A Step-by-Step Teaching Technique for Teachers With Adult Students of Mathematics

Abstract

Step-by-step teaching is a researcher-designed innovative process that takes the adult learner, step-by-step, from his present level of understanding to the required level. The technique is based on well-researched and accepted pedagogical practices set in their psychological, sociological, and andragogical perspectives. Using a convenience sample of 35 students and a pretest, posttest, control group quasi-experimental design with semistructured interviews and unobtrusive observation, the experimental group outperformed their untrained counterparts. They reported gains in their understanding of fundamental concepts that hitherto had presented them challenges for several years. There was an overall positive change in mathematical achievement, attitudes, beliefs, and self-confidence toward mathematics learning. This research could be a catalyst for more work in this understudied area.

Keywords

andragogy achievement attitude belief self-confidence adult learner

Although a number of students have had the privilege of finding their places in the world of mainstream mathematics learning from youth, some have been left out of the loop. The search for marketability has left educators with several challenges. One of these is making mathematics learning simple and understandable. Working in the only government-funded tertiary educational institution on a small island, I was inundated with learners who openly expressed their mathematics challenges. They knew that learning was mandatory for progress to the next level, and they could not graduate without it.

Learners claimed that they took mathematics classes because they were mandatory. Many were prepared to learn enough to “pass” the compulsory test and hopefully never have to take mathematics again. Others were prepared to “give it a try,” hoping that personal maturation would be in their favor. Faced with this challenge of preparing learners for the next level, I searched for innovative methods, resulting in what I refer to as step-by-step teaching (SBST).

Literature Review

This article details a researcher-developed technique for teachers. Safford-Ramus (2001) examined the content, scope, and methodology of adult mathematics education in North American dissertations. She observed that the most popular topics examined included success prediction, contributory methods for success, and learning anxiety capable of inhibiting success if not neutralized or minimized. She noted that only two dissertation abstracts focused on standards and the reform movement. Safford-Ramus found three dissertation abstracts that equated mathematics with problem solving. With no scholarly research mentioned about the effects of SBST on mathematics achievement, attitude to learning, beliefs about learning, and learner self-confidence, this article helps fill the gap.

Noncognitive factors such as attitudes to learning, beliefs about learning, and self-confidence are problematic in their precise measurement (Neale, 1969; Sam, 2002). Attitude is a hypothetical construct representing an individual’s degree of like or dislike. Attitudes may be positive or negative toward the attitude object (person, place, event, or thing). Ambivalence (having simultaneously both positive and negative dispositions toward the same person, place, event, or thing) is possible. Using Rosenberg and Hovland’s (1960) ABC (affect, behavior, and cognition) model, attitudes appear to be judgments where the affective is an emotional response, indicating the individual’s degree of preference for an entity. Behavior refers to the typical actions that the individual tends to exhibit, and cognition refers to a mental evaluation of what makes up the individual’s beliefs about the object, resulting primarily, but not exclusively, from experience or observational learning in the environment (Eagly & Chaiken, 1998). Eagly and Chaiken’s study helps the reader understand that generally there are challenges in attitude measurements from which this present study is not exempt.

Among the many attitudinal studies in learning, Akinsola and Olowojaiye (2008) looked at the effect of behavioral objective-based and study question-based instructional strategies on 312 students’ attitudes toward senior secondary learning. Using a quasi-experimental design with three groups and a Students’ Attitude Questionnaire, they found a positive significant effect of the two treatments on students’ attitudes toward learning compared with the control group. Such results imply that students’ attitudes and habits may be influenced by the classroom teachers’ instruction methods. These studies provide additional evidence for the study being reported here.

Ruffell, Mason, and Allen (1998) attest to the difficulty in precision of attitudinal research and testing the validity of attitudinal-based conjectures. They challenged cause-and-effect models underlying much research. They view attitude as a complex notion that appears to be more a construct of an observer’s need to provide an explanation for observations than a quality possessed by an individual.

