perception about mathematics essay

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Article Contents

1 background to the study, 2 theoretical perspectives, 3 methodology, 5 discussion, 6 conclusion, a. a. sample tasks from the writing intervention, b b. student questionnaire, students’ perceptions of mathematics writing and its impact on their enjoyment and self-confidence.

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Tandeep Kaur, Mark Prendergast, Students’ perceptions of mathematics writing and its impact on their enjoyment and self-confidence, Teaching Mathematics and its Applications: An International Journal of the IMA , Volume 41, Issue 1, March 2022, Pages 1–21, https://doi.org/10.1093/teamat/hrab008

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There have been universal endorsements of the benefits of writing as an effective medium of communicating mathematically. Writing and learning are seen as isomorphic to each other and writing can facilitate the comprehension of mathematical thinking through intrapersonal communication. Through a short writing intervention, this study investigates students’ perceptions on the use of writing in the mathematics classroom and explores the impact of writing on students’ affective domains of self-confidence and enjoyment levels in mathematics. A mixed-methods approach was employed using a pre-test, intervention, post-test design for the study. Quantitative data were collected through a questionnaire adapted from the Attitudes Towards Mathematics Inventory ( Tapia & Marsh, 2004 ), which was administered before and after the intervention. An analysis of the quantitative data revealed a significant increase in students’ mean scores for both enjoyment and self-confidence. Qualitative data collected in the form of students’ reflections of the writing intervention indicated that, overall, students had a positive perception of writing as a means of communicating in the mathematics classroom.

In India, the National Curriculum Framework (2005) describes mathematical communication as an important feature of any mathematical undertaking. It is recommended that mathematical communication which uses unambiguous and precise language is crucial for developing an appreciation of the subject. Matsuura et al. (2013) determined that such endeavours can be effective in the development of mathematical thinking and habits of mind. ‘These habits are not about particular definitions, theorems, or algorithms that one might find in a textbook; instead, they are about the thinking, mental habits, and research techniques that mathematicians employ to develop such definitions, theorems, or algorithms’ ( Matsuura et al. , 2013 , p. 736). Despite such recognized importance, there has been no research to date investigating the issue of mathematical writing in India. This present study is the first of its kind to be carried out in the Indian context.

Mathematics has evolved significantly over the past number of decades and so have the teaching practices associated with it. Various reforms have been implemented worldwide from time to time, with a focus on differing strategies to enhance deeper mathematical understanding and engagement among students. One feature of mathematical reform that has been under continuous scrutiny is ‘mathematical communication’. It has been suggested by the educational policies of almost all countries that it is necessary for students to be able to communicate mathematically and that mathematical communication should be at the heart of mathematical teaching. For example, in the USA, the National Council of Teachers of Mathematics (1989 , 2000) called for mathematical literacy for all, stressing the need for mathematical communication. Similarly, India’s National Curriculum Framework (2005) states that ‘Children see mathematics as something to talk about, to communicate through’ (p. 43). In Ireland, a recent reform of lower secondary level education has identified communication as one of the key skills of the entire curriculum. More specifically, communication also underpins a unifying strand across the Irish mathematics syllabus. The syllabus states that ‘Students should be able to communicate mathematics effectively in verbal and written form’ ( NCCA, 2018 , p. 9).

With specific reference to writing as the mode of communication, there has been increasing interest in recent years towards its role in mathematics classrooms. While participation in discourses and debates of mathematics for improved mathematical learning has been frequently emphasized ( Burton & Morgan, 2000 ), formal recommendations for mathematical writing are less explicitly available ( Casa et al. , 2016 ). Despite this, studies and interventions have consistently reported the benefits of writing in learning mathematics ( Fry & Villagomez, 2012 ; Knox, 2017 ; Kostos & Shin, 2010 ; Kuzle, 2013 ; Pugalee, 2001 ).

1.1 Mathematical writing—the perceived benefits

Bangert-Drowns et al. (2004) determine that writing and learning are isomorphic to each other. In all aspects of life, writing serves as a psychologically powerful instrument, providing a vent for thoughts through deep reflection and understanding ( Hacker et al. , 2009 ). In the classroom, writing assists students in constructing new knowledge through activities such as exploration, representation, investigation and justification ( Countryman, 1992 ). Being a planned and conscious process, it strengthens current knowledge while building new connections at the same time ( Kenney et al. , 2013 ; Kuzle, 2013 ). Pugalee (2001 ) notes that in order to express one’s thoughts through writing, inner speech has to be compressed maximally by connecting current knowledge to the new knowledge gained. With specific reference to mathematics, various studies have focused on the effects of mathematical writing in the mathematics classroom. Such research has revealed enhanced metacognitive thinking, self-confidence and enjoyment levels, which leads to increased mathematical achievement ( Knox, 2017 ; Kostos & Shin, 2010 ; Kuzle, 2013 ; Pugalee, 2001 ).

1.2 Writing and the affective domains

Emotions are fundamental to learning and can affect students’ thought processes as well as memory ( Hinton et al. , 2008 ; Westen, 1999 ). For example, threatening situations, such as peer competition, parental pressure, exam stress, image in front of teacher, etc., can affect learning in a negative way ( Wolfe & Brandt, 1998 ). Writing aids in dealing with such issues and may help bring positive changes in the affective constructs such as self-confidence and enjoyment ( Countryman, 1992 ).

Several studies have revealed that mathematics learning is highly influenced by learners’ mathematics-related beliefs, especially self-confidence ( Hannula & Malmivuori, 1997 ; Hannula et al. , 2004 ). In fact, self-confidence has been observed as the greatest non-cognitive predictor for academic achievement among other self-belief measures such as self-efficacy and self-concept ( Stankov et al. , 2014 ). On the other hand, a lack of self-confidence may negatively impact students’ motivation to learn ( Boekaerts & Rozendaal, 2010 ). Engaging students to communicate their mathematical ideas through writing may instil in them a higher level of self-confidence and critical thinking skills ( Quitadamo & Kurtz, 2007 ).

Another important construct is students’ enjoyment while learning. Many studies have evidenced a positive relationship between the level of enjoyment while performing a task and students’ attitudes and self-efficacy beliefs ( Ahmed et al. , 2010 ; Lorsbach & Jinks, 1999 ; Sakiz et al. , 2012 ). These effects are greatly enhanced if students enjoy the tasks they are working on ( Bramlett & Herron, 2009 ). Mathematical writing is one such task that may prove to be an enjoyable class activity, while also adhering to formal curricular demands.

However, little research has been conducted to examine the role of writing in relation to these affective domains ( Miller & Meece, 1997 ). With such a dearth of research in mind, this paper investigates students’ attitudes towards mathematics with regards to their enjoyment and self-confidence after a short writing intervention.

This study also aimed to capture students’ experiences of the writing intervention. It has been argued in the literature that learners’ perceptions of classroom events strongly influence their classroom behaviour and response to teaching approaches ( Marx, 1983 ; Roese & Sherman, 2007 ; Struyven et al. , 2005 ). Therefore, capturing students’ perceptions may provide useful insights for successful reform efforts. However, at the moment, such insights are not given due consideration ( Levin, 2000 ). This paper thus explores participants’ written reflections of the intervention and investigates their perceptions of it.

Keeping up with the goal of promoting mathematical communication, the Elementary Writing Mathematical Task Force from the University of Connecticut in the USA proposed the following four types of mathematical writing, with their purposes described as follows:

Exploratory (to personally make sense of a problem, situation or one’s own ideas)

Explanatory/informative (to describe, to explain)

Argumentative (to construct or critique an argument)

Mathematically creative (to document original ideas, problems and/or solutions, to convey fluency and flexibility in thinking, to elaborate on ideas)

( Casa et al. , 2016 , p. 4)

It has been suggested by the Task Force that all students should have exposure to all types of writing. However, there is an ambiguity in comprehending the context of writing in relation to mathematics ( Bossé & Faulconer, 2008 ). The Task Force highlighted two categories of writing that take place in mathematics classrooms—‘writing about mathematics’ and ‘mathematical writing’. Writing about mathematics stresses on the learning of literacy skills e.g. a mathematics autobiography, while mathematical writing emphasizes the use of mathematical symbols and vocabulary with an aim to develop mathematical reasoning.

The present study focused on the ‘explanatory’ type of mathematical writing where the purpose is to describe or explain ideas for mathematical reasoning. The study also explored the perceptions of participants for its use in mathematics class. The study aims to address the following research questions:

(1) What effect, if any, does a short mathematical writing intervention have on students’ self-confidence and enjoyment levels in mathematics?

(2) What are students’ perceptions about the use of mathematical writing in the classroom?

The study used a pre-test, intervention, post-test design to explore students’ perceptions and investigated the impact of a short writing intervention on students’ enjoyment and self-confidence in mathematics. A convergent mixed-methods approach was employed to look into the research phenomena from different viewpoints ( Creswell & Plano Clark, 2011 ), with both quantitative and qualitative data being collected. The instruments used for addressing the research questions were questionnaires and participants’ reflections. The analysis and triangulation of this data enabled a ‘complete, holistic and contextual portrayal’ of the research ( Clark & Creswell, 2008 , p. 109).

3.1 Study sample

The study was conducted in a secondary co-educational school in New Delhi, India, which is affiliated with the country’s Central Board of Secondary Education. This school caters for approximately 1000 students from Kindergarten to Grade 10. It was selected through a purposive sampling method as its location was in close proximity to one of the researchers and they had existing contacts in the school. Students from a Grade 7 and Grade 8 (aged 12–15 years old) class group were invited to partake in the study and 55 students agreed to participate. The sample was made up of 38 males (69%) and 17 females (31%), with a mean age of 13 years. For all the students, Hindi was their first language and English was their second language. However, the language of instruction for all students in the school was English.

3.2 The intervention

Realizing that students are the primary stakeholders in the field of education and that any educational reform affects them foremost, an intervention was designed to measure the impact of writing on students’ attitudes, especially the affective domains of enjoyment and self-confidence. Six sessions were administered for the intervention, in addition to an introductory session which detailed the purpose of the study. The methodology of the study was also shared at this opening session, ensuring that each student understood the research instruments that would be used. Each session was conducted for a duration of 40 min. These sessions took place during regular school study hours and in the periods allocated for extra-curricular activities. This meant that participants’ formal studies were not interrupted by any activities related to this research.

