In this section, we focus on learning mathematics, as this represents a somewhat different process than conceptual change in science understanding. In particular, we review findings from our laboratory on the links between general affect and learning mathematics for upper elementary and middle school students (e.g., Linnenbrink & Pintrich, 2003). We also discuss a number of studies that link affect to mathematics understanding and problem solving from the extant literature. Finally, in addition to considering the way that affect is linked to the processing of information for mathematics learning, we also consider how affect is linked to memory processes.
In a study conducted with middle school students, we examined the relation between students' affect and their scores on a computer math activity (Linnenbrink & Pintrich, 2003, study 1). In particular, middle school students worked in groups to learn how to solve number sequences. They then completed a similar series of math problems on the computer for 15 minutes. Immediately following the completion of the math problems, they reported on their current affect using single item indicators (sad-happy, tense-calm, tired-excited). Finally, after completing a series of word-recognition tasks, students were asked to report on their effort regulation and cognitive regulation during the computer math task.
Interestingly, the three indicators of affect (sad-happy, tense-calm, and tired-excited) were unrelated to students' scores on the math exam. However, affect was significantly related to students' effort and cognitive regulation during the math exam. For effort regulation, students who reported being more excited than tired reported higher levels of persistence even when they did not want to work on the task (P = .22, p < .001). For cognitive regulation, students who reported feeling more happy than sad (P = .13, p < .05) and more excited than tired (P = .16, p < .01) also reported that they planned, monitored, and checked their work as they completed the number sequences on the computer. What is interesting about these findings is that both valence (sad-happy) and arousal (tired-excited) were predictors of students' cognitive regulation while only arousal (tired-excited) significantly predicted effort regulation. This may mean that arousal is important in terms of motivation to engage in the task while both valence and arousal are important in terms of the quality of engagement (e.g., using higher level strategies). It is somewhat surprising, however, that the other measure of arousal, calm-tense, was unrelated to either type of regulation.
When interpreting these results, it is important to keep in mind that there were several limitations in the methodology used in this study. First, the affect measure was designed to assess students' affect while working on the computer math activity, but their affect may have changed as they completed the computer math test as a result of how well they perceived they were doing on the math problems. Second, the use of self-reported affect and self-reported regulation leaves one open to the possibility of a method bias, where shared variance may have more to do with similarities in measurement than with similarities in the underlying constructs (Winne & Perry, 2000). Third, the use of bipolar affect measures may be problematic if both ends of the scale (e.g., sad and happy) relate in the same way to the outcome. For instance, if both sadness and happiness are negative predictors of math performance, the use of a bipolar measure would not be able to detect a significant relation and would instead suggest that sad-happy and math performance were unrelated.
This third limitation was of particular concern in the current study in that all three measures of affect were unrelated to students' math performance. However, when we split the sample based on the bipolar indicators so that the scale assessed either end of the bipolar measure (e.g., neutral to happy or neutral to sad) for each of the three affect measures, the correlations between the affect measures and math performance were not significant suggesting that the use of the bipolar measure did not limit our ability to detect a significant relation. Nevertheless, it is important to note that for sad-happy, while the correlation was not significant, the correlations for sad and happy were both in the negative direction; this suggests that future studies may want to avoid using bipolar measures, especially when examining the relation between affect and math performance.
In another study conducted with upper elementary students (fifth and sixth graders) during a 6-week math unit on reading and interpreting graphs, we investigated the relation between students' affect and their learning during the unit (Linnenbrink & Pintrich, 2003, study 2). In order to examine how affect during the entire unit related to how much students learned in the unit, we regressed their post-test math score on self-reported positive and negative affect during the 6-week math unit. The measures of positive and negative affect included both high activation (e.g., energetic, agitated) and low activation (e.g., calm, sad) indicators of affect and asked students to rate how they felt during the entire mathematics unit. Therefore, they serve as indicators of valence but not arousal. The scales were initially designed to assess both valence and arousal, but the four dimensions did not separate in exploratory factor analyses, suggesting that younger children have a difficult time differentiating, or at least reporting, valence versus arousal. We also examined the relation of affect reported at the post-test to a follow-up measure of achievement given 6 weeks after the end of the unit and two self-reported measures of strategy use (effort and cognitive regulation).
Surprisingly, students' reports of both positive affect (P = -.24, p < .01) and negative affect (P = -.30, p < .01) were negatively related to how much students learned during the math unit and how much they retained 6 weeks later (positive affect: P = -.22,p < .01; negative affect: P = -.41,p < .001). For strategy use, positive affect was associated with higher levels of effort regulation (P = .22,p < .01) and cognitive regulation (P = .53,p < .001) while negative affect was unrelated.