Gal and Ginsburg (1994), in reviewing the role of affect and attitudes in statistics learning, critiqued current attitudinal and belief instruments in exploring assessment methods teachers can use to “gauge” students’ disposition toward the subject. Because a belief is a psychological state of mind in which an individual holds a proposition to be true, there would be variability in determining individual beliefs. Attitudes, beliefs, and self-confidence are so intertwined that it is difficult to easily disentangle them. Hence, studies measuring student attitudes, beliefs, and self-confidence in mathematics have limitations.

Step-by-Step Teaching

In SBST, information is explored in a step-by-step manner so that the learner has to show understanding of previous information before moving on. As an example, Mack (2007) showed how research on children’s geometric thinking was used together with the picture book The Greedy Triangle to understand children’s prior geometric knowledge of polygons (many-sided figures). According to the process he described, one might see the SBST technique of gathering useful information. Learners are forced to read information very carefully in order to understand what is required of them.

The researcher-designed SBST takes the learner step-by-step, in a gradual incremental manner to suit the learner’s pace, from the present level of understanding to the required level. SBST focuses on narrowing the student’s “zone of proximal development” where real learning takes place. That “zone of proximal development,” coined by Vygotsky (1978), represents the range of tasks that the student is able to complete without assistance and those that could be completed with the guidance/assistance of adults or more-skilled children. SBST literally takes learners by the hand and gently guides them step-by-step from beginning to end. Not being alone during the process, the learner progressively gains self-confidence. Learners received bonus points for making intelligent observations, making connections between concepts (past and present) and projecting futuristic ideas that could move them to the next level. The bonus points were meant to provide instant extrinsic motivation to learners who, because of their past experiences with learning, needed something to boost their morale. Bonus points were also given to control group members to ensure that no group had an added advantage over the other that could have affected the final results.

SBST embraces the basic concepts of adult learning. Knowles (1984) identified five andragogical assumptions of the adult learner:

Self-Concept: As a person matures, he or she moves from dependency to self-directness.

Experience: Adults draw on their experiences to aid their learning.

Readiness: The learning readiness of adults is closely related to the assumption of new social roles.

Orientation: As a person learns new knowledge, he or she wants to apply it immediately in problem solving.

Motivation: As a person matures, he or she receives motivation to learn from internal factors.

In guiding the student, I ensured that the atmosphere of all interactions with the student was social, friendly, and congenial. I befriended the learners, listened to their personal concerns, and offered to assist. Additionally, I showed the learners how their class activities would be beneficial for personal improvement and advancement. Most important, the work was presented as a personal interest activity and a viable escape route from boredom. In particular, one student who was having marital issues took advantage of SBST and reported on a daily basis how relieved she felt experiencing this manner of teaching. On no occasion did I operate in the role of dispenser of knowledge, instructor, or infallible director. There was always a relaxing atmosphere conducive to respectful exchanges.

In SBST, the learner understands what is involved in the assessment process. For each step in the SBST procedure, the learner can amass bonus points. This is in contrast to receiving an overall score at the end of a question as was customary for many of the learners who came from an era when accountability was not emphasized. The score by score breakdown is important because it allows learners to feel that all is not lost if they are unable to complete a question in its entirety. The learners realize that they are rewarded incrementally. They experience a feeling of fairness and satisfaction that appear to propel them to move on in self-confidence toward their final goals. Despite the usefulness of SBST, one cannot guarantee that it will always be used by all learners after the experiment. For example, Salman (2009), using a case study design, showed that one out of four novice teachers assigned to an urban high-poverty school implemented the Principles of School Mathematics during their mathematics instruction program. Using 36 unannounced observations of the participating teachers and an assessment guide, the research team found that in the implementation and direction of the principles the remaining three teachers fell short.