Tasks for the intervention were selected from the Trends in International Mathematics and Science Study (TIMSS) 2011 grade 8 mathematics assessment items. These items manifest various ways of measuring students’ understanding in several content and cognitive domains (see Appendix A for sample tasks). Six tasks were selected, one for each session of the intervention. As this study focused particularly on the ‘explanatory’ genre of mathematical writing, tasks based on cognitive domains of reasoning were specifically selected. The criteria for selection of tasks were based on students’ previous knowledge and their current class group curriculum. Two tasks were based on identification of patterns, one was a geometry problem and the remaining three tasks were multiple choice based (see Appendix A). Each session started with class discussion of a specific task wherein students came up with different ways of solving the problem at hand. This was then followed by students’ explanation of their understanding and thought processes through writing. The researcher was present throughout each session to offer any assistance if required. No formal assessment was done for the submitted writings. Oral feedback was provided, and suggestions given to improve their writing for the subsequent sessions. One of the tasks from the intervention is specified in Fig. 1 .

Here is a pattern:

What will the next line in the pattern be? (Item number M042186, Cognitive Domain—Reasoning).

Example of a multiple choice-based task (SOURCE: TIMSS 2011 Assessment. Copyright © 2013 International Association for the Evaluation of Educational Achievement (IEA). Publisher: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College, Chestnut Hill, MA and International Association for the Evaluation of Educational Achievement (IEA), IEA Secretariat, Amsterdam, the Netherlands.)

3.3 Data collection

Quantitative data were collected in the form of questionnaires which were administered both before and after the intervention. The questionnaires were adopted from the Attitudes Towards Mathematics Inventory (ATMI; Tapia & Marsh, 2004 ). The subscales of enjoyment and self-confidence were selected for the present study. The enjoyment subscale consisted of a total of 10 statements and the self-confidence subscale comprised of 15 statements (see Appendix B). Each statement had alternative response options which were based on a 5-point Likert-type scale. Every positive statement in the questionnaires was scored from 5 to 1, ranging from 5 = ‘strongly agree’ to 1 = ‘strongly disagree’. Negatively worded items were scored in the reverse direction ranging from 1 = ‘strongly agree’ to 5 = ‘strongly disagree’. The maximum score for enjoyment subscale was 50 and for the self-confidence subscale was 75, giving an overall total score of 125.

After the final session of the intervention, qualitative data were collected in the form of participants’ written reflections. This enabled participants to share their experience regarding the use of mathematical writing as a tool for learning. Participants were prompted to reflect and write about whether they liked the intervention and to include their reasoning. Varied opinions emerged as they expressed their perceptions about the use of writing in the mathematics classroom.

3.4 Data analysis

The quantitative data for the study were analysed using SPSS (Statistical Package for the Social Sciences) software. Mean scores for the pre- and post-intervention tests were compared by descriptive analysis. In addition, t -tests were conducted to determine the statistical significance of the findings.

In order to ensure the internal consistency of the quantitative scales, Cronbach alpha coefficients were calculated. The Cronbach alpha coefficients were recorded as 0.83 for the self-confidence subscale and 0.84 for the enjoyment subscale. The alpha coefficient for the overall scale was 0.95. High values of Cronbach alpha coefficients (suggested values greater than 0.70— Nunnally and Bernstein, 1994 ) indicate good internal consistency and ensure content validity for the data.

The qualitative data were analysed thematically by one of the authors. Following this, a convergence model of triangulation design ( Creswell et al. , 2007 ) was used to interpret the results from both sets (qualitative and quantitative) of analysis. This enabled the authors to examine the convergence, consistency or inconsistency of the overall data ( Ary et al. , 2010 ). Findings for both types of data are detailed in subsequent sections.

As discussed previously, both quantitative and qualitative data were collected in order to address the research questions. Findings from the quantitative data will be presented first followed by those from the qualitative data.

4.1 Quantitative findings

Descriptive statistics revealed an increase in the mean scores for both the enjoyment and self-confidence scales, from ‘pre-intervention’ to ‘post-intervention’. Figure 2 illustrates a comparison of mean scores on the enjoyment (EN) and self-confidence (SC) subscales.

The results of a long jump competition were reported as follows:

Average length

Team A 3.6 m

Team B 4.8 m

There was the same number of students in each team. Which statement about the competition MUST be true?

Comparison of mean scores on enjoyment and self-confidence.

As illustrated from Fig. 2 , the mean enjoyment scores increased from 35 (before the intervention) to 40 (after the intervention) from a total score of 50. The mean self-confidence scores rose from 49 (before the intervention) to 54 (after the intervention) from a total score of 75.

In addition, a paired-samples t -test was conducted to assess the statistical significance of the findings. The differences in enjoyment scores from pre-intervention ( M  = 35, SD = 7) to post-intervention ( M  = 40, SD = 5), with t (54) = −10.8, p  < 0.05 (two-tailed), were statistically significant. The mean increase in enjoyment scores was recorded as 5 with a 95% confidence interval ranging from 5.9 to 4.1. For the effect size, eta square was calculated and found to be 0.6. As suggested by Cohen (1988) , values greater than 0.1 indicate a large effect size. Therefore, it can be concluded that there was a large effect size with a significant increase in enjoyment mean scores recorded from before and after the intervention.

Similarly, for self-confidence, the increase in mean scores from the pre-intervention test ( M  = 49, SD = 9) to the post-intervention test ( M  = 54, SD = 9), with t (54) = −9.4, p  < 0.05 (two-tailed), was statistically significant. The 95% confidence interval ranged from 5.9 to 3.8 with an increase of 5 in the self-confidence mean scores. The eta square statistic (0.6) also indicated a large effect size, implying that participants felt greater levels of self-confidence in learning mathematics aided by the writing intervention.

4.1.1 Further analysis

A statement-wise analysis of the two subscales for both the pre- and post-intervention revealed some noteworthy findings. Six out of 10 statements on the enjoyment subscale showed noticeable changes in the responses marked by the participants at both testing points. As an example, for the statement ‘I really like mathematics’, 20 out of 55 students (36% of the participants) recorded their response as ‘strongly agree’ in the post-test as compared to 8 students (14%) in the pre-test.

In the self-confidence questionnaire analysis, 5 out of 15 statements showed clear changes in terms of responses obtained from participants. For example, for the statement ‘I am always under a terrible strain in a mathematics class’, 23 students (42% of the participants) recorded the response ‘strongly disagree’ in the post-test as compared to 9 participants (16%) in the pre-test. Interestingly, for the statement ‘When I hear the word mathematics, I have a feeling of dislike’, there were no responses in favour of ‘strongly agree’ or ‘agree’ in the post-test questionnaire as compared to 9 responses (2 for ‘strongly agree’ and 7 for ‘agree’) in the pre-test.

4.1.2 Gender-based findings

A descriptive analysis was performed to explore for any gender-based differences in the mean scores. Table 1 compares the mean scores of male and female participants in both the pre- and post-intervention tests and these are further illustrated by the line graph in Fig. 3 .

Comparison of mean scores gender-wise

Gender-based differences.

A general look at the line graph determines that females scored slightly higher on both the EN and SC subscales for the pre-intervention tests. While this trend continued in the post-intervention tests, the scores between both groups were closer.

To test the statistical significance of these gender-based differences, an independent-samples t -test was conducted on the mean scores of both males and females for each of the subscales, both before and after the intervention.

Regarding the enjoyment subscale, the tests recorded t (53) = −0.5, p  > 0.05 (two-tailed) for the pre-intervention and t (53) = −0.3, p  > 0.05 (two-tailed) for the post-intervention. For the self-confidence subscale, the tests showed the values as t (53) = −0.9, p  > 0.05 (two-tailed) for pre-intervention and t (53) = −0.4, p  > 0.05 (two-tailed) for the post-intervention. These findings revealed that differences between the scores obtained by the two gender groups were not statistically significant.

4.2 Qualitative findings

An analysis of participants’ reflections provided an interesting glimpse of their thoughts and opinions about writing. Based on responses that emerged from the qualitative data, participants’ reflections were coded into the six themes which are outlined in Table 2 . This table also shows the percentage of responses under each theme in the data sample.

% Distribution of themes for feedback responses

As evident from Table 2 , 29% of the participants related the writing activity to increased content knowledge and a greater understanding of the mathematics (either directly or indirectly). Many of these responses signalled the importance of connections to prior knowledge using phrases such as ‘ I need to know first what is average ’ and ‘ you have to remember what you have done before ’ e.g. a student wrote the following: ‘ for solving, I just need to remember the formula but for writing I have to remember the work of [the] previous class also and how it comes ’.

Support for non-gradation of the writing assignments ranked second in the list of themes that emerged from the data and was noted in 20% of the respondents’ reflections. Furthermore, 15% of participants expressed that it was easier to explain through writing as opposed to an oral explanation. For example, one student expressed ‘ I like it. I know all [the] answers, but I am shy to speak it to all my class. When [the] teacher tells me to explain it, I cannot do it. But by writing, I can explain. So, I like it ’. Moreover, 11% of participants also noted an ‘ improved efficiency in writing ’ in general.

However, the lengthy nature of writing activities was a cause of concern for 16% of participants, as could be found in their responses for feedback which included phrases like ‘ lengthy activity ’, ‘ It takes a long time to write ’ and ‘solving is quicker than writing ’.

Finally, 9% of the participants found the task of writing challenging due to a lack of vocabulary to express themselves. They mentioned that they were more comfortable in solving questions mathematically as opposed to writing explanations using the English language.

Sample responses from participants’ reflections under the above-mentioned themes are presented in Table 3 .

Sample responses for participants’ attitude towards writing

There were also some responses where the participants liked the activity provided some conditions were met. Their responses marked the presence of ‘If’, ‘but’, etc. For example,

‘…. if teacher is there to guide me how to write and explain’

‘… but not for big questions. You have to write a lot’

‘……only when it is for activity, not for exam’

A full excerpt from a student’s similar response is cited here for reference:

‘I think writing helps to understand more by explaining and also it is a good brain exercise when we have to remember what we done before in the previous class. But in the [examination] paper, we have to solve only and then we get full marks. So, we should not waste time in writing because in this time we can solve many questions if we remember the formula’.

Although this student acknowledged the benefit of writing in terms of understanding more, they preferred giving a procedural solution or stating the answer directly. This student recognized that memorizing formulae is enough for getting good marks in an examination (owing to the particular marking scheme adopted). This raises the question of whether current assessment systems and education policies are really in favour of developing students’ understanding or are merely ranking procedures that reward rote learning.

A detailed discussion of these findings with illustrative examples and links to the research questions and relevant literature is included in the next section.

In this section, each research question is addressed, and findings discussed with relevant supporting references from the literature.

5.1 Research question 1: what effects, if any, does a short mathematical writing intervention have on students’ self-confidence and enjoyment levels in mathematics?

5.1.1 impact of the intervention on students’ enjoyment.