It is somewhat surprising that positive affect was linked to higher levels of effort and cognitive regulation during the math unit, but this association did not seem to be beneficial for how much students learned during the math activity. In fact, positive affect was related to lower levels of achievement at the end of the unit and lower levels of retention 6 weeks later. One possibility is that the findings for strategy use may be influenced by the methodology used since both strategy use and affect were assessed using self-report measures. However, it is also possible that positive affect has a different relation with effort and cognitive regulation versus actual learning and achievement. For instance, positive affect may serve as a motivational tool, such that students who feel positively are more willing to engage and persist and even more willing to use effortful strategies such as those required for cognitive regulation. However, there may be another component of positive affect that is detrimental for learning mathematics in that it interferes with the storage or processing of the information. In this way, positive affect may help with engagement and strategy use, but if it interferes with cognitive processing, it will still hinder learning. This possibility needs to be explored in future research where either affect is experimentally manipulated or effort and cognitive regulation are not assessed with self-report measures in order to eliminate the possibility that the findings are based on a mono-method bias in assessment.
In summary, our work on the relation between affect and students' learning of mathematics material is consistent for effort and cognitive regulation but not for math performance or learning (Linnenbrink & Pintrich, 2003). As noted previously, it is possible that the discrepancy in the findings may be linked to the differences in the measurement of affect, although follow-up analyses indicated that this was not the case suggesting that we must consider other possibilities. For instance, the discrepant findings may have occurred because the tasks used were very different and the duration and context of the study differed.
Given these differences, it is somewhat surprising that affect did not alter students' performance on the math exam in study 1 since it was more similar both in terms of the task and the design to typical social psychology experiments. That is, students were tested outside of the regular classroom and asked to respond to tasks in an atypical manner (using a computer to record responses). Furthermore, the task was relatively short in duration, lasting 15 minutes. In contrast, study 2 was more similar to a typical classroom. The study took place during a 6-week math unit and students completed the posttest and follow-up tests as part of their regular classroom work. Furthermore, the affect measure was more general in that it was designed to assess affect during the 6-week math unit and examine the effects of that general affect or mood on their learning during the unit. In this sense, the relation of affect to learning was expected to take place over a longer time period and may not have influenced cognitive processing at the same level as was assessed in the first study. We consider these differences in applying the theoretical models to our findings.
It is also interesting that while study 1 and study 2 differed greatly in duration, the findings for engagement, as measured by effort regulation and cognitive regulation, were similar. This suggests that the differences in find ings between the studies may have had more to do with the types of tasks and processing of the information than students' motivation or willingness to engage. This similarity across studies in terms of engagement but discrepancy in terms of learning and performance needs to be more closely examined in future research.
Given the discrepancy in our work and some of the methodological limitations, it is important to consider other work on the relation between students' affect and learning in mathematics. Although there is not extensive research in this area, a few studies are relevant. For instance, Bryan and Bryan (1991) conducted a series of studies with upper elementary students with and without learning disabilities. They induced half of the students into a positive mood and half received no mood induction. Students were then asked to work on 50 subtraction and addition problems for 5 minutes. They found that students in a positive mood completed more problems correctly than students in the neutral mood. There was, however, no significant difference in the number of problems completed between the positive mood group and the control group. The authors replicated the first study with junior and senior high school students and found a similar pattern of results. Thus, the results from this study suggest that positive mood is beneficial for mathematics performance, at least in terms of computation. However, in a similar study, Yasutake and Bryan (1995) induced middle school students into a positive mood versus a neutral mood and asked them to complete a mathematics calculation subtest of the Woodcock-Johnson battery for 15 minutes. They found no significant effect of the mood condition on students' performance. Thus, the findings from the Bryan and Bryan (1991) study were not replicated, suggesting that the pattern linking positive affect to computation is not entirely consistent or may vary based on the length of the study (5 vs. 15 minutes).
Two studies also examined middle students' learning of shapes and symbols. While these tasks are not directly related to mathematics computation, they seem relevant in terms of understanding geometry and are therefore discussed here. In the same study described previously, Yasutake and Bryan (1995) compared middle school students' performance, working under a positive versus neutral mood condition, on a 2-minute task in which they needed to learn combinations of symbols and shapes and then make associations (Coding subtest from the Performance section of the WISC-R). The authors found that students in the positive mood condition outperformed students in the neutral mood condition. Masters, Barden, and Ford (1979) conducted a similar study examining how 4-year-old children performed on a shape discrimination task under three different mood conditions (positive, neutral, negative) and two different activation levels (active, passive). Children worked on the shape discrimination task until it was mastered (they could attempt up to 10 trial blocks consisting of 12 problems each). Pre school children induced into a positive mood learned the shape-discrimination task more quickly than children in a negative or neutral mood, as did those induced into an active rather than passive state. There was also an interaction of valence and arousal, with children in the negative mood condition taking longer to master the task when they were induced into a passive rather than active state. Taken together, these results suggest that positive moods are beneficial for learning shape discrimination tasks whereas negative moods are detrimental, especially when arousal is high.