Additionally, learning anxiety could have a crippling effect on student performance despite the use of SBST. For instance, Yüksel-Şahin (2008) used a sample of 249 fourth and fifth graders to investigate whether there was a statistically significant difference in students’ learning anxiety based on a group of variables. “The Mathematics Anxiety Scale for Elementary School Students” and “The Personal Information Form” were the data collection instruments used. Using independent samples t tests, one-way ANOVA, and Schefee test to analyze the data according to whether they liked mathematics class or not, whether they liked their mathematics teachers or not, and the achievement level in mathematics, the students’ learning anxiety differed significantly according to gender. Yüksel-Şahin (2008) noted that male students reported significantly lower learning anxiety than their female counterparts. Additionally, lower anxiety levels were found for high achievers and for students who liked their mathematics teachers and who liked mathematics class. Interestingly, with respect to their grade level and gender stereotypes about success in mathematics, there were no significant differences in students’ learning anxiety. With specific reference to adults, Benson (1989) found that those students who reported high levels of general test anxiety also reported high levels of statistical test anxiety.

Clearly, the extent of unraveling the myriad factors that could possibly interfere with the execution of SBST can be problematic. Therefore, evaluation of SBST can be challenging. Nevertheless, to evaluate SBST, the following four research questions were formulated:

Research Question 1: What changes in mathematical achievement could be detected in the learners?

Research Question 2: What changes in attitudes to learning could be observed in the learners?

Research Question 3: What changes in beliefs about learning occurred in the learners?

Research Question 4: What changes could be observed in the self-confidence of the learners?

Context and Participants

The participants for this present study comprised a total of 35 learners (ages 18-50 years) from the only tertiary educational institution campus on a small Caribbean island. Several of the learners are full-time employees on the island, primarily in the hospitality and financial sectors. However, with permission, they are allowed to attend classes after work hours and on weekends. With a population of 50,000, this apparently “small sized” convenience sample actually represents 100% of the cohort or the entire population for that semester (15 weeks) under study. Additionally, the population of the course reflects a small national population. A sample size of 35 was considered reasonable in that sampling error based on power analysis considerations would be minimized. Apart from being a useful tool in designing and evaluating research, Murphy (2003) posits that the relationships between the structure of statistical tests, the nature of the phenomenon under study, and the likelihood of consistent detection of the effects of treatments and interventions are involved in power analysis. One may view power as the conditional probability of rejecting the null hypothesis, given that treatments and interventions have some effect on the population. Murphy showed that there is variation in the sample size required to achieve a power of .80 as a function of effect size and the complexity of the hypothesis. Approximately 35 participants are reasonable to give the study a power of .80 (α = .05) when Murphy’s tables are used. These tables show effect size indexed both in terms of R² or the percentage of variance accounted for in the dependent variable and in terms of d or the standardized mean difference. Hence, the mean and standard deviation of the convenience sample were considered to be a reliable approximation of the mean and standard deviation of the population, thereby assuring reasonable generalizability.

The learners were taken from two mathematics groups that I taught. The groups were labeled F1 and F2 by the institution for administrative purposes. Mathematics is compulsory for all university students regardless of their major area of study. Degree programs offered at the tertiary institution include Business, Education, Nursing, and Management. Several ongoing personal enhancement programs such as cosmetology, woodwork, and culinary management, among others, are offered on a first-come first-served basis depending on class participation.

The Caribbean island is well-known internationally as a tourist destination of choice, especially for North American visitors and to a lesser extent visitors from the United Kingdom and Europe. The island offers a number of water sporting activities geared toward total relaxation in a “laid-back” atmosphere with welcoming, friendly, and helpful locals.

The learners perceived the attainment of a degree as a ticket to upward mobility in their workplaces and/or a stepping stone to emigration to the United States or Canada for further studies or for a better life. With time at a premium, coupled with personal, family, and community obligations, learners tend to trade deep understanding for superficial learning to the test. Accordingly, learners use a variety of “short-cuts” to meet their deadlines. Some of these include cooperative work “in sections” without a total appreciation of the whole activity and how that activity fits into the goals and objectives of their intended program.

