This blog post shares excerpts

from our 2021 presentation at the World Conference for Gifted and Talented Children (WCGTC) titled“Pathological” differentiation in Math is actually optimal differentiation in Math by K. Doherty, D. Graves, and M. Vetter.

Mathematically gifted children are often easy to identify “in the wild,” since they not only recognize and talk a lot about numbers from a very young age, but because they also know what those numbers are for much earlier than their age-peers. Some of them may understand 1-to-1 correspondence and ordinal numbers very early; some may be able to do complex arithmetic before they even go to school; and some of them immerse themselves completely in a world full of numbers, which seem to be their favorite toys (like these we’ve seen in a few PreK backpacks). 

It’s no accident that one highly-respected mathematics summer program for young gifted students helps parents decide if their camp is a good fit by asking the question, “Does your child take math books to bed?”  

Like many things about gifted children, someone in the general population would probably respond to such an idea with a quizzical head tilt, as though the questioner is an alien — but parents of such children instead experience a flash of recognition and relief: “ohhhhh — they know what it’s like to have a child flavored like mine (and phew, I’m not the only one with math books in the bed).”

summer21-math-games

MATHEMATICALLY GIFTED PROBLEMS?

When these children are old enough to go to school, their math skills are often perceived as a problem by their teachers and administrators. “If they love math so much, why won’t they ever sit still in math and listen?” they might ask.

The answer, of course, is that it’s impossible for a five- or six-year old — or a nine- or ten-year-old, for that matter — child to be patient and sit in compliant silence while they wait for their peers to “catch up.” Even aside from the question of the fairness of providing that child no instruction, it’s just not developmentally appropriate to expect that. Some gifted students may be as much as five or seven or more grade levels ahead of their contemporaries in mathematics. That is a long time to wait — for anything.

Patience is one thing; preternatural willingness to be bored is another thing, entirely.

What decades of research into mathematically gifted youth has shown is that these children actually need to be fed more math, which is exactly what they have wanted all along. However, the simplest interventions — just putting them into the nearest “gifted” math classroom or even accelerating them a year in math — may not be enough to challenge them and keep them growing and engaged.

A math-focused 3rd grader whose school sends her to the 5th grade classroom every day for math is still going to outpace those peers. Yes, she’s learning new material, but it’s not necessarily the right pace of instruction to keep that brain busy. Research has also shown that the number of repetitions required for mathematically gifted students to learn new material can be as low as one or two, while typical children may need ten or more repetitions to master a new topic. Those are dramatically different classrooms, as you might imagine.

In an ideal mathematical utopia, then, a teacher would first do a detailed inventory of what students know and start from there, teaching them things they do not know rather than repeating things they have already mastered.

The mathematically gifted are starving for more math

“RUNNING OUT OF MATH” is not a thing

Honestly, the most common protest that parents will hear from teachers and administrators regarding acceleration in math is this: “But what about when they run out of math?”

Meghan Vetter, a Lower School mathematics teacher at Grayson, chuckled when a parent shared this experience. Asked how she would handle this perspective, she replied, “If we have mathematicians out there doing math, how could a student ‘run out of math?’ We’ll just go find a mathematician and ask them what they’re working on!”

Kimm Doherty, Director of the Lower Schoo, echoes Vetter’s sentiment, explaining that a gifted program should  pay “obsessive attention” to each student’s trajectory, instead. “Not just where they are today, but where they are headed next year and the year thereafter.”

And that trajectory does not always have to be straight up as fast as possible. Instead, Derek Graves, an Advanced Math instructor at Grayson, says that introducing additional topics can challenge and change a student’s understanding of what math can be. “Advanced questions and topics broaden your sense of what is considered ‘mathematical”; deepen your understanding of topics that you didn’t even know existed; or introduce to you connections between various seemingly disparate parts of mathematics.”

What does a program for the mathematically gifted look like?

Doherty says teachers at all levels can include surprisingly sophisticated topics such as “combinatorics or statistics for little people, [which] can be much richer than what we normally offer in an elementary school. A lot of number theory that they don’t get, [or] logic and reasoning…can help expand their experience, and give them really meaty, juicy problems that sometimes are really different than what you might get in [a] traditional class.”

Vetter agrees, offering even more options that traditional curricula often don’t go anywhere near: “Teach them combinatorics. Teach them number theory. Teach them graph theory — have them do research in graph theory. This is a thing that young students can do. They can literally do research and solve — or work on solving — unsolved problems.”

“PATHOLOGICAL” DIFFERENTIATION IS ACTUALLY OPTIMAL DIFFERENTIATION

This kind of obsessive attention leads to what Grayson calls “pathological differentiation” in math groupings. The 2020-21 school year ended with 138 PreK-12 students grouped into 33 sections of math — some of which are one-to-one if a student is three years or more ahead of grade level with no peers in their grade cohort.

This kind of approach to placement creates very small classes, of course; for example, in 2020-21, the 17 students studying Algebra 2 were in three different groupings. The students are intentionally clustered tightly together so they can progress quickly together during the school year, collaborating with peers who are as engaged in math as they are. According to Graves, “The more heterogeneous you make that mix, the more diluted the experience becomes for all of the students in the room. But the more you can create groupings and differentiate so that each of the students is in a very similar band, the more you get out of every student, and the more meaningful the educational experience is every day for those students.”

mathematically gifted playground

The advantages of this practice for the mathematically gifted or math-passionate is obvious; however, there are also substantial gains for students whose first love is not math. “Those groups still benefit from the differentiation,” explains Graves. “It would be okay [to put both types of students in one classroom], but it is significantly better — for both groups — if we split them up.”

Graves actually experienced a perfectly illustrative moment in a Precalculus class. While learning the beginnings of trigonometry, the class had a mathematically magical moment: “Everyone in the room — every single student — gasped at the same time and said, ‘Ohhhhhhh!‘” he describes. “That is why we group them so carefully, so tightly: to have moments like those, where everyone has that “oh!” moment, when the teacher knows they all get it, and can move forward knowing that they’re all ready.”

Despite how much work is required, Doherty clearly believes in its value. “It’s an integral part of our research-based practices to continuously challenge our students,” she says. “They need it, and we offer it because it feeds them.”

Let us show you how we "get it."

Admittedly, this highly granular attention to placement may not easy for faculty or administrators to implement, but it is key to how we design our curriculum, train our teachers, and craft student schedules beginning in our Lower School.

If you would like to learn more about our Mathematics program, or other ways we nurture the unlimited potential of our students, please schedule a private tour of our school. We look forward to meeting you soon.

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