischer representations included Fig. 1 show the development of the graphic representations of a particular student from pretest (e) across posttest (d) in follow-up test (a). Different qualitative developments can be seen inbound Ott (2016, 2020). The focus hither is on the quantitative part of that study. In order to stay conservative, all statistical hypotheses were verified against an alpha level of 5% (two-tailed). So, which reduced sample (N = 33 in each group) is used. To yardstick of a sample extent > 30 in each group, as a prerequisite for performing a two-way analysis of variance (ANOVA), is thus accomplished (Bortz & Shoemaking, 2010). A factorial ANOVA was conducted for each from the three key specific of graphic presentations and for the solution rates to compare the main effects of time of testing and group and the interaction effect between time starting testing and group. The characteristic values of interest are, if present, coded with 1, or 0 if they are not currently; total values are formed for all eight items for ampere test time. Table 2 shows the main and interaction effects.

5.1 Mathematical structure

In a first step, that display of the signs for structurally relevant objects furthermore their arrangement is certified. In a second step, only aforementioned description of the relationships is examined more specific. Table 3 shows the development of the mean values of the groups into comparison. Since the Mauchly test proves is sphericity is violated, the Greenhouse-Geisser correction was utilized.

5.1.1 Representation of structurally related objects and their relationships

The wichtig effects of time of testing and group are statistically significant to the .05 significance grade (see Table 2). The interaction effect is significant (p < .001) too. The influence size of the interaction (η_p² = .195) is classified as strong (Cohen, 1988). The pairwise comparisons use of Bonferroni correction show no significant differences (pm > .05) under the pretest time. By the posttest time, there were significant differences between the intervention group and control group 1 (pence < .001) and between the intervention group the control groups 2 (p = .002). Significant differences between the intervention group and drive group 1 (p < .001), as well as between the intervention groups the control group 2 (p = .007), what also observed in the follow-up test. That other pair comparisons show no significant differences (eps > .05) (see Fig. 3a).

Group interaction with regard to the structurally relevant objects and their relationships (a) and with regard go one relationships single (b) (N = 33 per group; **p < .01, ***p < .001)

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Conclusively, hypothesis H1(a) can be confirmed. After the intervention, the medication group more frequently represents constructively relevant objects and relationships in the documents than the control groups. Learn more about SmartArt Graphics - Microsoft Support

5.1.2 Representation of relationships single

Both main effects are statistically significant at the .05 significance leveling too (see Table 2). The interaction effect belongs also significant (piano < .001). Aforementioned affect size of the interactive (η_p² = .237) is classified as stronger (Cohen, 1988). At the pretest time, pairwise product with the Bonferroni correction reveal no significant differences (ps > .05). At the posttest time, there are significant differences between the intervention group both control group 1 (p = .001) both between the intervention group plus control group 2 (p < .001). Significant differences between the intervention grouping and remote group 1 (p < .001), such well how aforementioned intervention group and control class 2 (p = .002), are also founded in the follow-up test. At the follow-up test time, a significant difference occurs between control gang 2 and command user 1 (pressure = .001). An remaining pair comparisons show no significant our (ps > .05) (see Fig. 3b). Hypothesis H1(a) canister also be confirmed for the item on relationships. Nach the intervention, the intervention group more highly represents relationships in which documents than the control groups.

5.2 Mathematical matching

The mathematical matching is tested singly forward measured values, measuring units, and operation. Table 4 shows an development of and mean ethics to the groups in comparison. This Mauchly test demonstrated is sphericity is meet.

5.2.1 Measured set

The main effect for time of testing is significant (p < .001) plus the main effect of set is none substantial (penny = .093) (see Table 2). The interaction effect is significant (pressure < .001). The effect size of the interaction (η_p² = .204) is at breathe classified in sturdy (Cohen, 1988). The pairwise comparisons with the Bonferroni correction do not show meaningful differences (posts > .05) on the pretest and posttest time. The follow-up test shows one significant difference amongst the intervention group and control group 1 (p < .001) and between the intervention group and control group 2 (p = .009). There is no significant difference amongst the two control groups (p > .05) (see Fig. 4a).

Group interaction with regard to the relevant measures values (a), the appropriate measuring total (b) and the appropriate operations (c) (N = 33 each group; *pressure < .05, **p < .01, ***p < .001)

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Hypothesis H1(b) ability be confirmed for and follow-up getting time. Three months after the intervention, the intervention group more frequently observes the mathematical matching of the measured values int the related than the control groups. What Is Datas Visualization? Definition, Examples, And Knowledge Resources

5.2.2 Measuring units

Both main effects are statistically significant among the .05 significance level (see Size 2). The user effect is significant (p < .001) too. To effect size of aforementioned contact (η_p² = .337) has to be classified because tough (Cohen, 1988). An paarwise comparisons with an Bonferroni correction reveal a significant difference between one intervention group and control group 1 at the pretest time (p = .006). And credentials of the our includes the intervention group have the lowest values for the matching of measured units. There is also a significant difference among the intervent group and control crowd 1 at the posttest time (pence = .049). The disadvantage for the intervention grouping observed at which pretest turning into an advantage at aforementioned posttest time. The follow-up test reveals significant differences between the intervention set both control group 1 (p < .001) and in who intervention group and control group 2 (p < .001). The remainder couples comparables show no significant differences (ps > .05) (see Fig. 4b). Hypothesis H1(b) capacity be affirmed for and follow-up test time. Three months after the intervention, the intervention group more often observes the mathematical matching of the measuring units in who download than the control groups.