Methodology

The overall research methodology may be described as “evaluation research” (Gall, Borg, & Gall, 2006). A mixed-methods approach (qualitative and quantitative methods) was used with the purpose of determining the impact of the 2-week intervention; it included a quasi-experimental method to provide comparative achievement data as well as qualitative methods to provide depth of understanding about the nature of the impact of the intervention. Students were randomly assigned to the groups by the tossing of an unbiased coin (head = control group). Having taught both groups before, extensive comparative data from the groups established that no group had an unfair advantage over the other prior to the experiment. A pretest confirmed that there was no difference between group scores on the topic “Inequalities” preceding the 2-week experiment that also covered topics other than “Inequalities.” Supporting statistics are in the Results section. Baseline data about attitudes toward learning, beliefs, and self-confidence were obtained from semistructured interviews, casual conversations, and unobtrusive observations of both groups during the entire course of the semester. Extended essays that served as a reflection exercise were done at the end of the semester and provided overall comments about SBST.

Topic Selection

The topic “Inequalities” was chosen by a panel of five subject experts who were familiar with the work of the learners, the nature of their challenges, the specific difficulties normally encountered by learners over the years with this topic, and its role in laying a foundation for the understanding of important topics such as Linear Programming, which is fundamental to an appreciation of basic principles of management, business, education, and so forth. The five experts worked at the institution at which the study was conducted. The process was iterative until consensus was attained. Given the required perquisites were satisfied, the topic “Inequalities” was taught in the fourth and fifth weeks of the semester. The conditional clause on the course outline that allows reorganization of the topics in the best interest of the learners facilitated this study.

Details of Step-by-Step Teaching

I introduced the experimental group to SBST during Week 4 of the semester.

Step 1 required the learner to form an equation from each inequality.

Step 2 required the learner to plot all equations on the same graph paper.

Step 3 required the learner to shade the required region.

The steps were displayed on flash cards, flip charts, the blackboard, overheads, and on a PowerPoint presentation. I played the role of instructor, learner, and evaluator. By repeatedly telling the learners that an SBST was in use, they appreciated that learning was not a “quick-fix” process but a gradual building-up process, precept on precept, toward a final result. During the process, I did not stifle the learners, but welcomed questions and suggestions in a friendly atmosphere. Learners were allowed to identify steps for themselves so that they could apply SBST to other topics even outside of mathematics and other aspects of their everyday lives. This made for interesting discussions that invigorated both learner and me.

Deciding on which side of the line to shade after plotting the line was a point of lengthy discussion among the learners. The use of test points to confirm the correct area to be shaded proved useful for many learners. In keeping with andragogy, adult learners were allowed to freely express themselves. A researcher-composed poem served to reinforce the three steps. The control group was taught the topic “Inequalities” without SBST. There was a lot of “chalk and talk” about accepted regions based on inequalities presented. I drew a number of diagrams on the board and explained the regions based on the specific inequalities. I questioned learners and offered additional explanations as required. I also gave the control group relevant homework as practice. I marked the homework in a timely manner, provided appropriate feedback, and encouraged the learners to reflect on their work. However, for the control group I did not follow teaching steps in a systematic manner, as in SBST. As mentioned earlier, the students were not aware of any pedagogical differences between the two groups.

Research Instruments

Figure 1 summarizes the data collection methods used in the study.

Figure 1.

Summary of data collection methods used.

A panel of experts established face and content validity for the teacher-made test. Interrater reliability was 90% for the comparison of test scores of both groups (experimental group = 19; control group = 16). A short 10- to 15-minute questionnaire soliciting closed- and open-ended responses from the experimental group provided additional information about the impact of the intervention. Semistructured interviews clarified information with respect to mathematics achievement, attitudes to learning, beliefs, and self-confidence in mathematics. Several unobtrusive observation techniques were employed before, during, and after the intervention.

Research Design

A pretest, posttest, control group quasi-experimental design was used with four research questions. By tossing a coin (head = control group), the F1 group was the experimental group and F2 the control group. Integrity of the experiment was not compromised by keeping the control group away from shared information.

Learners understood that although both groups had a common summative assessment (written examination toward the end of the semester), it was not uncommon for topics to be done at different times for a variety of reasons like differentiated student abilities or reasonable learner requests. One group was scheduled from 8 a.m. to 10 a.m., whereas the other group met from 6 p.m. to 8 p.m. three times weekly. Both groups rarely met for discussion during the early part of the semester. None was forced to participate against his or her will. Most learners saw this intervention as something positive for personal benefit and progress.

