By Michael Telek

Deep in the heart of Music City USA, Dr. Ricki Gibbs and his team at Warner Arts Magnet Elementary are engaged in some professional development with the Institute for Learning. Gathered in small groups, they are shown this picture and asked, “Which one does not belong?” (WODB) It seems like a simple question, but is it? Check it out and consider which you think doesn’t belong

Did you eliminate 4 because it is an unfamiliar diagram? Or, maybe you eliminated 3 because the expression contains parenthesis. You may have elimated 2 because it contains 3 factors. It’s possible that you struggled to eliminate any of them because they all equal 60.

So, what’s the point of asking a room full of educated adults, adults just like you, a question like this? A question that can also be posed to students. Well, it has to do with the cognitive effort employed to make connections among the different representations and reason how they may or may not be related. The mental acrobatics a person uses to connect and reason is unique to them because it is based on their existing knowledge and set of experiences. No matter which you eliminated or kept, the work of coming to a decision is the critical part!

“If we see entry points and how to solve this problem differently, think about how our scholars, who don’t have the knowledge background that we have, see it” said Dr. Gibbs “That ‘a-ha’ moment was, how are we going to find ways to onboard, give entry points, to scholars that show them how to attack the problem but not show them just one procedure to do it.”

IFL Math Fellow Kristin Klingensmith explains, “We use  Which One Doesn’t Belong? activities and other similar activities for a variety of reasons. The activity is not routinized or procedural in nature. Their design provides an opportunity for students to draw on their prior knowledge and from their experiences in and outside the classroom.” She goes on to say, “One of the most important things about activites like this one is that there is more than one way to reason about how the represenations are connected and which can be eliminated. The mental process that a person, school-aged or adult, uses to make a determination is one that is unique to them. The beauty of activities like this is the rich conversation that grows out of making connections between and among the four representations.”

More About “Which One Doesn’t Belong?” and Similar Activities

These types of activities are beneficial for many reasons, e.g., they can be used for formative assessment purposes before, during, and after the study of a concept.

Additionally, they

    • offer opportunities to apply existing knowledge.
    • provide low risk engagement opportunities with no/few wrong answers.
    • afford opportunities for every student to answer, to weigh in, to vote.
    • require the use and connection of representations.
    • require making and supporting a claim.
    • focus on the “why” of the claim, not agreement or disagreement with the claim.

By challenging the thinking of adults, the Warner team discovered new approaches that they can adopt and considered ways to refine their existing approaches to challenge the thinking of their students, or scholars, as they are known in the downtown Nashville campus

Purposeful questions, one of the eight effective teaching practices for mathematics named by the National Council of Teachers of Mathematics’s, is a specific instructional practice that was explored with the Warner team.

“Without purposeful questioning, activities such as these, though fun and relatively risk free, can fall short of being mathematically meaningful experiences,” said Klingensmith.

In a school where 93% of the learners are considered economically disadvantaged, a strong conceptual understanding of math can be part of the building blocks for a better tomorrow. That’s why Dr. Gibbs views math as “as a universal language that serves as a gateway to success” for scholars at the PK-5 Arts Magnet School. “Making numbers come alive” can be the key to opening that gateway.

“Most people only think of fun when it comes to recess, but the fun should happen in the academics as well,” said Dr. Gibbs. “It was one of the keys for us because often time school becomes so serious that everybody is worried about this student outcome, do you have this on track, that on track, and people forget about the true element—that you are educating children and children deserve to have fun.”

Our friends in the Volunteer State use an indicator called the Tennessee Value-Added Assessment System (TVAAS). For more than two decades, TVAAS has measured student growth year after year, regardless of whether the student is proficient on the state assessment. In calculating a TVAAS score, a student’s performance is compared relative to the performance of his or her peers who have performed similarly on past assessments.

Purposeful Questions and “Which One Doesn’t Belong?” Activities Using purposeful questions can help tie the discussion to the mathematical structures that are illustrated by the representations. Here are some that could be used with this WODB.
Questions to Elicit Student Thinking
  • Tell us about the one you think doesn’t belong and why it doesn’t belong.
  • Who else eliminated that representation and what was your reasoning?
  • Who eliminated a different representation and why did you eliminate it?
Questions to Connect Representations and Explain Mathematical Reasoning
  • Someone said that they all belong because they all represent 60. Do you agree or disagree with this claim, and why? Turn and talk.
  • How are Diagram 1 and 5 x 4 x 3 related?
  • Where do you see the factors of 5 x 4 x 3 in Diagram 1?
  • I heard someone say that all four of the representations can be shown with 5 x 12?
  • What connections can you make between (2 x 12) + (3 x 12) and Diagram 4?
  • Turn and talk. How could you change the blue line in Diagram 4 to more clearly show (2 x 12) + (3 x 12)?

Three years ago, Warner Arts Magnet Elementary was listed as a level 1 school in TVAAS terms. This meant students were not growing at all. That’s part of the reason the school looked to a partnership with the IFL “to expedite student growth and outcome in numeracy,” according to Dr. Gibbs.

Fast forward to August 2022, Warner now sits as a level 5 TVAAS school. It’s the highest level on the scale. This designates that the school was the “most effective” and that there’s “significant evidence that the school’s students made more growth than expected.”

A month later Warner earned more praise. They advanced from their state-designated “priority status” and are now considered a “reward school.” This recognizes Warner as one of the top-performing schools in Tennessee.

Klingensmith, who has collaborated with Gibbs and his team at Warner Elementary for several years, credits their “willingness to reflect on current practice and implement insights gained from our joint work to create spaces of more equitable instruction.”

“It gives them [students] that sense of pride. We saw that while taking our state assessments,” said Dr. Gibbs. “After the reading section, I had a couple kids walk up to me and say, ‘Man, Dr. Gibbs, that was really hard. But I’m ready for the math!’ So, we owe a debt of gratitude to some of the IFL work.”