Discovery Learning

I just came across an article from The Mathematical Association of America, 1999, by Keith Devlin.  The article is called The Greatest Math Teacher Ever, Part 2.  A couple of excerpts that relate to teaching tennis the un-academy way.

He developed a method of teaching that became widely known as "the Moore method". Its present-day derivative is often referred to as "discovery learning".

Discovery learning is popular in tennis today, especially with younger kids.  Guided discovery is another name.  How's it work?

Part of the secret to Moore's success with his method lay in the close attention he paid to his students. Former Moore student William Mahavier addresses this point:

" Moore treated different students differently and his classes varied depending on the caliber of his students. . . . Moore helped his students a lot but did it in such a way that they did not feel that the help detracted from the satisfaction they received from having solved a problem. He was a master at saying the right thing to the right student at the right time. Most of us would not consider devoting the time that Moore did to his classes. This is probably why so many people claim to have tried the Moore method without success."

Can this work in tennis?  Probably not with large groups of kids, and probably not without a lot of hard work and planning by the instructor.  It's easy to tell kids how "load their legs", how to "make a unit turn", and how to move their arms on a serve.  What's not easy is to guide their discovery.

Plan well in advance and be prepared to really get to know your students. Halmos puts it this way: "If you are a teacher and a possible convert to the Moore method ... don't think that you'll do less work that way. It takes me a couple of months of hard work to prepare for a Moore course. ... I have to chop the material into bite-sized pieces, I have to arrange it so that it becomes accessible, and I must visualize the course as a whole -- what can I hope that they will have learned when it's over? As the course goes along, I must keep preparing for each meeting: to stay on top of what goes on in class. I myself must be able to prove everything. In class I must stay on my toes every second. ... I am convinced that the Moore method is the best way to teach there is -- but if you try it, don't be surprised if it takes a lot out of you."

Shut Up!

As coaches and teachers we talk too much.  Just shut up and let the students learn.

 “The best way to learn is to do; the worst way to teach is to talk.”  -- P.R. Halmos

That's the first sentence of an article called "The Teaching of Problem Solving".  Paul Halmos was a mathematician.  He may as well have been a tennis coach.


Hat tip to Seth Roberts for mentioning the Halmos article on his blog.

What a Drag

When a ball moves through the air a force called drag slows the ball down.  The drag force acts in the opposite direction of the ball's movement.  Here's the equation for drag force:

F=Cd*A*d*(v^2)/2

I got this equation from page 362 of The Physics and Technology of Tennis by Brody, Cross, and Lindsey (BCL).  In the equation, Cd is the drag coefficient of the ball, A is the cross-sectional area of the ball, d is the density of the air, and v is the ball's velocity.  BCL give values of 1.21 kg/m^3 for d (sea level), and 0.55 for Cd for a new tennis ball.  The ball's cross sectional area is pi*r^2, where r is the radius of the tennis ball (roughly 33 mm).

Having fun yet?

There's an even more fun equation a few pages later (p. 388) in the appendix on ball trajectories.  In this equation we've ignored the relatively minor vertical component of a ball's velocity and are only looking at the horizontal component.  When you hit a ball toward the other side of the net, how much does it slow down?  Since I'm interested in the effects of lower density air on the flight of a tennis ball, these are the equations that I care about.  Here's the equation:

t = (e^(k*Cd*x)-1)/(k*Cd*v0)

This tells us how long it takes (t, measured in seconds) a tennis ball (with a coefficient of friction Cd) to travel a particular distance (x, measured in meters) when moving through air for a specific initial velocity (v0, measured in meters per second).  In addition to the terms just mentioned we have two other terms to define.  The term e is simply the base of the natural logarithm.  The term k is a bit more complicated.  We'll use an equation from BCL (p. 387) to define k,

k=(d*pi*R^2)/(2*m)

Argh.  More undefined terms.  We used d above, and it's the density of the air.  R is just the radius of the ball, roughly 33 mm (.033 m), and m is the mass of the tennis ball (57 g or 0.057 kg).  Pi is pi.

So for a sea level standard tennis ball,

k = (1.21*pi*0.033^2)/(2*0.057)

k = 0.0363 m^-1

Back we go to our more interesting equation, armed with a value for k,

t = (e^(k*Cd*x)-1)/(k*Cd*v0)

Let's plug in some numbers and see how long it takes a ball hit at 70 mph to travel 80 ft.  We're imagining a player standing a little bit behind his own baseline, perhaps, hitting a forehand to roughly the far baseline.

