Why don’t logarithms have horizontal asymptotes?

We’re studying logarithms in my Honors Algebra II class, and when looking at a logarithmic function for the first time, a very astute student asked “does it have a horizontal asymptote.” And my quick response was a slightly dismissive “no.” But the student was insistent, and pointed out that the slope seems to keep getting smaller and smaller, so shouldn’t it eventually reach zero, and won’t that mean that there is an asymptote?

I like this question because, if I let it, it pulls me away from simply repeating the same information I was handed back during my math education, and really gets me thinking about the nature of the logarithm and the meaning of horizontal asymptotes.

What I “know”…

So here’s what I “know”—the logarithm is just the inverse of the exponential function, and the exponential function doesn’t have any vertical asymptotes—you can always exponentiate a larger number. Thus, it should be that when you invert this function to form the logarithm, there shouldn’t be any horizontal asymptotes. As you take logarithms of larger and larger numbers, the output of the function should continue to increase.

A comparison:

Let’s compare the logarithm to a function that we know does have an asymptote, namely, the inverse function. Just to make it a bit easier to visualize, I want to work with 1/x reflected over the horizontal axis and vertically shifted by 3,

In many ways this function looks similar to the logarithm—it is increasing, and as it grows, its slope decreases. But this particular function has a horizontal asymptote at .

What about the rate of change?

Things get more interesting when I look at the rates of change for these functions.

For the logarithm:

for the inverse function:

As we can see from the graphs, both slopes decrease as x increases, and the inverse decreases at a bigger rate. Does this difference explain this behavior?

It does seem to tell me that just because the rate of change of a function goes to zero at infinity can’t be proof of a horizontal asymptote. But could it be that any function whose rate of change goes to zero faster than a particular rate (say will have a horizontal asymptote?

I’m flummoxed. How would you explain to a student why a function whose rate of change is always decreasing never reaches a maximum value, particularly when every similar function he’s previously encountered like this does have a horizontal asymptote?

Great interview with Cal Newport on Future of Education podcast

Cal Newport, author of How to Be a High School Superstar and more recently, So Good They Can’t Ignore You, just did an interview with Steve Hargadon for his Future of Education Podcast, one of my favorite podcasts.

The entire interview is excellent, but this one tidbit near the end of the interview grabbed me. It is one of the best bits of advice for a high school student I’ve heard. Here’s Cal’s elevator speech for current high school students on what his latest book is about.

There’s not one thing that you are hard-wired to do that you have to discover before you’ll be happy. You’re young—if you’re going to college, spend your next years exposing yourself to things, but more importantly, learning how to do hard intellectual things. Keep in mind that to build a life you love, what matters is how good you are at things, not whether or not you’ve found the one sort of true match you’re supposed to do. There’s no one true passion out there waiting for you, you have to go develop passion, and getting good at things is how you’re going to do it.

Talking Computational Thinking with some superstars

Last week, I was lucky enough to participate in a Google Edu On Air discussion on Computational thinking in the physics classroom. The discussion featured a bevy of teachers who are doing awesome things with computational thinking:

• Bruce Sherwood, author of Matter and Interactions
• Evan Weinberg, teacher at Hangzhou International School in China
• Danny Caballero, Postdoc at CU-Boulder in Physics Education Research
• Matt Greewolfe, physics teacher at Cary Academy
• Phil Wagner, googled, former physics and math teacher and now shepherd of Google’s Computational Thinking project.

Here’s the video. If you are looking for a good introduction to what Computational Thinking is, and why it is important for physics teaching, I can think of few better ways to learn about it than listening to these superstars.

Here’s the link to the google sites page.

Another point in the argument against timed tests…

Thanks to the awesome Grace Chen, I saw the following article yesterday:

The unscientific thinking that forever lingers in the minds of physics professors

Here are the two key paragraphs from the post (emphasis mine):

Deborah Kelemen and her colleagues presented 80 scientists (including physicists, chemists and geographers) with 100 one-sentence statements and their task was to say if each one was true or false. Among the items were teleological statements about nature, such as “Trees produce oxygen so that animals can breathe”. Crucially, half the scientists had to answer under time pressure – just over 3 seconds for each statement – while the others had as long as they liked. There were also control groups of college students and the general public.

