On Reading Mathematical Texts, Question-Asking and Cognitive Load

Abstract

In this study, we examined aspects relating to the impact of integrating question-asking activities and providing answers to these questions while reading historical mathematical texts on prospective mathematics teachers’ self-reported cognitive load. The research design of the study was quasi-experimental. The study participants included two groups of 20 students each (experimental and control). The experimental group was instructed to ask questions while coping with the texts, whereas the control group received no special instructions. The experimental group participants were asked to think aloud while coping with the texts and audio record themselves. These records were transcribed into written protocols. Both groups had to respond to a self-esteem index questionnaire in which they had to report the level of difficulty they experienced during their attempts to cope with the texts, as an indicator of their sense of cognitive load. This process was repeated at three time points, relating to three different texts. The data were analyzed using quantitative and qualitative methods. Two main observations were obtained: (1) A significant difference was found between the control and the experimental groups regarding the decrease in cognitive load along the time points. Because the only difference between the control and experimental groups was the activity of question-asking, it might be concluded that question-asking affects the reduction in cognitive load. (2) Question-asking supports the assimilation of new information up to a specific limit, depending on the gap between existing knowledge and new information.

1. Introduction

Reading and comprehending texts are among the main modes of individual learning in various settings [1]. Specifically in the case of mathematics, reading mathematical texts is an essential part of mathematics learning and the acquisition of mathematical knowledge because it includes the ability to interpret and understand these texts [2,3,4]. Nonetheless, the common high school mathematics textbooks in our country are mainly used as a resource for exercises and problems [5]. Therefore, how students understand and make sense of the textual content has an impact on their ability to solve these mathematical tasks. This also has significant implications for the multiple difficulties prospective mathematics teachers (PMTs) encounter when they need to read mathematical texts and understand and apply their content [6]. One of the courses we teach to middle and high school PMTs concerns the history of mathematics. The course is based on reading mathematical texts, including ancient solution methods and algorithms and their use for solving problems. Realizing the PMTs’ difficulties in reading mathematical texts, we looked for ways that would facilitate their coping with the texts.

While managing texts, readers apply various strategies, including encoding information and building a mental representation of the text content, making inferences for connecting different parts of the text, and employing prior knowledge to make sense of the textual information. In general, three factors affect the reading and comprehending of a text: text properties (e.g., content, structure), instructional context (e.g., the reading goals—educational purposes vs. entertainment), and reader characteristics (e.g., working memory—the personal capacity to manipulate information in memory) [1]. In the case of the specific course discussed in this paper, we intentionally did not change the text properties and the instructional context, trying to adhere to the authenticity of the historical texts. Therefore, we focused on finding ways to reduce the load of the working memory, as overloaded working memory impedes reading and learning (e.g., [7]), or, more generally, to decrease the cognitive load (CL) experienced by PMTs while managing with the texts. CL refers to the state of memory storage and processing of information while coping with a particular task [8,9] and is considered an indicator of the degree of difficulty learners experience while engaging in a task [10]. Recognizing that instructing students to ask questions related to a mathematical text they read and providing answers to these questions facilitates understanding complex texts [11], advances knowledge acquisition and mathematical thinking, and leads to the emergence of new mathematical insights [12], in the present study, we aim to explore the effect of question-asking while reading mathematical texts on a decrease in the CL experienced by PMTs. Given that CL can be measured directly using learner self-reports [13,14], changes in CL were assessed based on the PMTs’ self-rating of CL.

2. Theoretical Background

The concept of ‘cognitive load’ was coined by Sweller [8] to describe the state of memory storage and processing of information in a human’s mind while coping with a particular task or complex situation, and it can serve as an indicator of the degree of difficulty learners experience while engaging in a task [10,13,14]. Researchers distinguished between intrinsic, extraneous, and germane CL [15,16]. The intrinsic cognitive load (ICL) refers to the mental effort one invests in dealing with a task, which varies from person to person according to expertise in the field related to the task. The extraneous cognitive load (ECL) refers to how the task is formulated or presented, and thus can be controlled by modifying the task formulation. The germane cognitive load (GCL) refers to schemas’ processing, construction, and automation. Namely, the effort needed to use memory to transform information into schemas.

Sweller [17] found that learners’ self-report regarding the level of mental effort they have put into performing a particular task can serve as a reliable source of information. Namely, CL can be measured, among others, by employing learner self-reports using a self-esteem index measurement scale [13]. As indicated by Paas et al. [13], such measurement scales are relatively sensitive to small changes in CL and are valid, reliable, and non-invasive.

Relating to the reading of texts and CL, research shows that high working memory readers are more successful in adjusting their processing to their general reading goals [1], while overloaded working memory impedes reading and learning [7]. As mentioned in the Introduction, this led us to search for appropriate ways to reduce the CL of our PMTs while reading the historical mathematics text. To that end, we sought to examine the effect of question-asking (QA) by the PMTs while reading the mathematical texts and then answer these questions on the reduction in CL, as reported by the PMTs.

Asking mathematical questions is acknowledged as fostering mathematical thinking and knowledge acquisition and serves as a scaffolding tool. By asking mathematical questions and answering them, students are likely to think as mathematicians and develop mathematical insights [12]. This stands in contrast to the fact that, in school, most mathematical problems are presented by the teachers or the textbooks. When the learners are the ones who ask the questions, their mathematics knowledge is enhanced because asking a valid question necessitates exploring the relations embedded in the particular situation [6]. In addition, learning to ask questions also supports students’ ability to identify their difficulties and formulate appropriate questions for their instructor [12].

