# What Meaningfulness Means

Learning and schooling are not the same thing. There are children who are great learners but terrible students. These young people are full of ideas and questions, but they have not managed to connect their innate curiosity with their experiences in school. There are many possible reasons for this. Children may find school to be a hard place to inhabit, due to invisible expectations that leave them feeling alienated. Sometimes, school curriculum just seems irrelevant: their personal questions about the world do not find inroads in the work they are asked to do.

Although many parenting books extol children’s natural curiosity and emphasize its importance in their learning and development, schooling too often emphasizes compliance over curiosity. Thus, it is not surprising that children who are great learners and weak students have their antithesis: children who are great students but who are less invested in learning and sense making. Make no mistake: these students hit every mark of good organization, compliance, diligence, and timely work production, but they do not seek deep engagement with ideas. Given the freedom to develop a question or explore an idea, they balk and ask for more explicit directions. I have heard teachers refer to these children as “teacher-dependent.”

Too often, meaningfulness falls through this gap between learning and schooling. There is a fundamental contradiction at play: meaningfulness arises from and connects to children’s curiosity, yet “curious children” is not entirely synonymous with “successful students.” Meaningfulness comes about when students develop an appreciation for mathematical ideas. Rich and meaningful learning happens when students draw on prior knowledge and experiences to make sense of ideas and explore problems, invoke their own strategies, get to ask “what if…?” In short, meaningful learning happens when students’ activity connects to their own curiosity. To make meaningfulness central to math teaching, then, teachers need to narrow the gap between being curious and being a good student.

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Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.

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# Why Meaningfulness Matters

Every math teacher, at one one time or another, has been asked the question, “When are we going to use this?” While this question often gets cast as students’ resistance to learning, it can be productively reinterpreted as a plea for meaningfulness. When the hidden curriculum of math class –– the messages that are inadvertently relayed through classroom organization and activity –– consistently communicates that meaning does not matter, we end up with hordes of students who no longer reason when they are doing math. They instead focus on *rituals*, such as *following the worked example*, and *cues,* such as *applying the last learned procedure to the current problem*.

As researcher Sheila Tobias explained in her classic exploration of math anxiety, a lack of meaning exacerbates many students’ negative experiences learning mathematics. When math class emphasizes rituals and cues that rely on memorization over sense making, students’ own interpretations become worthless.

For instance, they memorize multiplication facts, and, in a search for meaning, they decide that multiplication makes things bigger. Then, they learn how to multiply numbers between 0 and 1. Their prior understanding of multiplication no longer works, so they might settle on the idea that mulitiplication intensifies numbers since it makes these fractional quantities even smaller. Finally, when they learn how to multiply negative numbers, all their ideas about multiplication become meaningless, leaving them completely at sea in their sense making. The inability to make meaning out of procedures leaves students grasping and anxious, as the procedures seem ever more arbitrary.

In contrast, when classrooms are geared toward supporting mathematical sense making, they reap multiple motivational benefits. First, students’ sense of ownership over their learning increases. Students see that multiplication can be thought of as repeated addition, the dimensions of a rectangle as related to its area, or the inverse of division. When they learn new types of multiplication, the procedures have a conceptual basis to expand on. Relatedly, their learning is more durable. Because they understand the meaning behind the mathematics they are learning, they are more likely to connect it to their own experiences. This, in turn, provides openings for their curiosity and questions. Beyond giving students opportunities for sense making, meaningful mathematics classrooms provide students chances to identify and explore their own problems. Indeed, in a systematic comparison of teacher-guided and student-driven problem solving, educational researchers Tesha Sengupta-Irving and Noel Enyedy found that the ownership, relevance, and opportunities to engage curiosity in student-driven problem solving supported stronger outcomes in student affect and engagement.[1]

The challenge, then, for teachers is how to help students engage in meaningful mathematical learning within the structures of schooling. I would love to hear your ideas about how to achieve this.