The analysis of the quantitative data revealed that participants’ mean enjoyment scores increased from 34.79 (prior to the intervention) to 39.82 (after the intervention). Results of a paired-samples t -test confirmed the statistical significance of this increase. In addition, these findings were supported through participants’ reflections of their perceptions of the writing intervention. Responses such as ‘ I like it ’, ‘ It was enjoyable ’, ‘… I like my maths class as this ’ indicate participants’ perceived enjoyment and positive attitude towards the writing intervention. A few students even expressed a desire for writing to be a regular feature of their mathematics lessons. For instance,

‘...Can we have it in our daily class also?’

‘…I think it should be weekly or on alternate days’

‘…We should have it in our daily class also’

This enjoyment was facilitated in a number of ways. For example, the role of a comfortable and interactive classroom environment was emphasized in the literature ( Firmender et al. , 2017 ; Hidi, 2000 ) and this was given due consideration throughout this study. The aim was to make students feel stress-free while writing their solution strategies.

5.1.2 Impact of the intervention on students’ self-confidence

Writing can develop greater confidence in mathematics by providing students with opportunities to grapple with mathematical ideas ( Powell, 1997 ). This assertion is particularly relevant to the present study, given the results of the self-confidence questionnaire. The increase in participants’ mean scores for self-confidence from 49.48 (pre-test) to 54.40 (post-test) is statistically significant as confirmed by the paired-samples t -test. The results are also supported by participants’ reflections of the writing intervention.

Participants felt an increase in confidence for a variety of reasons, for example, a greater understanding, improved efficiency in writing, clarity of thoughts, etc. Some excerpts from participants’ reflections are provided in Fig. 4 to highlight the findings specifically in relation to confidence.

Sample examples relating to confidence.

Sample of participants’ work.

Sample examples relating to non-gradation.

The use of phrases such as ‘ was sure’ , ‘ understood more’ and ‘ can improve’ in these excerpts are indicative of students’ increased confidence through writing and help to reaffirm the quantitative findings.

The reflection process which occurs during writing provides learners with an opportunity to look at their own thoughts and refine them in accordance with the information to be conveyed. ‘ Such acquisition of control and monitoring capabilities engenders in students’ feelings of accomplishment...students develop faith in themselves as learners who are capable of doing and understanding mathematics ’ ( Powell, 1997 , p. 23). According to Bandura (1977 ), performance accomplishments are the most powerful contributors to one’s self-efficacy beliefs. The confidence gained through these small accomplishments leads to motivation which brings further enjoyment for the task at hand.

5.2 Research question 2: what are students’ perceptions about the use of mathematical writing in the classroom?

A variety of student perceptions about the use of mathematical writing emerged from participants’ reflections. As detailed in Section 4.2, these responses were categorized into themes which will now be discussed considering similar studies from the literature.

5.2.1 Increase in content knowledge/understanding

As mentioned previously, 29% of participants noted an increase in their content knowledge and/or understanding. One participant wrote the following:

‘I don’t want to do maths for marks. This activity I enjoyed because you are doing it for your understanding, not as a paper (exam). I understood more when I solved my answers by writing’.

Participants’ feedback included statements such as ‘able to understand more by writing’, ‘writing helps to understand in a good way’, ‘by writing, there is less confusion for the answer’ and that ‘it will also clear any doubts that you had since you have to provide reasons on why your answer is correct’. Figure 5 presents a students’ work from the six sessions of the intervention, along with the feedback for how the student felt about the writing intervention.

These findings resonate with the results from many other studies which have reported the instrumental role of writing in a greater acquisition of content knowledge through a deeper engagement with the subject (for example, Borasi & Rose, 1989 ; Craig, 2016 ; Porter & Masingila, 2000 ; Pugalee, 2004 ).

A noteworthy observation was the reflection of thoughts through writing, as shown by the following response.

‘ When we write, we can revise it many times and we can know if we get a right or wrong answer. If I see that I am thinking wrong, I can start again with some other method but if we are just explaining orally, you have said all [the] words, and you cannot go back and change your answer. So, I like this part of writing that you can see what you are thinking and change it any time before submitting [the] final answer ’.

This excerpt is indicative of the importance of the reflection process that occurs while writing. Even though students may not recognize this on-going process, it is one of the many potential benefits of writing ( Craig, 2011 ; Ray Parsons, 2011 ). Effective learning occurs while resolving the cognitive conflicts in writers’ minds and results in metacognitive development ( Kuzle, 2013 ; Pugalee, 2004 ).

Sample examples relating to preference over oral explanations.

Sample examples relating to written efficiency.

5.2.2 Support for non-gradation of the writing assignments

In the qualitative data, many participants (20%) attributed the non-grading criterion of the written tasks as one of the reasons that they were comfortable with writing. They commented that while writing, they did not experience fear or anxiety of mathematics and enjoyed the activity without any stress as ‘it was not an exam’. Some participants’ reflections relevant to this are provided in Fig. 6 .

The literature provides evidence that anxiety regarding grading and assessment not only disrupts students’ capability to reason and understand but also causes a disliking for the subject ( Wells, 1994 ). In other words, the fear of being assessed may obstruct learning from taking place naturally and may hold back students from even attempting various mathematical tasks. This raises concerns regarding the adequacy of current assessment systems which often fail to assess the process and are more focused on the product of learning ( Little et al. , 2017 ). Another point of importance here is that participants were more comfortable because their wrong answers or mistakes were not highlighted. This is noteworthy, especially in the domain of mathematics where making mistakes can be an integral part of the learning experience. The fear of making mistakes may inhibit the brain’s growth and capacity to learn and understand ( Boaler & Dweck, 2016 ).

5.2.3 Ease of written explanation as opposed to oral

Among the various benefits of writing is its efficacy to reach out to diverse learners ( Bakewell, 2008 ). This assertion proved to be particularly true for the current study where approximately 15% of participants expressed being more comfortable with written explanations as opposed to oral. There were varied reasons for this response with many signalling a lack of confidence for class interactions. Some excerpts are provided in Fig. 7 to illustrate this.

These findings, as well as others from the literature, highlight that writing is a useful medium for empowering students who feel too shy to take part in class discussions. For example, a year-long study by Fry and Villagomez (2012) in the USA showed that writing helped introverts who seldom took part in class interactions. Students who participated in the writing-to-learn activities of that study showed an increased engagement with the course content. Furthermore, other research (for example, Pugalee, 2004 ) also notes the benefits of written explanations over oral, thus providing a rationale for writing to be an important vehicle for learning.

5.2.4 Improved efficiency in writing

Another feature perceived by participants in favour of writing was improved competence in their writing skills. Some students (11%) commented that they felt an improvement as they learned new words and gained confidence for writing. A few participants also expressed that they expect to improve further if they keep practicing. Figure 8 provides some of the participants’ reflections under this theme.

These findings are supported by existing literature in the field. It has been reported that an improved use of vocabulary (both in terms of formal mathematical vocabulary as well as the usage of complete sentences and linking words) is an associated advantage of providing reasoning in mathematical writing ( Cohen et al. , 2015 ). Rubenstein (2007 ) contends that in order to communicate mathematically, students must learn how to use correct mathematical language and this learning is supported through writing. Although participants in this study did not report an improvement in mathematics vocabulary, in particular, it could be expected that enjoyment and confidence gained through an improved comprehension in general may enthuse them with a liking for mastering discipline-specific language.

5.2.5 More time consuming

There were mixed responses from respondents in relation to the time-factor. Nine out of 55 students commented that writing takes a longer time and that they preferred giving a direct answer to the problem. Some of these contended that the aim of solving a mathematical task is getting the correct answer and thus viewed writing as a ‘waste of time’. A smaller number also felt that even though writing results in improved learning and is a good brain exercise, the lengthy nature of this activity trumps its benefits and thus, it may be ‘good for some problems but not for all’. Excerpts from some participants’ responses that fall into this category are presented in Fig. 9 .

The time-consuming nature of writing, as reported by the participants of this study, has also been reported by many others. In fact, the constraint of time acts as a potential drawback to the implementation of writing in regular teaching ( Baxter et al., 2005 ; McIntosh & Draper, 2001 ). On the contrary, Porter and Masingila (2000 ) assert that the success of writing in promoting a deeper mathematical understating might be primarily due to the increased time that is spent on writing for a given task. In fact, they consider whether the primary contributor in the process is the time spent on the task or the writing itself.

Sample examples relating to time consumption.

Sample example related to language.

5.2.6 Linguistic barriers

Although English was the language of instruction in the school where this study was carried out, 5 out of 55 students manifested a difficulty in using the English language for their explanations. For these students, a lack of language proficiency hindered their ability to explain their reasoning. For example, one student wrote the following: ‘… I know the maths of answer but not English words ’. Findings from other studies (for example, Craig, 2011 ; Porter & Masingila, 2000 ) confirm the prevalence of such linguistic difficulties for students.

The following reflection ( Fig. 10 ) is noteworthy and is worth mentioning with respect to this theme.

In this instance, although there were no rewards or incentives for the participants, this child wanted to be a good writer. She started learning new words to be more able to express herself. This one excerpt sets an example of how writing may instil a desire to learn more and implies that writing is a beneficial medium for inter-disciplinary learning.

This study sought to examine students’ perceptions and explore the impact of mathematical writing on students’ affective constructs of enjoyment and self-confidence. An analysis of the quantitative data revealed an increase in the mean scores for both enjoyment and self-confidence. Results of t -tests confirmed that these increases were statistically significant. A further analysis revealed there to be no gender-related differences.

A thematic analysis of participants’ reflections of the writing intervention also signalled a positive perception towards such activities. Overall, participants gave a positive response towards the intervention and reported an increase in content knowledge/understanding as the main reason. As well as supporting the non-grading of the tasks, some students also noted the ease of written explanations as opposed to oral and an improved efficiency in writing. At the same time, the time-consuming nature of the activity and a lack of proficiency in English language emerged as the factors of concern for a few participants.

In conclusion, findings from this study indicated a progressive shift in students’ attitude post-intervention. Hence, although this was a short intervention with a relatively small cohort, it can be inferred that mathematical writing has the potential to increase students’ enjoyment and self-confidence in mathematics and has a positive impact on their learning. In contrast with the traditional methods of teaching, writing activities in mathematics may serve as an effective medium for transforming students’ mindsets and fostering positive attitudes towards the subject.

However, the benefits of writing are contingent on a host of factors such as the nature of the writing tasks allotted to students, the intensity of intervention by instructors, the students’ ability to exploit its benefits, etc. It is necessary to keep these factors in mind as these may neutralize the positives that can be gained from writing. Furthermore, students and teachers may hold different views about mathematical writing which may affect the quality and nature of writing in a mathematics classroom. In addition, time-bound learning also constrains the integration of writing into classrooms. Future research might gauge the effect of other contextual factors, for example, classroom environment, motivational and constructive feedback, etc. that may aid in bringing out positive changes in students’ affective domains while learning. Additional research is also required to investigate how writing, if incorporated into the regular curriculum, may change learners’ as well as teachers’ beliefs about the nature of mathematics.