3. The Study

3.1. The Research Goal and Derived Research Questions

The present study sought to examine the effect of QA by PMTs while reading historic mathematical texts on reducing their CL, as self-reported by the PMTs.

The derived research questions were as follows:

• How do PMTs perceive the effect of question-asking while reading historic mathematical texts on their cognitive load?
• What are the differences between high- and low-achiever PMTs’ perceptions regarding the effect of question-asking while reading historic mathematical texts on their cognitive load?

The hypotheses we wished to confirm (H₁) were as follows: (i) repetitive use of QA while coping with historic mathematical texts reduces CL over time; (ii) high achievers perceive the decrease in cognitive load as greater than that perceived by low achievers.

3.2. The Study Participants

Forty female PMTs of an average age of 24 years and seven months participated in the study. The participants were in the third year of their undergraduate studies (out of four), qualifying to be middle and high school mathematics teachers. The research took place within an annual mandatory course dealing with selected topics from the history of mathematics until the beginning of the Christian era. The participants were assigned into two classes of 20 PMTs each (experimental and control groups) in accordance with the matching control technique [18]. This technique was applied first by matching pairs of students based on their average grades in the compulsory mathematics courses they attended during the first two years of their training and then assigned into two matching groups (control and experimental). The grade constituted the dependent variable. The rationale behind this grouping was that prior knowledge affects the decrease in CL while performing learning tasks [19]. Furthermore, we wondered whether QA has a different effect on self-reported CL among low- and high-achieving PMTs. Therefore, both PMTs groups (control and experimental) were divided into two sub-groups: high and low achievers. PMTs with an average grade of 80 or higher were assigned to the sub-group of “high achievers” (HA) and those with an average grade of 60–80 were assigned to the sub-group of “low achievers” (LA). It should be noted that, to pass a mathematics course, PMTs are required to receive a grade of 60 or higher. The sub-group of HA in both groups (control and experimental) consisted of 6 PMTs and the sub-group of LA consisted of 14 PMTs.

Both groups were engaged in the learning environment described in the following section, and the only difference between them was the fact that the experimental group was instructed to ask questions related to the text while reading the texts and address these questions. The same instructor taught both groups. It should be noted that the study participants had no prior experience in QA while reading mathematical texts.

3.3. The Learning Environment

The term “learning environment” refers to the diverse physical locations and contexts and how instructors manage educational settings to facilitate learning (The Glossary of Education Reform, https://www.edglossary.org/learning-environment accessed on 6 May 2023). In this paper, the context of the learning environment is the course contents, and the educational settings to facilitate learning are expressed in the format and the pedagogy of the learning and teaching:

The contents. The two-semester course deals with selected topics from the history of mathematics until the beginning of the Christian era and is divided into eleven chapters: Egyptian mathematics (three chapters), Babylonian mathematics (two chapters), prominent Greek mathematicians (three chapters), and Vedic arithmetic (two chapters). The readings include the background of the discussed period, mathematical texts that incorporate ancient solution methods and algorithms, and their application for solving problems. All solution methods and algorithms are a verbatim translation of the English version of the original texts into Hebrew, using the currently accepted mathematical language and notations.

The format and pedagogy. A new chapter was uploaded to the course website on the Moodle platform every two to four weeks, and the PMTs could read the contents of the chapter at any time convenient to them, within the allotted time for studying the specific chapter. The experimental group was instructed to ask questions and answer them while reading the texts. As they had no prior experience in QA, to facilitate this process, we proposed to formulate questions beginning with ‘What’, ’How’, ‘Whether’, or ‘Why’. The control group was instructed to cope with the historical texts using any method they deemed appropriate.

3.4. Research Method

The research design was quasi-experimental [20,21]. In quasi-experimental designs, there is no random assignment and the program is considered an intervention in which the treatment is assessed for how well it accomplishes its objectives. However, the control group should be as similar as possible to the experimental group [22]. As mentioned, in our case, the PMTs were matched based on their average grades in previous compulsory mathematics courses.

Within this design, we employed qualitative as well as quantitative methodologies [18]. We exploit the strength of the quantitative approaches for making statistical generalizations relating to the self-reported CL of both groups, and the advantage of the qualitative approach for gaining insights into processes related to QA, as manifested by the personal perspectives of the experimental group members and the subjective meanings they ascribe to their experience.

3.5. Research Instruments

CL can be measured by learner self-reports, using three techniques [13]: (i) real-time reporting, instructing subjects to produce think-aloud protocols while coping with a particular task; (ii) filling out a self-esteem index questionnaire upon completion of coping with a particular task; and (iii) retrospective reporting relating to the ranking of statements in the questionnaire. In the present study, we employed all three techniques, where the term ‘task’ refers to the deciphering of an ancient algorithm. The self-esteem index was measured through a quantitative tool (see Section 3.5.1), while the real-time reporting and the retrospective reporting were parts of the quantitative tools (see Section 3.5.2).

In what follows, the data gained from the quantitative tool relate to the experimental and control groups, whereas the data collected from the qualitative tools relate only to the experimental group.

