Knowing how to develop students’ mathematical processes and practices is one of the key indicators of preparing beginning teachers to deepen their understanding of students as mathematics learners (Association of Mathematics Teacher Educators, 2017). Well-prepared pre-service teachers (PSTs) should be able to guide their students to engage in the eight Common Core Standards for Mathematical Practice (SMPs), which range from reasoning abstractly through contextualizing and decontextualizing, to constructing arguments to communicate their thinking, to creating models to solve complex or everyday problems. Although there is widespread consensus that these practices are grounded in “important 'processes and proficiencies’ with longstanding importance in mathematics education” (CCSSI, 2010, p. 6), studies have shown a wide variation in how the SMPs are understood (Lloyd, 2024; Mortimer, 2018).

To this end, there is increasing movement to interpret SMPs as cross-curricular skills that can be extended to other content areas (Rosenfeld, 2020). One primary area of interest has been the complementary relationship between mathematical thinking and the concept of computational thinking (CT) (Barlow et al., 2023; Pérez, 2018). CT is a set of cognitive practices and processes drawn from the computer science field that can be used in problem-solving (Wing, 2006). Overlap between SMPs and CT concepts like sequencing, repetition, and conditionals have led scholars like Rich et al. (2020) to theorize nine general proto-computational thinking ideas that are common to mathematical and computational thinking. Many studies have also attempted to apply the SMPs to teach CT tasks, and vice versa (Wu & Yang, 2022). However, few studies have examined how CT can be applied to deepen teacher candidates’ understanding of the SMPs. This article explores how CT can be used as a framework of strategies to help PSTs interpret the SMPs. While our exploratory study covered all SMPs, we will focus this article only on analyzing results from SMP 1 “Making sense of problems and persevering” and SMP 7 “Looking for and making use of structure.”

Computational thinking

Our conceptualization of computational thinking is guided by the degree of convergence that has occurred around studying CT as a set of practices comprised of abstraction, decomposition, algorithms, and debugging (Angeli et al., 2016; Shute et al., 2017).  Unlike Pérez (2018) who created a theoretical framework for CT dispositions comprised of tolerance for ambiguity, persistence, and collaboration, we specifically focus on CT practices. We adapted Yadav et al.’s (2016) definition to operationalize CT as comprised of the following practices:

1. Decomposition - Breaking a problem into smaller, more manageable parts;
2. Pattern recognition - The ability to use prior knowledge to find patterns that will help solve problems more efficiently;
3. Abstraction - Removing unnecessary information and focusing on what is truly important in a given situation;
4. Algorithmic thinking - Developing a series of instructions to solve a problem; and
5. Debugging - Evaluating the solution to address any errors.

Intervention

Through a partnership with the New York Hall of Science, we designed and delivered a winter workshop on applying CT to elementary mathematics teaching for a cohort of eight undergraduate elementary education PSTs attending an urban, public university in the northeast. The PSTs had all completed one semester each of a mathematics content and methods course, had one semester experience student teaching in the field, and had already been exposed to the concept of CT in their prior teacher preparation coursework. The two-hour, online workshop began with a review of the aforementioned definition of CT, with examples first of applying CT to the everyday “problem” of cleaning your room, and second of the scenario of teaching a 1st grade mathematics task of 32+43 using a hundreds chart. PSTs quickly pointed out how the CT practice of decomposition can be used in breaking apart a number into its place value parts (e.g., 43 as 4 tens and 3 ones). We then went into more depth discussing how other CT practices can be applied to use a hundreds chart for addition. For instance, as shown in Figure 1, PSTs connected: pattern recognition to identifying rows and columns on the hundreds chart; abstraction to remove extraneous information like the need to move up or left in a hundreds chart when adding; and algorithmic thinking to plan step-by-step movements through the chart.

Figure 1. Example of applying CT practices to teach addition within 100 using a hundreds chart

The second part of the workshop focused on relating the SMPs to CT. After a brief review of the eight SMPs as a whole group, PSTs worked in breakout groups to use the Next Generation Mathematics Standards document (NYSED, 2019) – which includes a description of the eight SMPs – to create their own definition of two SMPs to present to the whole group. The workshop climaxed with an activity to create a crosswalk of the SMPs and CT practices. Specifically, we reviewed each SMP to identify which CT practices might be associated with the SMP.

