Looking at My (Real) World through Mathematics: Memories and Imaginaries of Math and Science Learning

Abstract

Critical math and science educators have argued for pedagogies that focus on equity, social justice, and the identities of learners. To inform debates about the purposes and values of math and science pedagogy, we need to better understand how different kinds of curriculum and instruction are taken up by learners over time. This study examines the ways one Latinx immigrant learner, Calvin (a pseudonym), constructed the values and purposes of his earlier math and science learning experiences, as an adult hoping to pursue a career in science. Narrative analysis is used to explore the ways Calvin made sense of his learning of math and science in high school, in community colleges, and on his own. Drawing on the construct of appropriation, we examine the conceptual tools Calvin took up from different math and science pedagogies (reform, critical, and traditional) in narrating his desire to explore and understand scientific phenomena. Narrative frames position Calvin with greater or lesser agency as he navigates different learning environments, imagines possible futures, and constructs the purposes of science and mathematics. The narratives of Calvin’s learning illustrate the conceptual tools he appropriates from reform and social justice pedagogies: real-to-my-life mathematics, learning as connecting and imagining, harder-but-easier math for a purpose, and the ways reading and writing the world can be applied to imagining planets and the universe. The analysis suggests the usefulness of restorying as a way to explore how our pedagogies interact with learners’ subjectivities, desires, values, and purposes for learning.     

中小学生数学知识观的调查研究

本研究采用12道假设性情境题目和包含数学涉及运算、思考、实用性三个维度的数学认识问卷,选取90名六年级小学生和106名初二学生作为被试,探讨中小学生的数学知识观。结果表明:1)中小学生数学知识观形成及发展与学校数学课程内容紧密相关。2)中小学生数学知识观存在差异,主要表现在初二学生对假设性情境的认同程度显著地高于六年级学生;在数学认识问卷的数学实用性维度上,六年级学生的肯定程度显著地高于初二学生。3)从总体上看,学生的学业水平与数学知识观的关系不大。     

Predictive model for early math skills based on structural equations

Early math skills are determined by higher cognitive processes that are particularly important for acquiring and developing skills during a child's early education. Such processes could be a critical target for identifying students at risk for math learning difficulties. Few studies have considered the use of a structural equation method to rationalize these relations. Participating in this study were 207 preschool students ages 59 to 72 months, 108 boys and 99 girls. Performance with respect to early math skills, early literacy, general intelligence, working memory, and short‐term memory was assessed. A structural equation model explaining 64.3% of the variance in early math skills was applied. Early literacy exhibited the highest statistical significance (β = 0.443, p < 0.05), followed by intelligence (β = 0.286, p < 0.05), working memory (β = 0.220, p < 0.05), and short‐term memory (β = 0.213, p < 0.05). Correlations between the independent variables were also significant (p < 0.05). According to the results, cognitive variables should be included in remedial intervention programs.     

Does calling it “Morgan's way” reduce student learning? Evaluating the effect of person-presentation during comparison and discussion of worked examples in mathematics classrooms

Mathematics textbooks sometimes present worked examples as being generated by particular fictitious students (i.e., person-presentation). However, there are indicators that person-presentation of worked examples may harm generalization of the presented strategies to new problems. In the context of comparing and discussing worked examples during extended classroom instruction, the current study compared the impact of person-presentation to strategy labels on students' posttest accuracy and ratings of strategy generalizability. Five algebra teachers and their 168 students used worked examples either presented using fictitious students or with a strategy label during a multiweek unit on equation solving, with teachers randomly assigned to condition. All students compared and discussed the worked examples. In this context, we found no effect of condition on student accuracy at posttest, nor on their ratings of the generalizability of the presented strategies. We discuss why previously found negative effects of person-presentation may not have extended to this context.     