Observation techniques allowed for detailed analysis of student behavior. Unitizing and categorizing (Glaser & Strauss, 1967) allowed me to sort and collect similar responses that could determine patterns. Student’s t test tested the null hypothesis that there was no difference in mean group scores. Cohen’s d was calculated to further determine treatment effect (Cohen, 1988).

Results

Research Question 1

Before the intervention, there was no statistically significant difference in academic performance between the experimental group and control group on any of the topics already covered, t(15) = 0.7, p > .05. Some topics included directed numbers, subject of the formula, and simultaneous equations. However, the performance of previous cohorts caused me to anticipate the need for assistance with the topic “Inequalities.” During the intervention, experimental group members appeared excited and interested in interactive class sessions. Control group members treated the experience like they did for any other topic studied for the previous 3 weeks.

The experimental group’s (n₂ = 19) posttest scores indicate that there was a statistically significant difference between the pre- and posttest scores. Using Welch’s unpaired t test there was no statistically significant difference between the academic achievement of two groups prior to the intervention (.5, not significant), but there was a statistically significant difference between the academic achievement of both groups after the intervention (34.3, p < .01). Mean differences between groups, favoring the experiment, was 6.4 ± 0.2. Assuming both populations were normally distributed, or both populations had the same variance, and each student’s score was sampled independently, we tested for population mean difference using Welch’s unpaired t test. Results indicate there was a significant difference between the means of two groups (34.3, p < .01).

A large Cohen’s d meant that the mean of the experimental group was beyond the 84th percentile of the control group. Another way of viewing these results is that there was a nonoverlap of more than 55.4% of the scores of the groups. This is significant because more than half of the scores of the experimental group did not overlap with those of the control group. It would seem that the mathematics achievement of the learners increased as a result of the use of SBST. The baseline data for individuals reveals the details of change scores (difference in the post- and pretest scores).

The open-ended responses from the experimental group were in various forms of narrative, requiring the use of a constant comparison form of analysis (Lincoln & Guba, 1985). This method of processing narrative data facilitated both descriptive and explanatory categories (Clandinin & Connelly, 2000; Riessman, 1993). By grouping similar responses together, three main categories of learners emerged. There were (a) those who found SBST childish and inappropriate for learners (CI; 5), (b) those who were tolerant of SBST because they needed it (TN; 4), and (c) those who were ecstatic about SBST and recommended that it be used for all mathematics topics (ER; 10). Information of this kind also provided some insight into the nature and extent of learners’ achievement, attitudes, beliefs, and self-confidence as would be discussed later.

The results obtained showed that the CI learners had average scores that were lower than any of the two other subgroups (CI: mean = 5.3, SD = 1.2; TN: mean = 7.2, SD = 1.1; ER: mean = 8.1, SD = 1.0, p < .01), possibly indicating that student attitude toward an innovative intervention correlated positively with their motivation in response to a method they found helpful or not. Approaching a “new” teaching method with the attitude that it is inappropriate seems to predispose learners to minimizing possible gains that could have accrued had those learners been willing to embrace something new. It may be that CI learners found SBST insulting and were prone to dropping out of the process.

Those who were tolerant of SBST because they needed it were found to have, on average, better scores than TN learners, but not as good as ER learners (CI: mean = 5.3, SD = 1.2; TN: mean = 7.2, SD = 1.1; ER: mean = 8.1, SD = 1.0, p < .01). In other words, the TN learners had scores lying between the other two subgroups. It is significant that tolerance of a new method could negatively affect student progress. Implications of this kind of attitude are discussed in the next section.

The ER learners were ecstatic about SBST and recommended that it be used for all mathematics topics outperformed their counterparts in achievement test scores (CI: mean = 5.3, SD = 1.2; TN: mean = 7.2, SD = 1.1; ER: mean = 8.1, SD = 1.0, p < .01). On average, their scores ranged from 5 to 10, with excellent presentations detailing all steps involved at arriving at their conclusions. An analysis of other factors such as age, school history, gender, and other academic scores that differed across subgroups could not be included in this article.

