Hello, everyone. Happy back-to-school!

The Fall semester of my second year as a doc student started Wednesday. This semester, I’ll be taking courses on cognitive development, intellectual history of educational psychology, and research design. I’ll continue to work on the CT4EDU project, helping our awesome cohort of teachers to integrate computational thinking into their mathematics and science instruction. In the spring, I will likely teach a course about integrating technology into mathematics instruction. Already, my brain is becoming awash with new ideas to explore and share on this blog.

One of my goals for this year is to make efforts to draw connections between the work I did last year and what I will do this year. When it comes to educational, intellectual, and even recreational pursuits, I have always been a bit of a dabbler. I like exploring lots of different things and struggle with feeling boxed in when I stay with one idea or activity for too long. I’m trying to find a happy medium this year where I don’t feel the need to perseverate on individual things, but also don’t entirely abandon ideas after playing with them for a while.

So as a step in that direction, I’m going to cheat just a little bit on the blogging this week, and share a piece of a course paper I wrote last year. The course focused on exploring educational technology as a field, and a piece of the final project was to make some connections between ed tech theorists and recent research in our particular domain of interest. The text below is my effort to connect the work of Seymour Papert and James Gee to recent research findings about virtual manipulatives. I think it contains some interesting ideas I could pursue this year and beyond.

It was useful for me to revisit it, and I hope you enjoy it too. Thanks for reading, and stay tuned for new blog posts each week this semester!

**Virtual Manipulatives as Transitional Objects with Embedded Knowledge**

Physical manipulatives are concrete objects designed to help children learn mathematical concepts (Uttal, Scudder, & DeLoache, 1997). Common examples of mathematics manipulatives include linking cubes, fraction circle pieces, and tangrams. Physical manipulatives have been used in mathematics education since at least the 1960s, but only in the last two decades have virtual versions of manipulatives had an increasing presence in classrooms. Moyer-Packenham and Bolyard (2016) recently defined a virtual manipulative (VM) as, “an interactive, technology-enabled visual representation of a dynamic mathematical object, including all of the programmable features that allow it to be manipulated, that presents opportunities for constructing mathematical knowledge” (p. 13).

Arguments for the use of manipulatives in elementary mathematics build on the arguments of theorists such as Piaget (1972) and Bruner (1960), who said that young children think concretely and thus have difficulty operating on abstract concepts. Researchers have touted manipulatives as tools that allow elementary children to think concretely about abstract mathematical concepts (Sarama & Clements, 2009; Uttal, Scudder, & DeLoache, 1997). Recent research suggests that VMs may have an edge over their physical counterparts when it comes to promoting learning. Moyer-Packenham and Westenskow (2013) conducted a meta-analysis of 66 studies comparing the use of VMs to other instructional treatments. They found an overall moderate positive effect on student achievement for the use of VMs in mathematics instruction — including in comparison to the use of physical manipulatives in similar instruction.

In an effort to make sense of this finding, Moyer-Packenham and Westenskow (2013) conducted a conceptual analysis on the papers included in the metaanalysis. The conceptual analysis focused on identifying “specific researcher-reported affordances of the virtual manipulatives that promoted student learning in mathematics” (p. 39). Two of the affordances they identified were focused constraint, or limitations placed on the actions that can be carried out on VMs, and simultaneous linking, or the ability to see how changes in one representation of a mathematical idea affect a related representation.

The above mentioned argument for the use of manipulatives, and for the particular utility of VMs, echo arguments by Papert (1980) for the utility of the LOGO environment for learning. Consider, for example, this excerpt from Mindstorms:

[I have] an interest in intellectual structures that could develop as opposed to those that actually at present do develop in the child, and the design of learning environments that are resonant with them. The Turtle can be used to illustrate both of these interests: first, the identification of a powerful set of mathematical ideas that we do not presume to be represented, at least not in developed form, in children; second, the creation of a transitional object, the Turtle, that can exist in the child’s environment and make contact with the ideas. (Papert, 1980, p. 161)

Papert’s discussion of the Turtle as a transitional object is well-aligned with the aforementioned claim that manipulatives help children connect their concrete knowledge to abstract mathematical concepts. In fact, Papert (1980) explicitly noted that the quotation above reflects his suggested extension of Piaget’s ideas about children’s concrete ways of thinking, just as advocates for manipulatives claim that manipulatives are an answer to Piaget’s research.