3.5.1. The Tool for Collecting Quantitative Data

For examining the PMTs’ self-esteem index of CL, they were asked to respond to a questionnaire at three points in time during the academic year: point 1—after deciphering the Egyptian algorithm (Text 1), point 2—after deciphering the Babylonian algorithm (Text 2) geometry), and point 3—after deciphering the Vedic algorithm (Text 3). Based on the cognitive measurement tool developed by Paas et al. [14], the participants were asked to report on their invested mental effort during their attempts to decipher the algorithm at hand. In a questionnaire consisting of a symmetrical scale, ranging from 1 (very very low mental effort) to 9 (very very high mental effort), reflecting the degree of difficulty the PMT experienced in coping with the task of deciphering the given algorithms. According to Paas et al. [13], the reliability and sensitivity of this scale and “moreover its ease of use have made this scale, and variants of it, the most widespread measure of working memory load within CLT [Cognitive Load Theory] research” (p. 68, [14]). Drawing on Paas et al. [14] and Paas et al. [13], the statements in the self-esteem index of the CL questionnaire were of two types. Below, statements ‘a’ to ‘f’ concern extraneous cognitive load (ECL), as they refer to the way the task is presented to the PMTs, while statements ‘g’ to ‘j’ concern intrinsic cognitive load (ICL), as they refer to aspects relating to the mental effort the PMTs invest while engaging with the historical algorithms. The statements are (a) the absence of explicit rationale of the algorithm; (b) the transition from step to step in the algorithm; (c) the verbal formulation of the algorithm; (d) the numerical examples presented throughout the algorithm; (e) the mathematical concepts that appear in the algorithm; (f) absence of proof for the algorithm; (g) applying the algorithm—solving computational problems based on it; (h) applying the algorithm—solving problems that require high order levels of thinking; (i) the need to generalize the algorithm; and (j) adapting the algorithm into new situations. These statements reflect the nature of the information the PMTs had to deal with or the mathematical operations they had to perform. It should be noted that none of the statements referred to GCL because it applies to the processing, construction, and automation of schemas, issues that were not addressed in the current study.

To examine differences between the results received after coping with each of the above algorithms, six variables based on the average value of items a–f and the average value of items g–j in the self-esteem index of the CL questionnaire were constructed. Details about the constructed variables are presented in

Table 1. Variables, items, and Cronbach’s alpha.

Variable Name	Variable	Items	Cronbach’s Alpha
ECL_text 1	Text 1	a–f	0.906
ICL_text 1	Text 1	g–j	0.858
ECL_text 2	Text 2	a–f	0.891
ICL_text 2	Text 2	g–j	0.862
ECL_text 3	Text 3	a–f	0.916
ICL_text 3	Text 3	g–j	0.874

. For example, the variable ECL_Text 1 refers to the average value of the reported level of difficulty for statements a–f (ECL) in the case of Text 1. ICL_Text 1 refers to the average value of the reported level of difficulty for statements g–j (ICL) in the case of Text 1. The other variables were calculated similarly. To check the internal consistency reliability of the variables, Cronbach’s alpha was calculated for the items of each variable.

To examine the differences between the results received from the questionnaire and given that our samples are small and there is no normal distribution of the data, the Friedman test [23] was found to be suitable. Namely, the Friedman test was conducted to identify differences in the results obtained from the three treatments (Text 1—Egyptian algorithm, Text 2—Babylonian algorithm, and Text 3—Vedic algorithm).

3.5.2. Instruments for Collecting Qualitative Data

For collecting the qualitative data, we employed the research instruments described below:

(i)

Think-aloud protocols. The real-time reporting [13] was obtained using think-aloud protocols, an approach for collecting data whereby people are asked to verbalize and document (whether in writing or audio/video recording) their thought processes as they manage a specific task [24,25]. Employing this tool, the experimental group members were asked to audio record the questions they asked, the answers they gave, and their thoughts while engaging with the mathematical texts at the same three points in time mentioned in Section 3.5.1. These recordings were transformed by the researchers into written protocols. It should be noted that, reviewing 94 studies (involving almost 3500 participants), Fox, Ericsson, and Best [26] found no reliable differences in cognitive performance with or without the employment of the think-aloud technique.

(ii)

Justifications of the self-esteem index of cognitive load. The retrospective reporting [13] was part of the self-esteem CL index questionnaire, where the PMTs were asked to justify their rankings. For this purpose, they were asked to recall relevant episodes that occurred while coping with the texts.

(iii)

Group interview. The group interview is intended to gather information from several individuals simultaneously. In educational settings, the group interview is beneficial for gaining insights into students’ reactions to certain aspects of a course [27]. Four times throughout the academic year (twice in each semester), we conducted a group interview intended to elicit the PMTs’ thinking about issues related to their engagement with the texts. The interviews were recorded and transcribed.

Owing to space limitations, in the results chapter, we will only refer to the think-aloud protocols.

3.6. Data Analysis

Table 2. Instruments for data collection and analysis.

Data Source	Data Type	Analysis Methods
Self-esteem index of cognitive load questionnaire	Quantitative	Friedman test [23]
	Qualitative	Coding (open, axial, selective) [28]
Transcripts of think-aloud protocols	Qualitative
Transcripts of group interviews	Qualitative

summarizes the instruments used for data collection, the data type, and the methods employed to analyze the data.

To analyze the think-aloud protocols, we employed three stages of coding [28]: Open coding, aimed at constructing the initial categories based on prominent words or phrases in PMTs’ protocols; Axial coding, in which the initial categories were grouped under more general categories based on their causal conditions; and Selective coding, in which we triangulated, refined, and defined the relationships among the categories that surfaced in the axial coding stage. We compared the categories created by each of the two researchers individually and by an external expert in mathematics education to increase the reliability of the categories resulting from the coding process until an agreement was reached [29].