Overall, the scientists endorsed fewer of the teleological statements than the control groups (22 per cent vs. 50 per cent approx). No surprise there, given that mainstream science rejects the idea that inanimate objects have purpose, or that there is purposeful design in the natural world. But look at what happened under time pressure. When they were rushed, the scientists endorsed 29 per cent of teleological statements compared with 15 per cent endorsed by the un-rushed scientists. This is consistent with the idea that a tendency to endorse teleological beliefs lingers in the scientists’ minds. This unscientific thinking is usually suppressed, but time pressure undermines that conscious suppression.

So let me get this straight—when professional scientists, who have years of training in a discipline, are forced to assess whether or not scientific statements are true under time pressure, their performance decreases by half and they blow almost one third of the questions, essentially earning a 71%, were this a test in school? That is, professional scientists become C- scientists when exposed to pressured multiple choice testing.

And we think that having novice students with almost no training in a particular science discipline take multiple choice tests that are filled with distractor responses designed to elicit misconceptions is an accurate, fair or useful way to judge students’ scientific understanding?

This also ties in beautifully to the some of the points Michael Pershan was making in his masterful discussion last night of the “Failure(?) of the Math Mistakes Blog”, which you should make a point to listen to. In his talk, Michael talks about why students make mistakes on ideas that they already know, such as why students will say , and yet he knows that students know this isn’t true, as they can explain why when confronted with this mistake.

Michael points to a possible explanation for this discrepancy from Daniel Kahneman, author of Thinking Fast and Slow. Khaneman writes:

If a satisfactory answer to a hard question is not found quickly, [the
intuitive capacity] will find a related question that is easier and will answer it. I call the operation of answering one question in place of another substitution.

So the quick, rapidly responding part of our brain that often succumbs to misconceptions and incomplete heuristics will too often leap at answering a question intuitively and incorrectly when faced with time pressure, before the slower, reasoning brain has a chance to process and come to a thoughtful and often correct response to the question when given time to think. My recollection of Kahneman also is that he said that this leaping to the wrong answer can never be eliminated—it can only be reduced by giving ourselves more time to process, often by deliberately introducing mental speed bumps (like drawing a diagram) designed to slow us down and allow our reasoning brain time to process.

So what’s the point of timed tests again? Is it to simply remind our students of how error prone their brains are when operating under high stress and time pressure?

I’m blogging tonight—developing a manliness curriculum

Ok, this blogging drought has to end sometime, right? I’ve got so many things I want to write about, I just am really having trouble finding the time and words to write them. But today, one awesome thing happened that I wanted to share.

This morning I saw a great post by Grant Lichtman about Presbyterian Day School, a k-6 boys school in Memphis, TN, and coincidentally, the school I attended for kindergarten and first grade.

In his post, Grant described how PDS has developed a list of key life skills they promise to teach every boy by 6th grade. Here they are:

• Doing a load of laundry (wash, dry, fold)
• Washing the dishes
• Ironing a shirt and pants
• Pump gas
• Cleaning a room (make bed, clean room, mop, dust, clean bathroom)
• Cook a basic meal
• Grill a hamburger
• Community service project
• Rules of dress (matching, dressing for occasions)
• Mowing, weeding, edging a lawn
• Tie a tie
• Start a fire
• Study for a test
• Drive a nail
• Write a handwritten thank-you note
• Conversation etiquette in person and on the phone
• Dating etiquette
• Basic first aid
• Give a speech

This got me thinking. My school has recently started a men’s affinity group at the behest of a few boys to talk about various issues related to masculinity. We’ve only met once, but I’m really impressed by the potential for this group. In addition, a colleague of mine who does duty on my dorm has a regular weekly men’s meeting on dorm where they talk about what it means to be a man at our school, and the guys love it.

After reading Grant’s post, I sent an email to my colleague wondering what would happen if we asked the boys at our school to make a similar list. What does a manliness curriculum look like for a guy who graduates from St. Andrew’s? What life skills should he have?