It is important to keep in mind that the absence of a comparison or control group may affect the generalizability of the results of this study and that the positive results may not be solely due to the intervention. Several other factors such as the non-routine nature of the mathematics tasks, the activity-based sessions, non-grading of work, absence of teacher, etc., could have contributed to the findings. Nonetheless, the results of the study point to the potential of mathematical writing to be used as an effective scaffolding tool for students’ mathematics learning.

Additionally, the effect of various other contextual factors cannot be ignored. There is a chance that participants’ enhanced enjoyment and self-confidence in this study could have been affected by the nature of the research context itself. For example, participants may be keen to impress and please the researcher which might also have acted as a restraining factor affecting the results of the study in a positive or negative manner. Activity-based intervention with no assessment involved may also have contributed to the heightened enjoyment and reduced levels of anxiety in participants.

Finally, the study has emphasized a domain of mathematics education where there is a dearth of research. It has thus added value to the existing body of research and is particularly enriching from an Indian context. In terms of future educational policy, it may prove to be helpful and act as a starting point for further research in India and indeed in other education systems around the world.

Ahmed , W. , Minnaert , A. , van der Werf , G. & Kuyper , H. ( 2010 ) Perceived social support and early adolescents’ achievement: the mediational roles of motivational beliefs and emotions . J. Youth Adolesc. , 39 , 36 – 46 .

Google Scholar

Ary , D. , Jacobs , L. & Sorensen , C. ( 2010 ) Introduction to Research in Education . USA: Wadsworth Cengage Learning .

Google Preview

Bakewell , L. L. ( 2008 ) Writing use in the mathematics classroom . Studies in Teaching 2008 Research Digest . Winston-Salem, NC : Wake Forest University Department of Education , pp. 7 – 12 .

Bandura , A. ( 1977 ) Self-efficacy: toward a unifying theory of behavioral change . Psychol. Rev. , 84 , 191 – 215 .

Bangert-Drowns , R. , Hurley , M. & Wilkinson , B. ( 2004 ) The effects of school-based writing-to-learn interventions on academic achievement: a meta-analysis . Rev. Educ. Res. , 74 , 29 – 58 .

Baxter , J. , Woodward , J. & Olson , D. ( 2005 ) Writing in mathematics: an alternative form of communication for academically low-achieving students . Learn. Disabil. Res. Pract. , 20 , 119 – 135 .

Boaler , J. & Dweck , C. ( 2016 ) Mathematical Mindsets: Unleashing Students' Potential Through Creative Math . USA : John Wiley & Sons .

Boekaerts , M. & Rozendaal , J. S. ( 2010 ) Using multiple calibration indices in order to capture the complex picture of what affects students' accuracy of feeling of confidence . Learn. Instr. , 20 , 372 – 382 .

Borasi , R. & Rose , B. ( 1989 ) Journal writing and mathematics instruction . Educ. Stud. Math. , 20 , 347 – 365 .

Bossé , M. & Faulconer , J. ( 2008 ) Learning and assessing mathematics through reading and writing . Sch. Sci. Math. , 108 , 8 – 19 .

Bramlett , D. & Herron , S. ( 2009 ) A study of African-American college students’ attitudes towards mathematics . J. Math. Sci. Math. Educ. , 4 , 43 – 51 .

Burton , L. & Morgan , C. ( 2000 ) Mathematicians writing . J. Res. Math. Educ. , 31 , 429 – 453 .

Casa , T. M. , Firmender , J. M. , Cahill , J. , Cardetti , F. , Choppin , J. M. , Cohen , J. & Zawodniak , R. ( 2016 ) Types of and purposes for elementary mathematical writing: task force recommendations . Retrieved from http://mathwriting.education.uconn.edu .

Clark , V. L. P. & Creswell , J. W. ( 2008 ) The Mixed Methods Reader . USA: Sage .

Cohen , J. ( 1988 ) Statistical Power Analysis for the Behavioral Sciences . New Jersey : Lawrence Erlbaum .

Cohen , J. , Casa , T. , Miller , H. & Firmender , J. ( 2015 ) Characteristics of second graders' mathematical writing . Sch. Sci. Math. , 115 , 344 – 355 .

Countryman , J. ( 1992 ) Writing to Learn Mathematics . Portsmouth : Heinemann .

Craig , T. ( 2011 ) Categorization and analysis of explanatory writing in mathematics . Int. J. Math. Educ. Sci. Technol. , 42 , 867 – 878 .

Craig , T. ( 2016 ) The role of expository writing in mathematical problem solving . Afr. J. Res. Math. Sci. Technol. Educ. , 20 , 57 – 66 .

Creswell , J. W. & Plano Clark , V. L. ( 2011 ) Choosing a mixed methods design . Designing and Conducting Mixed Methods Research , 2nd ed. Thousand Oaks, CA : SAGE. pp. 53 – 106 .

Creswell , J. , Hanson , W. , Clark Plano , V. & Morales , A. ( 2007 ) Qualitative research designs . Couns. Psychol. , 35 , 236 – 264 .

Firmender , J. , Dilley , A. , Amspaugh , C. , Field , K. , LeMay , S. & Casa , T. ( 2017 ) Beyond doing mathematics: engaging talented students in mathematically creative writing . Gift. Child Today , 40 , 205 – 211 .

Fry , S. & Villagomez , A. ( 2012 ) Writing to learn: benefits and limitations . Coll. Teach. , 60 , 170 – 175 .

Hacker , D. J. , Keener , M. C. & Kircher , J. C. ( 2009 ) Writing is applied metacognition . Handbook of Metacognition in Education , 154 – 172 . New York : Routledge .

Hannula , M. S. & Malmivuori , M. L. ( 1997 ) Gender differences and their relation to mathematics classroom context . The 21st Conference of the International Group for the Psychology of Mathematics Education , vol. 3 , 33 – 40 . University of Helsinki, Lahti Research and Training Centre .

Hannula , M. S. , Maïjala , H. & Pehkonen , E. ( 2004 ) On development of pupils’ self-confidence and understanding in middle grade mathematics . The XXI Annual Symposium of the Finnish Association of Mathematics and Science Education Research , 141 – 158 . University of Helsinki, Department of Applied Sciences of Education .

Hidi , S. ( 2000 ) An interest researcher's perspective on the effects of extrinsic and intrinsic factors on motivation . Intrinsic and Extrinsic Motivation. The Secret for Optimal Motivation and Performance ( C. Sansone & J. M. Harackiewicz eds). New York : Academic Press , pp. 309 – 339 .

Hinton , C. , Miyamoto , K. & Della-Chiesa , B. ( 2008 ) Brain research, learning and emotions: implications for education research, policy and practice 1 . Eur. J. Educ. , 43 , 87 – 103 .

Kenney , R. , Shoffner , M. & Norris , D. ( 2013 ) Reflecting to learn mathematics: supporting pre-service teachers’ pedagogical content knowledge with reflection on writing prompts in mathematics education . Reflective Pract. , 14 , 787 – 800 .

Knox , H. ( 2017 ) Using writing strategies in math to increase metacognitive skills for the gifted learner . Gift. Child Today , 40 , 43 – 47 .

Kostos , K. & Shin , E. K. ( 2010 ) Using math journals to enhance second graders’ communication of mathematical thinking . Early Child. Educ. J , 38 , 223 – 231 .

Kuzle , A. ( 2013 ) Promoting writing in mathematics: prospective teachers' experiences and perspectives on the process of writing when doing mathematics as problem solving . Cent. Educ. Policy Stud. J. , 3 , 41 – 59 .

Levin , B. ( 2000 ) Putting students at the centre in education reform . J. Educ. Change , 1 , 155 – 172 .

Little , G. , Horn , J. & Gilroy-Lowe , S. ( 2017 ) Maths is more than getting the right answer: redressing the balance through observation . Forum , 59 , 207 – 216 .

Lorsbach , A. & Jinks , J. ( 1999 ) Self-efficacy theory and learning environment research . Learn. Environ. Res. , 2 , 157 – 167 .

Marx , R. W. ( 1983 ) Student perception in classrooms . Educ. Psychol. , 145 – 164 .

Matsuura , R. , Sword , S. , Piecham , M. B. , Stevens , G. & Cuoco , A. ( 2013 ) Mathematical habits of mind for teaching: using language in algebra classrooms . Math. Enthus. , 10 , 735 – 776 .

McIntosh , M. E. & Draper , R. J. ( 2001 ) Using learning logs in mathematics: writing to learn . Math. Teach. , 94 , 554 – 557 .

Miller , S. D. & Meece , J. L. ( 1997 ) Enhancing elementary students’ motivation to read and write: a classroom intervention study . J. Educ. Res. , 90 , 286 – 299 .

National Council of Teachers of Mathematics ( 1989 ) Curriculum and Evaluation Standards for School Mathematics . Reston, VA : NCTM .

National Council of Teachers of Mathematics ( 2000 ) Principles and Standards for School Mathematics . Reston, VA : NCTM .

National Curriculum Framework ( 2005 ) National Council of Educational Research and Training . http://www.ncert.nic.in/rightside/links/pdf/framework/english/nf2005.pdf .

NCCA ( 2018 ) Junior Cycle Specification . https://www.jct.ie/perch/resources/maths/junior-cycle-mathematics-specification-2018.pdf .

Nunnally , J. C. & Bernstein , I. H. ( 1994 ) Psychometric Theory (McGraw-Hill Series in Psychology) , vol. 3 . New York : McGraw-Hill .

Porter , M. K. & Masingila , J. O. ( 2000 ) Examining the effects of writing on conceptual and procedural knowledge in calculus . Educ. Stud. Math. , 42 , 165 – 177 .

Powell , A. B. ( 1997 ) Capturing, examining, and responding to mathematical thinking through writing . Clearing House , 71 , 21 – 25 .

Pugalee , D. K. ( 2001 ) Writing, mathematics, and metacognition: looking for connections through students' work in mathematical problem solving . Sch. Sci. Math. , 101 , 236 – 245 .

Pugalee , D. K. ( 2004 ) A comparison of verbal and written descriptions of students' problem solving processes . Educ. Stud. Math. , 55 , 27 – 47 .

Quitadamo , I. & Kurtz , M. ( 2007 ) Learning to improve: using writing to increase critical thinking performance in general education biology . CBE Life Sci. Educ. , 6 , 140 – 154 .

Ray Parsons , M. ( 2011 ) Effects of writing to learn in pre-calculus mathematics on achievement and affective outcomes for students in a community college setting: a mixed methods approach . Doctoral Dissertation, Colorado State University. Libraries .

Roese , N. J. & Sherman , J. W. ( 2007 ) Expectancy . Social Psychology: Handbook of Basic Principles , pp. 91 – 115 . New York : The Guilford Press .

