When I comment on math, I always like to make it clear that I am not, in any way, a mathematician. Indeed, because many homeschooling parents requested it, I actually tried to write a mathematics program. Based on that experience, I can make one very definitive statement: While I use a lot of math in my field, I am definitely not an expert at it. When it comes to mathematics education, you need to take everything I say with a grain of salt.
Nevertheless, one thing I have always stressed when it comes to homeschooling is to emphasize math in the elementary years. Based on my experience with students from junior high all the way through graduate school, I have come to the conclusion that the best way to produce excellent science and mathematics students is to spend a lot of time on math in the elementary years. This not only includes learning new mathematical concepts, but it also means drilling your student in math. In my experience, the best science and math students are the ones who spent a lot of time on math drills when they were young.
An interesting study that seems to support this view was recently published in the Journal of Neuroscience.1
In the study, the researchers gave students very simple mathematics problems to work out. For example, they showed the students an equation like 7+2=8 and asked them to determine whether or not the equation was correct. In case you didn’t notice, that equation is incorrect. In all cases, the equations contained only 1-digit numbers. Students were also given digit-matching problems where they were shown three digits separated by double bars and were asked to determine whether or not the third digit was identical to either of the other two digits. The researchers kept track of the time it took the students to answer these very simple questions as well as whether or not their answers were correct.
The researchers did two more things. First, they got permission to track the PSAT scores of the students they were studying. The PSAT is a practice test for the SAT, which is one of the standardized tests used for college entrance. The researchers looked at the math score the students received on the PSAT as well as the critical reading score. They also scanned the brains of these students with an fMRI machine while the students were solving the simple problems discussed above. This allowed the researchers to identify which sections of the brain were most active while the students were solving very simple arithmetic problems. Now please understand that the fMRI process is not foolproof. Nevertheless, assuming that the fMRI scans were done and interpreted correctly, the results were rather interesting.
The researchers found that the accuracy of the students when it came to the simple arithmetic problems they solved had no relationship to their PSAT math scores. That makes sense. After all, the PSAT deals with higher math issues like word problems, geometry, and algebra. It doesn’t test a student’s ability with simple arithmetic. In addition, a detailed analysis showed that while the overall PSAT score was better for students who were able to solve the problems quickly, there was no direct relationship with the PSAT math score. Thus, whatever allowed the students to solve problems quickly helped them with both math and critical reading.
Now here’s the interesting result: While accuracy and time didn’t directly correlate with the students’ PSAT math scores: the region of the brain that was most active during the simple arithmetic problems did. Students whose supramarginal gyrus was very active during the simple math problems had higher PSAT math scores than students whose intraparietal sulcus was very active during the simple math problems.
What does this mean? Well, the supramarginal gyrus has been associated with the fact retrieval processes. When you are trying to recall a fact that you have memorized, your supramarginal gyrus becomes very active.2 However, the intraparietal sulcus has been associated with problem-solving strategies. When you are presented with a complex problem that requires a strategy, your intraparietal sulcus becomes very active.3 So, when solving simple arithmetic problems, students who rely on memory rather than problem-solving do better on more complex mathematical problems, such as those found on the PSAT. As the authors of the study say:
Thus, the current results suggest that individuals with higher PSAT Math standard scores are engaging neural mechanisms associated with memory retrieval to solve single-digit equations, while those with lower scores are engaging systems associated with processing numerical quantity, and likely relying on procedural computations.
What does this tell us from an educational point of view? It tells us that memorizing mathematical facts seems to help students with higher-level math problems. If students don’t have the simple math facts memorized, they don’t do as well on higher-level mathematical problems, at least not in a standardized testing environment. As the authors further note:
Thus, we suggest that successful encoding of arithmetic facts contributes, in combination with other factors not investigated in the present study, to the successful acquisition of higher level mathematical competence, affecting the ontogenetic construction of brain networks facilitating the learning of higher level mathematical skills.
If you want your student to do well in higher-level mathematics, then, you need to get the basic mathematical facts burned into their memory. That way, they can retrieve those facts when solving higher-level problems rather than working them out on the fly. In other words, don’t skimp on the math drills. While they may be dull and boring for the student, they end up encoding arithmetic facts into memory, which seems to help the student when he or she attacks more complicated mathematical problems.