Results

PSTs’ definitions of SMP 1 and SMP 7    

PSTs first presented a definition of what it means for their students to make sense of problems and persevere (SMP 1) by looking for entry points and solution pathways (see Table 1). As seen in the definition provided by PSTs for SMP 1, there is a strong emphasis on metacognition and persistence. They underscored the possibility of using concrete objects and pictures to help solve the problem, as well as trying simpler forms of the problem. They also highlighted the importance of reflection in sensemaking, during which students can check their answer using a different method.  They began seeing parallels to the CT practice of debugging, where iteration and correction are central through checking solutions and engaging in processes like trial and error.

Table 1. PSTs’ definition of SMP1 and associated CT practices

PSTs’ definition of SMP 7 was grounded in the idea of looking for a pattern (see Table 2). They noted that structures can vary by grade/age and take the form of everything from shapes, numbers, even/odds, expressions, to algebraic numbers/formulas. In explaining how a student “may see an equation and have it reconstructed in their heads,” a PST drew on the example of distributive property and how “7 times 8 kind of turns into like 7 times 5 and 7 times 3.” Students could thus “see complicated things as a single object composed of other objects.” This need not be captured in writing and could happen mentally “in their [the student’s] head.”

Table 2. PSTs’ definition of SMP7 and associated CT practices

Note: The grammatical and syntactic errors in the table presented reflect the original responses of our PSTs, which have been preserved to maintain authenticity. 

Applying CT practices to interpret the SMPs

One of the CT practices that PSTs associated most with the SMPs was pattern recognition. When PSTs began the discussion of SMP 1, the group collectively added pattern recognition to the list because students might try to make sense by drawing on prior knowledge and “looking at what they know already.” When looking for entry points to a problem or in trying analogous problems, they discussed how a student might identify and draw patterns across problems. Later when the group turned to discussing SMP 7, another PST was quick to add pattern recognition as a strategy because looking for structure, she argued, begins with looking for patterns. For a young child, this could take the form of intuitively finding common attributes to sort shapes.

PSTs also drew on abstraction as a key CT practice that overlapped with many SMPs. After agreeing upon pattern recognition as a practice that could be associated with SMP 7, a PST added that abstraction “complements pattern recognition in a way because you're removing unnecessary information that you might not need” when you look for the structure within a problem. Another PST argued, as she was explaining how she interprets SMP 1, “If you're trying to make sense of a problem like for me personally, it helps to kind of like filter out the unnecessary information and just focus on what I know and what I can do with it.”

Finally, PSTs drew on their understanding of decomposition to interpret SMPs. In the context of SMP 1, one PST said, “Because when you're trying to like, make sense of something, especially when you're doing like word problems…you're picking and pulling like the most important pieces to make sense of it all…so I feel like when you break it down to something that's more manageable and the important stuff, that's when you start to understand it.” As the group homed in on how SMP 7 entails seeing “complicated things as a single object composed of other objects,” one PST connected the example of distributive property to decomposition since you have to “break up 8 as like 2 separate numbers.”

Discussion

Implications for PST Learning and Teaching      

This article aimed to demonstrate examples of how PSTs can apply CT to interpreting the SMPs. Our results suggest that coupling SMPs with a CT framework enriched a discussion of the SMPs by providing PSTs another lens through which to understand the eight SMPs. Based on our past experience, many of our elementary education PSTs have found the Common Core/Next Generation definitions of the SMPs to be dense. PSTs often struggle both to understand and to know how to incorporate the SMPs into their teaching.

We argue that asking PSTs to interpret the SMPs through a CT framework made the SMPs themselves more accessible. In fact, one PST said in a pre-survey before the workshop that she “honestly need[s] more practice and support in helping students develop these.” In a post-survey however, she shared that she translated the workshop to plan a lesson where she “specifically targeted practices 1,2,3, and 8…through the use of ‘think alouds’ in which… [she] had [students] construct arguments through debugging.” She further detailed that she, “asked a question in which students would critique my reasoning and construct a viable argument with an explanation for how they would solve the problem.” Our findings thus suggest the utility of CT as a framework to help PSTs not only access the SMPs for themselves, but to potentially translate that into engaging students in mathematical practices.

