495 Final Project

This project was really fun for me. I started with the idea to recreate George Hart's tunnel cube puzzle, but with the added idea of having the different cards making patterns within the shape. Each step had a different challenge to it. First putting it together was the challenge and figuring out which cuts should connect to make the right solids. After I got the general shape I tried to make patterns on each of the faces. This was a bit frustrating because I was trying to move the least amount of cards since they were hard to move, but it wasn't at all obvious which cards would be best to move. As I was writing about the different symmetries I thought maybe it could become a puzzle in which the player is given the symmetries and is asked to recreate the solid. This could also just be interesting as an exercise to see the different combinations that could be made from the same clues.

These are the various observations I made about my card creations:

For every solid, each card has four sets of two cuts. No two cards can be connected at more than one place. Every corner cut is matched to a side cut.

My smallest shape made of 6 cards most closely resembles a Triangular Pyramid. This was the least amount of cards needed to make a shape that left no open cuts. Each face has one card from each suit. Four bigger triangles for each vertex and four smaller triangles in the middle of each face. There are four tunnels in which a small triangle matches with a big triangle. One face has all twos, but no face has all threes. The three threes are connected only with a vertex hole. Every vertex besides the one with all threes has two twos and one three. Each vertex and face has one of each suit. Each face besides the one with the three threes has two threes and a two.

The nine card solid most closely represents a Triangular Prism. I made this one on accident. I started trying to fit each cut together without realizing I had three cards left on the table. Six total faces, three square faces and two triangular faces on the ends. Six large triangular holes, three square holes, two small triangular holes. Each square face has two of the same number on one “diagonal” and two of the same suit on the other “diagonal.” Each triangular face has one of each suit and one of each number, these only came together once I figured out how to do the square face symmetries. On each vertex, two cards have the same number and one of those cards shares its suit with the other card. Looking at the top of each long edge, there's a diagonal of all one suit, and the other diagonal is all one number.

The shape that looks kind of like a cube has 12 cards. Possibly a cuboctohedron. Six faces, eight vertices. Six square holes, eight large triangular holes, no smaller triangular holes, which means there are 14 tunnels to the center. The top and bottom have all one color, one is red the other black. Every other face has both colors. Each face has two different suites and two different numbers, one number of each of the two suits. No number touches another card of its own number. Each of the eight vertices has a 9, 10 and Q. Four of the vertices have all the same suit and four have three different suits. There are six square symmetry axes, one from each face. There are six triangle axes from the vertices. I believe the two card symmetry axes connect the corners. So each number is in a two card symmetry with its number in the other color. Like the 10 of spades and the 10 of hearts. The exception is the 9s which match up with the 9 of the same color. This makes me wonder if somehow I could change the order of the 9s to make those two card symmetries alike with all of the ranks.

I also attempted to make an object with a pentagonal face, trying to use 15 cards. The pentagonal face did not work out though. The cards were too slippery to stay together. I also suspect the angle wasn't quite right to do more than four cards on one face.

Finally I made an object, trying to use 15 cards, but ended up using 12. It is sort of the reverse of the cube. It has 6 large square holes from the numbered corners coming together (like vertices). Then it has 8 small triangle holes from the sides coming together. I formed it just by following those two rules. Looking at a square hole it looks like a cube, but then looking at a triangle face it looks like a rhombus. I have no idea what kind of shape it actually is. I didn't attempt making patterns, but I'm sure it's probably possible. I think if I did clubs for example 4-7 on one square face, and on the opposite square face 4-7 in spades and 4-7 diamonds around the “circumference” through the middle, then hopefully make those middle squares have 4-7 on them. But then the triangles would have two suits and three different numbers around them. It wouldn't make as nice of a pattern for the triangle faces. I bet I could mimick the pattern from the cube, but with that one I used three cards of all four suits, and this shape has four numbers from three suits so I'm not exactly sure how it would work. Unfortunately I think the cards are a bit too worn out from trying to do a pentagonal face to take it apart and make a pattern right now.  

 

Overview

I've decided for my last blog post I will review MTH 495.  It's titled History of Math and I feel like we definitely covered that.  We started at the beginning to learn various milestones in math which seem so natural now.  We talked about the development of the number system and additions to this, from zero to irrational numbers and even infinity.  I found it very interesting at how deeply these concepts were debated at the time of their discovery/invention.  Something that stuck with me throughout the course was history repeating itself in the sense that every new idea had some adjustment time where it was debated then accepted.  It made me curious what parts of math we have yet to discover or invent and debate about.

The class also helped me discover the creative and artistic side of math.  Of course there was always graphs and some geometric drawings in math class, but we talked about new things I hadn't thought of as math.  Talking about tesselations was one of my favorite classes.  I also really enjoyed when we talked about topology because of the emphasis in visualizing how different shapes can mold into others.

I also had lots of fun with my individual project which led me on a journey trying to put together this puzzle by George Hart.  (You can see my whole project in my other post here.) Overall I really enjoyed this class as my final math class.  It gave a nice background to tie together all that we had learned over the past four years.

Fractals

Communicating Math

From IBM's tumblr

Do you know about Benoit B. Mandelbrot?  If you don't, you should!  Dr. Mandelbrot was born in Poland, but had citizenship in France and the US.  He worked for IBM for 35 years and dabbled in such vast areas as finance, aeronautics, physics, and of course math.  Oh yeah, and he invented a thing called fractal geometry.

