Sometimes new piano students and parents ask, "How long will it take me or my child to learn to play?"  I try to explain with a simple equation that it's a lot anything else we want to achieve.  

To G-E-T we have to G-I-V-E. 

At students' lessons I focus attention on setting weekly, monthly and yearly goals that are meaningful, and give kids effective learning habits and playing strategies to achieve these.  Then we add the human element - Artistry.  To help kids further, I've written a series of Piano Lesson Storybooks to inspire and motivate kids in piano.  I've also established a fun Piano Club where students record their polished pieces and publish them to personalized on-line Musical Memory Books.  Then I publish these recordings on iTunes through a podcast so kids can share their piano journey with family and friends.  Twice a year students play for their parents at recitals to earn awards.   Additionally, I've published many articles to help parents and students get the most out of piano lessons.  Why do I do all this?  Because I follow the simple equation above:  To G-E-T we have to G-I-V-E.  However, that being said, there is another source which indicates there may be a more scientific formula for caluculating students' progress to answer the question I'm often asked, "How long will it take me to learn to play piano?"  To learn more, keep reading.

Steven Strogatz is a professor of applied mathematics at Cornell University who writes articles for the New York Times.  In his recent column he laments that "Every year about a million American students take calculus.  But far fewer really understand what the subject is about or could tell you why they were learning it.  It’s not their fault, he says.  There are so many techniques to master and so many new ideas to absorb that the overall framework is easy to miss."  This is also true of piano, plus there are many traditional myths and misunderstandings that create unnecessary limitations that get in the way of learning.  For example, the assumption that describing = doing.  This statement is false.  Following through with an effective goal getting plan = doing/change.

"Calculus can be described as the mathematics of change.  As such, it can "describe everything from the spread of epidemics to the zigs and zags of a well-thrown curveball [and your child's piano progress].  Strogatz, explains that "every field has its own version of a derivative [Differential Calculus].  Whether it goes by “marginal return” or “growth rate” [or "learning rate"] or “velocity” or “slope...  Roughly speaking, the derivative tells how fast something is changing; the integral [Integral Calculus] tells how much it’s accumulating.   By plotting y versus x  on a graph to visualize how one variable affects another, scientists [can, for example] translate their problems into the common language of mathematics.  The rate of change that concerns them -- whether it be a viral growth rate, a jet’s velocity, or whatever -- then gets converted into something much more abstract but easier to picture, which calculus students are well familiar with: a slope on a graph.  Like slopes, derivatives can be positive, negative or zero, indicating whether something is rising, falling or leveling off." 

But there's something else, the rate of change is not constant.   "Things always change slowest at the top or the bottom [at the extremes]."  For example, explains Strogatz "In Ithaca, during the darkest depths of winter, the days are not just unmercifully short; they barely improve from one to the next.  Whereas once spring begins popping, the days are lengthening rapidly.  All of this makes sense, because change is most sluggish at the extremes precisely because the derivative is zero [where things stand still, momentarily.]"

This can be the way it seems in piano at times as students work to push through learning threshholds that can only be achieved with persistent effort over time.  The growth rate or piano learning rate is never constant.   At these times I like to re-direct students' attention to a different kind of picto-graph, their piano dream.  I ask students to draw a picture of their piano dream and post it on their refrigerator at home, or the wall over the piano.   Still, piano students could conceivably use calculus to measure their Piano Growth Rate or rate of progress.  Then they could graph this and stare at the slope.  So, for those who want a more scientific formula to answer the question, "How long will it take me (or my child) to learn to play piano?" I did some research and found a piano formula to answer this question that inquiring minds want to know.  Here it is below.

Note:  To understand the equation you must first accept the statement that learning is analogous to earning money, so that Income = Linear time and Earnings are how much knowlege a student retains over that time [of course, learning is not linear.]  

Piano Example: In practicing the same short segment for 4 hours, we are likely to gain a lot more during the first 30 minutes than during the last 30 minutes, and how much we learn or retain depends wildly on how we do it.  However, we are talking about an optimized practice session averaged over many practice sessions that are conducted over time intervals of years (optimized means, for example, we are not going to practice the same 4 notes for 4 hours.)  Averaging these learning processes gives the accuracy we need for approximation.  Note that linearity does not depend, to first approximation, on whether you are a fast learner or a slow learner; this changes only the proportionality constant. Thus we arrive at the first equation:

L = kT (Eq. 1.1) where L is an increment of learning in the time interval T and k is the proportionality constant. What we are trying to find is the time dependence of L, or L(t) where t is time (in contrast to T which is an interval of time). Similarly, L is an increment of learning, but L(t) is a function.