Finally, given the large emphasis on problem solving as part of the mathematics curriculum, research relating problem solving and mood seems relevant to this discussion. For instance, Isen et al. (1987) examined college students' performance on creative problem-solving tasks. In a series of studies, participants completed two types of creative problems solving tasks, Duncker's (1945) candle task and the Remote Associates Test, both of which lasted between 10 and 15 minutes under a variety of induced mood conditions. The results from these studies suggest that positive mood facilitates creative problem solving in comparison to a neutral or negative mood, but there are not differences in problem solving between negative and neutral moods. Finally, some of the studies included an arousal condition (exercise). Students in the positive mood condition scored higher than those in the arousal condition, while there was no difference between the arousal condition and the neutral mood condition. This suggests that valence, but not arousal, is important in terms of students' creative problem solving.
In summary, the research relating positive and negative affect to mathematics learning is not consistent. This may be due in part, however, to the broad range of tasks that fall under the purview of mathematics education as well as the context of the study, including the duration of the task. Therefore, in attempting to apply social psychological theories, we consider that the different tasks may require different processes and, accordingly, positive and negative affect may hinder or enhance cognitive processing in different situations. We also discuss whether differences in the contexts and lengths of tasks may help to account for the discrepancies in the results.
Bless (2000), Fiedler (2000), and Fredrickson (2001) all suggested that positive moods should result in broad, heuristic processing. Fiedler (2000) further suggested that positive affect is beneficial when active generation occurs. In terms of mathematics learning, we would therefore expect positive affect to enhance learning and performance when tasks require a broad perspective or active generation. For instance, learning and distinguishing shapes may require a broader perspective in that considering the whole shape rather than focusing on details of particular aspects of the shape may enhance performance. Furthermore, this information needs to be linked to prior knowledge, so Fiedler's suggestion that positive affect helps to activate prior knowledge should also enhance performance. Therefore, the empirical research suggesting that positive affect enhances shape discrimination (Masters et al., 1979; Yasutake & Bryan, 1995) lends support to these theories. In addition, consistent with Isen et al.'s (1987) findings, positive affect should enhance problem solving, particularly creative problem solving in that positive affect should help students move away from the details of the task and take a broader, perhaps more creative perspective.
In addition, we would expect that positive affect would enhance the interpretation or reading of graphs. That is, when interpreting graphs, students are often asked to look at general patterns, a process which should be facilitated by positive affect. However, the results from our research (Linnenbrink & Pintrich, 2003, study 2) suggest the positive affect hinders students' reading and interpretation of graphs. This unexpected finding may be because students in our study may have needed to use both heuristic and detailed-processing, as the types of tasks falling under the purview of graphing our quite broad. However, if this were the case, we would have expected positive affect to be unrelated to learning, as it might have enhanced learning for some aspects and hindered it for others.
Another possibility, is that our study assessed affect during a 6-week unit and looked at learning over 6 weeks while the prior studies and the studies on which the theories were developed assessed affect during a relatively short duration. Furthermore, we used measures of self-reported affect while prior research has manipulated mood. Thus, while our results regarding the relation between positive affect and graphing cannot be easily interpreted under the existing theories, they also differ in a number of ways from prior research suggesting that a variety of factors may account for the discrepancy. Nevertheless, we should note that our study on mathematics and graphing examined student learning in real school contexts; thus, in trying to understand how affect influences learning in school, the results may be quite relevant.
The results for computation and number sequences are also difficult to interpret in terms of the affect and cognitive processing theories, in part, because the findings are not consistent. In particular, Bryan and Bryan (1991) reported that positive moods enhanced performance on computation problems while Yasutake and Bryan (1995) and Linnenbrink and Pintrich (2003, study 1) found no relation between affect and performance on solving number sequences. One possible explanation for these discrepancies is the duration of time spent on the task. Participants in Bryan and Bryan's (1991) study had 5 minutes to complete the task while participants in the other two studies had 15 minutes. While time does differ among these studies, it seems unlikely that a 10-minute difference could account for the discrepant findings. Based on the theories presented in this chapter, it also seems plausible that the results might be mixed or inconsistent. That is, for typical number sequences or computation problems that follow the general patterns students have previously seen in math, positive affect may be beneficial in that students can activate the basic script for solving the problem (Bless, 2000) and it may be easier for students to access basic number facts to aid in solving the problems (Fiedler, 2000). Furthermore, the use of basic scripts should reduce the cognitive load making it easier for students to complete a series of numbers in the sequence or solve multi-digit computation problems in working memory. A positive mood may not, however, be beneficial when the number pattern does not follow the basic pattern that matches the activated schema or when a computation problem is unfamiliar. In this case, the student may take longer to solve the pattern because she must first try the pattern or solution suggested by the schema and then try other patterns when this one was not successful.
