Foundations of Learning – Constructivism in Mathematics

Last week in CEP 811, I used my Squishy Circuits kit and some repurposed items to make a totally awesome game to help my third graders practice rounding to the nearest tens/hundreds place. While rounding is a third grade learning goal, and it is worked on in second grade, I still have quite a few students who can’t master rounding to the nearest hundred. I created a self-checking game in which students worked in small groups (2-4 people) to make their way around a game trail, self-checking their rounding abilities to gain additional practice in rounding.

photo 4 (2)

I thought my game would be the perfect solution – while students would be earning “points” for correct answers on their own, they would be helping one another since they were working in a small group. Those who were getting incorrect answers could talk with their group members about how to arrive at the correct answer. Students would synergize to help one another learn, and they would be able to do so without the direct support of the classroom teacher (though I found myself sitting with each group the first time they played the game, just to monitor group work).

This week in class, we needed to “work backwards”, in a way. Now that we had designed this amazing lesson, what aspects of learning were already being supported by it? What research could we find that would either provide justification for what we had planned, or inform us of changes we needed to make?

We started by viewing a TED talk by Richard Culatta on reimagining learning. He discusses with viewers the implications of what he calls the “digital divide” – digitizing traditional learning practices, and knowing how to use technology to reimagine learning (Culatta, 2013). Too often, teachers simply digitize learning practices and call it “using technology” – the learning is not really revamped by the use of the technology (Culatta, 2013). His TED talk got me thinking about the game I had designed last week. It was using technology to reimagine learning (I created a new game and therefore used the Squishy Circuit in a new way to allow students to explore a previously-learned concept). What was it about my game that was helping my students to learn, though?

I began by researching Constructivism theory as it relates to mathematics. Constructivism is an approach to learning in which “the teacher…facilitates the students’ construction of meaning and the understanding of the content” (White-Clark, DiCarlo, & Gilchriest, 2004, p. 41). I first taught my students about rounding, and then I set up an opportunity for them to explore and master the concept on their own/in groups. In terms of mathematics, Constructivism allows students to make personal connections to the content, forming these connections based on work they are doing with peers in a small group (White-Clark, DiCarlo, & Gilchriest, 2004, p. 41). I had taught a couple of lessons on rounding about a week or two before I created my game, so my students already had a basic understanding of rounding to work off of. The game I created allows students to construct meaning and understanding of rounding by identifying patterns for rounding (they were seeking to understand why some LEDs light up and some do not). Though my game was designed for groups of two to four students, I found that it ran best with four students. Aligned with Constructivism, this allowed the most opportunities for students to make connections with their peers to further their understanding of rounding. While working with just one partner provides opportunities for connections, hearing ideas from three other peers offers more opportunities for students to make connections with one another. Students can learn better when their peers teach them. Having the chance to hear from three peers could significantly help my students who still aren’t quite grasping rounding yet.

This leads to the next point from Constructivism, tested by Kroesbergen, Van Luit, and Maas (2004) – does it hurt lower achieving students to allow them to use a more Constructivist approach to learning mathematics concepts? While the research of Kroesbergen et. al. (2004) focused on multiplication skills, the underlying debate of which type of instruction is more beneficial for low-achieving students really resonated with me in terms of my game. Just from sitting with a few groups of students who played the game, there always seemed to be at least one student who rounded incorrectly. What if my game was hurting their development of the concept of rounding, because they were being exposed to incorrect answers? Kroesbergen et. al. (2004) found in their research that Constructivist instruction positively influenced their low-achieving students’ motivation (p. 247). My lower-achieving students should not suffer any damage to their motivation by working with their peers. If anything, their peers are helping them to feel  better about their own skills, and helping them to discover what’s missing from their schema of rounding. My low-achieving students should be exposed to incorrect answers. While Kroesbergen et. al. (2004) bring a valid point to life – that “both correct and incorrect solutions… could produce confusion for low achievers” (p. 248), this is the only way to help students correct misconceptions. If they are never exposed to incorrect answers, how can they develop a full understanding of a concept? The best way for students to learn is by “doing”. My low-achieving students need the chance to discover on their own why their answer was incorrect.

So, what would I change about my game?

To help all students with this type of “incorrect answer turned correct” discovery, I should have allowed to students to use manipulatives. White-Clark, DiCarlo, and Gilchriest (2008) remind teachers that “hands-on activities… and manipulatives are elements that embrace the constructivist educational philosophy” (p. 42). I need to allow students to use open number lines, base-10 blocks, and paper/pencil to work their way towards the correct answer. I could check-in with groups playing the game and have them show me how they know their answer is correct. This will help my low-achieving students to do two things: learn from the work of their peers, and have a concrete model to rely on for future rounding problems. This modification to my original plan does not change the course of the game; rather, it allows me to update the materials needed to help ALL of my students to become successful mathematicians. Be sure to check out my updated lesson plan!


When students work together, learning becomes everyone’s weight to carry – not just one person’s. [photo credit – Rubin, Nancy. (4 July 2012). Connecting Critical Thinking to Online Learning [photo]. Retrieved from]


Culatta, Richard. (2013). Reimagining Learning: Richard Culatta at TEDxBeaconStreet. Retrieved from

Kroesbergen, Evelyn H., Van Luit, Johannes E. H., and Maas, Cora J. M. (Jan 2004). Effectiveness of Explicit and Constructivist Mathematics Instruction for Low-Achieving Students in the Netherlands. The Elementary School Journal, 104(3), 233-251. Retrieved from

Rubin, Nancy. (4 July 2012). Connecting Critical Thinking to Online Learning [photo]. Retrieved from 

White-Clark, Renee, DiCarlo, Maria and Gilchriest, Nancy. (Apr. – May, 2008). “Guide on the Side”: An Instructional Approach to Meet Mathematics Standards. The High School Journal, 91(4), 40-44. Retrieved from


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