Building engaging prompts that are “low floor, high ceiling.”
In her phenomenal book Mathematical Mindsets, Stanford professor Dr. Jo Boaler asks what makes a good math question. There are several characteristics. One, she says, is that the question is “low floor, high ceiling.” (Boaler, 62)
A question’s floor is its entrance point. Who can understand it? A question has a high ceiling if you can keep exploring the problem as it leaps and grows in complexity.
Fermat’s Last Theorem has a low floor. The question can be understood by a high school algebra student and takes up less than an inch of space. It has an extremely high ceiling. Proving the question took Andrew Wiles 7 years and 129 pages and involved a modularity theorem for semistable elliptic curves.
I went to work writing funny zero calculation prompts that were “extremely low floor and possibly high ceiling with a mezzanine area for dawdlers.” Because the mission is to make math funny, especially to those with math anxiety, everyone should be able to engage with the prompts regardless of age (within reason) or mathematical ability.
Let’s walk through two example cards from the picture above.
MAKE THE WORLD’S SKINNIEST TRIANGLE
Kids as young as 3 can identify shapes and kids as young as 4 can badly draw them. (Quinn, 65) This is low floor.
Triangles also represent curvature. If a triangle has more than 180 degrees, the surface it’s on has positive curvature. Less than 180 and the surface has negative curvature. We are now sort of exploring the Riemann curvature tensor. I will call this high ceiling.
Everything in between makes you play with the definition of a triangle. This vast middle is the mezzanine, where most of us regular folks live.
This prompt invites anyone to play around with triangles. No pretense, no calculation, no pressure.
WHAT’S HALF A COOKIE TIMES HALF A DUCK?
Fractions. They’ve been causing headaches since the Egyptians. Here’s a quote from Ian Stewart’s Taming the Infinite: “Fractions caused the Egyptians severe headaches.”
Here is another: “When the Egyptian scribe needed to compute with fractions, he was confronted with many difficulties arising from the restrictions of his notation.” (Gillings 1972)
Perhaps this card should us ask to give an Egyptian a cookie for all that stress.
Fractions are weird. Kids learn that multiplying two numbers makes a bigger number. But multiplying two fractions makes a smaller fraction! This is jarring and something we gloss over as adults.
Teaching fractions is inherently challenging because the operations are counterintuitive to what students already know about whole numbers.
— Ni & Zhou, 2005
This prompt is low floor because a duck-cookie crayon/marker/whatever mash is accessible to everyone. (The worse the drawing the better.) It is high ceiling because we can bring ourselves back to the Egyptians and ask what is really going on when we multiply fractions. The mezzanine for this card serves milk and cookies.
This leads to a question though. Instead of cookies and ducks…
WHY NOT JUST WRITE MATH QUESTIONS ON THE CARDS?
There are numerous incredible resources out there for those that are math-inclined, math educators or both. This deck was lovingly crafted for those, young and old, who see flashcards and math games and puzzles and recoil a bit into their shell.
Sometimes that math-phobe or math-hater is a student. Other times it’s a parent. The truth is there is usually someone in a student’s orbit who has expressed outright dislike of math.
I have come to the realization that, as their maths teacher, I am possibly the only positive mathematical role model in many of my students’ lives. Sadly, students are likely to be surrounded by math-haters or math-avoiders.
— Craig Barton, How I Wish I’d Taught Maths
There are no math questions on the cards because math questions exist all over the place. For the math-phobes, math haters and math avoiders: This is my present to you.