Still, Papert (1980) argued that computers, in particular, offer excellent opportunities for children to explore and get to know mathematical ideas:

Working in Turtle microworlds is a model for what it is to get to know an idea the way you get to know a person. Students who work in these environments certainly do discover facts, make propositional generalizations, and learn skills. But the primary learning experience is not one of memorizing facts or of practicing skills. Rather, it is getting to know the Turtle, exploring what a Turtle can and cannot do. (Papert, 1980, p. 136)

Recent studies of students’ use of virtual manipulatives – particularly those examining the focused constraints built into VMs – also contain arguments that technology provides opportunities for students to get to know ideas and explore what tools can and cannot do. Evans and Wilkins (2011), for example, compared students’ conversations when working with physical and virtual tangram pieces. Students could move the physical pieces any way that they wished, but when they worked with the virtual tangrams, flips, turns, and slides had to be handled separately via different controls. These focused constraints on movements in the virtual tangrams led to more explicit experimentation with how the pieces could be rearranged and more mathematical terminology being used in student conversations.

Relatedly, Hansen, Mavrikis, and Geraniou (2016) studied one teacher’s use of a VM that showed a numerical answer when students attempted to add two fractions with like denominators, but did not show a numerical answer when students attempted to add two fractions with unlike denominators. The teacher felt strongly that this constraint helped students grasp the reason and need for common denominators.

Researchers exploring the role of focused constraint in student learning with VMs do not typically reference Papert. However, connecting Papert’s ideas to the focused constraint affordance attributed to VMs gave me a new way to think about why focused constraint might be beneficial for learning.

Virtual manipulative research focused on the affordance of simultaneous linking echos arguments made by Gee (2013) in support of games as a context for learning. Consider, for example, Gee’s description of how video games distribute intelligence:

In the game, that experience – the skills and knowledge of professional military expertise – is distributed between the virtual soldiers and the real-world player. The soldiers in the player’s squads have been trained in movement formations; the role of the player is to select the best position for them on the field. The virtual characters (the soldiers) know part of the task (various movement formations), and the player must come to know another parts (when and where to engage in such formations). (Gee, 2013, p. 19)

I find this idea of knowledge distribution to be closely related to the idea of simultaneous linking in VMs. Moyer-Packenham and Westenskow (2013) give examples of forms of simultaneous linking as “linking two different dynamic pictorial objects” and “linking dynamic pictorial objects with symbols” (p. 43). Students may, for example, shade in one of four parts on a fraction VM’s area model and see the numerical representation of the fraction change to ¼. When considering the VM as a “virtual character” and a student using the VM as a “player,” this automatic updating of linked representations can be likened to the knowledge and skills offloaded to the VM. The VM takes care of updating the linked representation; the student is responsible for making the changes necessary to the manipulable representation in order for the linked representation to change to what the student desires it to be.

A recent study of students’ use of VMs with and without simultaneous linking found that students using the linked representations made more generalizations than students not using the linked representations (Anderson-Pence & Moyer-Packenham, 2016). Thus, it seems that through use of a tool that distributes knowledge between the student and a kind of virtual character, students do “come to know” (Gee, 2013, p. 19) particular mathematical ideas and relationships. Gee’s (2013) explanation of the learning potential of games helps to make sense of the learning potential of VMs with simultaneous linking.

In all, through my close reading of Papert (1980) and Gee (2013) this semester, I have come to a more nuanced theoretical understanding of why VMs might have advantages over physical manipulatives in supporting mathematics learning. VMs function as transitional objects that allow students to get to know mathematics ideas (Papert, 1980). They distribute knowledge between the tool and the student, allowing students to come to know particular mathematical ideas through simulation and experimentation (Gee, 2013).

**References**

Anderson-Pence, K. L., & Moyer-Packenham, P. S. (2016). The influence of different virtual manipulative types on student-led techno-mathematical discourse. *Journal of Computers in Mathematics and Science Teaching, 35*(1), 5–31.

Bruner, J. S. (1960). *The process of education.* Cambridge, MA: Harvard University Press.

Evans, M. A., & Wilkins, J. (Jay) L. M. (2011). Social interactions and instructional artifacts: Emergent socio-technical affordances and constraints for children’s geometric thinking. *Journal of Educational Computing Research, 44*(2), 141–171.

Gee, J. (2013). *Good video games + good learning* (2nd Ed.). New York: Peter Lang Publishing.

Hansen, A., Mavrikis, M., & Geraniou, E. (2016). Supporting teachers’ technological pedagogical content knowledge of fractions through co-designing a virtual manipulative. *Journal of Mathematics Teacher Education, 19*(2–3), 205–226.

Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the definition of a virtual manipulative. In P. S. Moyer-Packenham (Ed.),* International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives* (pp. 3–23). Switzerland: Springer International Publishing.

Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. *International Journal of Virtual and Personal Learning Environments, 4*(3), 35–50.

Papert, S. (1980). *Mindstorms: Children, computers, and powerful ideas.* New York: Basic Books, Inc.

Piaget, J. (1972). Intellectual evolution from adolescence to adulthood. *Human Development, 15,* 1–12.

Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. *Child Development Perspectives, 3*(3), 145–150.

Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. *Journal of Applied Developmental Psychology, 18*(1), 37–54.