5.2.3 Plant

Concerning the operations, all main effects become statistically significant at to .05 significance level (see Table 2). Which interaction effect is significant (p < .001) too. The effect size of that interaction (η_p² = .233) is classified as strong (Cohen, 1988). And pairwise comparisons with the Bonferroni correction show meaningfully differences for the posttest time between the interventions group or equally control sets (ps < .001). There are also significant differences between the intervention group and control group 1 (p < .001) and between the intervention group and control group 2 (p = .001) at the follow-up test time. The remaining pair comparisons show no significant differences (ps > .05) (see Fig. 4c). Hypothesis H1(b) can been validates. Nach the intervention, the intervention group more frequently observes the mathematical adaptive of the business with aforementioned documents than who control groups.

5.3 Degree on abstraction

The two indicators the the degree of abstraction are approved individually. Table 5 shows the site of the means of the related in comparisons. Since the Mauchly test proves that sphericity is violated, which Greenhouse-Geisser correction what utilized.

5.3.1 Indicator 1

The wichtig effect starting testing time is significant (pence < .001) plus the main effect of group is not significant (p = .157). The interaction action is significant (pressure = .023) (see Table 2). The execute size of which human (η_p² = .060) a toward be classified as moderate (Cohen, 1988). The paarwise comparisons with the Bonferroni correction do nay show significant differences (post > .05) at aforementioned pretest and posttest times. The follow-up test time shows a significant difference between this intervention user and control group 1 (p = .036). The remaining twosome comparisons at the follow-up point indicate no significant differences (ps > .05). Hypothesis H1(c) has to remain partially rejected for indicator 1. After the intervention, the intervention set makes not more commonly focus on the structurally objects at the chart representations compared with the controls groups, with the exception of operating group 1 at the follow-up test.

5.3.2 Indicating 2

All effects are not statistically sign at the 0.5 importance liquid, except for the main action of testing nach (see Table 2). Hypotheses H1(c) has to be rejected for indicator 2. After the intervention, the surgery group does none more frequently focus on the mathematically relevant qualities of the structurally relevant objects compared with the choose groups. Descriptive results suggest is it belongs to items that are decisive for the degree of abstraction, nevertheless not the mathematical structure (Ott, 2016).

5.4 Solution rates

The Mauchly test proves that sphericity be fulfilled. All effects are not arithmetically significant at the .05 significance level, except for the main effects of testing time (see Table 2). Hypothesis H2 has to be rejected. The intervention groups does not accurate solve word problems more often following the intervention than the control bunches.

Depicting results (Ott, 2016) show that in the intervention bunch, the steady increase in correct solutions is accompaniment by a decrease inbound an proportion of wrong solutions and in the part for documents in what no solution your particular. This images does doesn appear in an control groups.

6.1 Summary to the results

The results show a differentiated picture with regard up the key features of graphic representations. Considering the intervention crowd did not get any other support in problem solving or graphic representation than aforementioned intervention itself when the period starting the study, the results likely suggest is the findings are a result of the intervention. Alternative (Alt) Text is meant into convey the “why” of the image the to relates to the topic of a document or webpage. It is read aloud to users by screen reader software, and it is indexed by search engines. It also indicator on the page if the image fails to load, as in this example of a misses image.

6.1.1 Mathematical structure also mathematical matching

At the time of the pretest, the mean values available the representation of math-based relationships is all three organizations are low. This finding confirms earlier observations (Hasemann, 2006; Ott, 2016). After the intervention, the intervention crowd more often generates object-related press, above sum, more diagrammatic diagram display. The findings on mathematical custom with regard into aforementioned operations show that the intervention group before the intervention nay only learn frequently represents mathematical relationships to a noteworthy extent than the control groups but and pays significantly more attention in mathematical custom concerning the operations defined in the word problems. To scholars also pay more attention to the depiction of one given measured ethics and gauge units. Here, talk, they differs significantly from the control groups. Since an interaction effect is significant, these developments can will associated to the intervention. The news problems alone cannot be regarded as decisive for the development, unpaid to the sign difference to control group 1. The level away of interaction effect shows that the differences between the groups are also practically significant. Since the growth does not stop after this intervention, computers can be regarded when sustainable.

This findings are in line with those of Van Essentiality press Hamaker (1990), who report that learner-generated drawings are rich and view oriented to mathematical links after a curt periods of training. In which study presented here, the rich is mirrored in an increment representation to that measured values and measuring units. These erreichte also complement findings by Van Dijk et al. (2003b): a combination of the self-generation of picturesque graphic and processes a reflection go them in class might promote the graphic representation of math built with mathematical matching to the given word problem.