Now please note that there are several things you have to keep in mind when thinking about the results of this study. As I already noted, fMRI studies can be totally wrong. That’s always a possibility here. In addition, even the authors note that while they think their interpretation of the results is very reasonable, there are alternate interpretations. Also, the study shows only a correlation between the fMRI results and the PSAT math scores. It’s always possible that there is some unknown effect producing that correlation.
Finally, there is one more thing to keep in mind. Even assuming the results of this study are correct and the authors’ interpretation is valid, the key is that the students couldn’t solve the complex math problems on the PSAT if they didn’t know problem-solving strategies. So you can’t replace conceptual, problem-solving strategies with math drills. Instead, there seems to be a balance necessary. You need to drill your students well enough so that they get those simple arithmetic facts into their memory, but not so much that the students don’t spend time learning how to think mathematically so that they can utilize problem-solving strategies.
Like most things in life, then, teaching math to your students isn’t simple. You can’t rely on just “drill and kill,” and you can’t rely on just conceptual problem-solving strategies. Instead, you need to balance these approaches if you want your student to get the most out of his or her math education.
REFERENCES
1. G. R. Price, M. M. M. Mazzocco, and D. Ansari, “Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores,” Journal of Neuroscience, 33(1):156-163, 2013
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2. Grabner RH, Ansari D, Koschutnig K, Reishofer G, Ebner F, and Neuper C, “To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving,” Neuropsychologia 47:604–608, 2009
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3. Delazer M, Domahs F, Bartha L, Brenneis C, Lochy A, Trieb T, and Benke T, “Learning complex arithmetic–an fMRI study,” Cognitive Brain Research 18:76–88, 2003
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Funny. We have to do standardized testing in our state. My daughter (who is dyslexic, and has a hard time recalling math facts) does terrible on speed drills (and as a 7th grader I still have her do math drills every single day). But, the part of the tests that give a word problem and ask which equation will solve the problem, she does very well on. I wonder if they took into account learning disabilities like dyslexia and such. For some reason my two dyslexic kids really struggle with memorizing math facts, no matter how much I drill and kill. My one who is not dyslexic has no problems memorizing math facts.
Trish, the authors don’t mention whether or not any of the participants in the study had specific learning challenges such as dyslexia. They did choose from a broad range of achievement, however. They note, “we recruited a sample of these students representative of those with consistently deficient, low average, average, or above average levels of mathematics achievement from kindergarten to grade 9.” I suspect that the results are not directly applicable to those who have specific learning challenges.
I think this all makes perfect sense. If the basic facts are on tap for quick retrieval, then the two parts of the brain can work more effectively together to solve complex problems. If the problem solving section has to add to its workload by also figuring out the basic facts, it won’t be nearly as efficient.
Now if only I could figure out HOW to burn math facts into a brain that seems to have a block. My 15-yr-old daughter understands all the concepts, but after many years of every type of drilling i can find, still cannot seem to retain the basic multiplication facts. So frustrating, because every single problem is ten times harder when you can’t retrieve those numbers!
Have you see these resources, Ronda? I have heard from people who they have helped.
No, that’s a new one on me, but honestly, we have already tried interminable flashcards of different types, memory songs, Times Tales and other similar pictorial approaches(which this looks like,) Math-It, and numerous computer games. Their least expensive option seems to be $40, and I’m not very keen on sinking another forty bucks in yet another unsuccessful attempt. I’ve hoped so MANY times, and I’m starting to think that no program is going to help.
I’m feeling pretty pretty defeated about it. Besides, she’s going to react negatively to anything that looks juvenile. She already feels embarrassed enough to be so old and not know this stuff–using materials obviously made for little kids doesn’t go over well.
I know. I’m not really trying any more, am I? 🙁 Still feeling guilty and defeated about it, though.
Sorry, that was really negative. Thank you so much for the link. I appreciate your interest–I really do!
Ronda, I can understand your frustration. However, there is one very important thing to remember: every student is unique. As a result, every student has his or her strengths and weaknesses. Your daughter might never master math the way most other children do. That’s fine. I am sure she masters other things that are very difficult for other children. The key is to keep working. Education is a process by which we shore up students’ weaknesses and build on their strengths. When it comes to math, your goal might just be to shore up the weakness, while you use other subjects to build on her strengths.
Thank you. I need to hear that. She is a natural at grammar and spelling, and she writes extremely well. I need to quit obsessing about the math, I guess. I appreciate that encouragement.
Ronda, I am glad that I could help.