Contrasting two models of academic self‐efficacy – domain‐specific versus cross‐domain – in children receiving and not receiving special instruction in mathematics

In social cognitive theory, self‐efficacy is domain‐specific. An alternative model, the cross‐domain influence model, would predict that self‐efficacy beliefs in one domain might influence performance in other domains. Research has also found that children who receive special instruction are not good at estimating their performance. The aim was to test two models of how self‐efficacy beliefs influence achievement, and to contrast children receiving special instruction in mathematics with normally‐achieving children. The participants were 73 fifth‐grade children who receive special instruction and 70 children who do not receive any special instruction. In year four and five, the children's skills in mathematics and reading were assessed by national curriculum tests, and in their fifth year, self‐efficacy in mathematics and reading were measured. Structural equation modeling showed that in domains where children do not receive special instruction in mathematics, self‐efficacy is a mediating variable between earlier and later achievement in the same domain. Achievement in mathematics was not mediated by self‐efficacy in mathematics for children who receive special instruction. For normal achieving children, earlier achievement in the language domain had an influence on later self‐efficacy in the mathematics domain, and self‐efficacy beliefs in different domains were correlated. Self‐efficacy is mostly domain specific, but may play a different role in academic performance depending on whether children receive special instruction. The results of the present study provided some support of the Cross‐Domain Influence Model for normal achieving children.     

数学焦虑影响儿童数学任务表现的作用机制探析

研究设计了心算加法和图形旋转/平移两类数学任务,同时采用《儿童数学焦虑问卷》、工作记忆和抑制任务,以探讨数学焦虑影响儿童数学任务表现的作用机制.213名小学五年级儿童参与研究,其中89人参与正式实验.结果发现:(1)数学焦虑与数学成绩呈显著负相关;(2)高数学焦虑组在工作记忆和两类数学任务中的得分均显著低于低数学焦虑组,而两组在抑制任务指标上差异不显著;(3)言语工作记忆和视空间工作记忆是数学焦虑影响心算加法任务的完全中介变量;视空间工作记忆是数学焦虑影响图形旋转/平移任务的完全中介变量,言语工作记忆是二者关系的调节变量.研究结果支持了加工效能理论.     

Is the three-term contingency trial a predictor of effective instruction?

Two experiments are reported which test the effect of increased three-term contingency trials on students' correct and incorrect math responses. Experiment 1 included two junior high school students in need of special instruction in mathematics while in Experiment 2 two other students with similar instructional needs from the same classroom were studied. In the first experiment the increased rate of presentation (3 times the baseline rate) was done without regard to whether the antecedent teacher presentations to the student were presented vocally or in written form; in the second experiment, the rates of presentation were systematically rotated across written and vocal presentations. The dependent variables were rate per minute of correct and incorrect responses of students (vocal or written). The experiments used designs incorporating features of the multiple baseline (Students A and B) and reversal (Students B, C, and D). The data showed that increasing the number of three-term trials increased correct rates while incorrect rates remained relatively low. The second experiment replicated these findings with two other students and found that the treatment effect occurred independently with vocal and written antecedents when each was isolated. The results warrant further research to test whether or not rates of presentation of three-term contingency trials are predictors of effective instruction.     

First demonstration of effective spatial training for near transfer to spatial performance and far transfer to a range of mathematics skills at 8 years

There is evidence that spatial thinking is malleable, and that spatial and mathematical skills are associated (Mix et al. [2016] Journal of Experimental Psychology: General, 145, 1206; Mix et al. [2017] Journal of Cognition and Development, 18, 465; Uttal et al. [2013] Psychological Bulletin, 139, 352). However, few studies have investigated transfer of spatial training gains to mathematics outcomes in children, and no known studies have compared different modes of spatial instruction (explicit vs. implicit instruction). Based on a sample of 250 participants, this study compared the effectiveness of explicit and implicit spatial instruction in eliciting near transfer (to the specific spatial skills trained), intermediate transfer (to untrained spatial skills) and far transfer (to mathematics domains) at age 8. Spatial scaling and mental rotation skills were chosen as training targets as previous studies have found, and proposed explanations for, associations between these skills and mathematics in children of this age (Journal of Experimental Psychology: General, 145, 2016 and 1206). In this study, spatial training led to near, intermediate and far transfer of gains. Mental visualization and proportional reasoning were proposed to explain far transfer from mental rotation and spatial scaling skills respectively. For most outcomes, except for geometry, there was no difference in the effectiveness of implicit (practice with feedback) compared to explicit instruction (instructional videos). From a theoretical perspective, the study identified a specific causal effect of spatial skills on mathematics skills in children. Practically, the results also highlight the potential of instructional videos as a method of introducing spatial thinking into the classroom.     