Rubenstein , R. N. ( 2007 ) Focused strategies for middle-grades mathematics vocabulary development . Math. Teach. Middle School , 13 , 200 – 207 .

Sakiz , G. , Pape , S. J. & Hoy , A. W. ( 2012 ) Does perceived teacher affective support matter for middle school students in mathematics classrooms? J. Sch. Psychol. , 50 , 235 – 255 .

Stankov , L. , Morony , S. & Lee , Y. P. ( 2014 ) Confidence: the best non-cognitive predictor of academic achievement? Educ. Psychol. , 34 , 9 – 28 .

Struyven , K. , Dochy , F. & Janssens , S. ( 2005 ) Students’ perceptions about evaluation and assessment in higher education: a review . Assess. Eval. High. Educ. , 30 , 331 – 347 .

Tapia , M. & Marsh , G. E. ( 2004 ) An instrument to measure mathematics attitudes . Acad. Exch. Q. , 8 , 16 – 22 .

TIMSS ( 2011 ) Assessment . International Association for the Evaluation of Educational Achievement (IEA) . MA : TIMSS & PIRLS International Study Center .

Wells , D. ( 1994 ) Anxiety, insight and appreciation . Math. Teach. , 147 , 8 – 11 .

Westen , D. ( 1999 ) Psychology . New York : John Wiley & Sons .

Wolfe , P. & Brandt , R. ( 1998 ) What do we know from brain research? Educ. Leadersh. , 56 , 8 – 13 .

Tandeep Kaur is a research associate and a doctoral student at the Institute of Education, Dublin City University. She has extensive experience in teaching mathematics at the secondary and senior secondary level. Her research interests lie in exploration of best practices in mathematics education, initial teacher education and students’ mental health and well-being. E-mail: [email protected]

Mark Prendergast is a senior lecturer in Education in the School of Education at University College Cork. His teaching and research interests include mathematics education, teacher education and working with non-traditional students. E-mail: [email protected]

(All tasks were selected from TIMSS (2011) 8th-Grade Mathematics Concepts and Mathematics Items SOURCE: TIMSS 2011 Assessment. Copyright © 2013 International Association for the Evaluation of Educational Achievement (IEA). Publisher: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College, Chestnut Hill, MA and International Association for the Evaluation of Educational Achievement (IEA), IEA Secretariat, Amsterdam, the Netherlands.)

Each student in team B jumped farther than any student in team A.

After every student in team A jumped, there was a student in team B who jumped farther.

As a group, team B jumped farther than team A.

Some students in team A jumped farther than some students in team B.

(Item number M042269, Cognitive Domain—Reasoning)

EN1. I get a great deal of satisfaction out of solving a mathematics problem.

EN2. I have usually enjoyed studying mathematics in school.

EN3. Mathematics is dull and boring.

EN4. I like to solve new problems in mathematics.

EN5. I would prefer to do an assignment in mathematics than to write an essay.

EN6. I really like mathematics.

EN7. I am happier in a mathematics class than in any other class.

EN8. Mathematics is a very interesting subject.

EN9. I am comfortable expressing my own ideas on how to look for solutions to a difficult problem in mathematics.

EN10. I am comfortable answering questions in mathematics class.

SC1. Mathematics is one of my most dreaded subjects.

SC2. My mind goes blank and I am unable to think clearly when working with mathematics.

SC3. Studying mathematics makes me feel nervous.

SC4. Mathematics makes me feel uncomfortable.

SC5. I am always under a terrible strain in a mathematics class.

SC6. When I hear the word mathematics, I have a feeling of dislike.

SC7. It makes me nervous to even think about having to do a mathematics problem.

SC8. Mathematics does not scare me at all.

SC9. I expect to do fairly well in any mathematics class I take.

SC10. I am always confused in my mathematics class.

SC11. I have a lot of self-confidence when it comes to mathematics.

SC12. I am able to solve mathematics problems without too much difficulty.

SC13. I feel a sense of insecurity when attempting mathematics.

SC14. I learn mathematics easily.

SC15. I believe I am good at solving mathematics problems.

(ATMI; Tapia & Marsh, 2004 ).

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ISSN 2719-146X (Print) 2719-1478 (Online)

Attitude and Perception towards Mathematics: A study of School Students

Rekha mahajan, volume 2 july 2021, presented in 2nd international conference on multidisciplinary industry and academic research (icmiar), 2021-07-31.

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For over thirty years, researchers have been investigating students’ attitudes and beliefs towards mathematics. In the present study, 113 responses were received from math students from Class 6 to Class 12. The Mathematics Attitudes and Perceptions Survey (MAPS) was used to assess the perception of the students’ right at school level and identify the difficulties faced by school children in understanding mathematics. The demography of the respondents showed more female students (58.4%) who did not get any awards in mathematics (73.4%). The results indicated that most of the mathematics students in school had an attitude for mathematics, persistence on problem solving especially their use in daily life and high motivation and interest in mathematics. The relationship among the components indicated strong correlation between age of students and class. There is a significant and positive correlation between component ‘Answers’ with all other components except ‘Real World’ and ‘Interest’. ‘Sense Making’ had significant and positive correlation with all other components. Similarly, ‘Persistence in problem solving’ too had significant and positive correlations with all the components except ‘Real World’. There is strong correlation among ‘Persistence’, ‘Sense making’, ‘Interest’ and ‘Answers’. The present study helps in identifying and guiding students to pursue mathematics based on their attitude and beliefs towards the subject.

Keywords: attitude towards mathematics, confidence, growth mindset, mathematics education, perceptions of mathematics, problem solving, real world

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Students’ Perception towards Mathematics and Its Effects on Academic Performance

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Mathematics Applied to the Economy and Sustainable Development Goals: A Necessary Relationship of Dependence.

Mobile computer-supported collaborative learning for mathematics: a scoping review, modeling student’s interest in mathematics: role of history of mathematics, peer-assisted learning, and student’s perception, does taking additional maths classes in high school affect academic outcomes, influence of students’ perception of mathematics on junior secondary school students’ academic performance in yala local government area of cross river state, nigeria, teachers' beliefs and conceptions: a synthesis of the research., beliefs a hidden variable in mathematics education, assessing the relationship between attitude toward mathematics and achievement in mathematics: a meta-analysis., the role of attitudes in learning mathematics., beliefs, attitudes, and emotions: new views of affect in mathematics education, trending questions (3).

The provided paper does not specifically mention the academic performance of Humss strand students or how their perception of Mathematics affects their performance. The paper focuses on the perception of students towards Mathematics and its effect on academic performance in Ghanaian senior high schools.

The paper states that students have a positive perception towards mathematics, but they also find it difficult. However, the study found that students' perception towards mathematics does not have a significant influence on their academic performance.

The paper does not provide specific information about students' perception about the nature of mathematics.

High School Mathematics at Work: Essays and Examples for the Education of All Students (1998)

Chapter: introduction, introduction.

Society's technological, economic, and cultural changes of the last 50 years have made many important mathematical ideas more relevant and accessible in work and in everyday life. As examples of mathematics proliferate, the mathematics education community is provided with both a responsibility and an opportunity. Educators have a responsibility to provide a high-quality mathematics education for all of our students. A recent report of the National Academy of Sciences (NAS) entitled Preparing for the 21st Century: The Education Imperative (National Research Council [NRC], 1997) neatly summarizes this point:

… today, an understanding of science, mathematics, and technology is very important in the workplace. As routine mechanical and clerical tasks become computerized, more and more jobs require high-level skills that involve critical thinking, problem solving, communicating ideas to others and collaborating effectively. Many of these jobs build on skills developed through high-quality science, mathematics, and technology education. Our nation is unlikely to remain a world leader without a better-educated workforce. (p. 1)

These economic and technological changes also present an opportunity for providing that high-quality education. Specifically, there is rich mathematics in workplace applications and in everyday life that can contribute to the school curriculum. Thus, today's world not only calls for increasing connection

between mathematics and its applications, but also provides compelling examples of mathematical ideas in everyday and workplace settings. These examples can serve to broaden the nation's mathematics education programs to encompass the dual objectives of preparing students for the worlds of work and of higher education. Furthermore, such programs can provide students with the flexibility to return to higher education whenever appropriate in their career paths. By illustrating the commonalities among the mathematical expectations for college, for work, and for everyday life, and by illustrating sophisticated uses of mathematics taught in high schools as well as in community colleges, this document aims to offer an expanded vision of mathematics. Mathematics based in the workplace and in everyday life can be good mathematics for everyone.

High School Mathematics at Work is a collection of essays and illustrative tasks from workplace and everyday contexts that suggest ways to strengthen the mathematical education of all students. The essays are written by a wide range of individuals who have thought deeply about mathematics education and about the futures of today's students, from mathematics educators to business leaders, from mathematicians to educational researchers, from curriculum developers to policy makers. The essays and tasks in High School Mathematics at Work not only underscore the points made in The Education Imperative (NRC, 1997), but also begin to explore connections between academic mathematics and mathematics for work and life.

As a step toward examining ways in which our schools and colleges can better serve the needs of both academic and vocational education, the National Research Council (NRC) of the National Academy of Sciences hosted a workshop in 1994 that resulted in a report entitled Mathematical Preparation of the Technical Work Force (NRC, 1995). Participants discussed questions such as

• How can mathematics content and technical applications of mathematics be integrated into educational programs?
• Should algebra continue to be the ''critical filter" used to determine whether or not students will be admitted into youth apprenticeship programs?
• Is the mathematics included in technical education programs consistent with emerging educational and occupational skills standards?
• Is it possible (or desirable) to design a core mathematics curriculum for the high school and community college levels that prepares students both for further formal education and for immediate employment in the technical work force? (p. 6)

High School Mathematics at Work continues discussion of these questions, and considers in particular how workplace and everyday mathematics can enrich mathematics teaching and learning.

Though the nominal mathematical content of this volume is high school mathematics, consideration of the above issues will lead to implications for colleges as well. For example, some two-year colleges have moved toward programs that include contextual learning and work-based experiences to enhance academic learning, often through articulated 2+2 partnerships that combine two years of course-work in high school with two years at a community college. The movement toward work-based learning has gained momentum in recent years through the School-to-Work Opportunities Act of 1994, administered jointly by the Departments of Education and Labor, and through the Advanced Technological Education program at the National Science Foundation. Both programs emphasize high academic expectations and require strong connections among schools, two-year colleges, businesses, and industry. By bringing these issues to the attention of the broader college and school communities, and by promoting higher mathematical expectations for all students, this document might provide an opportunity for schools and colleges to reconsider the mathematics courses before calculus, perhaps leading to new conceptualizations of their remedial, developmental, and "liberal arts" courses.