We have to convert feet to meters and mph to m/s.  Google will do that for us, spitting out 31.3 m/s for 70 mph, and 24.4 m for 80 ft. 

We already said that Cd for a new tennis ball is 0.55 according to BCL.  Let's plug and play:

t = (e^(0.0363*0.55*24.4)-1)/(.0363*0.55*31.3)

t = 1.004

Let's round that to 1.0 sec.  How cool is that?  Must have picked those numbers to come out that way!

So how much does the air slow down a ball moving horizontally?  Well, without air we know how long a ball leaving the racquet at 31.3 m/s would take to travel 24.4 m by dividing the second number by the first.

t = 24.4/31.3 = 0.78 sec

So this forehand took 28% longer to go 80 ft through air than though a vacuum.  By playing at sea level instead of outer space (or the moon), we bought ourselves 0.22 sec to get to this ball.

Now for the big question:  How much time do we lose in Denver?

The density of air in Denver is not 1.21 kg/m^3.  It's about 0.84 times that, or roughly 1.02 kg/m^3.

Back to our equation.  Where does the density of air come in?  It comes in via that pesky k.  Recall

k=(d*pi*R^2)/(2*m)

There's d in there.  So we need to recalculate k for Denver rather than sea level.

k = 1.02*pi*0.033^2/(2*.057)

k = 0.0306

Should we call that kd for Denver?  Back to the time equation:

t = (e^(kd*Cd*x)-1)/(kd*Cd*v0)

Plug in some numbers and let Google give us a value for long it takes a 70 mph forehand to travel 80 ft in Denver:

t = (e^(0.0306*0.55*24.4)-1)/(0.0306*0.55*31.3)

t = 0.964 sec

We've lost some time by moving up to Denver.  But not much.  We've only lost about 0.036 sec.  That's a loss of 3.6% (again note the easy math because of the 1.0 sec transit time at sea level!).

Trouble.  In earlier posts, I've been assuming that the ball slows down 16 to 17% less slowly in Denver than it does at sea level.  All good players say the ball travels faster through the air up here than down there.  What gives?

What gives, indeed!

It could be completely due to the extra bounce of the ball.  The high-altitude balls do tend to bounce a bit more up here than balls bounce at sea level.  That 9% livelier ball, plus this roughly 4% less time due to thinner air, may add up to a difference that we all notice.  If you can simply add these effects, then 13% or so livelier game probably is a big deal.

Since I screwed up so badly on my prior estimation of the effects of our air on the ball, I'll keep my mouth (or fingers) shut until I've given this some more thought.

A New Can of Balls

UPDATE: The other tests I referred to in this post were flawed. I dropped balls from a lower height and assumed, incorrectly, that the coefficient of restitution would be the same as if I'd dropped them from the 100 inch ITF specification. Turns out the COR decreases as velocity increases. So the prior tests over-estimated the 100 inch rebound.


I just opened a new can of Penn Championship High Altitude tennis balls.  The balls had been sitting at 68 F degrees for a week.  They averaged about 57 to 58 inches of rebound when dropped from 100 inches (68 F degrees).  That's within the sea level specification of 53 to 58 inches.  Nice.

But I've tested other balls that rebounded 62 inches.  Well, that makes sense.  The specification range for high altitude balls is 48 to 53 inches at sea level.  So that means my range at 5,300 ft is roughly 57 to 63 inches.  So if I happen to get a high altitude ball that's at the low end of the specified range, I'll have a ball in Denver that is within the sea level specification.

That's just luck, of course.  We're still stuck with balls that, on average, exceed the mid-range of the sea-level specifications by about 8%.  Since the balls slow down less up here once hit, that's a big problem.

I've tested roughly 9-month-old balls and they're at the bottom end of the sea level spec for rebound.  Nobody seems to like the sound and feel of those balls up here.  Maybe nobody likes balls at the bottom of the spec at sea level either.  I don't know.  But up here, balls that meet the rebound spec, still leave us with a game that's about 16% faster through the air.

Lighter balls slow down more, but they feel odd, too.  Balls with fluffier naps may be the answer.

The Prince Tournament ball fluffed up a lot.  Maybe I'll have to track down more of those.

