Teaching children to use spatial reasoning cues could help them understand how numbers relate to one another, a fundamental concept in early math education.
Two Lynch School researchers are exploring this possibility, with the goal of improving children鈥檚 math fluency throughout their school years and beyond.
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If you learned to add and subtract using a pencil and paper, it may come as a surprise to watch elementary school students solving equations by moving colorful wooden sticks of different sizes. But developmental psychologists Elida Laski and Marina Vasilyeva, associate professors at the Boston College Lynch School of Education and Human Development, are creating learning aids like these to help kids develop math fluency faster and more effectively. The tools encourage them to use spatial reasoning to understand how numbers relate to one another in size, a fundamental math concept known as numerical magnitude. Research shows that early mastery of these kinds of math basics promotes greater achievement later in life, according to Laski and Vasilyeva, yet many children in the United States struggle with the subject.
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Spatial reasoning learning aid (Photo: Caitlin Cunningham)
A math problem
The United States has a math problem. The most recent Trends in International Mathematics and Science Study (TIMSS), conducted by the National Center for Education Statistics, found that U.S. students鈥� math proficiency has declined dramatically in recent years. The COVID-19 pandemic contributed to this alarming drop, but Laski notes that the issues with math achievement existed before school lockdowns and the temporary switch to online learning.
Associate Professor Elida Laski (Photo: Caitlin Cunningham)
鈥淭he United States persistently underachieves in math relative to other countries,鈥� she says. For example, TIMSS found that U.S. fourth graders鈥� average math scores were lower than those of their peers in 21 countries.
Laski and Vasilyeva say part of the reason some kids struggle with math is that they don鈥檛 develop an early understanding of the nature of numeracy. 鈥淏efore children can move on to more complex mathematical concepts, they need to understand what numbers are about,鈥� says Vasilyeva. 鈥淭hey need to develop a sense of how different numbers represent different magnitudes鈥攖hat six is bigger than three, for example.鈥�
According to the 2023 Trends in International Mathematics and Science Study (TIMSS), which is conducted by the National Center for Education Statistics:
U.S. fourth graders鈥� average math scores dropped 18 points since 2019
U.S. eighth graders鈥� average math scores dipped 27 points since 2019
Source:听The ,听housed at Boston College鈥檚 Lynch School, where the study鈥檚 item development, data analysis, and interpretation are conducted
A shared interest
Laski, who taught kindergarten for five years and also worked as an instructional coach in Boston Public Schools鈥� early learning centers, is drawn to working with young children. When she joined the Lynch School鈥檚 Applied Developmental and Educational Psychology program and met Vasilyeva in 2009, the two discovered that they shared an interest in cognitive science and the role of spatial reasoning in math learning. Their collaboration is inspired by other research indicating that early mastery of numbers and math has a profound influence on later achievement鈥攁nd even health.
Associate Professor Marina Vasilyeva (Photo: Christopher Soldt)
鈥淓arly math skills are among the strongest predictors of a wide variety of outcomes that all people should care about,鈥� says Vasilyeva. For example, she and Laski point out that young children who master basic mathematics concepts score higher on standardized tests and go on to take more challenging math courses in high school. They鈥檙e also more likely to attend college and earn higher salaries after they graduate, and they might even be healthier in later life than children who take longer to embrace math concepts. 鈥淚n theory, if you have better functional numeracy, you make better risk decisions and calculations about health outcomes and health care,鈥� says Laski.
A persistent听achievement gap
Studies also show that struggles with math are most acute among students of lower socioeconomic status (SES) in the United States, who tend to perform worse in school than their higher-income peers. For example, low-SES students are behind in reading and math proficiency by 20 to 26 percent compared to high-SES students, according to a 2015 study by the Rand Corporation.
鈥淭he gap is very large and persistent,鈥� says Vasilyeva, noting that a typical low-SES student is several grade levels behind high-SES students in math fluency. Earlier attempts to narrow this achievement gap in math have largely failed, she says: 鈥淵ou can鈥檛 just fix the problem simply by throwing money at it.鈥� Instead, she argues, understanding the fundamental skills children use to learn math is essential for developing effective interventions.
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Mind Your Pasture, a learning game developed by Lynch School faculty Elida Laski and Marina Vasilyeva (Photo: Caitlin Cunningham)
Numbers, big and small
Laski and Vasilyeva found that a significant body of prior research suggests that kids who understand numerical magnitude early on perform better in math later in their academic careers. Numerical magnitude is the absolute and relative size of one number compared to another鈥攕even is more than two, for instance, and 70 is much more than two.