Several learners in the ER subgroup were observed to have completed the 30-minute test in less than 15 minutes. In contrast, those in CI and TN subgroups spent on average 25 and 30 minutes, respectively, completing the identical test. Additionally, their countenance at the end of the exercise appeared to manifest stress unlike the countenance of their colleagues in CI subgroup. The anecdotal information gleaned from my many encounters with the learners before, during, and after SBST seemed to offer support for this significant difference in mathematics achievement between groups as a result of the use of SBST. Coding, tagging details, and studying emerging themes arising from data supplied by participants in the experimental group supported gains in mathematics achievement by the learners. Some comments were as follows: “I never dreamed that this SBST would change my grades by so much”; “I would recommend SBST to anyone interested in improving their mathematics scores”; “It’s not just about having better grades as much as it is about the desire to continue achieving”; “Now I want to learn mathematics and actually enjoy the experience”; “Looking for better mathematics scores.”

Research Question 2

By sorting, selecting, and grouping similar responses, we found that learners in the CI subgroup chose not to change their attitudes. They explained that they were in a tertiary institution, that they had no time for activities that reminded them of children’s play, and they expected sophisticated explanations that could assist them in understanding the topic. Comments like the following surfaced: “I am a busy mother and I have no time to go step by step”; “Get to the point and let me leave to attend to my business”; “If I needed this I would not enroll in a tertiary institution . . . a kindergarten would serve this purpose well I think”; “I need a quick and easy way of getting this thing.” Accordingly, their attitudes regarding neat presentation of graph work manifested no change.

In the TN subgroup, learners who were tolerant of SBST because they needed it reported a positive change in their attitudes towards learning. Although there appeared to be ambivalence about the value or appropriate use of SBST, TN learners seemed to have a positive attitudinal change. The following comments seem to support my observation: “I wish I had known this method before”; “From today I promise myself to be more open to new ideas and be willing to give them a try”; “It’s amazing what happens when I am between a rock and a hard place . . . my tolerance for a new technique has saved me from dropping out of university”; “My attitude gives me my altitude.”

Learners of the ER subgroup who were ecstatic about SBST and recommended that it be used for all mathematics topics reported only positive comments on their completed questionnaires and from responses to their semistructured interviews. Such comments included statements such as “Now I actually love mathematics . . . I really surprise myself”; “This new method should be used not only in mathematics but as many other subjects as possible . . . I am sure it would make a big difference to my grades”; “My attitude toward learning will never be the same again.” Furthermore, learners in this subgroup distinguished between deep and surface learning. Generally, they felt that SBST allowed for better understanding than conventional methods to which they were previously exposed. The notion of lifelong learning surfaced as a bonus for using interesting techniques like SBST. One student said it empowered him to seek alternative ways of conceptualizing mathematical thought. From conversations with control group members before and after the experiment, there appeared to be no change in attitude towards mathematics learning.

Research Question 3

The CI learners reported that their beliefs about learning changed from conceptualizing it as an impossible subject to one that could be manageable provided that people were willing to embrace innovative methods that promise to assist student learning. Faced with the results of the test and realizing that their colleagues who had a more positive approach fared better, they claimed that mathematics was manageable and could be enjoyed, provided the correct stimulus is available.

The TN learners confirmed that need forces them to learn. Several of them recalled trying to understand this topic without much success while at high school. At that time, they felt that the topic had no bearing to anything they could possibly use in later life. Now that they were faced with linear programming at their workplaces and this topic was a prerequisite for their understanding of linear programming, they realized that learning had relevance to everyday life. Other learners emphasized the urgency for educators to adjust their pedagogical practices to suit current research in andragogy. For example, learners made statements such as “Create the need to learn and student will respond positively” and “What I have to do I am prepared to do.”