Gender Differences in Interest and Knowledge Acquisition: The United States,Taiwan, and Japan

The relationship between interest and knowledge was investigated in a representative sample of 11th grade students from cultures that differ in the strength of their gender-role stereotypes and their endorsement of effort-based versus interest-based learning. Among 11th graders from the United States (N = 1052), Taiwan (N = 1475), and Japan (N = 1119), boys preferred science, math, and sports, whereas girls preferred language arts, music, and art. General information scores were comparable across the three locations; however, boys consistently outscored girls. Gender and interest in science independently predicted general information scores, whereas gender and interest in math independently predicted mathematics scores. Cultural variations in the strength of the relationship between gender, interest, and scores indicate that specific socialization practices can minimize or exaggerate these gender differences.     

A Q-Methodological Study of Women's Subjective Perspectives on Mathematics

The objective of this study was to explore, using Q-methodology, women's subjective reactions to and experiences with mathematics. Ninety-six undergraduate women from a private university conducted Q-sorts on items that related to their personal experiences, attitudes, and belief/awareness of gender stereotypes for math. On the basis of the Q-factor analysis, three unique perspectives toward math emerged. The perspectives were classified as Successfully Encouraged, Stereotypically Discouraged, and Mathematically Aversive. These groupings were differentiated by their variety of experiences, attitudes, and awareness of stereotypes about math. Measures of math self-schema, math anxiety, and self-reported math ability were also used to interpret the groupings. This research revealed the women's various experiences with and attitudes toward math. The findings are integrated with previous theories in order to understand women's underrepresentation in math-related fields.     

The Instructional Characteristics of Inclusive Classes for Elementary Students with Severe Disabilities: An Exploratory Study

This exploratory study examined several instructional characteristics of elementary classrooms serving students with severe disabilities including the individuals who provided instruction to students, the instructional behaviors of these teachers, and grouping structures. The analysis focused on the impact of these variables on the academic responding of students and the extent to which they were the exclusive target of instruction by the various teachers. Observation data were collected on six students with severe disabilities participating in general elementary classrooms for reading or math, and at least one other subject area. The academic responding of students was similar across general education teachers, special education teachers, paraprofessionals, and peers; and the instructional behaviors of each of these groups of instructors was similar. Students had higher rates of academic responding during one-on-one and small group instruction, but they also were engaged in instructional tasks during whole class instruction. Finally, academic responding was positively associated with instruction focused on the student, one-on-one instruction, and instructional interactions with teachers. Academic responding was negatively correlated with whole class instruction and instructional behaviors that were not directed at the student.     

The role of planning in different mathematical skills

Despite the fact that planning has been found to be a significant predictor of reading (particularly of reading comprehension), much less is known about its contribution to mathematics. The purpose of this study was to examine the role of two levels of planning (operation planning and action planning) in three mathematical skills (calculation fluency, math problem-solving, and math reasoning). Eighty Grade 2 children from Shanghai, China were assessed on measures of nonverbal cognitive ability (nonverbal matrices), working memory (digit span backwards and N-back), operation planning (matching numbers, planned codes, and planned search), action planning (crack the code), and mathematics (calculation fluency, math problem-solving, and math reasoning). The results of regression analyses showed that both levels of planning accounted for unique variance in mathematics over and above the effects of nonverbal cognitive ability and working memory. The effects of action planning were particularly strong in math problem-solving. These findings suggest that measures of planning could be used along with measures of working memory to detect children at-risk for mathematics disabilities and that intervention programmes targeting planning could be developed to boost children's mathematics performance.     

