At Math-U-See, we have always proclaimed the truism that understanding the “why?” of math is important for long-term success. It is always worth taking extra time to make sure your student understands and is mastering each concept before moving on, because without a solid foundation in the basics, students won’t be as able to succeed when they get to more advanced concepts.

A survey conducted by Roberg Siegler at Carnegie Mellon University appears to confirm that understanding basic math may have a significant impact on future learning.

The study shows that, while they may be similar in other ways (for example, similar grades in multiplication, or from similar family backgrounds), students who get poor grades in either fractions or division are more likely to get poor grades in algebra than students who got good grades in fractions and division.

“We suspected that early knowledge in these areas was absolutely crucial to later learning of more advanced mathematics, but did not have any evidence until now,” said Siegler, the Teresa Heinz Professor of Cognitive Psychology at CMU.

The clear message is that we need to improve instruction in long division and fractions, which will require helping teachers to gain a deeper understanding of the concepts that underlie these mathematical operations.” [. . .] At present, many teachers lack this understanding. Because mastery of fractions, ratios, and proportions is necessary in a high percentage of contemporary occupations, we need to start making these improvements now.

Carnegie Mellon University

In the video below, Siegler explains more about the study and mentions that there have been several additional studies comparing teachers in the US and teachers in East Asia. The difference between the groups is their level of understanding. United States teachers are asked “Why is it a legitimate thing to invert and multiply when dividing fractions?” and can rarely give even one reason, while East Asian teachers can usually give two or three reasons why the procedures work that way. So if the teacher doesn’t know why the formulas work, all the teacher can say is, “Just do it because it’s the right way to do it.” Unfortunately, both children and adults have difficulty remembering rote information that they don’t understand.

Equipping teachers and helping them and their students understand the concepts is what Math-U-See does. We value the understanding and mastery of each concept, which is why we have an entire book dedicated to fractions, building a strong foundation to prepare students for upper-level math. Our Epsilon level focuses on fractions and emphasizes understanding “why“ the procedures and formulas work the way they do. Epsilon also uses our plastic Fraction Overlays which illustrate the abstract concepts in a hands-on, concrete way to help the students see and understand math.

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