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Math Chat: Effective Thinking Through Mathematics, Puzzles, & NetLogo

In this "Math Chat" we will explore three topics. First, Michael Starbird's methods and vision for "effective thinking through mathematics". Second,  using puzzles to explore and practice effective thinking. We will examine four example puzzles he and his student Scott work through (and a bonus puzzle by Derek Muller). Finally, we will discuss NetLogo, an agent-based programming language and modeling environment, based on several introductory videos by Melanie Mitchell. 

The featured videos total an hour and a half, but if you follow all of the links there is an additional hour if you are feeling more ambitious. Each video is short: the longest one is only 18 minutes long. As always, watching the videos is optional, but will give you more grist for the discussion mill.

• In this video Starbird introduces his five strategies to effective thinking: 

http://www.youtube.com/watch?v=OiFbaj84G2Y

• In this exquisite video Starbird relates a story about a trumpet player to explain the first strategy of effective thinking, namely, to understand deeply, in particular, to understand simple things deeply: 

http://www.youtube.com/watch?v=clX_AS38yL8

• In this video Starbird powerfully explains the importance of making mistakes and learning from them: 

http://www.youtube.com/watch?v=dC47fQDFxqM

• Link to the full 27 minute playlist for "Effective Thinking Through Mathematics" (includes 2 videos not highlighted totaling 5 minutes).

• Starbird introduces puzzles as a tool for practicing and evaluating strategies for effective thinking: 


http://www.youtube.com/watch?v=zPSyAfJPS_Q

• The Meanie Genie Puzzle: 

http://www.youtube.com/watch?v=2tfXdjw1aPU

• The Pirates and Admirals Puzzle: 

http://www.youtube.com/watch?v=d_FfJqxvaJ0

• Whom do you trust Puzzle: 

http://www.youtube.com/watch?v=UPkX5X8WRhw

• A Shaky Story Puzzle: 

http://www.youtube.com/watch?v=98oTOBFgz5Q

• This playlist includes the four introductory puzzles and other introductory material but no solutions (14 minutes).

• This playlist includes the four puzzles above plus videos where Starbird guides Scott in the practice of effective thinking. Warning: two of the puzzles are solved in the videos. Try to solve the puzzles on your own before watching the whole playlist. 52 minutes.

• Bonus puzzle by Derek Muller (can you guess the rule before Derek gives the answer?): 

http://www.youtube.com/watch?v=vKA4w2O61Xo

• Melanie Mitchell's Introduction to NetLogo (Download NetLogo through this linkDownload Mitchell's "Getting Started with NetLogo" (pdf, 863k), Download the code for AntsNew.nlogo):  

http://www.youtube.com/watch?v=dKUQ6FuPrwc

• A simple NetLogo model: 

http://www.youtube.com/watch?v=Ddt72deIUlw

• A more advanced NetLogo model: 

http://www.youtube.com/watch?v=yETrB84oNN0

• This is the complete playlist for an Introduction to NetLogo with Melanie MitchellIt includes four additional videos plus the three above (1 hour, 11 minutes). Additional resources needed for the extra videos: 1) Homework (pdf, 68k), 2) Download OneAnt.nlogo (12k), and 3) Download MultipleAnts.nlogo (12k).

Here are some of the questions we might discuss:

• What do you think of Michael Starbird's five strategies for effective thinking?

• Can we learn effective thinking from studying mathematics?

• With the five strategies for effective thinking can anyone learn how to do mathematics better? If not, what additional strategies are required? Or is mathematics the purview of only those with certain cognitive capabilities?

• Is reviewing and mastering the basics essential for high performance thinking in any field?

• Is learning from mistakes important for effective thinking? Do mistakes guide our work in solving problems and thinking effectively?

• How is one supposed to use "raising questions", following the "flow of ideas" and change or growth as tools for effective thinking?

• What did you learn from solving the four Starbird puzzles and the bonus Derek Muller video? Did you use the strategies for effective thinking? What strategies did you use? Can puzzles help us understand the qualities of effective thinking?

