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This is a somewhat infamous expression used for research in Liping Ma’s classic book, Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States—
1¾ ÷ ½
Most adults can do the arithmetic. The answer is 3½. Writing a story that matches the expression is harder, and the distance between those two tasks is the whole point.
In Liping Ma's 1999 book, she presented a large group of American teachers and another group of Chinese teachers with the expression 1¾ ÷ ½. She asked them to create a word problem that would require this expression to solve it. Almost none of the American teachers could write a correct one. Nearly all of the Chinese teachers could. The American teachers had more years of schooling. They still could not do it.
Most of the American teachers wrote some version of this. "I have 1¾ pizzas and I split them in half." It sounds right. The words "divide" and "half" are both in there. But splitting 1¾ in half is multiplication. It gives ⅞. The expression 1¾ ÷ ½ asks a different question. It asks how many halves are in 1¾. The answer is 3½, the same way 6 ÷ 2 asks how many 2s are in 6.
The distinction matters more than it might seem.
One reading gives ⅞. The other gives 3½. The teachers who wrote the pizza problem were not bad at math. They had been taught the way most of us were taught. Invert and multiply. Multiply across. They learned the steps and never learned what the steps meant.
That is a tidy story. The harder question is whether it matters for kids.
In 2005, Heather Hill, Brian Rowan, and Deborah Ball asked whether teachers' mathematical knowledge actually improves student achievement. They followed first and third graders over a year. They used a statistical model that accounted for students grouped within teachers and teachers grouped within schools. They controlled for the things you would want controlled, including student background and teacher characteristics. Teachers' mathematical knowledge for teaching, what teachers know about math at their grade level, still predicted how much students gained, in both grades.
The measure they used is the part that matters most. It was not a test of advanced math. It was a test of the specialized knowledge teachers use on the job. They looked at teacher’s knowledge of why a procedure works and which representation (or mathematical model) would help. Hill and Ball also looked at whether teachers knew what a particular student error is telling you. This is the knowledge the pizza problem exposes.
Other research points the same way. Teachers with stronger conceptual understanding and a better grasp of how ideas connect tend to teach in ways that help students make sense of math (Baumert et al., 2010). Weak understanding does the opposite. It narrows what a teacher can offer the moment a student gets confused. When we aren’t sure of a concept, we tend to hold tightly to what we understand.
Not every kind of knowledge, though, helps equally. Conceptual knowledge and knowledge of connections tend to be a stronger predictor of student performance. More abstract knowledge of formal models and generalizations does not, at least not consistently (Tchoshanov et al., 2017; Yang & Kaiser, 2022). The goal with improving teacher content knowledge is not to turn teachers into mathematicians. The goal is for them to understand a specific, usable version of the math they actually teach. That is what "knowing the math deeply" means. It is narrower and more practical than it sounds.
The teachers who do the most demanding math teaching are often special educators. The research base for them is thinner, but what exists needs attention.
One study found no difference in mathematical knowledge for teaching between K-12 special educators and K-6 general educators before professional development (Faulkner & Cain, 2013). But both groups made significant gains after five-day days of PD around math content. The need for this is especially clear in the upper grades. Special educators are frequently asked to teach math well above the level their preparation covered. That is not a personal failing. It is a system that hands people a job and skips part of the training.
A recent synthesis focused on students with learning disabilities draws the same conclusion (Johnson & Roy, 2026). The knowledge that helps teachers be more effective for students with learning differences is not more math coursework. It is knowledge of common misconceptions, of multiple ways to represent an idea, of how to read a student's error, and of how to build understanding that carries over to new problems. That is pedagogical content knowledge. It is concrete, and it is teachable.
The teachers in Ma's study were not stuck. Neither are the rest of us.
Faulkner's groups moved in five days. That is the part to hold onto. The reason most people struggle with multiplying and dividing fractions is that they were taught procedures without meaning, and that can be repaired. It gets repaired the same way children learn it well the first time. You work with models. You make conjectures. You talk through your thinking with other people. You apply what you know to problems that matter.
Teachers deserve that kind of learning as much as students do. When they get it, their relationship to the math changes. So does what they can do for the child in front of them who is sure, one more time, that they are just bad at math.
That child is the reason any of this is worth doing. Knowing the math is how we reach them.
Baumert, J., Kunter, M., et al. (2010). Teachers' mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 47(1), 133–180.
Faulkner, V. N., & Cain, C. R. (2013). Improving the mathematical content knowledge of general and special educators. https://consensus.app/papers/improving-the-mathematical-contentknowledge-of-general-faulkner-cain/84a74dc197e251b78a1cba6cb773c6b0/
Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371–406. https://doi.org/10.3102/00028312042002371
Johnson, & Roy. (2026). Mathematics teachers' pedagogical content knowledge for students with learning disabilities. https://consensus.app/papers/mathematics-teachers-pedagogicalcontent-knowledge-in-johnson-roy/750c5fbc5f7c5472865b8138db2a7f06/
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Lawrence Erlbaum.
Tchoshanov, M., & Cruz, M. D., et al. (2017). Examination of lower secondary mathematics teachers' content knowledge and its connection to teaching. https://consensus.app/papers/examination-of-lower-secondary-mathematics-teacherstchoshanov-cruz/4714feda45fc50368c3ff1215d6874c6/
Yang, X., & Kaiser, G. (2022). The impact of mathematics teachers' professional competence on instructional quality and students' achievement. https://consensus.app/papers/the-impact-ofmathematics-teachers-professional-yang-kaiser/ca6b2eccb5b75ee398d30d2625c23c7a/
All Learners Network is committed to supporting pedagogy so that all students can access quality math instruction. We do this through our online platform, free resources, events, and embedded professional development. Learn more about how we work with schools and districts here.