Other research has revealed a strong association between spatial reasoning, which is the ability to think about objects in space and understand how they relate to each other, and the understanding of numbers and math. Humans are born with varying degrees of spatial reasoning, but it鈥檚 a skill that can be honed and developed, says Vasilyeva. She and Laski are now testing several interventions that embed spatial reasoning cues into instruction materials that help children understand numerical magnitude.
Children who used learning aids that included spatial reasoning clues developed by Laski and Vasilyeva had nearly twice the improvement in math test scores compared to students who used conventional learning aids.
Using learning aids
One of the essential learning aids Laski and Vasilyeva are studying is a set of rectangular sticks of different lengths. Each stick is labeled with a numeral from one to 10, with lengths proportional to their numeric value: The stick representing 10 is the longest, nine is slightly shorter, and so on. A child asked to solve 鈥�3 + 4 = ?鈥� learns to place the three stick in line with the four stick. 鈥淭hen they look at the other sticks and figure out which one has a length that corresponds to the total length of these two sticks,鈥� explains Vasilyeva. Comparing this new formation with the seven stick solves the equation.
Spatial reasoning learning aid. (Photo: Caitlin Cunningham)
Over time, children transition to using visual images of numbered rectangles and must solve more challenging equations. For example, they might be shown 鈥�5 + 5 = 10.鈥� But if the first number is changed to three, how do you solve for 10? 鈥淭hey see that if the first part gets smaller, the second part gets bigger,鈥� says Laski. 鈥淭hey start thinking about magnitude relations using spatial cues, such as bigger, smaller, a lot bigger, a little bigger.鈥澨�
Laski and Vasilyeva tested these learning aids in a study funded by the Institute of Education Sciences. Their findings were published online in 2025 in Developmental Psychology. The study included 205 racially and ethnically diverse first graders from low-income homes. Some kids were given instructional math materials that used spatial cues, while others, who served as controls, used learning aids that didn鈥檛 feature spatial cues (such as equal-sized square tiles, each imprinted with a numeral).听
After a pretest of their math skills, students trained with the learning aids in eight 30-minute sessions. Post-intervention testing found that children who trained with spatial cues increased their accuracy by an average of 11 points, compared to an average increase of six points in the control group. Laski and Vasilyeva hope to build on this finding to develop a classroom instruction program based on spatial cues.
Parents and math talk
The way parents talk to their kids about math and numbers at home has a significant impact on how well young children embrace mathematics, research shows. Family play time is an opportunity for these interactions, but Laski and Vasilyeva found that most commercially available math-education games don鈥檛 employ spatial reasoning strategies. 鈥淪o we began thinking about how we can create materials for parents to engage with in the home that also take advantage of these kinds of spatial cues,鈥� says Laski.
Laski and Vasilyeva developed two games that they will begin testing this summer in a two-part study involving 80 Boston-area families with kindergarteners, work supported by grants from the Caplan Foundation for Early Childhood and 天美传媒app鈥檚 Ignite Program. One game, Mind Your Pasture, asks players to build fences of prescribed lengths using rails of different sizes to keep sheep from straying. The other, Alien Invasion, calls for players to make lasers of different lengths to zap Martian invaders.
Mind Your Pasture learning game (Photo: Caitlin Cunningham)
Alien Invasion learning game (Photo: Caitlin Cunningham)
Half of the participating families will receive these games, and the other half will be given commercially available math games that lack spatial cues. Researchers will encourage parents to play the games with their children a few times a week under different conditions鈥攁s partners or as opponents. Pre- and post-intervention testing will measure whether games using spatial cues were more likely to improve children鈥檚 math scores.
For the second part of the study, Laski and Vasilyeva will ask parents to videotape some game sessions so they can analyze how parents talk about math with their kids.
鈥淲e are interested in the kind of math language, or math talk, that the parents use as they鈥檙e playing these games,鈥� says Laski. 鈥淭here is quite a bit of research indicating that math talk in the home is important for children鈥檚 math learning, but that not all math talk is equal.鈥� Most parents, she says, simply label numbers鈥斺�渢his is four, this is six鈥濃�攂ut rarely use spatial language, such as 鈥渟ix is bigger than four.鈥� Yet recent research shows that those kinds of statements 鈥渕ight be some of the most powerful kind of talk,鈥� says Laski.
Laski and Vasilyeva say they鈥檙e building on 50 years of research aimed at understanding the cognitive underpinnings of how children learn math, which they argue has not yet been adequately incorporated into curriculum development and classroom instruction.
鈥淭he through line in our work,鈥� says Laski, 鈥渋s thinking about how we can use the science of math to develop effective instructional interventions.鈥