Tag Archives: Mathematics

So close . . . and yet so far.

One of my science students, PK, recommended a game to me called Doodle God, from JoyBits Ltd.  My student said, “since we are studying chemistry, which is about combining things, check out this game, where you try to combine things.”

I was excited to give it a try and have now been playing it off-and-on for about a month.

Here’s what I like

  • Easy to learn to play.
  • Very rewarding to see two things swirl and create something new.
  • Some combinations are very creative, not immediately obvious, but logical.

Here’s what I don’t like

  • From the start, the portrayal of a monotheistic, creative deity as a grandfatherly, tinkering, bumbling, bug-eyed, wild-haired, white-bearded, tongue-protruding simpleton, seems borderline blasphemous for at least three major religious traditions of the world.  But, hey, whatever sells…
  • When I let my 8-year-old play—ignoring the 13+ rating of the game—I should not have been too surprised when certain risqué themes emerged.  Alcohol, drugs, sex, [censored], rock-n-roll are all there, and were they all necessary?  But, hey, whatever sells…
  • From the beginning I was a little peeved when I tried to combine certain things—that made logical sense to me—but they didn’t combine.  Similarly, when I saw hints which led to things that did wondrously(?) and improbably combine, I almost put the game down.  (And how was I supposed to guess that?)

Seeing that the dislikes for me seem to outweigh the likes, why am I writing this blog post?  I feel this game is *so close* to being something of real and amazing educational value.  Imagine something like “Chemistry Zeus” [any similarities between deities living or dead is purely coincidental], where students start with a few elements and either bombard them to make new elements (nuclear physics) or combine them with other elements or molecules to make compounds, or whole families of substances.  I think I would play that game, and if it taught a little science or history of science along the way, cool!

JoyBits, if you need a scientific consultant, you can contact me.  Smile

Some math fooling around

You start with 4 elements and are asked to deduce pairings that create successively more complex elements.  Since the deduction part is flawed in my opinion, I believe most successful game-players resort to trial and error.  Let me explain.

If I give you N items and tell you some of them might pair, by trial and error you would take the 1st item and try to pair it with the 2nd, 3rd, 4th, etc. up to N.  When you are done with the 1st item you then take up the 2nd item, and try some pairings, but you don’t have to test it with the 1st item, since you already did that, so you test 2nd+3rd, and 2nd+4th, and 2nd+5th, etc.  one way to visualize this is with a grid.

  1 2 3
1 1+1 1+2 1+3
2 X 2+2 2+3
3 X X 3+3
Doodle God game with 3 elements (1-3)
Table shows all the combinations you need to check.
You do not need to check combinations marked “X”

The square with 1+1 means you are taking the 1st element in the game and trying to pair it with itself.  The 1+2 means that you are trying to pair the 1st element with the 2nd element.  Notice that the square that would be 2+1 in this example is marked with “X”.  That means you don’t need to test that combination because in the game 1+2 is the same as 2+1, the order you click on elements to pair them in the game doesn’t matter.  (If someday it did, the following analysis would be invalid.)

The formula for how many pairings you have to check for N total elements is

Total Pairs You Need to Check = N2-(1/2)(N-1)(N)
(thanks to Wolfram Alpha for helping me evaluate a sum)

We can verify that this formula is correct, by checking for N=3, plugging that into the formula and then counting in the table above to see if the results agree.  For N=3, from the table I would need to check 6 pairings to exhaust all possible combinations in the game.  The formula predicts

Total Pairs You Need to Check = N2-(1/2)(N-1)(N)

Total Pairs You Need to Check = 3*3-(1/2)(3-1)(3)=9-(1/2)(2)(3)=9-(1/2)(6)=9-3=6.

Now, the object of the game is to find successful pairings so let’s say 1+1 is successful.  But that would produce a 4th element.  That means we have to check more potential pairs.  (Note that some pairings produce two elements, that happens pretty rarely so the analysis so far and following is not invalidated.)

If successful, then you create a 4th element, and the table would now look like this:

  1 2 3 4
1 1+1      
2 N      
3 N N    
4 N N N  
Doodle God game with 3 elements
But the 1st element paired with 1st element produced a new 4th element.
The table of combinations thus adds a row and a column
 
Notice that although the 1st element paired with the 1st element made a 4th element, you haven’t tested any combinations of that 4th element yet.  You will have to test those, so we add a column to the table.
 
