Some cold hard facts on the dangers of rote memorization of math facts without understanding math facts:
When measured throughout the world, students who memorize were among the lowest performing among their age groups (FYI: The US ranked in the top third of offenders)
Memorizers were often benchmarked to be up to six months below grade level averages than peers
That being said, a classic example of kids who memorize vs kids who understand is asking your student to mentally solve a problem that should not require scratch paper. Kids who can articulate their logic verbally and maybe need to jot down some of their data are light years ahead of the kids who can’t reason through the process and must go back to a formula exclusively.
See how flexible your student is by asking these multiplication questions. Ask them the fact and see if they can quickly and confidently provide you with the solution.
3rd grade example: 99x3 can be figured out by understanding multiplication is merely repeat addition. So I need 99 three times, 99s are like 100s, so three 100s is 300. I need to take away the three extras, though. So my answer is 297.
4th grader example: 12x13. They have probably memorized 12x12, but are they able to swiftly add just one more twelve. 144+12 does not require regrouping, so they should be able to make that leap.
5th grader example: 41x12 also not a stretch through mental math and, better yet, there are two ways to solve this one! 40x12 plus one more 40 so 480+12=492 or 41x10+41x2=410+82=492.
Kids being able to flexibly move numbers around and reconfigure them in ways that maintain the integrity of the problem, but with critical thinking creativity. We achieve this every day with our Numeric Fluency program. We’d be happy to work with your student.
Source: https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/