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[1] Tesha Sengupta-Irving & Noel Enyedy (2014): Why Engaging in Mathematical Practices May Explain Stronger Outcomes in Affect and Engagement: Comparing Student-Driven With Highly Guided Inquiry, Journal of the Learning Sciences, DOI: 10.1080/10508406.2014.928214

I really appreciate the concrete multiplication example you have provided as well as the acknowledgement that students may be seeking relevance for some other reason than to “resistant” or “difficult”. I know that for me the procedures were very hard to memorize outside of any context. While I haven’t tested this idea on a large scale yet, one strategy that we will be practicing with next year is to start with the big picture and move into the procedures after this exploration has taken place. This goes against the frequently held belief that procedures must be learned before students can experience the application. To me this is like saying that you can’t read unless you know every vocabulary word in the text. We learn the words as they come up either through context or by looking them up. I contend that students will learn the procedures once they have a purpose to learn them. I hope others have ideas about how to achieve meaningfulness in math so we can all start addressing this serious need for today’s learners.

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In Naveen Jain’s own words, the purpose of education is to breed enthusiasm, with the materials merely being a proxy for that purpose. And this doesn’t only apply to math, but to any other subjects as well.

At the K to 12 level though, many school teachers simply don’t have enough mathematical insight to impart the “why” onto the students. Truth be told, at the exploratory / proof-based level, mathematics is inherently valuable, and does not require justification for its usefulness, much like the way playing music is inherently a liberating experience — hence the mathematical experience. Along with the lack of customized help in classroom setting, it’s unclear whether a service-math-based curriculum — which is geared towards applications that student might or might not find appealing — is the right way to execute do it. Meaningfulness is in the eye of the beholder so to speak.

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Thank you for these insights – especially about the hidden math (but perhaps not restricted to math) curriculum conveyed through rituals and cues. This is very much how I remember and think of math even now. Also this notion of narrowing the gap “between being curious and being a good student” is extremely compelling for me because I know how I tend to operate in that space. Compliance tends to win out over curiosity – that is my teaching truth. But the way that you’ve named it here makes me hopeful that I can (I must) go against my own grain, so to speak, and create more space for exploration & wondering, making it a good thing to ask lots and lots of questions.

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Hi

“Meaningfulness: When students connect their own curiosity and experience to ideas, thereby developing an interest in and appreciation for mathematical content.”

This is a nice description. It feels circular: meaningfulness creates, and is created by, curiosity, a want to know. I love circularity. I don’t think this is all there is to meaningfulness, but I don’t have a better definition.

Could we ask the children in our classrooms: “What do you want to know (about mathematics)? What would make it meaningful for you?”

I wonder: does The Question ‘when are we going to use this?’ appear as a response to the disconnect between what or how the child *wants* to be (what they want to know), and what is deemed necessary for them to be (know)?

Thanks for this thought provoking post,

Danny

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I like that question Danny, “What do you want to know (about mathematics)?” – and, remarkably, I don’t think I’ve asked it. I must try!

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“Meaningfulness” is constructed by the student and his/her expectations of what is next. In high school, mathematics beyond Algebra 1 and Basic Geometry are gatekeeping for college entrance. Those students focused on climbing up the ladder in colleges (from community to state to elite private) find “meaning” in these mathematics courses but many not in the mathematical content. Ensuring that these courses are ticked off on the college requirements and with an appropriately signaling grade is the meaning. I fault the ratchet effect of pushing mathematical courses with little appreciation for content into high schools. Generally, these sequences dampen even the most ardent student.

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Very good post. Follow as well as now

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It’s a great question, Ilana, and worth pondering a lot.

I think one of the weapons against falling-into-line (institutionalization, passivity, opposition, etc) is divergence. It seems so much easier in writing and art, but I think it’s possible in mathematics too. Whether it be the small-scale and frequent, “how do you total up this dot pattern?”, “which one doesn’t belong?”, or the bigger “make your own ant hotel within these constraints”, “make your own growing pattern with cubes”. There has to be room for the open-middle, open-ended or even open-begun.

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Hey, Lani!

I am intrigued by your line about the disconnect between the values promoted by parenting books and those ultimately desired by too many teachers. I wonder if there is a way to leverage all those parents who love to get up in arms about their child’s education, to actually get up in arms to demand that teachers value curiosity over compliance?

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I liked that you said that participating in s math program is beneficial to becoming motivated to enter a career field that is geared towards mathematics. I would agree that school systems generally struggle to motivate students to learn and become well educated in mathematics. I would be sure to enroll my children in a math program in order to teach them proper mathematics and to encourage them to pursue careers in this field.

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