The number of total combinations we needed to check when we only had 3 elements was 6.  We tried one pairing of elements and we were successful so now we only need to check 5 plus the new pairings we potentially created.  It turns out that when you add 1 new element to an N-element game, you add N+1 more pairs to check.  In this case, i.e. N=3, we get 6-1+4=9.
 
Can we write a formula for how many pairs we still need to check on the 12th turn of the game?  Sure!  First let’s define a few things.
 
N = total number of elements in the game that you start with.
t = the turn of the game that you are on, in other words how many pairs you have tried already
s = successful matches already
u = unsuccessful matches already
Note that one relationship we can spot right away is
t = s + u
Which just says that the number of unsuccessful + successful matches you have made is equal to the number of turns you have been playing.  But the relationship we are after is “How many more matches do I need to test after t turns in the game?”  I believe this works, let’s try it out.
Potential Matches Left = (N+s)2-(1/2)(N+s-1)(N+s)-t
In the example above, N=3, t=1, s=1, u=0, the number of matches left to test, i.e. the number of blank squares in that grid is:
(3+1)2-(1/2)(3+1-1)(3+1)-1=16-(1/2)(3)(4)-1=16-(1/2)(12)-1=16-6-1=9.
Notice that the function goes like a quadratic in the total number of elements, which means the game gets progressively harder as it goes along.  Even if you don’t blindly try all combinations, you still have to remember which combinations you have made and the combinations you haven’t or review all those elements you have not yet combined for “reasonable suspicion” of being able to combine to form new element.  We say the order of that comparison is O(N2), O() means “order of”.
 
The tradeoff that becomes important in the game is that if every turn in the game produces a new element (s=t), then the number of new combinations increases quadratically.  But that is the reward of the game, producing a new element.  The frequency of reward needs to be traded-off with the rate of increase in complexity of the game.  You can make the game less complex (s << t) by not letting any elements combine, but then who would play it?
 

Back to the game Doodle Farm (Free)

 
Meanwhile…a game that would allow players to combine things in ways that are accurate given physical laws, e.g. chemistry, would be an amazing pedagogical tool.  None of the flavors of Doodle God to date seem to represent any even remotely accurate view of the physical world.
 
I played Doodle Farm (Free) and used a Google Sheet to keep track of my pairings, much like the table above.  I was able to solve the game fairly systematically that way.  But what was annoying (and a deal-breaker for me, sadly) is that two of the first 4 pairings were completely illogical.  Not that the game makes any pretense of teaching accurate animal husbandry, but the whole point of this post is that the game would be used by Teachers if it were more accurate.
 
image
Doodle Farm Free initial elements and successful pairings.
How does Mouse+Mouse=Rat+Cat?
How does Worm+Mouse=Ant?
 

Crazy Idea: Negative Numbers

I’ve been thinking a little bit about adding and subtracting negative numbers.  Here’s my idea.

You have 4, which everybody understands, and then you have –4, which I am going to call “anti-four”.  It is defined as the number that annihilates 4 so that

4 + (-4) = 0

In my scheme there is no such thing as subtraction, but only the addition of annihilators.

The subtraction of annihilators is merely the addition of an anti-anti-number.

4 – (-4) = 4 + (-1)*(-4)

Which means you have to define the anti- operator to be multiplication by (-1).

By reducing subtraction to the addition of the anti-operator times a number, you are helping to remove some of those rules like:  subtracting a negative number is adding a positive and adding a negative number is like subtracting a positive (which seems silly).

What this helps clear up is difficulty around adding and subtracting, since there is only adding.  And it helps give a name to anti-four such that the basis could some day be laid for anti-matter in physics which annihilates its counterpart when they interact.

This also helps clear up confusion that may occur in accounting where all is summing, just sometimes you sum debits (anti-income) and sometimes you sum credits and similarly you can take away credits and take away debits.

Just wanted to get this off my chest.

