Author: pythagitup

1 Step, 2 Step

I’ve been working with R, a former student, this summer. A few days ago, I gave him an NRICH problem called 1 Step, 2 Step.

Liam’s house has a staircase with 12 steps. He can go down the steps one at a time or two at a time.

Our goal is to determine the number of ways Liam can walk down the stairs. I gave R no instructions or guidance; I actually stood in the yard while he worked. After around ten minutes, R told me he’d made some progress but might be stuck. I asked him to tell me what he’d come up with so far.

R told me that he’d decided to start by looking at smaller staircases. If Liam’s staircase had only two stairs, for example, he could walk down two different ways: take both steps one at a time or take the pair of steps two at a time. R listed out these possibilities for staircases with less than six steps. He used 1 to represent Liam walking one step at a time and 2 to represent Liam walking two steps at a time. Here’s the list for a staircase with four steps:

Case Approach for 4

R felt confident that he’d listed all of the possibilities for the first five staircases, but he felt like he might have missed one for the staircase with six steps. We decided that I’d say all of the different ways I could, and R would check them off his list as we went along. R’s intuition proved correct: he had missed one of the ways. The table below shows the number of ways for staircases with one to six steps. I encourage you to take a close look before you keep scrolling.

Case Approach part 1

Do you see it? R noticed that the number of ways looked a lot like the Fibonacci sequence. He couldn’t see, however, why this problem would have anything to do with the Fibonacci sequence. Frankly, neither could I. Actually, I couldn’t even begin to consider why the Fibonacci sequence appeared in the table because R’s thinking had impressed me so much. (As you’ll see below, I took a totally different approach to this problem.) R and I decided to test his hypothesis that the number of ways matched the Fibonacci sequence by looking at the staircase with seven steps. First, R added 8 and 13 to predict 21 ways. Then, R listed out all of the different ways to walk down that staircase. When R only came up with 20, he suspected he’d again missed one. We went through the ways together and found the missing one. His prediction for the seven-step staircase having proven correct, R confidently filled out the table below:

Case Approach part 2

R asked and I confirmed that the correct answer was indeed 233. But the mystery remained: What did this staircase problem have to do with the Fibonacci sequence? We managed to come up with this explanation. To walk down the twelve-step staircase, you could walk down a ten-step staircase and then go down the last two in one step. Or you could walk down an eleven-step staircase and then go down the last one in one step. Thus, you’d add the number of ways for ten- and eleven-step staircases.

As mentioned above, I solved the problem a totally different way. I viewed the problem as involving several different cases, each of which I could deal with by using combinations. The table below shows my reasoning for the twelve-step staircase:

Combinations Approach

Adding the numbers in the last column gives the 233 total ways that R calculated earlier.

Okay, now it all gets a little crazy. R and I haven’t talked about this yet, but I spent some time today thinking about it. Combinations got me thinking about Pascal’s Triangle, and it turns out there’s an interesting connection between it and the Fibonacci sequence. Take a look at the following image of a left-justified Pascal’s Triangle:

Fibonacci_in_Pascal_triangle
Image credit: Phan Yamada; Link to original file

Each diagonal adds up to a Fibonacci number. We’ll number the diagonals starting with 0. Let’s take a look at the 7th diagonal:

7th diagonal

Now, take a look at the approach I used to calculate the number of ways to walk down a seven-step staircase:

Combinations Approach 7 stairs

See the similarity? The entries in Pascal’s Triangle are binomial coefficients. If you want the second column of the fifth row, for example, you’d use the following binomial coefficient:

5C2

Comparing the “Combination Version 2” column of the table to the left-justified Pascal’s Triangle confirms that our calculation matches the diagonal. This is a pretty awesome result, one that I definitely wasn’t expecting when I gave R this problem!

Here are a few takeaways:

  1. Giving R the freedom to attack the problem without any guidance allowed him to develop a unique approach.
  2. Examining different approaches to the same problem reveals much more than focusing on a single approach.
  3. NRICH problems are awesome!

As I prepare for what promises to be a strange and challenging school year, I keep reminding myself of the power of problem solving. I haven’t quite figured out how to make this sort of exploration and discussion happen during remote learning, but I’m more committed than ever to finding a way to make it work.

One Sentence #5: Uniquely Human

One Sentence #5: Uniquely Human

I recently finished Dr. Barry Prizant’s wonderful Uniquely Human: A Different Way of Seeing Autism. In the first chapter – entitled “Ask Why?” – Prizant details the “behavioral-assessment approach” to autism. He describes this approach as “using a checklist of deficits” and “treat[ing] the person as a problem to be solved rather than an individual to be understood.” Prizant does not consider this approach effective, instead offering the following recommendation:

What’s more helpful is to dig deeper: to ask what is motivating these behaviors, what is underlying these patterns.

For Prizant, only by first understanding a person with autism can you begin to help that person. Uniquely Human offers a great overview of the challenges that people with autism face and how to support them through these challenges, but Prizant never treats autism as something to be overcome. Instead, Prizant shows incredible respect and love for the people with autism that he’s known and shares so many examples of amazing individuals who have thrived as much because of their autism as in spite of it. This is a powerful book, and I strongly recommend you check it out.

One Sentence #4: Breakfast of Champions

One Sentence #4: Breakfast of Champions

I love Kurt Vonnegut, and although there’s a lot to love in Breakfast of Champions, I have mixed feelings about the book.

His high school was named after a slave owner who was also one of the world’s greatest theoreticians on the subject of human liberty.

This sentence illustrates what I love about Vonnegut: his ability to put in clear terms the absurdities and paradoxes of our world. Earlier in the book, Vonnegut puts the lie to a fact known to so many American children – that America was discovered in 1492. Of course, as Vonnegut writes, “millions of human beings were already living full and imaginative lives” in America then. Instead, 1492 “was simply the year in which sea pirates began to cheat and rob and kill” the indigenous Americans.

For all it does to counter false narratives and question American racism, however, I think Breakfast of Champions ultimately falls flat because of problematic language (especially use of the “N” word) and underdeveloped characters of color (Wayne Hoobler). And even if I could ignore those flaws (which I really can’t), neither the plot nor the theme are strong enough for me to consider Breakfast of Champions a great book.

Current Vonnegut Rankings:

  1. The Sirens of Titan
  2. Slaughterhouse-Five
  3. Player Piano
  4. Breakfast of Champions
  5. Cat’s Cradle
One Sentence #3: Untangled

One Sentence #3: Untangled

Here’s a quote from Untangled: Guiding Teenage Girls through the Seven Transitions into Adulthood by Lisa Damour:

Discipline should always come with the opportunity to make things right again.

As the title suggests, Untangled is all about supporting girls as they develop into adults. A girl who is contending with adult authority, one of Damour’s seven transitions, might disregard an adult’s instructions and do something inappropriate. A correct response will, of course, involve some sort of discipline, but the discipline needs to be more than simply punishment. It needs to include a learning opportunity, a chance to correct a mistake and grow.

I really enjoyed Untangled. Damour’s perspective as a clinical psychologist who works with teenage girls helped me think about my students in a different way. I look forward to applying new ideas and insights in my future work with my students.