In the meantime, I still think the Stage 1 "green" ball is a better compromise than the current high-altitude ball.  We used them for an "elite" junior group today and the play was more fun, I thought.  Then I hit with some on an artificial grass court and again the play was fun.  It's nice to have time to set up and it's nice to be able to take a cut at the ball.  Both are not too common at this altitude.

What's Wrong With French Tennis?

A friend recently attended a USTA High Performance Coaches Workshop.  At the workshop he was told that France has 10x as many juniors than the US playing tournament tennis.  The message was that we're doing something wrong in the US.  Could be.  Probably.

But the most recent ATP World Tour rankings show the US with nine men in the top 100, while France has eight.

Spain leads the way with fourteen.  No word from the USTA on how many juniors play tournaments in Spain.

Once proud Sweden has one, same as Australia and Great Britain.  Fewer than Kazakhstan.

Maybe it's a prosperity problem.  More prosperity means higher opportunity cost of becoming a tennis champion.  You'd think rich countries would produce more great players since it costs a lot to become great at tennis.  But maybe it doesn't cost much to become great at tennis and the more important factor is how much are you giving up to become a great tennis player.

Maybe rich countries can afford more coaching and more coaching ruins players.

Defense, Defense

The two best defenders in the draw have advanced to the 2011 Wimbledon gentleman's final.  Nadal beat Murray (the third or fourth best defender) in four sets in one semi, while Djokovic beat Tsonga (not a very good defender) in four sets in the other.

A quick and crude look at the stats shows simply that the guy with the least unforced errors advanced in each match.  Nadal had just eight unforced errors in for sets.  Murray had forty-four.  That's an enormous advantage for Nadal.  Djokovic made just fourteen unforced errors in his match compared to thirty-four by Tsonga.

This doesn't mean that Nadal and Djokovic were just pushing, far from it.   Nadal hit seventy-four winning shots (winners plus forcing Murray errors) and Djokovic hit 107 (winners plus forcing Tsonga errors).  Dokovic's ratio of winning shots to unforced errors was 7.6 to 1.  That's awesome.  But not as awesome as Nadal's ratio of 9.25 to 1.

The losing semi-finalists had much worse winning to losing shot ratios, naturally (3 to 1 for Tsonga and 2 to 1 for Murray).

Neither Nadal nor Djokovic hit as many outright winners as their opponents today, but both obviously hit a lot of winning shots, in Djokovic's case, more winning shots than Tsonga hit.

The defenders won, but they had to win the matches.  Their opponents didn't just hand them the matches.  You wouldn't expect that in a Wimbledon semi.  But for players well below this level who do not aspire to this level (and if you aspire to this level you don't need my advice!), the simple take-home message is the guy who gave away the fewest points won both matches.

Over Training?

Now that it's clear we are playing a much tougher, faster game of tennis at high altitude, is it possible we have an advantage up here when we go to lower altitudes?  Since the ball comes at us much faster and bounces much higher, are we training ourselves to play higher speed, higher quality tennis?  Since we're forced to hit through much smaller vertical acceptance windows, are we becoming more precise than our low-altitude counterparts?

I think the answer is yes to all the above.  I think players ages about 13 to 20 years old have a hard time learning and mastering the game up here, because it's so much harder.  We don't see many great players who were born, raised, and trained exclusively at high altitudes, so I think it's reasonable to assume that the thin air hinders development for the reasons I've discussed in previous posts.  But...

If you mastered the game at lower altitudes, learned to be patient, develop points, play with high racquet speed generating high ball velocities and spin rates, then at some point you may benefit from playing at high altitude.

Those of us who've lived up here for a long time certainly have adjusted to the game.  It no longer feels crazy to me like it did when I came here thirty years ago.  If I play a bit at lower altitudes, the ball seems hard to control at first when I come back, but it's nothing a few minutes of hitting doesn't cure.  And it's hard for me to get any depth on the ball at first down at sea level, but again, it's nothing that a little bit of hitting won't cure.

So, if we can avoid the impatience that this altitude can breed, if we can avoid the illusion that we can end points quickly, and if we can train ourselves to go for balls we don't think we can get to,  I think we can benefit from playing up here.

I think we've had plenty of players do well in senior age group play on a national basis.  That's evidence that playing up here can make you a better player at sea level.

I still think we need to replicate the low-altitude game with a much slower ball at this altitude.  But it's nice to know we can benefit from cracking open a can of jumpers when we feel the need.