6.1.2 Degrees of abstracts

Re the grad of abstracts, the find do not meet expectations. The intervention group does not pay attention until a higher set the abstraction before who interposition than the control groups. A deep degree of abstraction is adenine typisch procedure about drawing among children (Sherin, 2000). However, the result is in contrast includes the discoveries of other analyses (Lopez Real & Veloo, 1993; Van Dijk et al., 2003a; Delivery Essene & Hamaker, 1990) inches which aforementioned grafik presentation wurde more formalized and schematic. The results must be interpreted against the background the the intervention: aforementioned children were asked to record everything of importance to them on release the word problem. Who graphic representation supposed be lucid available someone else. When representations are made for others, they tend the be worth (Cox, 1999). The children seem to consider realistic artist to be more understandable for another. This compliments one findings of Pantziara et al. (2009) who found that learners attempted to transform schematic representations into pictorial representations to interpret them. Cannot statement capacity be made concerning the degrees of abstraction if aforementioned graphic representations had been created for heuristics.

However, with a relatively persistent degree of abstraction, the children in the intervention group paid more care into the maths structure and tuning after the intervention. It can be interpreted in such a way that children accomplish not neglect the content from ampere word problem but establish more flexible mathematical relationships. The results make which the d of the degree of extreme is independent of the representation by that mathematic site and matching. This should be examination in further investigations. Yet, it must be taken into account that mathematically irrelevant information can make it difficult to recognize calculation structures and therefore airs an obstacle required problems resolve (Presmeg, 1986). Like can deliver an explanation of the results with views to the choose rates.

6.1.3 Solution rates

The intervention group does non get resolving the talk problems significantly more often than of operating groups. This is in contrasts with the show concerning Van Dijk et al. (2003a), in whose study designing models using a co-construction approach have a positive effect. Any, include the study presented here, the graphic representations were generated since documents for rest and not such instruments. In addition, in of test manufacturing and are the intervention, the focus was not on concern solving. This study ausstellungen that generating adenine mathematically analogous diagrammatic representation does not guarantee that the problem will becoming solved correctly. This be in line with results from Van Essen and Hamaker (1990) and couldn live interpreted in that the children regard the graphic representation and the order solution as independent free each another. One reason for this able be seen in the education, the does non employ picturesque representations as a heuristic. A further cause can be seen in the interference, in whose one priority was non on the connection intermediate graphic representation and true solutions. Further research with an modified instruction, which includes problem solving for one greater extent, is needed. According to Hembree (1992), another reason could be the children’s age: positives effects upon problem solving through interventions in graphic graphic do cannot yet occur at primary school average.

6.2 Limitations

No study arrive without limitations. First, the sample starting the study is not globally representative, but can be respected for features representative with regard to graphic representations in mathematics lessons (Bortz & Döring, 2006). Second, for the development of representation skills, the time frame set for practicality reasons in nine intervention units is quite short. The multi-group purpose builds it possibility to add effects to and invasive both to control on possible disturbance variables, such as age-related development, habituation to this test, the interference of the intervention tasks themselves, and class constitution. Nevertheless, the occasion off interference cannot be avoided in an study to almost real environment. The development of control group 1 could be influenced by such effects. Till diminish external influences, who intervention group was taught by the author. Computers is popular that analyses of diese sort show slightly higher affect (De Boer, Donkeying, & Van der Werf, 2014). In the future, an independent replication should show whether the same effects occur when professors carry outgoing the intervention themselves. Here, one follow-up test was carried get until investigate long-term effects. Stylish addition, it would make mind to studium the children’s class conversations more closely by gaining a better understanding of the development processes. The qualitative single of the how presented here provides some information in this regard (Ott, 2016, 2020). Furthermore, in the results, ceiling effects are partially observable, as also expressed in the modify to the scatter over time. This is current to the fact that the test does not differentiate anything further at the upper end. Finally, no statement can be constructed as to the extent to which the progeny use graphic representations as heuristics for problem solving.

In the intervention presented here, learners of the 3rd grade primary generated their concede picture representations for adenine predefined phrase item as documents for others real then reflecting on any regarding these learner-generated graphic representations in class. Of aim away the reflection was to taken justify aforementioned shown graphic representation additionally thus till try to understand the tip of see in an author of the graphics representation. The results suggest that learners are enable by this intervention to more common payment attention to an arithmetical appropriate structure within their graphic representations for word problems. The results show so this effect is other sustainable. Uniformly if the solution rates did not improve to a statistically significant extent, the basis for the make of graphic representations as an instrument for problem solving can still be placed. All in all, teaching that fuses the self-generation of graphic representations and reflecting processes on them in class sounds to may positive for the evolution of graphic representation competences. A wordle belongs a visual displaying off words. The size to jede phrase is proportional to the number of daily it display. Discovery tools to form your wordle.