Fundamentally, High School Mathematics at Work is about mathematics. Its view of mathematics and mathematics learning recognizes a potential symbiotic relationship between concrete and abstract mathematics, each contributing to the other, enhancing their joint richness and power. This view is not new. Historically, much mathematics originated from attempts to solve problems from science and engineering. On the other hand, solutions to many problems from science and engineering have been based on creative ways of applying some mathematics that until then had no known applications. Mathematics can help solve problems, and complex workplace problems can help stimulate the creation of new mathematics.

Embracing this connected view of mathematics requires more than addressing content issues. In this document, the essays and tasks are organized according to four themes, each considering a different aspect of the many challenges involved in creating an enriched mathematics education for students. Each theme is introduced by an overview that provides a context for and a summary of the essays and tasks that follow. The first theme, Connecting Mathematics with Work and Life , sets the stage for the document as a whole, examining why and how "real world problems" can be used to enhance the learning of mathematics. With that premise, the remaining themes emphasize implications for various components of the educational system. The Roles of Standards and Assessments highlights the roles of standards and assessments in maintaining and also changing a vision of mathematics education. Curricular Considerations explores ways of designing curricula that attend to the needs of a diverse citizenry. Finally, Implications for Teaching and Teacher Education underscores the background and support teachers must have to respond to the needs of today's students.

Many of the issues raised by these essays are quite complex; no single essay provides a definitive resolution for any of these issues, and in fact, on some matters, some of the essayists disagree. Collectively, these essays point toward a vision of mathematics education that simultaneously considers the needs of all students. High School Mathematics at Work , however, unlike many documents produced by the National Research Council, is not a consensus document. The intent of this document is to point out some mathematical possibilities that are provided by today's world and to discuss some of the issues involved—not to resolve the issues, but to put forward some individual and personal perspectives that may contribute to the discussion.

Under each theme, the essays are accompanied by several tasks that illustrate some of the points raised in those essays, though many of the tasks could appropriately fit under several of the themes. The tasks serve as examples of where today's world can provide good contexts for good mathematics. They never were intended to represent, or even suggest, a full menu of high school mathematics. They provide possibilities for teaching. They exemplify central mathematical ideas and simultaneously convey the explanatory power of mathematics to help us make sense of the world around us. This book offers an existence proof: one can make connections between typical high school mathematics content and important problems from our everyday lives. And, it makes an important point: that the mathematics we learn in the classroom can and should help us to deal with the situations we encounter in our everyday lives. But High School Mathematics at Work is not only about relevance and utility. The mathematics involved is often generalizable; it often has aesthetic value, too. Mathematics can be beautiful, powerful, and useful. We hope you will discover all three of these virtues in some of the examples.

At a time when analysts of the Third International Mathematics and Science Study (TIMSS) have characterized the K-12 mathematics curriculum as "a mile wide and an inch deep" (Schmidt, McKnight & Raizen, 1996) this report does not advocate that tasks like the ones in this volume merely augment the curriculum. Rather, it suggests that tasks like these can provide meaningful contexts for important mathematics we already teach, including both well-established topics such as exponential growth and proportional reasoning, as well as more recent additions to the curriculum, such as data analysis and statistics.

Collectively, these essays and tasks explore how mathematics supports careers that are both high in stature and widely in demand. By suggesting ways that mathematics education can be structured to serve the needs of all students, the Mathematical Sciences Education Board (MSEB) hopes to initiate, inform, and invigorate discussions of how and what might be taught to whom. To this end, High School Mathematics at Work is appropriate for a broad audience, including teachers, teacher educators, college faculty, parents, mathematicians, curriculum designers, superintendents, school board members, and policy makers—in short, anyone interested in mathematics education.

For those who teach mathematics, the essays might provide new ways of thinking about teaching and learning; the tasks might provide ideas for the classroom. For parents, this book can give a sense of how mathematics can be powerful, useful, beautiful, meaningful, and relevant for students. And for those who influence educational policy, this book might motivate a search for curricula with these virtues.

As with all of the recent published work of the MSEB, High School Mathematics at Work is meant to be shared by all who care about the future of mathematics education, to serve as a stimulus for further discussion, planning, and action. All those who contributed to this report would be delighted if teachers gave copies to school board members, college faculty gave copies to deans, curriculum developers gave copies to publishers, employers gave copies to policy makers, and so on. Only through continued, broad-based discussion of curricular issues can we implement change and raise our expectations of what students know and are able to do.

National Research Council. (1995). Mathematical preparation of the technical work force . Washington, DC: National Academy Press.

National Research Council. (1997). Preparing for the 21st century: The education imperative . Washington, DC: National Academy Press.

Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1996) . A splintered vision: An investigation of U.S. science and mathematics education . Dordrecht, The Netherlands: Kluwer Academic Publishers.

Traditionally, vocational mathematics and precollege mathematics have been separate in schools. But the technological world in which today's students will work and live calls for increasing connection between mathematics and its applications. Workplace-based mathematics may be good mathematics for everyone.

High School Mathematics at Work illuminates the interplay between technical and academic mathematics. This collection of thought-provoking essays—by mathematicians, educators, and other experts—is enhanced with illustrative tasks from workplace and everyday contexts that suggest ways to strengthen high school mathematical education.

This important book addresses how to make mathematical education of all students meaningful—how to meet the practical needs of students entering the work force after high school as well as the needs of students going on to postsecondary education.

The short readable essays frame basic issues, provide background, and suggest alternatives to the traditional separation between technical and academic mathematics. They are accompanied by intriguing multipart problems that illustrate how deep mathematics functions in everyday settings—from analysis of ambulance response times to energy utilization, from buying a used car to "rounding off" to simplify problems.

The book addresses the role of standards in mathematics education, discussing issues such as finding common ground between science and mathematics education standards, improving the articulation from school to work, and comparing SAT results across settings.

Experts discuss how to develop curricula so that students learn to solve problems they are likely to encounter in life—while also providing them with approaches to unfamiliar problems. The book also addresses how teachers can help prepare students for postsecondary education.

For teacher education the book explores the changing nature of pedagogy and new approaches to teacher development. What kind of teaching will allow mathematics to be a guide rather than a gatekeeper to many career paths? Essays discuss pedagogical implication in problem-centered teaching, the role of complex mathematical tasks in teacher education, and the idea of making open-ended tasks—and the student work they elicit—central to professional discourse.

High School Mathematics at Work presents thoughtful views from experts. It identifies rich possibilities for teaching mathematics and preparing students for the technological challenges of the future. This book will inform and inspire teachers, teacher educators, curriculum developers, and others involved in improving mathematics education and the capabilities of tomorrow's work force.

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Active Learning in Mathematics, Part IV: Personal Reflections

By Benjamin Braun, Editor-in-Chief , University of Kentucky; Priscilla Bremser, Contributing Editor , Middlebury College; Art Duval,  Contributing Editor , University of Texas at El Paso; Elise Lockwood,  Contributing Editor , Oregon State University; and Diana White,  Contributing Editor , University of Colorado Denver.

Editor’s note: This is the fourth article in a series devoted to active learning in mathematics courses.  The other articles in the series can be found here .

In contrast to our first three articles in this series on active learning, in this article we take a more personal approach to the subject.  Below, the contributing editors for this blog share aspects of our journeys into active learning, including the fundamental reasons we began using active learning methods, why we have persisted in using them, and some of our most visceral responses to our own experiences with these methods, both positive and negative.  As is clear from these reflections, mathematicians begin using active learning techniques for many different reasons, from personal experiences as students (both good and bad) to the influence of colleagues, conferences, and workshops.  The path to active learning is not always a smooth one, and is almost always a winding road.

Because of this, we believe it is important for mathematics teachers to share their own experiences, both positive and negative, in the search for more meaningful student engagement and learning.   We invite all our readers to share their own stories in the comments at the end of this post.  We also recognize that many other mathematicians have shared their experiences in other venues, so at the end of this article we provide a collection of links to essays, blog posts, and book chapters that we have found inspirational.

There is one more implicit message contained in the reflections below that we want to highlight.  All mathematics teachers, even those using the most ambitious student-centered methods, use a range of teaching techniques combined in different ways.  In our next post, we will dig deeper into the idea of instructor “telling” to gain a better understanding of how an effective balance can be found between the process of student discovery and the act of faculty sharing their expertise and experience.

Priscilla Bremser:

I began using active learning methods for several reasons, but two interconnected ones come to mind.  First, Middlebury College requires all departments to contribute to the First-Year Seminar program, which places every incoming student into a small writing-intensive class. The topic is chosen by the instructor, while guidelines for writing instruction apply to all seminars.  As I have developed and taught my seminars over the years, I’ve become convinced that students learn better when they are required to express themselves clearly and precisely, rather than simply listening or reading.  At some point it became obvious that the same principle applies in my other courses as well, and hence I was ready to try some of the active learning approaches I’d been hearing about at American Mathematical Society meetings and reading about in journals .

Second, I got a few student comments on course evaluations, especially for Calculus courses, that suggested I was more helpful in office hours than in lecture.  Thinking it through, I realized that in office hours, I routinely and repeatedly ask students about their own thinking, whereas in lecture, I was constantly making assumptions about student thinking, and relying on their responses to “Any questions?” for guidance, which didn’t elicit enough information to address the misunderstandings around the room. One way to make class more like office hours is to put students into small groups. I then set ground rules for participation and ask for a single set of problem solutions from each group. This encourages everyone to speak some mathematics in each class session, and to ask for clarity and precision from classmates.  Because I’m joining each conversation for a while, I get a more accurate perception of students’ comprehension levels.

This semester I’m teaching Mathematics for Teachers, using an IBL textbook by Matthew Jones . I’ve already seen several students throw fists up in the air, saying “I get it now!  That’s so cool!” How well I remember having that response to my first Number Theory course; it’s why I went into teaching at this level in the first place.  On the other hand, a Linear Algebra student who insists that  “I learn better from reading a traditional textbook” leaves me feeling rather deflated. It seems that I’ve failed to convey why I direct the course the way that I do, or at least I haven’t yet succeeded.  The truth is, though, that I used to feel the same way.  I regarded mathematics as a solitary pursuit, in which checking in with classmates was a sign of weakness.  Had I been required to discuss my thinking regularly during class and encouraged to do so between sessions, I would have developed a more solid foundation for my later learning. Remembering this inspires me to be intentional with students, and explain repeatedly why I direct my courses the way that I do.  Most of them come around eventually.

Elise Lockwood:

I have a strong memory of being an undergraduate in a discrete mathematics course, trying desperately to understand the formulas for permutations, combinations, and the differences between the two. The instructor had presented the material, perhaps providing an example or two, but she had not provided an opportunity for us to actively explore and understand why the formulas might make sense. By the time I was working on homework, I simply tried (and often failed) to apply the formulas I had been given. I strongly disliked and feared counting problems for years after that experience. It wasn’t until much later that I took a combinatorics course as a master’s student. Here, the counting material was brought to life as we were given opportunities to work through problems during class, to unpack formulas, and to come to understand the subtlety and wonder of counting. The teacher did not simply present a formula and move on, assuming we understood it. Rather, he persisted by challenging us to make sense of what was going on in the problems we solved.