Developing a Learning Progression for Curriculum,Instruction, and Student Learning: An Example from Mathematics Education

Learning progressions have been demarcated by some for science education, or only concerned with levels of sophistication in student thinking as determined by logical analyses of the discipline. We take the stance that learning progressions can be leveraged in mathematics education as a form of curriculum research that advances a linked understanding of students learning over time through careful articulation of a curricular framework and progression, instructional sequence, assessments, and levels of sophistication in student learning. Under this broadened conceptualization, we advance a methodology for developing and validating learning progressions, and advance several design considerations that can guide research concerned with engendering forms of mathematics learning, and curricular and instructional support for that learning. We advance a two-phase methodology of (a) research and development, and (b) testing and revision. Each phase involves iterative cycles of design and experimentation with the aim of developing a validated learning progression. In particular, we gathered empirical data to revise our hypothesized curricular framework and progression and to measure change in students. thinking over time as a means to validate both the effectiveness of our instructional sequence and of the assessments designed to capture learning. We use the context of early algebra to exemplify our approach to learning progressions in mathematics education with a focus on the concept of mathematical equivalence across Grades 3-5. The domain of work on research on learning over time is evolving; our work contributes a broadened role for learning progressions work in mathematics education research and practice.     

父母教育卷入与小学儿童数学焦虑的纵向联系:数学态度的中介作用

为探讨父母教育卷入与学龄期儿童数学焦虑之间的纵向联系及内在作用机制,从山东省聊城市两所普通小学选取1734名三、四年级学生,对其进行为期一年的追踪研究。结果发现：（1）T1父母教育卷入能够显著负向预测T2儿童数学焦虑,但T1儿童数学焦虑对T2父母教育卷入的预测不显著;（2）在同一时间点上,儿童数学态度在父母教育卷入与儿童数学焦虑之间发挥显著中介作用;（3）在不同时间点上,儿童数学态度的中介作用仍然成立,表明儿童数学态度的中介作用具有跨时间的稳定性。该结果强调了父母教育卷入对学龄期儿童数学焦虑变化的重要作用,亦为从父母教育卷入和数学态度角度降低儿童的数学焦虑水平提供了初步有力证据。     

Discounting the Difficult: How High Math-Identified Women Respond to Stereotype Threat

In this study, we examined how math identity moderates women's response to gender-related stereotypes in the domain of mathematics. Male and female college students with varying degrees of math identification took a challenging math test with a gender-related stereotype either activated (i.e., stereotype threat) or nullified. Consistent with previous research, women performed worse than men in the stereotype threat condition, but equal to men in the stereotype nullification condition when performance was adjusted for math SAT scores. Moreover, when faced with stereotype threat, high math-identified women discounted the validity of the test more than did less math-identified women or men in general. We discuss potential benefits and drawbacks of a discounting strategy for women who are highly identified with math.     

数学学业不良初中生的工作记忆特点：领域普遍性还是特殊性?

研究选取上海111名(男生48名, 女生63名, 平均年龄11.97岁)初中学生(数学学业不良学生55名与数学学业优秀学生56名), 根据国家数学课程标准, 将数学划分为数与代数、空间与几何两部分, 又从空间与几何领域中选取初一数学学业知识点—— 轴对称和中心对称图形, 分析在不同内容知识领域的解答过程中所涉及的工作记忆成分。结果发现：(1)数与代数学习需要中央执行系统、视觉-空间模板、语音环路三个成分的共同作用; 空间与几何学习主要受到视觉-空间模板和中央执行系统的影响, 但不存在语音环路的影响。(2)对轴对称与中心对称图形任务的成绩影响最大的是视觉-空间模板, 其次是中央执行系统, 语音环路对该任务作用不明显。两个研究说明工作记忆在初中不同年级的各类数学学业任务中具有不同的作用, 中央执行系统和视觉-空间模板相对更具有普遍性作用, 语音环路具有特殊性作用, 并且随着年龄增高, 视觉－空间模板对数学学业任务的作用更为突显。     

