I’ve been thinking a lot this week about the role of constraints in learning.

I first got interested in this idea a few years ago after reading a review paper on virtual manipulatives. Moyer-Packenham and Westenskow (2013) conducted a meta-analysis of studies of virtual manipulatives (VMs), finding an overall moderate positive effect on learning. They also examined the studies to identify specific affordances of VMs that seem to contribute to the positive effect. They described one such affordance as follows:

One affordance identified during the conceptual analysis was *focused constraint.* Constraining and focusing features included: bringing to a specific level of awareness mathematical aspects of the objects which may not have been observed by the student; and, applets focusing student attention on specific characteristics of mathematical processes or procedures. (Moyer-Packenham & Westenskow, 2013, p. 42)

Moyer-Packenham and Westenskow’s (2013) pointed me to a few nice examples of the ways constraints can influence learning, and I’ve found a few more within the VM literature. For example, constraints can promote efficient problem-solving strategies. Manches, O’Malley, and Benford (2010) found that when a VM constrained students to move only one counter at a time, they were more likely to use compensation strategies to find number combinations. That is, rather than starting from scratch when finding number combinations, students were more likely to make small adjustments to the combinations — transforming 6 and 3, for example, to 5 and 4 by moving one counter. Students were less likely to do this when using physical manipulatives, when they could move many counters at once. Maschietto and Soury-Lavergne (2013) described how they made design decisions with a VM to promote efficient strategies. For example, they sometimes removed a button that allowed for adding 1. This constraint was meant to encourage students to instead add 10s to complete a task.

Constraints can also draw attention to particular elements of mathematics that tend to pose difficulties for students. Evans and Wilkins (2011) found that the tools students used to manipulate the pieces of a virtual tangram, while somewhat restrictive in that they separate rotation from other moves, focused their attention on the underlying geometry. By contrast, students using physical pieces, where movement was not restricted, did not discuss the underlying geometry. Hansen, Mavrikis, and Geraniou (2016) described a virtual fractions manipulative that shows the numeric sum when students add two fractions with common denominators, but does not show the numeric sum when they add fractions with unlike denominators. Students see a visual representation of the two combined fractions, but the lack of a numeric answer prompts them to think about how to express the sum numerically. The authors described one teacher’s thoughts about how this constraint helped a student overcome a tendency to add fractions by adding numerators and adding denominators.

All in all, these articles really pushed my thinking about constraints. The word *constraints *tends to carry with it some negative connotations, and I found it really interesting to think about the positive effects they can have on learning. They can promote efficiency and help student think and multiple levels about a problem — about the overall problem solving task or procedure as well as the underlying mathematical ideas.

With that summary, my train of thought switched tracks a bit. Efficient strategies and multiple levels of abstraction — what does that sound like? Computational thinking! I’m deeply interested, now, in how VMs can play a role in transforming elementary mathematics to support CT and get kids ready for computer science. That’s something I hope to explore further in my remaining years in graduate school. Stay tuned for further posts on that.

But staying with the idea of constraints a bit longer, connecting constraints to CT did make me remember a conversation I had with the LTEC team a while back. We were developing a learning trajectory for sequencing, and discussing this particular learning goal: “Choose from a limited set of instructions a valid set to accomplish a particular task.” We came to realize we had differing ideas about the effect of the constraint, “a limited set of instructions.” I had been thinking about it as a scaffold: choosing among a few options can be easier than coming up with the answer out of nowhere. But other team members pointed out the constraint can actually add difficulty: It’s easier to express something using any words or actions you want than it is to express the same idea using only a limited set of options.

So what’s the difference? Why are some constraints helpful and others not? I think key lies in the source of the constraint. When constraints are intentionally built in to an educational artifact or task by a thoughtful designer, they can be really helpful to learning. When tasks are constrained by the real-world context of the problem — for example, the particular commands available in a programming language — those constraints pose learning challenges. Still, they are challenges we need to help students overcome. Designers of educational interventions would do well to keep both kinds of constraints in mind.

**References**

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. https://doi.org/10.2190/EC.44.2.b

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. https://doi.org/10.1007/s10857-016-9344-0

Manches, A., O’Malley, C., & Benford, S. (2010). The role of physical representations in solving number problems: A comparison of young children’s use of physical and virtual materials. *Computers and Education*, *54*(3), 622–640. https://doi.org/10.1016/j.compedu.2009.09.023

Maschietto, M., & Soury-Lavergne, S. (2013). Designing a duo of material and digital artifacts: The pascaline and Cabri Elem e-books in primary school mathematics. *ZDM – International Journal on Mathematics Education*, *45*(7), 959–971. https://doi.org/10.1007/s11858-013-0533-3

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. https://doi.org/10.4018/jvple.2013070103