• What do you think of NetLogo? Is computer simulation a tool for effective thinking? Is agent-based modeling mathematics? Is building a mathematical model, in NetLogo for example, a way to make mathematics and effective thinking more broadly accessible?

• What examples from the NetLego models library did you explore?

• Is NetLogo a programming environment that you might consider using? For what projects?

Join or login to comment.

  • A former member
    A former member

    Hey guys, working through this course now. Just wrote a short python script to solve the "Meanie Genie" bonus problem:

    find 2 different ways to express the number 1729 as the sum of 2 cubes

    https://gist.github.com/FiCode/8926047fa5568f0f1914

    1 · June 11, 2015

  • Martin C.

    Here is another puzzle that you might like to work on. The generalizations of this problem have some fairly extensive math behind them. Can you arrange the numbers from 1 to 100 so that there is no ascending or descending subsequence of length greater than 10? Recall that members of a subsequence need not be contiguous. For example, the sequence 7,2,5,3,4,1,6 has as one of its ascending subsequences: 2,5,6 and as one of its descending subsequences: 7,5,1. It may help to work with smaller numbers. What is the relationship between 10 and 100?

    December 14, 2014

  • Martin C.

    For those who have worked out the recreational math problems for next week, I have 2 more challenges that you can work on.

    The first is based on the Ask Marilyn column in the Parade Magazine newspaper supplement. A few weeks ago she answered a question that you can read about athere:http://parade.condenast.com/308009/marilynvossavant/308009/ The answer she gave is incorrect and she has since retracted it. More interesting than the specific problem is to come up with an argument as to why her reasoning was incorrect. That is, given that Angelina can finish the project faster than Brad, show without the use of algebra why the sum of their project completion hours must be than 24. Can you also give an algebraic proof: That is, given that 6/a + 6/b = 1 and 0 < a < b, can you prove that a+b > 24?

    Here is a problem that I thought of. Using simple calculations that you can do in your head, show that the 36th century (3500 to 3599) is the first not to have a year that's a perfect square.

    July 19, 2014

    • Martin C.

      Jeannie, Exactly right, which puts you ahead of Marilyn, who reputedly has the highest IQ score ever recorded. Here is the argument again for why the completion times add up to more than 24. Working together, Angelina and Brad can do 2 projects in 12 hours. The sum of their times is 24. Angelina does more than one project in that time and Brad does less. If we transfer the work over one project from Angelina to Brad, we subtract Angelina's time and add Brad's to the sum. Since Brad takes longer to do the same amount of work, the sum will increase and thus be > 24.

      July 29, 2014

    • Martin C.

      The algebra for Marilyn's problem is easier if guided by the intuition. When they work together, Angelina does 12 - a extra hours. The work she does in that time is (12-a)/a. To get Brad's extra time for the work, divide by his rate of 1/b to get (12-a)b/a. Intuitively, it seems that we get b=12+(12-a)b/a.
      Assuming for the moment this is true, we get b > (12-a) + 12, since b/a>1 and that gives a+b >24. To show that b = 12+(12-a)b/a, multiply both sides by a and simplify to get 2ab=12a + 12b, then divide both sides by 2ab to get 6/a +6/b=1, the original equation. We could work backwards to prove b = 12+(12-a)b/a.

      July 29, 2014

  • CJ F.

    For Saturday's "Math Chat", we will discuss among other topics Michael Starbird's vision for "Effective Thinking Through Mathematics". The full playlist of 5 videos totaling 27 minutes: http://www.youtube.com/playlist?list=PLkCiNL_gZp2cd5L3wk57TFFq8uWeJa5zs

    My favorite was the trumpet player video: http://y2u.be/clX_AS38yL8

    But the Fire video was also very interesting: http://y2u.be/dC47fQDFxqM

    In prior comments, Marty and I discussed these videos. Marty thought they were too vague to be useful. That may be true, but I think Starbird has a profound pedagogical point: in addition to heuristics, the math student needs to master metacognition. These metacognitive strategies of effective thinking as Starbird calls them may help fill in a gap for many math students who struggle.

