In prior work, we fit the mixture Rasch model to item responses from a fractions survey administered to a nationwide sample of middle grades mathematics teachers in the United States. The mixture Rasch model located teachers on a continuous, unidimensional scale and fit best with 3 latent classes. We used item response data to generate initial interpretations of the reasoning characteristic of each latent class. Our results suggested increasing facility reasoning about fraction arithmetic from one class to the next. The present study contributes two further arguments for the validity of our initial interpretations. First, we administered the same survey to a new sample of future middle grades mathematics teachers before and after 20 weeks of instruction on multiplication, division, and fractions, and we found that from pretest to posttest future teachers transitioned from one latent class to another in ways consistent with increased proficiency in fraction arithmetic. Second, we interviewed 8 of the future teachers before and after the instruction and found that future teachers’ reasoning during interviews was largely consistent with our original interpretation of the 3 latent classes. These results provide further support for our original interpretation of the mixture Rasch analysis, demonstrate the utility of our approach for capturing growth and change in future teachers’ reasoning during teacher education coursework, and contribute innovative applications of psychometric models for surveying teachers’ reasoning at scale.

在之前的工作中，我们将混合 Rasch 模型与分数调查的项目响应相匹配，该分数调查是针对美国中年级数学教师的全国样本进行的。混合 Rasch 模型将教师定位在一个连续的、一维的尺度上，并且最适合 3 个潜在类别。我们使用项目响应数据来生成对每个潜在类的推理特征的初步解释。我们的结果表明，从一类到另一类，增加关于分数算术推理的便利性。本研究为我们最初解释的有效性提供了两个进一步的论据。首先，我们在接受乘法、除法和分数教学 20 周前后，对未来中学数学教师的新样本进行了同样的调查，我们发现，从前测到后测，未来的教师从一个潜在类别过渡到另一个潜在类别的方式与分数算术熟练程度的提高相一致。其次，我们在教学前后采访了 8 位未来教师，发现未来教师在采访中的推理与我们最初对 3 个潜在类别的解释基本一致。这些结果进一步支持了我们对混合 Rasch 分析的原始解释，展示了我们的方法在教师教育课程中捕捉未来教师推理的增长和变化的效用，并为大规模调查教师推理贡献了心理测量模型的创新应用.