Some people think anybody can learn mathematics and succeed in math class.
Others are convinced they are doomed to fail at learning math, no matter the effort they invest in practicing the skills required of a master mathematician.
Who is right?
According to my informal research, there are three significant determinants of success when it comes to learning mathematics. Students need all three elements to perform well.
One of my former tutoring clients shared an excellent perspective on this topic last week when I asked my readers about what they believe about learning mathematics.
Throughout this post, I will include direct quotes from this reader because they encapsulate the whole issue of learning mathematics. They mention all three of the elements that I consider to be important determinants of success.
A successful student leverages intrinsic ability, sustained interest, and rigorous practice. Recognize the importance of these three ingredients for academic success and seek to optimize each one.
Let’s assume that school grades in math class are a good measure or indication of one’s ability in mathematics. The math grades you earn are directly proportional to your math ability.
Success in this context is defined as earning good grades. It’s up to personal standards to decide what is considered a “good grade,” but it’s better to receive a higher grade than a lower one.
The “talented” students with high levels of natural ability in mathematics should be able to earn good grades almost automatically. They may not need high standards of interest or practice.
For a while (until they encounter a sufficiently tricky math class) these students may be able to sleep through classes, pay their dogs to do their homework, and finally show up to ace the final exam.
Even though they were uninterested in the classes and did not invest time to practice their skills, they end up earning an A+ as their final grade.
Another student in the same class may have shown up highly engaged in every lecture and diligently completed their practice problems and study guides. They ended up improving their abilities dramatically over the course of the semester.
After all this time and energy, they end up earning a B+ on the final exam, a full letter grade lower than the student who didn’t spend nearly as much time studying.
This is an exaggeration, but the point is to illustrate that ability, not just interest or practice, is what is ultimately required for earning higher grades.
If you’re able to perform math skills well, you will probably earn high grades, because grades are meant to assess one’s ability when it comes to a given topic.
It’s a beautiful thing that God gave us all unique talents and gifts. There is positively such a thing as natural ability. We all have different degrees and areas of strength from the beginning.
You are a unique individual. Your own set of natural propensities has never existed in this exact combination in another human being, ever since time began!
In mathematics, as in almost any learnable skill, some people are initially more talented and capable. Here’s how one of my readers puts it:
“I do believe some people are more able to grasp math concepts quicker and deeper than others. It seems to be a talent or skill of sorts. Just like some people can hear the tones in music and piece together songs or artists can draw/paint/build like no other. We are all given different talents and abilities, thank goodness. I do not believe we all should be masters in every skill set.”
Nobody has to be good at everything. In fact, we’re all terrible at many things. And that’s okay! It works out just fine because each of us shares in the collective gifts of humanity.
As long as everybody contributes to society from their areas of strength while mitigating their weaknesses, so they don’t get in the way, everything necessary in life just seems to get done.
Museums are filled with beautiful paintings. Bridges are built durable to last. Hospital patients receive proper care and treatment. Planes fly to their destinations without incident. Cakes and pastries are continuously available, fresh at the local bakery.
Free markets are a beautiful thing. Nobody needs to know everything there is to know or to be able to do everything there is to do. Exchanging our best work ultimately enriches us all.
When it comes to learning mathematics, however, every American must spend a considerable amount of time and effort in learning the subject. This isn’t true of painting, cake-baking, or many other topics of general interest.
Each student shows up to math class with different levels of interest and ability. Is it unfair to judge everybody by the same set of objective academic standards?
It would seem silly to require that every graduating high school senior pass an exam on baking skills, repeating a grade in school if they burn their chocolate cupcakes or use the wrong color of sprinkles.
We require that students pass math exams, along with examinations in other essential subjects, because we deem learning and mastering those areas of study to be more universally relevant.
While math ability can be improved over time by focused effort and practice, some people are initially less talented at math, and still, others are frankly uninterested in math.
Since grades are valuable to most students and parents, it would be useful to know how to improve one’s math abilities regardless of talent or interest.
How do a person’s ability and interest relate to each other? I did some research into this, and the results were surprising. There is a correlation, but it may not be what you expect.
And the question of “which comes first?” is even more difficult to answer….
Another point my reader makes is that people experience different levels of interest and enjoyment across different skill sets. Here’s their quote:
“I am a believer in having a predisposition toward certain skill sets, in being able to recognize what your strengths are and in honing them into a viable skill set for a job that can sustain you. I think you do your best at what you enjoy. Some just don’t enjoy math! Imagine that. ?”
There is published research in applied psychology suggesting that “in predicting success, the greater the time interval involved, the more important becomes interest and the less important becomes ability.”
This reminds me of the classic story of the tortoise and the hare.
If we think of completing a math class as finishing a footrace, the student with high levels of natural ability might be like the hare.
The student who maintains interest while practicing over a more extended period of time, according to this research, is more likely to succeed than the talented but lazy student.
Another study published in 2007 called School Achievement, Perceptions Of Ability, And Interest Change As Children Age confirmed what you might suspect: as children grow older, they tend to remain interested in the areas of ability in which they perform best.
The study tracked approximately a thousand children as they progressed from elementary to high school. They found a remarkable pattern that carries a valuable lesson:
“Children in early grades may like a subject in which they don’t feel very competent, or they may feel competent in a subject in spite of poor grades. But by the end of high school, children generally feel most interested in subjects in which they feel they are the strongest.”
I find this rather sad, don’t you? At some point along the way of growing up and “adulting,” we learn to hate those subjects that we are not competent at doing.
We learn to remain inside our intellectual comfort zone. We give up on learning something when we don’t feel like we are performing well. We give up even if we liked it at first.
That sense of competence or incompetence must come from assessments and evaluations. How else besides good or bad grades do students form their beliefs about their abilities?
The first study shows that regardless of natural ability, students can succeed in something like K-12 mathematics if only they will maintain their interest in the subject. The second predicts that students will just remain interested in the matter if they succeed, but not otherwise.
Does this sound like an unsolvable chicken and egg problem to you? It almost does to me.
Most people acknowledge that practice also plays a vital role in developing mastery. Nobody ever got worse at math by doing their homework. But, as my reader explains, there are practical limitations:
“Practice certainly helps sharpen and improve certain skills, but I will probably never be a mathematician who enjoys crunching away at solving formulas no matter how much I practice. So, yes, repetition while practicing math (homework) will improve your abilities in math, but a certain group of us will reach a limit with our interest and stamina.”
This reader recognized the importance of practice and would probably agree that training is accessible to anyone, regardless of intrinsic interest or natural ability.
Here’s the rub: your abilities are not fixed. Your skills can be changed and improved upon. And this process has a lot to do with practicing and maintaining interest.
Yes, each of us comes with a cornucopia of natural talents and abilities. Yes, there are some things you are able to do well even if you don’t try hard at learning them. And yes, there are some (or in my case, many!) things that each of us naturally sucks at doing.
But you can definitely get better. You can learn anything, including mathematics.
The fact of the matter is that you are a human being with a human brain. That biological machine between your ears is the most complex system in the universe.
Widely accepted conclusions about the neuroplasticity of the brain combined with Carol Dweck’s concept of growth mindset paint an encouraging picture of our ability to acquire skills through practice.
More on that in future blog posts. For now, let me know if you agree with my three determinants of success!