For example, we once were discussing a counting problem in class (I can’t recall if it was an in-class problem or a problem that had been assigned for homework). During this discussion, it became clear that students had answered the problem in two different ways — both of them seemed to make sense logically, but they did not yield the same numerical result. The instructor did not just tell us which answer was right, but he used the opportunity to have us consider both answers, facilitating a (friendly) debate among the class about which approach was correct. We had to defend whichever answer we thought was correct and critique the one we thought was incorrect. This had the effect not only of engaging us and piquing our curiosity about a correct solution, but it made us think more carefully and deeply about the subtleties of the problem.

Now, studying how students solve counting problems is the primary focus of my research in mathematics education. My passion for the teaching and learning of counting was probably in large part formed by the frustrations I felt as an undergraduate and the elation I later experienced when I actually understood some of the fundamental ideas.

When I have been given the opportunity to teach counting over the years (in discrete mathematics or combinatorics classes, or in courses for pre-service teachers), I have tried my hardest to facilitate my students’ active engagement with the material during class. This has not taken an inordinate amount of time or effort: instead of just giving students the formulas off the bat, I give them a series of counting problems that both introduce counting as a problem solving activity and motivate (and build up to) some key counting formulas. For example, students are given problems in which they list some outcomes and appreciate the difference between permutations and combinations firsthand. I have found that a number of important issues and ideas (concerns about order, errors of overcounting, key binomial identities) can emerge on their own through the students’ activity, making any subsequent discussion or lecture much more meaningful for students. When I incorporate these kinds of activities for my students, I am consistently impressed at the meaning they are able to make of complex and notoriously tricky ideas.

More broadly, these pedagogical decisions I make are also based on my belief about the nature of mathematics and the nature of what it means to learn mathematics. Through my own experiences as a student, a teacher, and a researcher, I have become convinced that providing students with opportunities to actively engage with and think about mathematical concepts — during class, and not just on their own time — is a beneficial practice. My experience with the topic of counting (something near and dear to my heart) is but one example of the powerful ways in which student engagement can be leverage for deep and meaningful mathematical understanding.

Diana White:

What stands out most to me as I reflect upon my journey into active learning is not so much how or why I got involved, but the struggles that I faced during my first few years as a tenure-track faculty member as I tried to switch from being a good “lecturer” to all out inquiry-based learning.  I was enthusiastic and ambitious, but lacking in the skills to genuinely teach in the manner in which I wanted.

As a junior faculty member, I was already sold on the value of inquiry-based learning and student-centered teaching.  I had worked in various ways with teachers as a graduate student at the University of Nebraska and as a post-doc at the University of South Carolina, including teaching math content courses for elementary teachers and assisting with summer professional development courses for teachers.  Then, the summer before I started my current position, I attended both the annual Legacy of R.L. Moore conference and a weeklong workshop on teaching number theory with IBL through the MAA PREP program.  The enthusiasm and passion at both of these was contagious.  

However, upon starting my tenure track position, I jumped straight in, with extremely ambitious goals for my courses and my students, ones for which I did not have the skills to implement yet.  In hindsight, it was too much for me to try to both switch from being a good “lecturer” to doing full out IBL and running an intensely student centered classroom, all while teaching new courses in a new place.  I tried to do way too much too soon, and in many ways that was not healthy for either me or the students, as evidenced by low student evaluations and frustrations on both sides.

Figuring out specifically what was going wrong was a challenge, though.  Those who came to observe, both from my department and our Center for Faculty Development, did not find anything specific that was major, and student comments were somewhat generic – frustration that they felt the class was disorganized and that they were having to teach themselves the material.  

I thus backtracked to more in the center of the spectrum, using an interactive lecture  Things smoothed out and students became happier.  What I am not at all convinced of, though, is that this decision was best for student learning.  Despite the unhappiness on both our ends when I was at the far end of the active learning spectrum, I had ample evidence (both from assessments and from direct observation of their thought processes in class) that students were both learning how to think mathematically and building a sense of community outside the classroom.  To this day, I feel torn, like I made a decision that was best for student satisfaction, as well as for how my colleagues within my department perceive me.  Yet I remain convinced that my students are now learning less, and that there are students who are not passing my classes who would have passed had I taught using more active learning. (It was impossible to “hide” with my earlier classes, due to the natural accountability built into the process, so struggling students had to confront their weaknesses much sooner.)

It is hard for me to look back with regrets, as the lessons learned have been quite powerful and no doubt shaped who I am today.  However, I would offer some thoughts, aimed primarily at junior faculty.  

Don’t be afraid to start slow.  Even if it’s not where you want to end up, just getting started is still an important first step.  Negative perceptions from students and colleagues are incredibly hard to overcome.

Don’t underestimate the importance of student buy-in, or of faculty buy-in.  I found many faculty feel like coverage and exposure are essential, and believe strongly that performance on traditional exams is an indicator of depth of knowledge or ability to think mathematically.

Don’t be afraid to politely request to decline teaching assignments.  When I was asked to teach the history of mathematics, a course for which I had no knowledge of or background in, I wasn’t comfortable asking to teach something else instead.  While it has proved really beneficial to my career (I’m now part of an NSF grant related to the use of primary source projects in the undergraduate mathematics classroom), I was in no way qualified to take that on as a first course at a new university.

I have personally gained a tremendous amount from my participation in the IBL community, perhaps most importantly a sense of community with others who believe strongly in active learning.  

My first experience with active learning in mathematics was as a student at the Hampshire College Summer Studies in Mathematics program during high school.  Although I’d had good math teachers in junior high and high school, this was nothing like I’d seen before: The first day of class, we spent several hours discussing one problem (the number of regions formed in 3-dimensional space by drawing \(n\) planes), drawing pictures and making conjectures; the rest of the summer was similar.  The six-week experience made such an impression on me, that (as I realized some years later) most of the educational innovations I have tried as a teacher have been an attempt to recreate that experience in some way for my own students.

When I was an undergraduate, I noticed that classes where all I did was furiously take notes to try to keep up with the instructor were not nearly as successful for me as those where I had to do something.  Early in my teaching career, I got a big push towards using active learning course structures from teaching “ reform calculus ” and courses for future elementary school teachers.  In each case, this was greatly facilitated by my sitting in on another instructor’s section that already incorporated these structures.  Later I learned, through my participation in a K-16 mathematics alignment initiative , the importance of conceptual understanding among the levels of cognitive demand , and this helped me find the language to describe what I was trying to achieve.

Over time, I noticed that students in my courses with more active learning seemed to stay after class more often to discuss mathematics with me or with their peers, and to provide me with more feedback about the course.  This sort of engagement, in addition to being good for the students, is very addictive to me.  My end-of-semester course ratings didn’t seem to be noticeably different, but the written comments students submitted were more in-depth, and indicated the course was more rewarding in fundamental ways.  As with many habits, after I’d done this for a while, it became hard not to incorporate at least little bits of interactivity (think-pair-share, student presentation of homework problems), even in courses where external forces keep me from incorporating more radical active learning structures.

Of course, there are always challenges to overcome.  The biggest difficulty I face with including any sort of active learning is how much more time it takes to get students to realize something than it takes to simply tell them.  I also still find it hard to figure out the right sort of scaffolding to help students see their way to a new concept or the solution to a problem.  Still, I keep including as much active learning as I can in each course.  The parts of classes I took as a student (going back to junior high school) that I remember most vividly, and the lessons I learned most thoroughly, whether in mathematics or in other subjects, were the activities, not the lectures.  Along the same lines, I occasionally run into former students who took my courses many years ago, and it’s the students who took the courses with extensive active learning, much more than those who took more traditional courses, who still remember all these years later details of the course and how much they learned from it.

Other Essays and Reflections:

Benjamin Braun, The Secret Question (Are We Actually Good at Math?), http://blogs.ams.org/matheducation/2015/09/01/the-secret-question-are-we-actually-good-at-math/

David Bressoud, Personal Thoughts on Mature Teaching, in How to Teach Mathematics, 2nd Edition , by Steven Krantz, American Mathematical Society, 1999.   Google books preview

Jerry Dwyer, Transformation of a Math Professor’s Teaching, http://blogs.ams.org/matheducation/2014/06/01/transformation-of-a-math-professors-teaching/

Oscar E. Fernandez, Helping All Students Experience the Magic of Mathematics, http://blogs.ams.org/matheducation/2014/10/10/helping-all-students-experience-the-magic-of-mathematics/

Ellie Kennedy, A First-timer’s Experience With IBL, http://maamathedmatters.blogspot.com/2014/09/a-first-timers-experience-with-ibl.html

Bob Klein, Knowing What to Do is not Doing, http://maamathedmatters.blogspot.com/2015/07/knowing-what-to-do-is-not-doing.html

Evelyn Lamb, Blogs for an IBL Novice, http://blogs.ams.org/blogonmathblogs/2015/09/21/blogs-for-an-ibl-novice/

Carl Lee, The Place of Mathematics and the Mathematics of Place, http://blogs.ams.org/matheducation/2014/10/01/the-place-of-mathematics-and-the-mathematics-of-place/

Steven Strogatz, Teaching Through Inquiry: A Beginner’s Perspectives, Parts I and II,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1,  http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-2

Francis Su, The Lesson of Grace in Teaching, http://mathyawp.blogspot.com/2013/01/the-lesson-of-grace-in-teaching.html

2 Responses to Active Learning in Mathematics, Part IV: Personal Reflections

In response to Priscilla Bremser, I feel as though it is almost elementary that students who are able to precisely express themselves are better to understand the information conceptually. What I mean by this is that the students who are able to interact with the information will get a better idea of what that information means conceptually rather than the students who simply listen to lecturing.

In regards to your second point, I also find this point to be important, even though it may seem obvious. Similarly to your first point, students who get more personal interaction with the instructor will probably be more likely to understand the information that is being presented. Since I am still in school, we have been discussing the best ways to prompt questions from students. Asking “are there any questions” is not a good way to do this. Breaking up into groups is a good way to see where the students are at conceptually.

However, this may prove to be tricky at the college level because of class size. One way to battle this is to ask for thumbs (either up, down, or in the middle) as to whether they understand the information being presented. This practice will give you a good idea at where the class is as a whole in a quick snapshot and students will be less likely to feel as though they are being singled out.