    The paper "Polya, Problem Solving, and Education" by Alan H. Schoenfeld makes a similar point. It is accessible if you create a free account at http://www.jstor.org/stable/2690409

    July 24, 2014

    • CJ F.

      Marty, he also says in the first video "These strategies, by the way, are not strategies by which you can do things with great efficiency. I don't believe in efficiency in education. ... If you really want to change and become a better thinker and more successful, you need to develop habits of thought over time and through exertion."

      He is basically saying that math is hard: "there is no royal road to geometry". But he's wrapping it in a positive, can-do panache.

      The question is whether his strategies are effective. In our marketing-intensive society, we are overwhelmed by excessive positivity. We need to look past it to assess "is there something useful here". I think Starbird delivers some value. He is not just marketing panache.

      Would you prefer he said: math is terribly difficult, if you're not super-human smart don't bother? Instead he offers hope and a bridge (his strategies) so that ordinary people can develop their thinking faculties through mathematics!

      July 25, 2014

    • CJ F.

      There are some very interesting insights into the limitations of Polya's approach to heuristics in Alan H. Schoenfeld's paper "Polya, Problem Solving, and Education". I highly recommend the paper which can be read for free on-line if you create a free jstor account: http://www.jstor.org/...­

      I highly recommend the article. It is short: 9 pages with several pictures.

      The original article in paper format was published in Mathematics Magazine
      Vol. 60, No. 5 (Dec., 1987), pp. [masked]

      July 26, 2014

  • Martin C.

    Here is a simple proof of a question I asked earlier about showing that the 36th century is the first one not to have a year that is a perfect square.

    Note that for a century not to have a perfect square, there must be over 100 years between two successive perfect squares, the one before the beginning of the century and the one after. The difference between two successive squares is (x+1)^2 - x^2 = 2x+1. Before 2500 = 50^2, the difference between successive squares <= 2*49 + 1 = 99 < 100, so all centuries before 2500 have at least one perfect square.

    3600 = 60^2, and we know that 60^2 - 59^2 > 100. 59^2 < 3500 and so the 36th century is squareless. Now all we need to do is show that every century from 2500 to 3600 has a square. 3600 -2500 = 1100, so 11 centuries from 2500 to 3600, and 10 squares, 50^2 to 59^2. Only the 36th century is squareless, since after 2500 a century has at most one square, meaning the other 10 centuries have exactly one square each.

    July 26, 2014

  • Roy B

    Hi - I'm tentative for this at the moment, and may not be able to stay for the entire time if I am able to go at all.

    1 · July 25, 2014

    • Roy B

      Sorry I was not able to make it. Maybe next time

      1 · July 26, 2014

  • Lynn

    Lost possessions found, but taken ill. And it's a real horrid pity because you guys are going to be talking about agent-based modeling today. If you have any peculiar questions for me about that subject matter, fire away.

    1 · July 26, 2014

  • Sam B.

    The last puzzle I'm going to post about is the Meanie Genie puzzle. An archeology student is doing a job in Egypt, trying to find the lost McGuffin Crystal. Instead, she finds a magic lamp. A genie appears, gives her three wishes, and being a unimaginative buffoon, she wastes them on the crystal. The genie gives her nine crystals to choose from for her first wish, and gives her two one-use-scales for her next two wishes. The McGuffin crystal only weighs slightly more than the other crystals. Can she find the McGuffin crystal? And what is the best technique to find the crystal?
    Another classic logic puzzle. Personally, I think two steps are missing from his problem solving steps. One is from Polya, reflect on the problem what worked and what didn't. And the second is, is there another way to solve it? A creative answer you haven't seen? See you tomorrow.

    July 25, 2014

    • CJ F.

      For anyone who wants to spend the whole evening working puzzles. There are two more puzzles from Starbird's course hidden in the playlist in the description. Here they are:

      Whom do you trust? http://www.youtube.co...­

      A Shaky Story: http://www.youtube.co...­

      Both videos are only 1 minute long. But it will take you more than 1 minute to solve them!