3.7. Ethical Issues

As the researchers and the study participants had an authority relationship (lecturer–student), we received the approval of the college research authority to carry out the study. Before the course started, the PMTs were informed about the researchers’ intention to conduct a study that would follow their experiences by using their learning products. We clarified to the PMTs the distinction between their participation in the study and their participation in the course. They were free to choose whether or not to participate in the study, while we guaranteed that their grade in the course would not be harmed in any way if they chose not to participate in the study. All of the PMTs expressed their consent to take part in the study while keeping their anonymity.

4. Results

4.1. Empirical Results Obtained from the Self-Esteem Index of the Cognitive Load Questionnaire

In this section, we present the results obtained from the self-esteem index of the CL questionnaire, distinguishing between groups (control and experimental) and sub-groups (HA and LA). In addition, we distinguished between statements referring to ECL (statements a–f) and ICL (statements g–j) (see Section 3.5.1).

4.1.1. General Empirical Results

At each of the three points in time mentioned above, immediately after completing the coping with the mathematical text to avoid forgetting, both PMT groups responded to the CL questionnaire, in which they had to rank the level of difficulty they had experienced (which implies on the CL) while coping with the specific task. Figure 1 demonstrates the distribution of the PMTs’ average self-esteem index of ECL and ICL at the three points in time (indicated as 1, 2, and 3 for Text 1, Text 2, and Text 3, respectively) for the control (c) and the experimental (e) groups. The y-axis denotes the range of the self-reported index of CL, 1–9 [13]. Thus, for example, the left column in the graph represents the average ECL as self-reported by the control group while referring to Text 1.

As can be seen from Figure 1, the self-reported CL index (both extraneous and intrinsic) of the control group remains almost the same for the three texts, while in the experimental group, there is a moderately decreased tendency of the self-reported ECL and ICL along with the three texts. Given that the self-report regarding CL indicates learners’ sense regarding the degree of difficulty they experienced while coping with a text (or the degree of cognitive effort they invested) [14,30,31], and the fact that the only difference between the two groups was the act of QA, it might be said that QA affected the decrease in the PMTs’ reported CL.

The PMTs’ general justifications supported this observation in their ranking of the self-esteem index of CL. For example: “For statement ‘a’ I marked 9 because I could not understand the Egyptian algorithm. There was no justification attached to what they did”; “Since visual illustrations supported the Babylonian algorithm, it was quite easy to follow its steps, so for statement ‘b’ I marked 2”.

Particular attention should be given to the reduction in ECL as reported by the experimental group. Sweller [32] indicated that, to reduce the ICL, educators need to first reduce the ECL. However, in the case of the discussed historical texts, we did not make any changes in their design that might affect the ECL, as they were presented in their original formulation (translated into Hebrew). Nevertheless, there was a moderate decrease in both the ECL and ICL among the experimental group. This observation might indicate the effect of QA on both types of CL while coping with mathematical texts.

4.1.2. General Results Related to High and Low Achievers

Figure 2 shows that, in both groups (control and experimental), the LA’s self-reported CL (extraneous and intrinsic) is higher than that of the HA. In addition, referring to the control group, both CLs (intrinsic and extraneous) remain almost the same, while for the experimental group, a moderate decrease in CLs (both extraneous and intrinsic) along the three texts is observed.

4.1.3. Examination of Differences between Groups

Table 1 shows that Cronbach’s alpha of the variables is greater than 0.7, which indicates good internal consistency reliability. The results of the Friedman test are summarized in

Table 3. Results of the Friedman test.

		Control Group						Experimental Group
		HA (N = 6)		LA (N = 14)		Total (N = 20)		HA (N = 6)		LA (N = 14)		Total (N = 20)
		M	SD	M	SD	M	SD	M	SD	M	SD	M	SD
Extraneous	Text 1	5.75	0.57	7.48	0.36	6.96	0.91	5.86	0.31	7.12	0.35	6.74	0.68
	Text 2	5.75	0.50	7.40	0.40	6.91	0.88	5.64	0.19	6.75	0.32	6.42	0.59
	Text 3	5.72	0.53	7.51	0.47	6.98	0.96	5.33	0.21	6.44	0.28	6.11	0.58
	χ2 (df = 2)	0.50		3.26		2.68		9.36		19.09		27.61
	p	0.779		0.196		0.262		0.009		<0.001		<0.001
Intrinsic	Text 1	6.50	0.27	8.39	0.32	7.82	0.94	6.50	0.22	8.13	0.40	7.64	0.84
	Text 2	6.50	0.39	8.29	0.34	7.75	0.91	6.08	0.30	7.88	0.34	7.34	0.90
	Text 3	6.58	0.34	8.32	0.32	7.80	0.88	5.83	0.34	7.64	0.29	7.10	0.90
	χ2 (df = 2)	1.14		4.17		1.64		8.82		12.53		20.87
	p	0.565		0.124		0.441		0.012		0.002		<0.001

.

From Table 3, it can be seen that, while in the control group, the differences between the three measurements are not significant, neither for all students nor for HA or LA (ECL: p = 0.262, 0.779, 0.196; ICL: p = 0.441, 0.565, 0.124, in accordance), in the experimental group, significant differences were found between the three measurements, either for all students or for HA and LA (ECL: p < 0.001, 0.09, <0.001; ICL: p < 0.001, 0.012, 0.02, in accordance). In addition, it can be seen that, in the experimental group, the change is more significant for LA students than for HA students.

4.2. Qualitative Data Obtained from the Think-Aloud Protocols

Reviewing the think-aloud protocols of the LAs revealed that their protocols were short and did not include meaningful information that would shed light on their learning process. Hence, we could not learn much from these protocols about the effect of QA on the change in their sense of CL, namely, the degree of difficulty they experienced while dealing with the algorithms. Nevertheless, we extracted most of the information about the low achievers’ experience from the group interviews. Owing to space limitations, this information is not included in the current article.

