In Sarah Cuddihy’s 1st grade class at Driscoll School here, math specialist Jenna Laib brings a statistics lesson into science class. In 3rd grade, it’s social studies. In 4th grade, it’s a math class pop-up conversation about the Oscars.

If you want to introduce statistics and data-science concepts in elementary school, Laib says, it’s best to be flexible. Name a subject area, and Laib can craft a statistics lesson for it.

As part of her job, Laib rotates through two-week co-teaching sessions with each teacher in this K-8 school in Brookline. Along the way, she helps teachers learn to introduce statistics concepts in any subject area, spread throughout the year.

While states from Georgia to Utah are trying to improve student performance in statistics and geometry, teachers say the concepts often end up getting crunched at the end of an overfilled school year, and instructional holes that start in elementary and middle grades often carry over to high school. These pacing problems have worsened decade-long declines in student achievement in statistics and geometry across grade levels.

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“For math teachers, oftentimes statistics is the last unit in the year, and it’s something that it’s easy to just say you don’t have time for and not get to, because it’s presented as a stand-alone in a lot of the curriculum,” said Luke Wilcox, a math teacher at East Calwood High School in Grand Rapids, Mich., and a former Michigan teacher of the year. “And certainly in the last two years, we’ve had even more of a struggle getting through all of our content within a year … so while that’s been a problem for a long time for math teachers, after the struggles with COVID, that has likely happened even more than it did before.”

## Gaps ‘accumulate’ across grades

Pacing is challenging, because teachers say simply following a textbook often falls short. Studies of geometry courses in 2014 and 2018 found that teachers relying on textbook units often skip lessons on applying the concepts and give students less complex tasks.

If a teacher in one grade misses a few concepts, students can go years before the gaps are filled. Sal Khan, the head and founder of the online tutoring program Khan Academy, said he has found that when students learn math concepts in discrete units, they are more likely to “accumulate gaps” over time.

Researchers Julie Booth and Allie Huyghe discovered that firsthand when they developed GeometryByExample, a collection of standards-based lesson plans for high school teachers. The pair had already developed similar caches of teacher-vetted lessons for elementary math and algebra but were hamstrung in geometry by instructional gaps in lower grades.

High school teachers repeatedly told the researchers that they wouldn’t get to computation or proofs in geometry until late April or May, because they had to cover “the basics” that students had not been exposed to in earlier grades.

“I asked, what do you mean by ‘basics,’ and [the high school teachers] are like, ‘solid geometry, volume, and perimeter and area,’” recalled Huyghe, an assistant director at the Strategic Education Research Partnership Institute.

“I was in person with one of our teachers and I just happened to have a MathByExample book for 4th and 5th grade. I pulled it out and said, ‘You mean this stuff?’ And she said, ‘Yeah, that’s what we’re covering now,’” Huyghe said. “This was in November of 10th grade. And [the teacher] didn’t even consider it remedial; it was just what they had to do in high school geometry, because of all the content students had never been exposed to.”

About 80 percent of the lesson plans that Booth and Huyghe ultimately developed for geometry teachers covered geometry content typically at the middle school level, Huyghe said.

Dashiell Young-Saver, a San Antonio statistics teacher who also develops math lessons, said he sees the same pacing problem in his Advanced Placement statistics courses. “One of the strange things about [AP] stats is the first three units really are all things that students could learn in middle school and probably should be learning in middle school,” Young-Saver said, “but because we push off stats so long, it ends up only taught to a group of students who, as juniors and seniors, decide to take it as an elective.”

The lack of attention to data and statistics early on in textbooks means it’s up to teachers to develop their own lessons or find ways to squeeze in needed math content anywhere they can.

For example, Debi Hays, a 3rd grade teacher at Explorer Elementary School in Kentwood, Mich., uses a math curriculum with a two-week unit on data and statistics concepts, which generally falls after the state tests in spring. To cover data and statistics before those tests, Hays said she had to use multiple class periods for math each day.

To give students earlier exposure, Hays also asks her students to come up with statistical questions for a poster fair in the first semester of the school year. In small groups, students collect and analyze data on topics like student exercise routines or opinions about school.

“The poster contest gave a vehicle to practice creating data stories to be shared with other people,” Hays said. “If I just teach straight from the math book, the kids learn what I’ve asked them to learn, but it’s not meaningful to them and it doesn’t stick with them. But when we try these creative ways to learn, they are so engaged and they remember all year.”

## Leveraging ‘connectedness’

In Cuddihy’s 1st grade class at Driscoll, Laib, the math specialist, shows students a complicated but unlabeled pictograph chart of many columns of hearts. It’s part of a lesson called a “slow reveal,” in which students slowly puzzle out the meaning of a complicated graph in stages, to help them understand things like scale, axes, and data sampling.

“You know, you can’t do logarithmic skills in 5th grade, but even 1st graders are able to understand some of the ‘mathy concepts’ in something like the line graphs,” Laib said.

Over the course of the class, Laib gradually layers information onto a graph depicting columns of hearts, while discussing students’ changing understanding of what the graph means. After seeing two column labels, for example, the students were able to calculate how much each individual heart represents. In another layer, Laib revealed pictures of a chicken and a rabbit, each with their weights in pounds, above each column.

One student guessed the hearts measure “beats per minute,” which prompts a classmate to weigh in. “The heavier the animal, the bigger it is but the fewer its heartbeats,” said Billy Boland.

Using that as their hypothesis, the students label a cat, a horse, and a mouse in the appropriate columns. The final layer of the chart shows the children all data with full labels, on animals from a hummingbird to a blue whale. The class confirms Billy’s correct theory that heart rate was inversely connected to an animal’s size, and discuss the biological reasons for different heart rates.

The slow reveal engaged students more than simply presenting the chart all at once, Cuddihy said, and gave the class time to discuss why heart rate relates to an animal’s size.

A 2023 report by the National Academies of Science found that while data science and statistics most often show up in state math standards, a few states also include the topics in science, computer science, and career and technical education standards. Integrating projects that use statistics or geometry in other subjects can both help teachers find time to teach the concepts and help students understand math applications across different fields.

“One of the powers of mathematics is the interrelatedness of it, but we often segment pieces of it,” said Trena Wilkerson, a professor of math education at Baylor University and a past president of the National Council of Teachers of Mathematics. “We’ll teach this particular thing, and we’ll teach this thing, and then this thing, rather than necessarily looking at the bigger picture. If we could view mathematics from its connectedness, then students as well as teachers would understand more deeply.”

Joel Bezaire, a middle school math teacher at the independent University School of Nashville, created his own “statistical investigations” class. When the school in Tennessee needed him to teach an additional algebra course instead, Bezaire said he incorporated some of those statistics projects, such as a comparison of different marathon runners’ times, to present algebra concepts.

“I’m not going to pretend it’s easy to just do this along with everything else,” Bezaire said, “but it’s good to break that sort of death march toward calculus that we seem to be on, to look at a different side of math.”

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