      July 25, 2014

  • Lynn

    Much as I'd like to go, we lost our strongbox (!!!) and I'm stressing out about it. The thing needs to be found.

    July 24, 2014

    • Martin C.

      I don't know how much of the NetLogo videos that I will finish. My initial feeling is that the NetLogo is a nice environment for doing simulations, particularly for those who are not programmers. The Logo language itself is something of a dinosaur. The two main programming paradigms are object oriented and functional programming. NetLogo does not support either, which means that programs may require much more code than they might otherwise, especially if you want to be able to work with multiple types of agents. I also did not see any way of creating breakpoints in the code or for being able to step through it one statement at a time. This makes debugging a bit awkward.

      July 25, 2014

    • CJ F.

      Watching the Netlogo videos is optional. But the first video is an overview (11 minutes): https://www.youtube.co...­

      The second video is a very elementary tutorial (17 minutes): https://www.youtube.co...­

      The third is a more advanced tutorial (18 minutes): https://www.youtube.co...­

      We will mostly discuss the overview. But if you see how easy it is to program ants to eat food, the power and utility of NetLogo becomes clearer.

      Another approach would be to just watch the introduction and play with this ants model: http://s3.amazonaws.c...­

      July 25, 2014

  • Sam B.

    Michael Starbird includes 4 puzzles in his series to help practice effective thinking. The problem I will focus on today is a variation on an old problem that you may be familiar with. Three pirates and Three admirals are out walking and they come to the same side of the river with only one boat. The boat can only carry two people at a time(1 ferryman, 1 passenger) and if the pirates outnumber the admirals, they will kill them.
    Starbird points out in one of the videos that the objective of doing the puzzle is to gain experience. His videos walk the viewer through his process of problem solving. https://www.youtube.com/watch?v=X0pU_diL_sc&list=PLkCiNL_gZp2fvxUFaJ0H2WxfQMxyLflMI&index=10 The first video of the problem solving process. Can you solve the problem? To mix it up a bit, what if the admirals and pirates started on opposite sides of the river (P on the left, and A on the right, with the boat). What's the least amount of trips you can come up with?

    July 23, 2014

    • Martin C.

      Who is on the near shore and who is on the far shore? Which side is the boat on? If the boat is on the near shore then the problem cannot be solved if the pirates are on the far shore. If the pirates are on the near shore, then it takes two crossings. If the boat is on the far shore, we need to have the pirates also on the far shore. If one pirate crosses, we are at one the states of the solution to the original problem. As I mentioned elsewhere, if you want a more challenging river crossing problem with an interactive interface, see http://www.robmathiow...­

      July 23, 2014

    • Martin C.

      Sam, Maybe what you meant was to have the pirates and admirals on opposite banks and to reverse their positions. Notice that once again the final position is the mirror image of the initial position, allowing us to again make use of symmetry. The boat needs to initially be on the side with the pirates. One pirate rows across and two admirals return. We have two admirals and pirates on one side and one admiral and pirate on the other. Just as in the original problem, we get a new position that is the mirror image of the current one by having a pirate and admiral row across. We now simply reverse our previous two moves. Two admirals row across and then one pirate.

      July 25, 2014

  • Martin C.

    Regarding the coin weighing problem, suppose the number of coins was 12 instead of 9. Do you see a simple calculation you could make that shows that it would not be possible to find the counterfeit coin in two weighings? How can the problem be generalized? How many coins could be handled in 3 weighings? In n weighings for any whole number n?

    July 23, 2014

  • CJ F.

    For this month's "Math Chat" there are 26 short videos (2 and a half hours in all). Here is a quick overview:

    These five videos introduce Michael Starbird's vision for "Effective Thinking Through Mathematics" (are these heuristics or metacognition, both, or neither? Helpful?):
    http://www.youtube.com/playlist?list=PLkCiNL_gZp2cd5L3wk57TFFq8uWeJa5zs

    In these 13 videos Michael Starbird and his student Scott explore four puzzles to demonstrate effective thinking through Mathematics (please try to solve the puzzles before watching the videos that give the solutions: challenge yourself! None of them is too difficult, none of them is trivial.): http://www.youtube.com/playlist?list=PLkCiNL_gZp2fvxUFaJ0H2WxfQMxyLflMI

    In this video by Derek Muller, can you guess the rule before he explains it at the end? https://www.youtube.com/watch?v=vKA4w2O61Xo

    In these 7 videos Melanie Mitchell (and John Driscoll) on NetLogo: http://www.youtube.com/playlist?list=PLkCiNL_gZp2f9NB7Woaod9kPJjjq-sweK

    July 22, 2014

  • Martin C.