As to the HAs, reviewing their think-aloud protocols revealed several similar thinking processes, as demonstrated below. To that end, we present excerpts taken from the think-aloud protocols of Rina (pseudonym). Rina’s protocols were chosen because they were more detailed than the other HAs. Nonetheless, we present merely excerpts that reflect the spirit of other HAs’ protocols. All of the protocols were analyzed through the three stages of coding—open, axial, and selective [28], as elaborated in Section 3.6. Below are three excerpts that were taken from Rina’s think-aloud protocols, where each excerpt is separated into numbered sentences. In the analysis of the sentences, S_i means statement number i.

4.2.1. Think-Aloud Protocol No. 1—Egyptian Multiplication

[First, Rina reads the entire text related to the Egyptian multiplication algorithm]

1:

The Egyptian multiplication looks unclear.

2:

I read it several times and can’t figure out why they multiplied the numbers by 2.

3:

I keep thinking about why they did not use the method we use today, and I must admit, this distracts my mind, and obviously, I can’t answer it.

4:

O.K., so before giving up, it seems the right time to start asking questions.

5:

I noticed that in the two examples you presented, the multiplicand was an odd number, while the multiplier was an even number.

6:

Since I can’t figure out the underlying rationale, the question I ask is whether I can repeat the process with the other three combinations of multiplying natural numbers—even-odd, odd-odd, even-even.

7:

While checking these combinations, I feel I am acting automatically, just reproducing the method steps without understanding why I am doing that.

8:

Now I just want to make sure the products are correct.

9:

Yes. The calculator verifies it.

10:

O.K. These 4 examples convinced me that the Egyptian method is valid for any multiplication of two natural numbers.

11:

The next obvious question is why? How can I justify the method? To be honest, I have no idea where to start.

12:

So, I’ll just keep asking questions about the validity of the method as regards additional types of numbers.

13:

For example, fractions. Perhaps this will help me to figure out the rationale.

14:

I know from the introduction section that the Egyptians dealt with unit fractions, so I’ll try to reproduce the method for multiplying $15\frac{1}{2}·18$…. hmmm… I’m stuck with the $\frac{1}{2}$.

15:

Obviously, I am missing something.

16:

I don’t get to the bottom of the rationale of the method.

17:

I merely replicate procedures.

18:

Just for the protocol—I have no more questions to ask to help me get into the rationale of the method.

19:

I feel like reaching the end of the road. So, let me think of another direction.

20:

What is the connection between this method and the method we learned at school?... I don’t know what to say, because I don’t find any connection.

21:

Perhaps I should rephrase my question and ask—is there a connection between the methods at all? If there is any connection, I can’t see it.

22:

So, at this point, I give up and choose to present the questions I asked at the class meeting, to get feedback that will help me to proceed.

After reading the text that includes the Egyptian algorithm for multiplying two natural numbers, Rina declares it is unclear (S1–S2). Then, she recognizes that she was distracted by something she could not resolve whatsoever (S3). It might be that, without the instruction to ask questions, Rina would quit and give up (S4). She starts with a distinction between the types of numbers that demonstrate the algorithm and decides to check the other three variations (S5–S6). In doing so, she feels like she acts automatically without paying attention to the essence of the algorithm (S7), which immediately leads her to the need to verify the correctness of the result (S8–S9). From S10, it can be inferred that, up to this point, she was not convinced as to the correctness of the algorithm for any two natural numbers. Having no idea how to justify the algorithm (S11–S12), in an attempt, ostensibly, to “gain time”, she wants to re-engage herself in checking the correctness of the algorithm and now chooses to incorporate unit fractions as well (S13). However, because she does not understand the essence of the algorithm (and, in any case, at this point, it has not yet been clarified in the text how to act in the case of fractions), she feels stuck even before she starts (S14). S15–S19 indicate that Rina is aware that she does not understand the mathematics behind the algorithm, and she recognizes that what she has done so far will not help her move forward. Then, with what appears to be a continuation of S3, she asks herself what the connection between the Egyptian algorithm and the currently accepted method (S20) is and immediately continues asking whether there is such a connection at all between the two (S21). Then, Rina stops struggling with the text (S22). Looking at S3, S20, and S21, it appears that Rina is trying to identify the connection between the Egyptian algorithm and her existing schema of multiplication. Because she does not immediately see any such connection, she decides to quit. From the above protocol, it can be seen that, although a process of asking questions occurred, one set of questions focused on the validity of the algorithm for rational numbers in general, questions that remained at the technical level of algorithm testing, and the other set focused on the relationship between the Egyptian algorithm and the multiplication algorithm known to Rina, which is, in fact, the schema she holds for the essence of multiplication.

The term ‘schema’ refers to the description of a pattern of thought or behavior that organizes categories of information and the relationships among them [33]. It seems that her existing schema of multiplication generated difficulty in deciphering the algorithm, which prevented Rina from striving to understand the algorithm. Such difficulty is indeed reflected in Rina’s self-esteem cognitive load questionnaire, reporting high cognitive load (6.38).

4.2.2. Think-Aloud Protocol No. 2—Babylonian Method for Solving a Set of Two Equations

1:

After reading the Babylonian method of solving two equations with two unknown variables, I must say that I was fascinated.

2:

The method is so different from the way we learned in math lessons.