Now comes the first interesting new concept. We have control over L; if we want 2L, we simply practice twice. But that is not the L that we retain because we lose some L over time after we practice [AND OVER THE SUMMER IF SUDENTS DON'T TAKE LESSONS OR PRACTICE].  Unfortunately, the more we know, the more we can forget; that is, the amount we forget is proportional to the original amount of knowledge, L(O). Therefore, assuming that we acquired L(O), the amount of L we lose in T is: L = -kTL(O) (Eq. 1.2) where the k’s in equations 1.1 and 1.2 are different, but we will not re-label them for simplicity. Note that k has a negative sign because we are losing L. Eq. 1.2 leads to the differential equation dL(t)/dt = -kL(t) (Eq. 1.3) where “d” stands for differential (this is all standard calculus), and the solution to this differential equation is L(t) = Ke(exp.-kt) (Eq. 1.4) where “e” is a number called the natural logarithm which satisfies Eq. 1.3, and K is a new constant related to k (for simplicity, we have ignored another term in the solution that is unimportant at this stage). Eq. 1.4 tells us that once we learn L, we will immediately start to forget it exponentially with time if the forgetting process is linear with time. [THIS IS WHY ROUTINE PRACTICE IS NECESSARY] Since the exponent is just a number, k in Eq. 1.4 has the units of 1/time. We shall set k = 1/T(k) where T(k) is called the characteristic time. Here, k refers to a specific learning/forgetting process. When we learn piano, we learn via a myriad of processes, most of which are not well understood [EXCEPT BY YOUR PIANO TEACHER]. Therefore, determining accurate values for T(k) for each process is generally not possible, so in the numerical calculations, we will have to make some “intelligent guesses”. In piano practice, we must repeat difficult material many times before we can play them well, and we need to assign a number (say, “i”) to each practice repetition. Then Eq. 1.4 becomes L(i,t,k) = K(i)e(expt.-t[i]/T[k]) (Eq. 1.5) for each repetition i and learning/forgetting process k. Let’s apply this to a relevant example. Suppose that you are practicing 4 parallel set (PS) notes in succession, playing rapidly and switching hands, etc., for 10 minutes. We assign i = 0 to one PS execution, which may take only about half a second. You might repeat this 10 or 100 times during the practice session. You have learned L(0) after the first PS. But what we need to calculate is the amount of L(0) that we retain after the 10 minute practice session. In fact, because we repeat many times, we must calculate the cumulative learning from all of them. According to Eq. 1.5, this cumulative effect is given by summing the L’s over all the PS repetitions: L(Total) = Sum(over i)K(i)e(expt.-t[i]/T[k]) (Eq.1.6)

Now let’s put in some numbers into Eq. 1.6 in order to get some answers. Take a passage that you can play slowly, HT, in about 100 seconds. This passage may contain 2 or 3 PSs that are difficult and that you can play rapidly in less than a second, so that you can repeat them over 100 times in those 100 seconds (method of this book). Typically, these 2 or 3 difficult spots are the only ones holding you back, so if you can play them well, you can play the entire passage at speed.  For this quick learning process, our tendency to “lose it” is also fast, so we can pick a “forgetting time constant” of around 30 seconds; that is, every 30 seconds, you end up forgetting almost 30% of what you learned from one repetition. Note that you never forget everything even after a long time because the forgetting process is exponential -- exponential decays never reach zero. Also, you can make many repetitions in a short time for PSs, so these learning events can pile up quickly. This forgetting time constant of 30 seconds depends on the mechanism of learning/forgetting, and I have chosen a relatively short one for rapid repetitions; we shall examine a much longer one below. Assuming one PS repetition per second, the learning from the first repetition is e(expt.-100/30) = 0.04 (you have 100 seconds to forget the first repetition), while the last repetition gives e(expt.-1/30) = 0.97, and the average learning is somewhere in between, about 0.4 (we don’t have to be exact, as we shall see), and with over 100 repetitions, we have over 40 units of learning for the use of PSs.  The 30 second time constant used above was for a “fast” learning process, such as that associated with learning during a single practice session. There are many others, such as technique acquisition by PPI (post practice improvement). After any rigorous conditioning, your technique will improve by PPI for a week or more. The rate of forgetting, or technique loss, for such slow processes is not 30 seconds, but much longer, probably several weeks. Therefore, in order to calculate the total difference in learning rates, we must calculate the difference for all known methods of technique acquisition using the corresponding time constant.  [OR TAKE PIANO LESSONS OVER THE LONG SUMMER AND CONTINUE PRACTICING].   The scientific approach ensures that errors are corrected as soon as possible, [ IF YOU DIDN'T TAKE LESSONS LAST SUMMER THIS CAN BE CORRECTED BY TAKING LESSONS THIS SUMMER.]

This formula is an excerpt from pianofundamentals.com [THE WORDS IN BRACKETS ARE MINE].

Don't Wait to Enroll for Summer Lessons!

Best Wishes,

Cynthia Marie VanLandingham