Tonight, my colleague had that discussion with our dorm, and here’s the list they generated:

• How to cook

• How to survive a night in the wilderness
• How to Hunt
• How to respect all people (be empathetic)
• How to tie a bow tie
• How to chop firewood
• How to jumpstart a car
• How to cold start a chainsaw
• How to change a tire
• How to change a diaper
• How to negotiate
• How to fish
• How to cut hair
• How to shave
• How to manage finances
• How to catch mice
• How to discipline children
• How to treat a woman

Our thought is to do this with a couple of more dorms, and then bring the guys group together to compare the lists and see how this might be the first step in building a manliness curriculum. Once we get the list, it shouldn’t be too hard for the guys to self organize lessons on many of these topics and teach one another.

All of this has me pretty excited, and it also has me thinking if it is seemingly this easy for students to design their own curriculum for life skills, why can’t students have more say in the shape and design of the more traditional academic curriculum?

A MOOC is born

I apologize for my one month hiatus from blogging. I’ve got a lot of things I’d like to write about, but it’s likely going to have to wait until Thanksgiving break next week.

But I do want to take a moment to give some huge props to my colleagues at the Georgia Tech Physics Education Research Group, who just received a Gates Foundation grant to develop a Massively Open Online Course in Introductory Mechanics. This course will be launched this spring on Coursera.

Here’s the proposal video the group created :

I am most excited that this course promises to be one of the first (if not the first) MOOC to offer a rich laboratory experience for students, something I think is essential to any science course.

I’m really excited to play a small part in this venture, and I’ll have a lot more to say in the months ahead.

My colleague finds a mistake in Feynman

Since I first saw it on Minds of Our Own, I loved the seed and a log question, where Harvard and MIT graduates are given a seed and a log, and asked how the seed becomes a log. Students wax scientific about photosynthesis, but then when pushed to explain where all that mass comes from, is it the air, the ground, or water, almost all of them default to thinking it’s the dirt that makes up the mass of the tree. Minds of our own goes on to show how this shows students are missing both a very basic scientific idea—that most of the mass of the tree comes from Carbon Dioxide, a gas, and this is because they don’t have a real understanding of the idea that gases has mass.

Recently, I stumbled upon a video of Richard Feynman giving his own answer to this question, via the excellent blog, It’s Okay to be Smart.

I forwarded this video to my colleague, who shares a similar love for this question (we were both indoctrinated in its value at the Klingenstein Summer Institute (Have you been teaching in an independent school for less that 5 years? If so, apply now).

Not to long after I sent the email, he wrote me back.

Sorry to say this about such a great speaker and great scientist, but Feynman gets things badly wrong here.

He suggests that the oxygen generated during photosynthesis is the product of a splitting apart of carbon and oxygen in carbon dioxide. (See min. 3:39)
In fact, all the oxygen produced by photosynthesis is generated by the splitting of water to make H+ e- and . The role of carbon dioxide in photosynthesis is to get reduced, to have hydrogen atoms added to it, not to have oxygen removed. This is easy to demonstrate experimentally by using radioisotopically labeled and . Only when the label is in the is the produced radioactive.

I’d like to think I would have caught this error if I’d paid closer attention to the video as I was watching it and trying to decide whether to forward it, but honestly, my recollection of the AP chemistry class I took 20 years ago is now rather faded, and redox reactions are definitely one of the gaping holes in my understanding.

But I see this as a really exciting interdisciplinary moment—here’s a great physicist, stepping pretty far outside his area of expertise to talk about a topic in biology, photosynthesis, only to be called out on incorrect chemistry by a biologist. It’s all the more exciting that Feynman made this mistake, since it shows we all make mistakes, which is a great lesson for my students to see.

This is exactly the type of interdisciplinary learning I’d like to be setting my students up to do. However, I’m not sure I am setting them up to understand ideas like this, since they see most topics in year long courses completely isolated from one another. Could a biology student hear an explanation from a physicist about photosynthesis and bring up his understanding of redox reactions from the previous year’s study of chemistry to check Feynman’s work? I’m skeptical. And I’m sure than none of them would be able to push to the extra layer of thinking about how we might be able to know this experimentally by using radioisotopes. So how do we teach science students to think in this interdisciplinary manner?

If you want even more information about this, my colleague wrote this excellent article in The American Biology Teacher: Dust Thou Art Not & unto Dust Thouh Shan’t Return: Common Mistakes in Teaching Biogeochemical Cycles.