Limitations and Future Directions

Despite the affordances that CT seemed to offer in deepening PSTs’ analyses of SMPs, there were notable constraints mainly around the accuracy of PSTs’ interpretations. For instance, in identifying CT practices that are associated with SMP7 (making use of structure), one PST added algorithmic thinking to the list based on the argument that “you sometimes solve the problem in like a specific order, like PEMDAS, for example…[where] you have to do the multiplication first and then you add.” However, one might question how applying PEMDAS to solve a problem shows a direct connection between algorithmic thinking and making use of structure. While there may be an ancillary link between the two, the basic act of solving a problem through steps that follow a sequential order does not illustrate seeing structure within a number or exploiting a pattern and structure within a problem (Bleiler et al., 2015).

Nevertheless, this exploratory study suggests that CT can be a promising way to guide PSTs to understand the SMPs in order to cultivate these practices in their students. Future research can further explore how CT can help PSTs attend to developing their students’ mathematical thinking, as well as how the explicit introduction of CT directly to elementary students can affect students’ engagement in mathematical practices.

Acknowledgement: This research was supported by a City University of New York Computing Integrated Teacher Education (CITE) grant. We gratefully acknowledge the financial support that made this study and its publication possible. We would also like to thank the editor and reviewers of Connections for their suggestions which helped to strengthen this publication. 

References
 

Angeli, C., Voogt, J., Fluck, A., Webb, M., Cox, M., Malyn-Smith, J., & Zagami, J. (2016). A K-6 computational thinking curriculum framework: Implications for teacher knowledge. Journal of Educational Technology & Society, 19(3), 47-57.

Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of Mathematics. Available online at amte.net/standards.

Barlow, E. K., Barlow, A. T., & Nadelson, L. S. (2023). Computational thinking: Perspectives of preservice K-8 mathematics teachers. Contemporary Issues in Technology and Teacher Education, 23(2). https://citejournal.org/volume-23/issue-2-23/mathematics/computational-thinking-perspectives-of-preservice-k-8-mathematics-teachers

Bleiler, S. K., Baxter, W. A., Stephens, D. C., & Barlow, A. T. (2015). Constructing meaning: Standards for mathematical practice. Teaching Children Mathematics, 21(6), 336-344.

Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Retrieved from http://corestandards.org

Lloyd, M. E. R. (2024). Mathematical practices are everywhere: The intersections of pre‐service teacher claims, non‐mathematics‐education faculty claims, and observable actions. School Science and Mathematics, 124(5), 340-358.

Mortimer, J. (2018). Pre-Service Teachers' Understandings and Interpretations of the Common Core State Standards for Mathematical Practice (Doctoral dissertation).

NYSED. (2019). New York State Next Generation Mathematics Learning Standards. Retrieved from https://www.nysed.gov/sites/default/files/programs/curriculum-instruction/nys-next-generation-mathematics-p-12-standards.pdf

Pérez, A. (2018). A framework for computational thinking dispositions in mathematics education. Journal for Research in Mathematics Education, 49(4), 424–461. https://doi.org/10.5951/jresematheduc.49.4.0424

Rich, K. M., Spaepen, E., Strickland, C., & Moran, C. (2020). Synergies and differences in mathematical and computational thinking: Implications for integrated instruction. Interactive Learning Environments, 28(3), 272-283.

Rosenfeld, R. (2020). Extending mathematical practice standards across the curriculum. WestEd. https://www.wested.org/blog/mathematical-practice-standards-across-the-c...

Shute, V. J., Sun, C., & Asbell-Clarke, J. (2017). Demystifying computational thinking. Educational Research Review, 22, 142-158.

Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35. https://doi.org/10.1145/1118178.1118215

Wu, W. R., & Yang, K. L. (2022). The relationships between computational and mathematical thinking: A review study on tasks. Cogent Education, 9(1), 2098929.

Yadav, A., Hong, H., & Stephenson, C. (2016). Computational thinking for all: Pedagogical approaches to embedding 21st century problem solving in K-12 classrooms. TechTrends, 60, 565-568.