3:

We learned to solve the equations using an algebraic method, while the Babylonians used geometric considerations.

4:

In the algebraic method, one option is to isolate one of the variables from one of the two equations and substitute it in the second equation to get an equation with one unknown variable and then solve it using the quadratic roots formula.

5:

The Babylonian method uses geometry, and I have to recall my geometry knowledge regarding the area and perimeter of squares and rectangles.

6:

O.K., Now to asking questions. First, does it work with other numbers? For example: a + b = 16 and a·b = 90… hmmm… [checking]. Something is not working here.

7:

what happened here? Why doesn’t the method work for these numbers? I got a negative difference (64–90). You can’t create a square with a negative area!

8:

Let me check it using the algebraic method.

9:

O.K., I got a quadratic equation with no real roots. This is why the numbers are not good.

10:

They, I mean, the writers of the historical text, forgot to mention that the multiplication of the two numbers must be smaller than the area of the square.

11:

But why does it have to be smaller?

12:

I have no idea. I think I leave the problem for a while to refresh my mind.

13:

[a day after] The problem kept bothering me and I think I have a breakthrough.

14:

I remember that we learned that of all rectangles with a given perimeter, the square has the maximum area.

15:

Hence, the area of the rectangle in the second equation must be smaller than the area of the square whose side equals 16/2. So now it’s OK.

16:

My next question is- does the method works for any rational numbers?

17:

I am skipping the checking of the case that one, the first equation, is a natural number and the other, the second equation, is not, and get to the case that both numbers are not natural numbers.

18:

I am now checking the case of a + b = 15.7 and a·b = 58.3.

19:

O.K. The a = b = 15.7/2 = 7.85 is the side of a square with which to start.

20:

So, to get the area of 58.3, I have to remove a square whose area is 61.6225 − 58.3 = 3.3225. Hence the side of this square is √3.3225.

21:

Now I need to put the left rectangle, as you can see in the next draw, to the side of the square and get a rectangle.

22:

The rectangle’s sides are: 7.85 + √3.3225 and 7.85 − √3.3225, and the area of this rectangle is 58.3.

23:

Wow! It works!

24:

Now, how can I prove the general case?

25:

Will the algebra help to get the proof? Let’s see.

$$x·y=m\Rightarrow y=\frac{m}{x};x+y=n\Rightarrow x+\frac{m}{x}=n\Rightarrow x^2-nx+m=0;x_{\mathrm{1,2}}=\frac{n\pm \sqrt{n^2-4m}}{2}$$

26:

What now? How can I connect the above expression to the Babylonian method?

27:

I have no clue.

28:

I feel I’m stuck!

29:

This is frustrating since I like the Babylonian method, but I can’t show why it always works.

As in the previous algorithm, Rina first reads the entire text that relates to the algorithm. She then expresses her favorable impression of it (S1). Similar to the case of the Egyptian algorithm, the first thing that comes to her mind concerns the comparison between the schema Rina holds regarding the solution of a system of two equations with two unknowns and the Babylonian method, in particular—an algebraic vs. a geometric approach (S2–S3). After recalling an algebraic schema and specifying the algorithm stages (S4), she tells herself what knowledge she needs to follow the Babylonian algorithm (S5). At this point, Rina starts to ask questions. Similar to the previous protocol, she starts with verifying the algorithm for other numbers (S6). However, as she picks two random numbers without considering any constraints, she ends up with a negative area (S7). In an attempt to understand what went wrong, she turns back to her existing (algebraic) schema (S8), finds the algebraic reason for the problem (S9), and “blames” the Babylonian writers for omitting the information about the constraint from the algorithm (S10). However, she cannot figure out why the constraint is needed (S11–S12). At this stage, she stops her probing to refresh her mind (S12), and a day later, she feels that she had reached a breakthrough (S13). She recalls the extremum problem related to rectangles having a fixed perimeter (S14) and employs it to justify the missing constraint (S15). Again, like in the case of the Egyptian algorithm, in order to progress, Rina’s new question refers to other types of numbers (S16), aiming to generalization the Babylonian algorithm for rational numbers. She realizes that it works (S17–S23). In her attempts to provide formal proof of the correctness of the algorithm, she turns back to the algebraic schema she possesses (S25), and because Rina fails to find the connection between the Babylonian algorithm and the schema Rina is holding, she again feels stuck and gives up (S26–S28). Nonetheless, as she was impressed by the method (S1), she was frustrated that she could not bridge the two algorithms (S29).

In the second protocol, it is evident that, although Rina did not fully understand the Babylonian algorithm, the activity of QA enabled her to monitor her attempts to understand the Babylonian algorithm. We might attribute her meaningful progress to experiencing less cognitive load during her coping with the algorithm, as reflected in Rina’s self-esteem cognitive load questionnaire (5.98).

4.2.3. Think-Aloud Protocol No. 3—Vedic Method for Squaring Two-Digit Numbers

The third think-aloud protocol concerns the finding of the square number of a two-digit number according to the Vedic mathematics method.

[reading the Vedic algorithm for squaring two-digit natural numbers]

1:

In our school mathematics, we did not learn a particular method for squaring a given number.

2:

So, it is interesting to learn about the Vedic method for finding such a number.

3:

Let’s try other numbers to see if it also works for them.

4:

${36}^2=3·10·\left(36+6\right)+6^2=1296$

5:

O.K. I’ll check the result with the calculator… It works! Now I’ll try to generalize it.

6:

${\left(10\mathrm{a}+\mathrm{b}\right)}^2=100·{\mathrm{a}}^2+20·\mathrm{a}·\mathrm{b}+{\mathrm{b}}^2$

7:

What now? How is it connected to the Vedic method?