    CJ, I do not mean any offense, but I find these so called strategies too vague to be of much use. Here are some problem solving strategies which, while common sensical, can be effective if deliberately applied.

    1. Divide and conquer. Break up a larger problem into one or more smaller problems. This is something that all of us have done at one time or another.

    2. Work backwards. When I tutor, I refer to this as strategic thinking. If you want to prove two triangles congruent and you have two pairs of congruent angles then try to find a pair of congruent sides so you can apply ASA or SAA.

    3. Try to solve a special case. This may provide insight or lead directly to a solution. To determine the formula for the area of a triangle, first do it for a right triangle and then notice that every triangle is made of two right triangles.

    July 18, 2014

    • Martin C.

      CJ, What you are describing sounds like a motivational speaker - good at getting people fired up, but not helpful in actually showing how to do anything. You say that Starbird's methods make you a better math student. Could you give one example of how his method got you to be able to do something you otherwise could not have done?

      July 20, 2014

    • CJ F.

      Starbird gives me confidence that by going with the flow of my investigation of a problem, I can learn something even if it proves to be a muddled path that leads nowhere. Mainly he gives confidence.

      He shows that by increasing awareness of what I am trying to do, I can keep on the trail of working through the problem. These are the aspects of mathematics that always defeated me: losing confidence, getting muddled, trying to memorize the basic facts instead of reexploring the basics so that I could master them.

      Before heuristics can be helpful, one needs to have confidence and good meta-cognition. Starbird is, I think, trying to teach the meta-cognitive skills that defeat so many math students like me.

      See "Polya, Problem Solving, and Education" by Alan H. Schoenfeld http://www.jstor.org/...­

      July 20, 2014

  • Martin C.

    Being able to exploit the symmetry of a problem can help in the solution. The Pirates and Admirals problem has a symmetry. The final position is the mirror image of the initial one and the reverse of any move is also a legal move. That means that if you reverse all the moves of the solution, you get another solution, except in this case you get the same solution: The last position is the mirror image of the first and the next to last position is the mirror image of the second and so on. Therefore the two middle positions are mirror images. If you see the chance to get from a position to its mirror image, you know that you are done. Just do all the moves in reverse, with respect to opposite sides.

    For a more advanced and interactive river crossing problem see http://www.robmathiowetz.com/
    Note that this problem has an internal symmetry. Do you see it. With respect to the internal symmetry, the solution is symmetric, though getting to the symmetry point is a bit tricky.

    July 20, 2014

  • Martin C.

    Following from my previous post.

    4. Consider a generalization of the problem. Applying theorems in abstract algebra can give solutions to problems involving numbers as a special case of a group, ring or field.

    5. Look for symmetries. The river crossing problem strategy that I mentioned elsewhere is an example.

    6. Try to solve a simplified version of the problem. I solved the handshaking problem by considering the cases of 2 and then 3 couples.

    7. Try to apply recursion. You can create an example that satisfies the handshaking problem constraints by working in stages and recursively do what was done in the previous stages.

    July 18, 2014

  • Martin C.

    That nine coin problem should be familiar to anyone who has done any recreational math. Here is a fairly easy variation that gives some insight into the nature of this problem and a lot of related ones. Devise the weighings so that the coins chosen for the second weighing is independent of the results of the first weighing. Note that this means that the order of the weighings does not matter. There is a much more interesting classic recreational math problem involving 12 coins. I give the problem and a rather interesting approach to solving it that I read about, on my Web site at http://www.mathed.org/12coinproblem.html

    July 13, 2014

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