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.     

Equivalence class formation as a function of preliminary training with pictorial stimuli

The present experiment investigated the effects of preliminary training with pictorial stimuli on the subsequent formation of three 5‐member equivalence classes (A?B?C?D?E) in 84 university students assigned to seven groups of 12. In the Abstract (ABS) group, all stimuli were abstract shapes. In the Picture (PIC) group, the C stimuli were pictures, and the remaining stimuli were the same abstract shapes as in the ABS group. For the remaining five groups, all stimuli were the same abstract shapes as in the ABS group. However, across groups, preliminary training involved either the establishment of conditional relations with simultaneous (SMTS) or delayed (DMTS) matching‐to‐sample with 0 s, 3 s, 6 s, or 9 s between the abstract C stimuli and the meaningful pictures. For the ABS and the PIC groups, 16.7% and 83.3% of participants formed classes, respectively. Preliminary training with SMTS and DMTS with 0 s, 3 s, and 6 s produced a linear increase in the likelihood of equivalence class formation, 41.7%, 50%, and 75%, respectively. Increasing the duration of delay further from 6 s to 9 s produced a substantial decline, 50%. This experiment extends knowledge about how including meaningful pictures enhances equivalence class formation.     

Measurement invariance of the Satisfaction With Life Scale (SWLS) by gender and age in Angola

Subjective well-being is a research arena that has grown almost exponentially: over the last 20 years, the number of publications on subjective well-being has increased approximately 16-fold (Diener, 2009). The cognitive aspect of subjective well-being or life satisfaction is referred to a conscious cognitive judgment of life (Diener, Emmons, Larsen, & Griffin 1985), in which person’s quality of life is globally assessed (Shin & Johnson, 1978). The Satisfaction With Life Scale (SWLS, Diener et al., 1985) is the most widely used instrument for its measurement. A reliable, valid and invariant measurement is critical for meaningful comparisons. The aim of this study is to examine the configural, metric and scalar invariance across age in the Portuguese version of the SWLS with a sample of 5630 Angolans. A standard measurement invariance procedure has been applied both across gender and age. Results shown that scalar invariance of the SWLS held across gender and age. This strong invariance allowed for meaningful latent mean comparisons. There were latent mean differences due to gender and age, but while gender differences were modest, the age differences were larger. Results are discussed in their relation with existing literature.     

USING COMPLEX AUDITORY—VISUAL SAMPLES TO PRODUCE EMERGENT RELATIONS IN CHILDREN WITH AUTISM

Six participants with autism learned conditional relations between complex auditory‐visual sample stimuli (dictated words and pictures) and simple visual comparisons (printed words) using matching‐to‐sample training procedures. Pre‐ and posttests examined potential stimulus control by each element of the complex sample when presented individually and emergence of additional conditional relations and oral labeling. Tests revealed class‐consistent performance for all participants following training.     

Semantic false memories in the form of derived relational intrusions following training

Contemporary behavior analytic research is making headway in characterizing memory phenomena that typically have been characterized by cognitive models, and the current study extends this development by producing "false memories" in the form of functional equivalence responding. A match-to-sample training procedure was administered in order to encourage participants to treat groups of unrelated English words as being interchangeable. Following training, participants were presented with a list of words from within one of the groups for a free recall test and a recognition test. Results showed that participants were more likely to falsely recall and recognize words that had been assigned to the same group as the list words during prior training, relative to words not assigned to the same group and relative to words that co-occurred with list words. These results indicate that semantic relatedness can be experimentally manipulated in order to produce specific false memories.     

混合模式调查下的测量不变性：跨文化视角

混合模式调查收集数据的有效性需以模式间测量等同为前提.以往调查模式间的测量等同研究尚未考虑文化因素的影响.本研究通过两个自我创新能力评价量表,基于中美两国数据,运用多组确认性因子分析法考察了将由网络调查和纸笔调查构成的混合调查模式应用于自我能力和行为评估时的测量等同问题.结果在美国的样本中发现创新能力评价量表在两种调查模式下具有测量不变性,而在中国的样本中只发现强因子测量不变性,提示混合模式调查下测量不等同的出现是调查模式和文化因素共同导致的结果.我们认为,集体主义文化扩大与强化了纸笔调查和网络调查之间的匿名感知差异,进而受访者可能在网络调查中呈现出“满意决策”作答行为或在线下的调查中呈现出“社会称许性作答行为”,从而增加了出现测量不等同的可能性.     

Interplay of Biomechanical Constraints and Kinematic Strategies in Selecting Arm Postures

In this study, the authors examined the interplay between biomechanics and control strategies in the resolution of excess degrees of freedom at the joint level. Seven participants made aimed arm movements from 30 starting points and several starting postures to targets. Final arm postures for movements to a target exhibited substantial joint angle variation. Through regression modeling and by comparing observed final arm postures with biomechanically plausible postures, the authors identified 3 kinematic strategies: (a) Maintain deviations from the average angle at the starting point to the joint's final posture; (b) make torso rotations that are a fixed proportion of shoulder rotations; and (c) adopt a characteristic combination of 4 wrist-positioning approaches. The results demonstrated that kinematic strategies can account for substantial variance in final arm postures, if one takes into account 2 types of individual differences—those that arise inevitably from biomechanical constraints and those that reflect choices in movement strategy.