8:

Maybe I have to write it differently?

9:

I’ll now return to one of the examples from the text and try to write it differently:

10:

${12}^2=(1·10+2{)}^2=(1·10+2)·(1·10+2)=1·10·1·10+1·10·2+2·1·10+2^2$

11:

So, what in this expression might imply about the Vedic method?

12:

Hmmm…. I’ll try to write it like this: $1·10\left[\left(1·10+2\right)+2\right]+2^2=10\left[\left(1·10+2\right)+2\right]+2^2$

13:

O.K. what do we have here? 10 (the number + the unit number) + the square of the unit number.

14:

Does it look clear now? Hmmm… Of course! I got it!

15:

Now I can try to generalize it.

16:

I have to replace the tens and unit digits with a and b, in accordance.

17:

${\left(10\mathrm{a}+\mathrm{b}\right)}^2=\left(\mathrm{a}·10+\mathrm{b}\right)\left(\mathrm{a}·10+\mathrm{b}\right)=\mathrm{a}·10·\mathrm{a}·10+\mathrm{a}·\mathrm{b}·10+\mathrm{a}·\mathrm{b}·10+{\mathrm{b}}^2$

18:

$\mathrm{a}·10·\left(\left(\mathrm{a}·10+\mathrm{b}\right)+\mathrm{b}\right)+{\mathrm{b}}^2$

19:

Wow! They are genius!

20:

It would be interesting to see if I can adapt it for 3-digit numbers.

The process of reading the whole algorithm and expressing the initial evaluation of it repeats itself. Moreover, Rina specifies the absence of a similar method to the Vedic one in today’s mathematics (S1–S2). Similar to the previous algorithms, she raises the question of whether the method also works for additional examples to convince herself of its correctness (S3–S4), and for double-checking, she verifies the result using a calculator (S5). Her next step is raising the question of how it can be generalized (S5). Rina turns to her existing schema regarding short multiplication formulas (S6) and asks herself how to continue (S7). She turns back to the Vedic method and uses one of the given numerical examples to follow the Vedic method logic (S8–S13). At this point, Rina feels like she is having a breakthrough (S14). Next, without being aware of it, she applies the transparent proof technique [34,35] (S16–S18). “A Transparent Proof is a proof of a particular case which is small enough to serve as a concrete example, yet large enough to be considered a non-specific representative of the flow of arguments in the proof of the general case; one can see the formal proof through it since nothing specific to the particular case enters the transparent proof” (p. 29, [34]). Then, she proudly declares her success in proving the Vedic algorithm (S19). Motivated by her success, Rina raises the question of whether this method can be adapted for three-digit numbers (S20). Rina’s success in deciphering the Vedic algorithm is reflected in her decreased self-reported intrinsic cognitive load, as reflected in her self-esteem index questionnaire (5.74).

5. Discussion and Conclusions

In this study, we examined the effect of QA on the reduction in CL. Based on Paas et al. [14], we distinguished between ICL (the mental effort one invests in dealing with a task) and ECL (the mental effort one invests as a result of the design of the task). It should be noted that, because we intentionally did not change the mathematical texts’ formulation, trying to adhere to their authenticity, we did not make an effort to reduce the ECL. Based on Paas et al. [14], the mental effort reported in the questionnaire ranged from 1 (very very low mental effort) to 9 (very very high mental effort), reflecting the degree of difficulty the PMTs experienced in coping with the texts.

5.1. PMTs’ Perceptions Regarding the Effect of Question-Asking While Reading Historic Mathematical Texts on Their Cognitive Load and Differences between High and Low Acheivers

As apparent from the quantitative data presented in Figure 1, while the ECL reported by the control group remained almost the same (6.96, 6.98, and 6.98), there was a minor gradual reduction among the experimental group (6.74, 6.42, and 6.11). As evident from Table 3, applying the Friedman test [23] revealed that, while in the control group, the differences between the three measurements were not significant, in the case of the experimental group, significant differences were found between the three measurements. Therefore, we might infer that the QA affected the ECL. Further research is needed to explore this result and, in particular, to focus on the relationship between the ICL and the ECL in the context of QA.

Moreover, it can be seen that, while the ICL of the control group remained almost unchanged (7.83, 7.75, and 7.8), the members of the experimental group reported a gradual decrease in the ICL (7.64, 7.34, and 7.1). Table 3 shows that, in the case of the control group, the differences between the three measurements were not significant, while for the experimental group, significant differences were found between the three measurements.

It is worth noting that, in the case of the control group, the changes in the ECL and ICL indices were not significant for both the LA and HA sub-groups, while in the case of the experimental group, the changes were significant for both the HA and LA sub-groups.

Thus, the first observation is that QA reduces the ICL and ECL in learning situations where task characteristics remain unchanged. As evident from Figure 2, this observation is more prominent among the HA experimental group than among the LAs. One might attribute the resulting differences to the use of the think-aloud technique as well. However, researchers showed that the use of this technique has little or no reliable effect on the difference in performance relating to cognitive activities [25].

