"Spontaneous focus on quantitative relationships is not something explicitly taught in school. It refers to how people describe various everyday situations," says Tallinn University doctoral student Triinu Kilp-Kabel. For example, someone looking at a cabinet at home might say that one shelf is half as full again as another or that one contains a lot of items while another has few.

Although this kind of descriptive thinking isn't directly taught in school, it's where differences in students' math skills begin to emerge. In her soon-to-be-defended dissertation, Kilp-Kabel explores the connection between noticing mathematical relationships and motivation to learn math. Her focus is specifically on basic school students. "Highly motivated students tend to identify numerical patterns in their environment without being told to focus on such relationships," the doctoral student notes.

Mathematics must be able to spark interest

According to Triinu Kilp-Kabel, spontaneous focus on quantitative relationships begins with whether children notice numbers in their surroundings at a young age and describe what they see in numerical terms. "At some point, this develops into recognizing quantitative relationships — like fractions or multiplication — that they begin to articulate," she explains. For example, a student might realize that one drawer is too full to fit anything more, while also noticing that another drawer is three times emptier and can still hold items.

As part of her doctoral research, Kilp-Kabel investigated how this kind of number sense relates to students' motivation to learn. "We found that students who are more motivated than average are the ones who tend to identify quantitative relationships in their environment. In contrast, there's no significant difference among students with average, low or otherwise distinct motivation profiles," she says.

In describing these differences in student motivation or motivation profiles Kilp-Kabel drew on the expectancy-value theory of motivation. She considered students' expectations of success, how important and useful they found strong math skills, whether they were interested in engaging with math and how costly they perceived it to be. "For cost, we looked more specifically at effort cost — that is, how much students feel math demands effort from them," the researcher explains.

Taking all this into account, one group stood out: students with high motivation who believed they performed very well in math. According to Kilp-Kabel, these students saw good math skills as both useful and important, were interested in the subject and didn't feel it required much effort. "There was also a contrasting group who didn't believe they could succeed in math. They weren't interested in it, didn't see math skills as important or useful and felt the subject demanded a lot of effort. We referred to them as students with low motivation," she says.

About 6 percent of students formed a distinct group: they considered good math skills just as important and useful as the highly motivated students did, but they didn't believe they were good at math, weren't interested in it and felt it required a lot of effort. "These students actually performed worse in math than those in the low-motivation group. That suggests that simply seeing math as important or useful isn't enough to support a student," Kilp-Kabel concludes.

Where will I need it in life?

As part of her dissertation, Triinu Kilp-Kabel conducted an intervention with 7th-grade students focused on supporting math skills in the context of linear functions. "The national curriculum for basic school emphasizes that students should be able to apply their math knowledge outside the classroom. We wanted to see whether real-life context problems, discussion and hands-on data collection could support the development of math skills," she recalls.

During the intervention, students were given tasks such as one involving hair growth. First, they were presented with an explanation of how quickly hair grows at different life stages and during various seasons. Then, based on descriptions of different people, they had to identify on a graph the line representing each person's hair growth. For homework, they were asked to measure the length of their own hair and calculate when it would grow to, for example, one meter in length.

Since the intervention lasted only two weeks, its overall impact was modest, Kilp-Kabel notes. Still, it proved to be beneficial. "The discussions, real-world context and opportunity to collect their own data helped support students. Afterwards, they were better able to identify components of a linear function — such as the y-intercept and the coefficient of the linear term — represent the function on a graph and find the correct solution," she explains.

Students who participated in the intervention were also better able to articulate why linear functions are useful. However, by the start of 8th grade, these differences in perceived usefulness had disappeared. "For example, they said that functions make it possible to represent a linear relationship over time: if you save the same amount of money every month, you can quickly figure out in which month you'll reach a certain total," Kilp-Kabel notes.

These kinds of examples are valuable, she says, because they're concrete and grounded in real life and they reflect students' motivation. "Many students, on the other hand, don't truly grasp the benefits of math unless they've worked through topic-based scenarios that show why these skills are useful," she adds. In such cases, students might know that strong skills are useful, but they struggle to explain why.

A single task per lesson enough

In light of her research, Triinu Kilp-Kabel offers several recommendations for how teachers can better support students' math skills. "One key suggestion is that, in order to maintain motivation, students should experience as many moments of success as possible. That means identifying and addressing any gaps in their prior knowledge," she says. Without a solid foundation, it becomes difficult for students to progress, and those experiences of success may never come.

Her second recommendation is to give students meaningful feedback on what needs improvement, how to develop their skills and what steps they can take next. "Instead of simply saying, 'You're not very good at this topic yet,' a teacher might say, 'You did a great job identifying the y-intercept, but had some trouble with the coefficient of the linear term — let's work on how to calculate that,'" she explains. In other words, feedback should be substantive, not just a percentage score.

Kilp-Kabel also emphasizes the importance of actively involving students in classroom work. While teachers often assign five or six problems per class, she believes a single problem examined in depth can be just as effective. "We found that when a class spent the entire period on one problem, students didn't lose out on skills. The reduced time pressure created more space for discussion and helped identify where students were struggling. Taking a deep-dive approach is definitely supportive," she explains.

Her dissertation also assessed students' math anxiety. While this is often seen as a separate issue, Kilp-Kabel argues that the strategies mentioned above also help reduce anxiety. "The same recommendations that support skills and motivation — working at a calm pace, giving thoughtful answers to students' questions and offering solution-oriented feedback — also reduce anxiety," she says.

Triinu Kilp-Kabel, a doctoral student at Tallinn University's School of Natural and Health Sciences, will defend her dissertation, "Students' Math Motivation and Math Skills in Basic School," on November 26 at Tallinn University. Her advisor is senior researcher Kaja Mädamürk of Tallinn University. The opponents are Professor Marja-Kristiina Lerkkanen of the University of Jyväskylä and Lecturer Kerli Orav-Puurand of the University of Tartu.

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