From extended essays from this ER subgroup, learners reported a positive change in their beliefs about learning as evidenced from the following comments: “Belief kills or belief cures, my Mom used to say . . . now I actually experience the truth of this local saying”; “It’s really all about what a person chooses to believe . . . it’s pretty much like the self-fulfilling prophecy that I tell my own children about every day. . . . If you believe that mathematics learning is difficult then it will always be difficult” I heard affirmation of positive change from the following comments: “Now I believe that anyone, given the right stimulus, can do mathematics well” and

If I can learn mathematics then anyone else can. . . . I have come from a long way . . . I am not going back that way. . . . I will encourage anyone to snap out of the learning depression and challenge themselves to move forward.

It must be noted here that from conversations with control group members before and after the experiment there appeared to be no change in their beliefs about learning.

Research Question 4

All subgroups, CI, TN, and ER, reported an increase in self-confidence in mathematics. All learners of the experimental group passed the test attaining more than 5 out of a total of 10 marks. Besides, their graph work was neat, emphasizing their psychomotor skill improvement. Neat graph work is important in mathematics, especially in minimizing errors. Since these test results were unusual, all the learners of the experimental group realized that attempting innovative pedagogical methods has its merits regardless of the age of the target audience. Several learners reported their willingness to use SBST with their own children at home and with anyone under their mentorship. Their responses to interview questions indicated that the self-confidence attained in learning was transferable to all other disciplines. Learners reported how they used the SBST method in their daily routines like getting dressed for classes, doing tasks at their work places, organizing their own children, and structuring their many chores and responsibilities. Further analysis of the responses in all subgroups provides deeper insight into the nature and extent of change in self-confidence that occurred. Clearly, the extent to which this innovation has effect on the whole individual is yet to be measured, especially because conversations with control group members before and after the experiment revealed that there was no change in their self-confidence.

Discussion and Conclusion

In what initially appeared to be a hopeless situation, I attempted to find ways of assisting learners in the four areas previously identified. To this end, a class of 35 learners laden with responsibilities, both personal and community and family, comprised the subjects for this present study. As a foreigner in a new culture, I was personally challenged to make a difference in the learners’ lives of in a number of areas. As mentioned earlier, this present article seeks to explain the findings from work with the learners and provide teachers with a tried-and-tested, hands-on guide to presenting difficult-to-grasp mathematical concepts to learners. Research in this area is limited coupled with the present drive to increase mathematics knowledge to ensure completion of postsecondary education and training. This fact makes the study current and significant.

The SBST method is especially helpful for those learners who like to think through the solution in their heads and write the result without providing adequate written evidence of their analysis; a norm among them. Examination instructions usually indicate that all working must be shown to justify answers. By reinforcing a step-by-step procedure learners got accustomed to detailing information in a step-by-step manner. That discipline appears to be easily transferable to other areas of operation and real-life decision-making situations as the learners reported. In summary, SBST, where the words “step-by-step” are repeatedly vocalized, lays the foundation for a systematic, methodical strategic way of imparting knowledge to learners as they become active participants in the learning process.

There are numerous assumptions and limitations attached to this present study. In addition to the apparent small sample size, the study was done on a single cohort from one campus on one island with only a brief 2-week intervention, so that generalizations need be made guardedly. There may be potential issues of neutrality, since I was both teacher and evaluator; an important concern considering a mixed-methods approach was used. Certain learners may be inclined to perform at their optimum when they believe that a “new” method is being used to teach a topic. The Hawthorne or Halo effect (Jones, 1992) and the John Henry effect (Saretsky, 1972) may have surfaced. The facilitator may also approach the SBST activity with greater enthusiasm than is usual, and this could send signals to the learners who usually weigh the importance of a topic by a number of variables including repetition and teacher enthusiasm. Accordingly, the recorded test scores may not be a “true” reflection of the learners’ real achievement. Despite trying to counteract this anticipated behavior with up-front advisories that all work was equally important and all scores from assessments would be treated equally, there may still be some learners who ignored any advisory. Although every possible effort was made to keep information about SBST from the control group, it is not impossible that certain learners may have secretly obtained information that could have been used to undermine the integrity of the experiment. Issues of power relations between the students and me and queries on trustworthiness and objectivity of qualitative data are possible limitations to this study. Then too, while the expertise of the panel of experts is to be respected, perhaps much more time could have been spent verifying that SBST satisfied the fundamental principles of andragogy. This could be an area for future research.