5.2. Insights Gained from the Think-Aloud Protocols

To gain insights into the process of QA and its effect on the deciphering of the algorithms, we instructed the PMTs to audio record their think-aloud process [23,24] while reading the texts. The researchers transcribed these records of the think-aloud processes, representing real-time reporting [13]. As the protocols of the LA were relatively short and did not include factual information from which one could follow their learning process, we were unable to extract valuable information from them regarding the issues under study. A follow-up study is proposed to examine whether there is a connection and, if so, of what nature it is, between the quantitative data (as shown in Figure 2) and the ability of students to produce more detailed protocols, which indicate the use of QA as a means of coping with mathematical texts. Therefore, the qualitative analysis focused on the think-aloud protocols of HAs. Specifically, we have presented representative quotes taken from Rina’s protocols, one of the HAs, which express ideas similar to the other HAs. Rina was chosen because her protocols were more detailed than those of the other HAs.

The analysis of Rina’s protocols reveals the existence of two of the reading strategies addressed by [1]: making inferences and employing prior knowledge. As revealed from the three protocols, this was done by asking two types of questions, indicating a two-tier gradual process. The first type of question is related to trying to understand the rationale of the algorithm by examining other numerical examples merely by substituting numbers in the algorithms. From Rina’s protocols, it is evident that she relates to this operation as an intuitive action done casually, one that does not involve any cognitive effort (in other words—it does not generate an ICL). However, this type of question did not lead Rina to comprehend the rationale underlying the algorithms. The second type of question, which follows the first type of question, is related to her desire to find compatibility between familiar methods (existing schemas) and the new methods appearing in the texts. This is done through a consistent comparison between the methods. As evident from Rina’s protocols, this comparison is made by making inferences and employing prior knowledge [1]. From the analysis of Rina’s utterances, one can learn about the high intrinsic cognitive effort that was invested to that end. This can be supported by the results described in Figure 2 in which the ICL is higher than the ECL along the three texts.

The activity of QA (both types of questions) served as an engine for the whole process. Through the process of QA from the first type, Rina progressed in her attempts to understand the texts and not give up right from the beginning. However, realizing it was not enough, she asked the second type of questions during her attempts to assimilate new information into existing knowledge (schema). In case the assimilation was problematic, she experienced difficulties that were reflected by her self-report of high CL. In the last group interview, Rina said:

“The questions I had to ask prevented me from giving up right after the first reading of the texts. Since I was unfamiliar with this technique [QA], I started with questions that helped me repeat the algorithms without necessarily understanding them. Then it occurred to me that I have to cope with different questions that will lead to a breakthrough in my attempts to make sense of the algorithms”.

Thus, it can be concluded that the first type of question neither adds to the CL nor promotes understanding of the texts. Whereas the second type of question promotes understanding, but at the same time, adds to the CL. In other words, an increase in CL and the development of insights seem to be interwoven. Nonetheless, questions of the first type help preserve coping with the task and not giving up right away. Although it is evident that QA, and in particular questions of the second type, helps reduce the CL, their effect on the reduction is limited. A possible explanation for this limitation might be attributed to a gap between existing schemas and new information [36,37].

From the three think-aloud protocols, it is evident that, when Rina had a solid schema (as in the case of multiplication), she had difficulties understanding the new knowledge (Egyptian multiplication algorithm). Rina’s existing schema relates to multiplying two numbers as a whole, while in the Egyptian multiplication method, the multiplicand is decomposed into a sum of numbers represented in powers of 2. Rina’s adhering to her solid schema prevents her from deciphering the new algorithm. As to the second think-aloud protocol, there is also a conflict between the existing schema—the algebraic method for solving two equations with two unknown variables, and the Babylonian method, for which the solution is geometric. In this case, the gap between the existing (algebraic) schema and the new (geometric) knowledge has a different nature than the one manifested in the first text, a fact that allowed Rina, through asking questions, to progress further in her process of understanding the text at hand. In the third protocol (Vedic method for squaring two-digit numbers), Rina understood the algorithm and provided formal proof. This success can be attributed to the fact that she had no existing schema for squaring two-digit numbers. This brings into discussion the connection between bridging existing schemas and new information and the CL that learners experience. Neumann and Kopcha [36] indicated the importance of the existence of schemas in learners’ minds for the absorption and assimilation of new knowledge. However, studies show that the existence of solid schemas can also interfere with the assimilation of new knowledge [37], as, in fact, happened in the present study. The above results are consistent with Van Kesteren and Meeter [37], who claimed that, while schemas are presumed to help memory encode and consolidate new data, solid schemas can also lead to unwanted side effects, as found in the present study. In the think-aloud protocols, we can see both the positive and negative effects of schema. The positive effect is expressed in recalling data from memory (e.g., how to calculate the area and perimeter of rectangles and squares, short multiplication formula), knowledge that was crucial for understanding the historical algorithms. The negative effect is expressed in Rina’s attempts to prove the new algorithms (Egyptian and Babylonian) holding onto her solid existing schema with no success. For example, referring to the third text (Vedic method for calculating a square of two-digit natural numbers), the fact that Rina did not have an existing schema for calculating the square of two-digit natural numbers helped her, by using QA, to come up with alternative ways of thinking and eventually to provide a formal proof to the Vedic method. Thus, the second observation is that QA supports the process of assimilating new information up to a certain point. Because solid schemas might interfere in bridging existing knowledge and new information, although QA of the second type might be efficient in reducing the CL, its effect is dependent on the nature of the gap between existing schema and new information.

To conclude, the findings indicate the potential of QA to reduce CL in the case of dealing with mathematical texts. To be able to optimally implement QA in mathematics educational settings or any other discipline, further research is needed to characterize the appropriate questions for reducing the CL generated by the gap between existing schemas and new information. Further research is also needed to identify strategies that will support LAs to cope with mathematical texts. Finally, to examine whether the content of the text influences the CL, further research should be conducted in which the texts will be given in random order to different PMTs.