The relation of negative affect to mathematics learning and performance should also vary depending on the type of task involved. For instance, we would expect negative affect to be beneficial for detail-oriented tasks, as negative affect should focus students on the appropriate aspects of the task. That is, both Fiedler (2000) and Bless (2000) suggested that negative affect should focus students on the details of a particular task or situation and Fiedler further noted that negative affect is beneficial for processing new stimuli.
In terms of mathematics, we would expect that negative affect might be particularly beneficial for computation problems, in which students must focus on the details of processing each aspect of the problem. For instance, a student in a negative mood may be more successful on unusual, atypical number patterns as he will begin by focusing on the details of the pattern and may easily detect the pattern based on this focus. This notion is not clearly supported by the empirical data; however, the findings also do not clearly refute this idea. That is, Bryan and his colleagues (Bryan & Bryan, 1991; Yasutake & Bryan, 1995) did not examine how negative mood conditions related to computation, and Linnenbrink and Pintrich (2003, study 1) found no significant relation between negative affect and performance on number sequences. One possibility is that while negative affect may focus students on the details, there is a cost to this focus that may be detrimental for overall performance. That is, a focus on details may overwhelm working memory as suggested by Ellis and Ashbrook (1988). Indeed, in a study conducted with college students, we found that negative affect was associated with lower levels of working memory functioning (Linnenbrink, Ryan, & Pintrich, 1999).
Negative affect should be detrimental for tasks such as problem solving and shape discrimination in that a focus on details may distract students from the broader perspective. While this idea is supported in terms of shape discrimination (Masters et al., 1979), it is not supported by Isen et al.'s (1987) study on problem solving in which the negative and neutral mood conditions did not significantly differ. For graphing, a focus on details may be beneficial in certain situations, such as plotting data on graphs, calculating statistics, or interpreting misleading graphs. In addition, if students do not have prior experience with graphs, a focus on the new stimuli, which Fiedler (2000) suggested is associated with negative moods, should facilitate learning. In this case, it is not necessary to link the new information to prior information, as students may not have relevant prior information to which they would link the new information. However, our research suggests that negative affect is negatively related to reading and interpreting graphs (Linnenbrink & Pin-trich, 2003, study 2). As noted previously, however, our tasks were rather complex and occurred over a 6-week period, which may help to explain why our findings our not consistent with the theoretical predictions.
Finally, similar to our results for conceptual change in science (Linnenbrink & Pintrich, 2002b), we found that while positive affect did not enhance performance in mathematics, it was related to high levels of effort and cognitive regulation during the solving of number sequences and during graphing. This provides further support for the notion that positive affect does not signal a lack of motivation (Bless, 2000).
It is also important to consider how the storage and retrieval of information is linked to affect for mathematics learning. Based on Forgas' (2000a) model, we would expect affect to be relevant to long-term memory under certain conditions. For instance, it seems likely that computation tasks, where students are simply retrieving strategies or number facts from long-term memory and applying them, should not be influenced by affect. That is, this type of processing involves direct retrieval, a type of processing in which affect should not infuse thinking. In contrast, other mathematical tasks such as problem solving, graphing, and shape discrimination may involve more substantive processing. Students engaged in these tasks may be learning new information or trying to link new information to prior knowledge. In these situations, it is likely that the affective state is encoded along with the relevant mathematical material. Therefore, this may be a situation in which a congru-ency between the encoding and retrieval states will facilitate recall. However, none of the studies reviewed in this section tested this idea.
In summary, the research relating affect to cognitive processing in mathematics presents a varied and complex view of the way in which affect influences performance and learning. This is due in part to the wide variety of tasks that fall under the domain of mathematics. Nevertheless, even within a type of task, the results are not consistent, making it difficult to clearly analyze the findings based on the proposed social psychological models of affect and cognitive processing. We have suggested that part of the discrepancy in the findings may be due to the duration of the task, in that affect may have different effects on students' processing depending on whether they must work on the task for a long or short period of time. Other possible sources for the discrepant findings are the complexity of the task (whether it requires both heuristic and detail-
oriented processing) and the manipulation of mood versus self-reported affect. Finally, the few studies that examined arousal versus valence provide a mixed view of whether it is important to consider both dimensions of affect. Therefore, we urge researchers to conduct carefully designed experimental and correlational studies that directly examine how mood influences cognitive processing, keeping in mind that the type of task, the duration of the task, the way in which affect is measured or induced, and the distinction between arousal and valence may be important variables to consider.
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