THE EFFECT OF TOKEN REWARDS ON “INTRINSIC” MOTIVATION FOR DOING MATH

This study used a multielement baseline design to analyze the effects of token rewards delivered contingent upon completion of math problems by 2 middle-school boys. Time spent on math and number of work pages completed increased (with high accuracy) during reward conditions and were maintained during fading and withdrawal. At follow-up, time spent and work pages completed remained well above baseline for 1 boy and fell below for the other, while accuracy remained high and ratings of liking math were the highest possible for both boys. Overall, the results are inconsistent with warnings about use of token rewards to motivate children.     

Teaching simple addition to children with mental retardation using a microcomputer

Computer assisted instruction (CAI) was used for improving the efficiency of simple addition (single-digit without carrying) computation in three children with mental retardation. The training program, conducted on a portable personal computer, consisted of demonstrations and repetitive practices on simple addition presented in a vertical format. A number of behavioral instruction techniques such as direct feedback, corrective demonstration, verbal association, and token reinforcement were also included. To evaluate the effects of the CAI package, a multiple-baseline across subjects design with followup was employed. Results showed that performances of children improved with training and were maintained at 1-week, 4-week, and 12-week followups. Generalization of the learned skills was evident when children were assessed on a paper-and-pencil addition test in a vertical format. The skills were also transferable, with initial prompting, to a computerized addition test in a horizontal format. These findings reaffirmed the applicability of CAI and behavioral techniques for teaching math skills to children with mental retardation.     

Distributing mathematical practice of third and seventh graders: Applicability of the spacing effect in the classroom

We examined the effect of distributed practice on the mathematical performance of third and seventh graders (N = 213) in school. Students first received an introduction to a mathematical topic, derived from their curriculum. Thereafter, they practiced in one of two conditions. In the massed condition, they worked on three practice sets in 1 day. In the distributed condition, they worked on one practice set per day for 3 consecutive days. Bayesian analyses of the performance in two follow‐up tests 1 and 6 weeks after the last practice set revealed a positive effect of distributed practice as compared with massed practice in Grade 7. In Grade 3, a positive effect of distributed practice was supported by the data only in the test 1 week after the last practice set. The results suggest that distributed practice is a powerful learning tool for both elementary and secondary school students in the classroom.     

‘MusiMath’ and ‘Academic Music’ – Two music‐based intervention programs for fractions learning in fourth grade students

Music and mathematics require abstract thinking and using symbolic notations. Controversy exists regarding transfer from musical training to math achievements. The current study examined the effect of two integrated intervention programs representing holistic versus acoustic approaches, on fraction knowledge. Three classes of fourth graders attended 12 lessons on fractions: One class attended the ‘MusiMath’ holistic program (n = 30) focusing on rhythm within the melody. Another class attended the ‘Academic Music’ acoustic program (Courey et al., Educ Stud Math 81:251, 2012) (n = 25) which uses rhythm only. The third class received regular fraction lessons (comparison group, n = 22). Students in both music programs learned to write musical notes and perform rhythmic patterns through clapping and drumming as part of their fraction lessons. They worked toward adding musical notes to produce a number (fraction), and created addition/subtraction problems with musical notes. The music programs used a 4/4 time signature with whole, half, quarter and eighth notes. In the math lessons, the students learned the analogy between musical durations and fractions, but also practiced fractions other than . Music and math were assessed before, immediately following, and 3‐ and 6‐months post‐intervention. Pre‐ to post‐intervention analyses indicated that only the ‘MusiMath’ group showed greater transfer to intervention‐trained and untrained fractions than the comparison group. The ‘Academic Music’ group showed a trend on trained fractions. Although both music groups outperformed the comparison group 3‐ and 6‐months post‐intervention on trained fractions, only the ‘MusiMath’ group demonstrated greater gains in untrained fractions. Gains were more evident in trained than in untrained fractions. A video abstract of this article can be viewed at https://youtu.be/uJ_KWWDO624