skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on February 13, 2026

Title: A case study of a seven-year old’s reasoning with and understandings of function graphs
This study consisted of Grades 1 and 2 (ages 6-8) classroom teaching experiments (CTE) with a lesson sequence focused on graphical representations of algebraic relationships. We interviewed students before, during, and after the CTE. Here we report on the progression of one Grade 2 (age 7) student’s thinking across the CTE. Prior to the CTE, the student had not previously interacted with representations of algebraic relationships. By the end of the CTE, the student was able to generalize about the functional relationships using graphs. This study illustrates how a learning trajectory modeling students’ understandings of function graphs can be used to characterize one children’s learning and provides evidence that young students are able to reason with function graphs.  more » « less
Award ID(s):
2201095
PAR ID:
10635672
Author(s) / Creator(s):
; ; ; ; ;
Publisher / Repository:
Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14)
Date Published:
Format(s):
Medium: X
Location:
https://hal.science/hal-05199163v1
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; ; (, Proceedings of the 19th International Conference of the Learning Sciences - ICLS 2025)
    Rajala, A; Cortez, A; Hofmann, H; Jornet, A; Lotz-Sisitka, H; Markauskaite, L (Ed.)
    This study investigated the seeds of algebraic thinking that Kindergarten students use when engaging with function tables and graphs. Through interviews with three Kindergarteners, we explored how they reasoned about functional relationships. Our results illustrate how the Kindergarteners used seeds of algebraic thinking when using function tables and graphs to represent and reason about functional relationships. Building on the seeds of algebraic thinking and Knowledge in Pieces frameworks, we categorized these seeds as either strategies (classify, pair, and compare) or ideas (seeds of covariation). Strategy seeds were goal-oriented, and seeds of covariation were elicited without any goal and reflected a broader understanding of change between quantities. 
    more » « less
  2. ; ; ; ; ; (, PME)
    Evans, T; Marmur, O; Hunter, J; Leach, G; Jhagroo, J (Ed.)
    This case study of one first grade student involves the analysis of three interviews that took place before, during, and after classroom teaching experiments (CTEs). The CTEs were designed to engage children in representing algebraic concepts using graphs. Using a knowledge-in-pieces perspective, our analysis focused on identifying students’ natural intuitions and ways of thinking algebraically about a functional relationship represented using graphs. Findings reveal four seeds, two of which were identified in prior studies, and how the activation and coordination of these seeds results in students' production of function graphs. 
    more » « less
  3. ; ; (, Journal of Research in Science Teaching)
    Abstract Demand for engineering‐interested and proficient high school graduates continues to grow across the nation. However, there remains a severe gap in college participation and employment in engineering fields for students with learning disabilities (SWLDs). One potential way to encourage SWLDs to consider engineering as a profession and promote the development of key science attitudes may be through engineering and technology career and technical education (E‐CTE) coursework. In this study, we address the following research questions: Do SWLDs take E‐CTE courses in the early years of high school at different rates compared to students without learning disabilities? What is the relationship between early E‐CTE coursetaking and science attitudes (self‐efficacy, utility, identity), and does this differ for students with and without learning disabilities? How do specific engineering career expectations change with respect to enrollment in early E‐CTE coursework, and do these differ for students with and without learning disabilities? We utilize the High School Longitudinal Study of 2009 (HSLS) to respond to the research questions through moderation models and a student fixed effects methodology. Ultimately, we found no evidence of SWLD underrepresentation in E‐CTE in high school. However, SWLDs were expected to benefit more than the general population from E‐CTE participation with respect to higher levels of science self‐efficacy and science identity. Implications from these findings include how to encourage persistence along the engineering pathway, the growth of career pathway policies at the state level, and how to incorporate E‐CTE practices in academic courses. 
    more » « less
  4. ; ; (, Chemistry Education Research and Practice)
    Education in organic chemistry is highly reliant on molecular representations. Students abstract information from representations to make sense of submicroscopic interactions. This study investigates relationships between differing representations: bond-line structures, ball-and-stick, or electrostatic potential maps (EPMs), and predicting partial charges, nucleophiles, and electrophiles. The study makes use of students’ answers in hot-spot question format, where they select partially charged atoms on the image of a molecule and explanations. Analysis showed no significant difference among students when predicting a partially positive atom with each representation; however, more students with EPMs were able to correctly predict the partially negative atom. No difference was observed across representations in students predicting electrophilic character; while representations did influence students identifying nucleophilic character. The affordance of EPMs was that they cued more students to cite relative electronegativity indicating that such students were able to recognize the cause for electron rich/poor areas. This recognition is central to rationalizing mechanisms in organic chemistry. This study offers implications on incorporating EPMs during instruction and provides evidence-based support in how EPMs could be useful in promoting learning on topics that relate to an uneven charge distribution. 
    more » « less
  5. (, Journal of career and technical education)
    The legacy of career and technical education (CTE) in years past, stretching into the vocational education era, was lower college enrollment after participating in CTE. More recently, others have concluded that college enrollment is not affected but that CTE students enroll more frequently in 2-year degrees over 4-year degrees. I revisit the question with more recent data than has been used by other researchers. I adjust my estimates by the level of academic preparation that students have undergone in high school, known as curricular intensity. I use transcripts from the High School Longitudinal Study of 2009 (HSLS:09) to determine participation in meaningful CTE course sequences, known as programs of study, and then follow the students via the HSLS:09 survey administered in the fall after high school. I find that participation in CTE has no association with a student’s probability of enrolling in college. Further, participating in CTE has no association with a student’s decision to enroll in a two-year or four-year program, after adjusting for curricular intensity. These results are exciting because they allow for the general recommendation of CTE to any interested student, without fear of adverse effects on later educational attainment and the no-harm finding opens the door to many new CTE policy conversations. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email