A few points in this post resonated with me particularly well. First, when Priscilla said that she was more helpful in office hours than in lecture because she asked students about their own thinking in the former, I agreed with it from a student’s perspective. Making class feel more like office hours, with more one-on-one time, helps students feel more like individual learners in the classroom. By suggesting small group work in order to facilitate more participation and allow for more analysis of each student’s performance, I feel that Bremser is acknowledging the ineffectiveness of using the phrase “Any questions”, which is something I try not to use, and hate to hear in my college classes. I also can relate to what Diana White says about trying to switch teaching styles as you would flip a switch. Not having the skills necessary to be at the level you want will be frustrating, and I know that as a future teacher, I will want to be successful right out of the gate. I know that this is unreasonable, and largely impossible, but this is more of a personality flaw that I will have to suppress. When it comes to being evaluated by others, I will have to recognize that many of my evaluators were once young teachers themselves, with the same aspirations, the same experience, and probably the same results as me. I will have to be patient, and use their feedback (and my own) to improve my teaching over time, rather than overnight. I wonder if this is a good assessment of what I should expect of myself when I begin teaching.

Comments are closed.

Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.

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TOM ROCKS MATHS

Maths, but not as you know it…, does the way we speak about maths affect our perception of numbers.

Arushi Ramaiya – Commended entry in the 2021 Teddy Rocks Maths Essay Competition

Mathematics is often thought of as a universal language. Perhaps it even represents the harmony of numerous and diverse cultures. We have adopted a Hindu-Arabic counting system; taken letters from the Greek and Latin alphabets; and created a wealth of symbols to go in between. Yet, despite this, perception of numbers and mathematical understanding seems to vary from person to person. There are probably a few factors that contribute to this, but one of them may be the way in which we talk about maths, and subtle linguistic differences.

By the time you reach secondary school, counting becomes second nature. In English, that is. The moment you enter the French classroom, and learn that sixty is ‘soixante’ , seventy is ‘soixante-dix’ (sixty-ten), eighty is ‘quatre-vingt’ (four-twenty), and ninety is ‘quatre-vingt-dix’ (four-twenty-ten), counting becomes a chore again. Reflecting on this somewhat arbitrary peculiarity, you come to the realisation that English has its numerical quirks too. Why do we use eleven and twelve and then go to thirteen, fourteen and fifteen – why not oneteen and twoteen? Or the mystery of where the ‘ u ’ in forty goes? There is an inclination to say that although this is slightly bizarre, it does not affect our understanding of numbers in any meaningful way. Yet, other languages demonstrate more regular counting systems, and there is evidence that shows the greater efficacy of these, comparatively. In certain languages, one of them being Mandarin, numbers such as 21 are written as two-ten-one, and 92 is written as 9-10-2. In one study, first grade students were told to represent numbers such as 42 using blocks of ten and blocks of one. Children from the US, France or Sweden ended up being more likely to use 42 unit blocks, whereas those from Japan or Korea (where the counting system is like the Mandarin system) were more likely to use four blocks of ten and two unit blocks. While language seems a likely cause of this outcome, it would have been hard to control other factors, such as teaching style, during this experiment, which may make the link between language and mathematical understanding seem slightly tenuous. However, an experiment carried out in Wales proves the hypothesis much more convincingly. In Wales, around 80% of students are taught in English, and the other 20% are taught in modern Welsh, but they still study the same curriculum and are taught in a similar way to one another. Modern Welsh has a counting system that resembles Mandarin, rather than English. Six-year-olds taught in Welsh and English were asked to estimate where a number would lie on a number line from 0 to 100, and the Welsh children did significantly better. A professor at Oxford, Ann Dowker, explains that it is likely that it was because Welsh children had a more precise representation of two digit numbers, due to the linguistic difference in their counting system. The organisation of the number system has such a noticeable impact on the way children view numbers due to the fact that it allows children to see the relationships between numbers – viewing all natural numbers as a web rather than isolated values. It makes basic mathematical functions, such as multiplication and addition, implicit through the language itself.

Visualisation

At times, mathematical processes are described in the way they physically look on the page. Although natural and intuitive, doing so at a young age can hinder understanding of various concepts. The first and most basic example of this is with fractions. It has become fairly common to refer to fractions as ‘x over y’, for example ‘one over three’ instead of ‘one by three’ or ‘one third’. Rather than describing the mathematical meaning of the fraction, we are describing the way it looks on the page, encouraging us to view the fraction as independent numbers separated by a line, instead of a ratio or division. In a study done (Boulet 1998), when students were asked to illustrate a fraction they were given, they ended up illustrating the numerator, denominator and the fraction bar (vinculum) as separate entities, highlighting a blatant misunderstanding (through no fault of their own) of what a fraction actually is.

A similar instance of describing the visual representation rather than mathematical meaning occurs when doing addition, multiplication, and even division. Phrases such as ‘carry the one’, although simple and even useful, do not actually give a full picture of what is happening. In the same way, during long multiplication (with two two-digit numbers), we are told to ‘put a zero’ in the second row without explanation, instead of grasping the fact that we are multiplying by the tens unit. Perhaps this is linked to our inexplicably confusing counting system – I doubt that it would be as difficult to comprehend if the language itself insinuated a multiplication by ten. Similar issues arise during division and subtraction. Perhaps teaching style is partly to blame for this, but it is clear that we have coined misleading terms with which we describe mathematical actions, and that is a purely linguistic problem.

The previous issue was taking the depiction of the numbers on the page too literally; a second issue, is slightly the opposite – not focussing enough on what mathematical concepts actually look like. The first instance of this, is the number ‘a billion’, or maybe more precisely, the lack of understanding of the true disparity between a million and a billion, through just the terms themselves. Often, to fully explain the difference, comparisons need to be drawn. A clear example is time – a million seconds is 12 days; a billion seconds is 32 years (and a trillion seconds is 31,688 years!). However, with regards to this misunderstanding, there are some relevant, although damaging economic outcomes, such as the fact that fiscal policy does not discriminate between millionaires and billionaires, although it probably should, but that’s a hefty discussion for another time. It is likely that language is somewhat to blame here – a million, a billion, and even a trillion are very similar words – giving the impression that the three are looped together under the subsection of ‘large numbers’ and not leaving space for distinguishing between them.

Exponents and factorials tend to have the same effect. A simple symbol, for example the ‘!’ for factorials does no justice to the sheer size of numbers such as 50!. The brevity of the sign is what seems to be deceiving. The lack of a language-based description, and the use of solely a single punctuation mark is what alters the perception of the number, and makes it hard to visualise. Unlike the lack distinction between a million and a billion, there are no real issues that stem from this inability to visualise, but it is entertaining to see how surprised people are with facts about folding paper to the moon.

Maths is filled with seemingly mundane words which have an entirely different mathematical meaning. Looking back on this essay, I’m sure I’ve used at least five or six. Some obvious examples include the number sets – ‘complex numbers’ and ‘imaginary numbers’. I must confess – I have no studies to prove this hypothesis, but having talked amongst my peers and with my teachers, I genuinely believe that people have preconceptions about these topics even before approaching them, due to their everyday meaning: they believe ‘complex numbers’ to be complicated, and imaginary numbers to be a hard concept to grasp. Even labels like ‘rational’ and ‘irrational’ are misleading – it is not necessarily explicit that ‘rational’ refers to ‘a number that can be expressed as a ratio’, as ‘rational’ now has the more common meaning of ‘sensible’. The issue, in this case, is etymology. Words have mutated over time, yet old-fashioned terms have remained, warping our perception of numbers.

In some cases of mathematical homonyms, the distinctions between the meaning of the two words is slightly more nuanced. Take infinity, for example: mathematical infinity and colloquial infinity both refer to a similar concept, which differs in a discreet manner. A study was done to investigate whether languages which differentiate between mathematical infinity and colloquial infinity give rise to a better understanding and more sophisticated way of communicating on the topic. Korean is a language where the two infinities are separate terms, and so English and Korean students were asked to have monitored discourse on the topic of ‘infinity’. Data showed that despite similar mathematical capabilities, the approach to the discussion on ‘infinity’ was evidently dissimilar. The discourse of the English speakers was more processual; meanwhile, the Korean speakers talked of infinity in a more structured and mathematically astute way. Again, unfortunately, not all the variables can be controlled, but it seems intuitive to say that language does play a role in this observed difference.

With certain terms, most prominently, physical terms such as ‘weight’, the colloquial version is used ‘incorrectly’, if you take the mathematical definition of the word to be ‘correct’. ‘Weight’ during casual conversation refers to the definition of ‘mass’ – unchanging and measured in kilograms. In terms of physics, ‘weight’ refers to a force (measured in Newtons), and can change depending on gravitational pull. Naturally, having to almost ‘unlearn’ your definition of weight can prove quite confusing. Another example is of the phrase ‘in general’, which, in mathematics, is used to describe a result where there are no exceptions. The colloquial use of the phrase is quite the opposite – often denoting a claim that is mostly true but has a few exceptions. To correct this, mathematicians sometimes use ‘generically’, which perhaps complicates things further.

The intersection between linguistics and mathematics is a fascinating one; maybe even one that is not researched enough. Perhaps this is because our mathematical capabilities do not suffer detrimentally, even if our counting system makes little sense or if we have the same word for ‘infinity’. Yet it is ironic that, arguably, the most universal of all languages is still more variable than it seems – not just from culture to culture, but from one individual to the next. Maybe certain phrases should be avoided, as they perpetuate a less mathematical or logical way of thinking. But what I think is potentially more important, is encouraging delving more into the etymology of technical mathematical terms (even from a young age, explaining that ‘fifty’ is a modified version of ‘five-ten’); describing certain functions and concepts in a more qualitative way so that become easier to visualise; and an increased understanding of the (subtle) differences between mathematical homonyms. Exploring language, especially mathematical language will be interesting, useful and could bring to light certain strategies to improve learning. After all, we think in language – there is no doubt it influences us immensely in everything – maths is no exception.

• http://www.translatemedia.com/translation-blog/mysterious-interactions-language -mathematics/
• http://www.sciencedirect.com/science/article/abs/pii/S0883035512000055
• http://www.bbc.com/future/article/20191121-why-you-might-be-counting-in-the- wrong-language
• http://www.researchgate.net/publication/268395038_How_Does_Language_Impact_the_Learning_of_Mathematics_Let_Me_Count_the_Ways
• http://www.naldic.org.uk/Resources/NALDIC/docs/resources/documents/The%20Role%20of%20language%20in%20mathematics.pdf
• eprints.nottingham.ac.uk/14002/1/395630.pdf
• http://www.jstor.org/stable/1169995?seq=1#metadata_info_tab_contents
• scholarworks.wladenu.edu/cgi/viewcontent.cgi?article=5962&context=dissertations

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Exploring the Relationship Between Students’ Perception, Interest and Mathematics Achievement

2022, Mediterranean Journal of Social & Behavioral Research

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