This present study made no attempt to investigate the differential effect of SBST on males compared with females. Since, as mentioned earlier, there already exists a body of research on gender in learning, a follow-up study with data collected from a larger sample on other islands could provide new insights into appropriate methods to facilitate enhanced achievement in mathematics.

The study could have been enhanced if the intervention were not limited to a short period of time and focused on only one topic in mathematics (inequalities). With a longer time frame and the use of a variety of topics, one could better appreciate the impact of SBST. For instance, one could determine if learners confuse sequences of steps over time, they could figure out what to do when problems vary, whether their enthusiasm for a new strategy dissipate over time, instructions vary for different topics as well, the learners’ conceptual understanding of the topics is enhanced and the learners’ discussions show depth of understanding and deep as opposed to surface learning.

Additionally, SBST forces learners to understand previous steps before proceeding to the next step and so allows for reflection. Reflection is powerful in cementing ideas, revisiting concepts, planning on/for action, and planning for next steps in any process (Gagnon & Collay, 2001; Schön, 1991). It allows learners of all disciplines, and in particular, for the purposes of this present study, to review processes in the comfort of their “spare” time when they are not constrained by deadlines, peer pressure, and other factors undermining personal advancement and self-actualization. Learners are encouraged to do cooperative learning as they discuss step-by-step algorithms. Using SBST principles, learners are better able to accurately calculate their own test scores in advance of obtaining the teachers’ score. This “foreknowledge” serves to minimize anxiety, thereby causing learners to feel more relaxed. In summary, the study found positive gains in achievement, attitudes, beliefs, and self-confidence in most of those in the experimental group experiencing the SBST intervention. Three distinct groups emerged among the experimental group: those who felt SBST childish and inappropriate for learners (CI), those who were tolerant of it because they needed it (TN), and those who were ecstatic about it and recommended (ER) its use. These groups indicate how different learners have different perceptions of the same activity. It is hoped that this research would serve as a catalyst for more work in this understudied area and that educators in challenging situations with reluctant learners would be encouraged and assisted.

Recommendations

Research into the effect of SBST usage could further authenticate the process. Longitudinal studies into the effect of SBST on learners could be useful, since this study was done for a brief 2-week period. It is not clear precisely which topics would be suited to SBST, so more research work in this area could be useful.

It may be beneficial to see if the CI, TN, and ER categories could be confirmed using surveys or other methodologies like confirmatory factor analysis with future cohorts. Finding out if SBST makes any lasting impact on the attitudes of the learners or is limited to only the class where SBST is practiced could verify or confirm lifelong learning habits.

More in-depth research into the subgroups might provide useful data for curriculum planners, policy makers, and other stakeholders. Further follow-up with the CI learners might reveal other concerns including perceptions of “childishness” correlated with age, school history, gender, or other academic scores especially when one thinks of teaching inequalities without the use of test points. The CI learners may find SBST insulting and be prone to dropping out of the process. Herein lies the call for attention to be given to these affective feelings when teaching. Teachers whose learners have had a generational history of learning phobia would find this article particularly useful, assuming that participants do not respond differently because of perceived effectiveness. SBST allows learners to justify scores obtained and demystify the assessment process. The results of this study could demonstrate how difficult-to-understand mathematics topics could be simplified to meet the immediate learner needs. An important component in operationalizing SBST is the upfront preparatory work needed by the facilitator. A question emanating from this and other personal experiences is whether faculty can invest enough time and energy to ensuring that SBST meets specific needs of their learners. Clearly, success is moderated by facilitator training, experience, and exposure.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Author Biography

Betty McDonald, PhD, Manager, Professional Development Unit, The Learning Centre at the University of Trinidad and Tobago has more than 40 years academic experience and is widely published. She is an associate editor and reviewer for several international journals. Local and international graduates continue to benefit from her supervision. Measurement, assessment, problem-based learning, service learning, teaching and learning, applied statistics, math education, project management, and professional development are her research interests.

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