National Assessment of Educational Progress (NAEP) 2011 Results

Read up a little on finding of the latest NAEP.  Here some data on teacher degrees and test results.

The National Assessment of Educational Progress (NAEP) is a continuing and nationally representative measure of trends in academic achievement of U.S. elementary and secondary students in various subjects. For nearly four decades, NAEP assessments have been conducted periodically in reading, mathematics, science, writing, U.S. history, civics, geography, and other subjects. By collecting and reporting information on student performance at the national, state, and local levels, NAEP is an integral part of our nation’s evaluation of the condition and progress of education.  (from http://www.nagb.org/publications/frameworks/math-2011-framework.pdf)

Dig deeper into the Mathematics results

To investigate the relationship between students’ achievement and various contextual factors, NAEP collects information from teachers about their background, education, and training. One of the questions on the teacher questionnaires for grade 4 and grade 8 asked them to indicate the highest degree that they held. Explore this contextual factor by navigating through the slides below for the nation, states, and the districts participating in the Trial Urban District Assessment (TUDA).

Comments
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image One way to interpret is that impacts to scores due to Master’s degrees have been virtually flat for 6 years, while teachers with Bachelor’s degrees seem to be improving their scores.
image You need a Master’s to teach 4th grade anymore.
image You need a Master’s degree to teach 8th grade anymore.
image Washington has a high percentage of 4th Grade teachers with Master’s degrees.
image Washington has a high percentage of teachers in 8th grade with Master’s degree.

Conrad Wolfram: Teaching kids real math with computers (TED.com, TEDGlobal 2010)

Here’s the link.  Here’s the summary.  Of course, he would advocate using his company’s (Mathematica) product to teach mathematics.

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imageMain thesis.

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1. Technical jobs

2. Everyday living

3. Logical mind training, logical thinking

image

1. Posing the right questions

2. Real world –> math formulation

3. Computation

4. Math formulation –> real world, verification

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image “Should you force people to learn it?”

imageObjection #1.

imageObjection #2.

imageObjection #3.  Teach procedures but use computer programming.

imageThe opportunity, make math…

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imageStop testing by hand, get computers into examinations.

imageComputational knowledge economy.

imageSuggestion:  Completely renewed, changed re-written math curriculum, based on computers being there.

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Dan Meyer on TED.com

Dan Meyer is a math teacher in NYC.  Here’s a link to his TED talk.  This talk gets me fired up so I wanted to blog it so I can return to it next time I need some encouragement.

Quotable quotes:

This is an amazing time to be a math teacher.

Five signs that you are doing math reasoning wrong in your classroom:

  1. Lack of initiative
  2. Lack of perseverance
  3. Lack of retention
  4. Aversion to word problems
  5. Eagerness for formula

No problem worth solving is that simple…I am going to retire into a world that my students will run.

I believe in real life.  And ask yourself, what problem have you solved, ever that was worth solving, where you knew all of the given information in advance, or where you didn’t have a surplus of information and you had to filter it out, or where you didn’t have sufficient information and you had to go find some.

The math serves the conversation, not the conversation serving the math.

Einstein:  “The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.

So 90 percent of what I do with my five hours of prep time per week is to take fairly compelling elements of problems like this from my textbook and rebuild them in a way that supports math reasoning and patient problem solving.

I encourage math teachers:

  1. Use multimedia
  2. Encourage student intuition
  3. Ask the shortest question you can
  4. Let students build the problem
  5. Be less helpful

And from all this, I can only conclude that people, not just students, are really hungry for this.  Math makes sense of the world.  Math is the vocabulary for your own intuition.

Dan Meyer’s web site is http://mrmeyer.com

Key takeaway:  beware impatient problem solving…

OK, if you haven’t checked out Vi Hart, you really should.

[Thanks Elizabeth Alford for the tip!]

Elizabeth sent me the NPR article containing the below video.  Here’s how Robert Krulwich (NPR Science Blogger) introduces Vi Hart.

Vi Hart calls herself "a recreational mathemusician currently living on Long Island." She talks faster than a machine gun, loves math, and draws like a dream. Her newest video: "Doodling in Math Class: Snakes + Graphs" is eye-popping.

Vi Hart’s web site is here [permalink]

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