Backward transfer, the relationship between new learning and prior ways of reasoning, and action versus process views of linear functions

ABSTRACT

Backward transfer is defined as the influence that new learning has on individuals’ prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students’ prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from low-socioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions.

KEYWORDS:

• Transfer
• functions
• linear
• quadratic
• action view
• process view

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Breidenbach et al. (1992) also proposed two other categories, namely reasoning about functions as objects and as schemas. However, in our study we did not see evidence of these ways of reasoning about functions. Also, our focus on just action and process views is consistent with the focus of Breidenbach et al.

2. An abbreviated buildup process could be used apart from a linear equation or as part of using a linear equation.

3. Other problems on the assessment were designed to look at correspondence vs covariational reasoning about functions (Confrey & Smith, 1995), and at levels of covariational reasoning (Carlson et al., 2002). Since this was an exploratory study, we used an array of types of problems to capture as many influences on students’ reasoning as we could. We reported findings from the other problems elsewhere (Hohensee et al., 2021).

4. The following link provides an overview of quadratic function topics covered in each lesson: (https://drive.google.com/file/d/1IVukvuOni8WfNvC-iEgT_GwHjxGA0faX/view?usp=sharing).

5. Note that Abby completed the same version of the assessment pre and post.

Additional information

Funding

The research reported in this article was funded by the National Science Foundation (DRL 1651571).