I may have undersold it.  Fractal geometry is a big deal.  It can be applied to coastlines, bronchi in lungs, clusters of galaxies, and financial price increments, to name a few.  The basic idea of fractal geometry is that as you zoom in on a picture, it looks pretty much the same as when you were zoomed out.  Also, while zooming in, the curve doesn't get simpler.  For example, zooming in on a section of a parabola makes it eventually look like a line segment (Jordan Ellenberg actually discusses this in his book "How Not to be Wrong: The Power of Mathematical Thinking" when introducing the logic behind calculus).  But when you zoom in on a section of a fractal it continues to be as detailed as it was before, if not more so.  Benoit referred to this detail as "roughness" which he would calculate.  He discusses an instant in this TED talk of a colleague giving him a picture of a curve and asking him to estimate the roughness.  His estimate was off by .02.  This idea of roughness is interesting though because it can describe so many things from mountains to cauliflower.

Anyway, fractals have a little wiggle room in their definition.  How else could these both be fractals?

Sierpinski Carpet

Mandelbrot Set

Both of these have some sort of self-similarity--that idea of the picture being about the same at different "zooms."  Plus, fractals have "fine or detailed structure at arbitrarily small scales. irregularity locally and globally that is not easily described in
traditional Euclidean geometric language, and simple and 'perhaps recursive' definitions" (https://en.wikipedia.org/wiki/Fractal).

The Mandelbrot set is a bit "complex" ;) but can be written as a simple recursive rule.  The Mandelbrot set is the set of complex numbers c for which the sequence (c, c² + c, (c²+c)² + c, ((c²+c)²+c)² +c, (((c²+c)²+c)²+c)² + c, …) does not approach infinity.  It's defined recursively as .  Some numbers like c=1 do not belong to the Mandebrot set because the rule would go to infinity.  However, c= -1 is in the set because that sequence oscillates between 0 and -1 so it does not approach infinity. (https://en.wikipedia.org/wiki/Mandelbrot_set)

Anyway, fractals are fun to look at and play with (check out John Golden's Geogebra file) and Mandelbrot was a very interesting man.  He was highly revered for this discovery that is so applicable to nature.  Plus he has some great quotes:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
-The Fractal Geometry of Nature

Bottomless wonders spring from simple rules, which are repeated without end.

-TED talk "Fractals and the art of roughness"

Euler's Square

Nature of Mathematics

This week we learned about the incredible Euler.  One of the activities we could do was to make an Euler Square of 16 playing cards. Elaine Young describes it in a MAA post, "In 1694, Jacques Ozanam posed the problem of arranging 16 playing cards in a 4 x 4 array such that no row or column contained more than one card of each suit and each rank. The solution forms an Euler square."  It's an Euler square because it combines two different Latin rectangles.  A Latin rectangle is like Sudoku, no number (or rank and suit in this case) is in the same row or column.  

Below are my two solutions to Euler's Square.  The first one is interesting because it has symmetry along the diagonals with the ranks of the cards.  It was really hard to get started on this one.  I started off by trying to make diagonals with one kind of number.  That didn't really work except for the Aces on the big diagonal.  From there I just tried to randomly move around cards to get something started.  The breakthrough was when I not only had the Aces on the diagonal, but also the Jacks on the other diagonal.  The second picture shows that each row, column, diagonal, and 2x2 square satisfies the requirements of no repeating rank or suit--including the middle 2x2 square.  I struggled with this one a bit too at first.  I knew I had to change up the diagonals so I attempted that first.  Then I tried to place each group of numbers as in a sudoku, with one number in each row or column and just filled in the gaps with the other numbers.  This worked a little, but then I had to modify and switch some based on suit.  I found that moving one usually results in changing three more.  But looking at it like a sudoku and filling in the blanks was helpful.

What I think is more interesting than just forming these Euler squares is thinking about if doing this is really math.  On one hand it's hardly more than arranging the cards in a puzzle type of way.  I certainly wasn't using any mathematical formula to figure out the arrangement.  However, there are many patterns to be seen in these squares and I believe finding patterns is a large part of mathematical principles.  For instance, if these squares had missing cards, you should be able to complete them based on the patterns within them.  Incidentally my Honors thesis is on rook polynomials which have similar properties to Latin rectangles, so I know there are ways to calculate the number of combinations of the cards.  There is a whole section of math dealing with combinatorics, which is counting combinations of elements.  So these squares can use math to describe and generalize them.  

Next question, if I think these are a part of math, was I doing math while I was making them?  This is a tough question.  I admitted I didn't use any of those formulas that count how many arrangements can be made.  But perhaps recognizing the patterns and following those pattern rules to fill out my square was doing math.  

It's hard to nail down exactly what is math and what is considered "doing math."  Instead of determining if this exercise was math, one could consider if this exercise is math, then what is math in general.  In that case I would definitely consider strategic pattern finding as a major part of math, in addition to the ability to numerically find conclusions that would be hard to find without math.  

Since I've decided to consider this math, what similar things could possibly be considered math?  Sudoku?  Jigsaw puzzles?  What examples can you think of?

Review: The Math Book

For class we each read a book of our choosing.  I chose the Math Book by Clifford Pickover because it looked like an interesting quick read.  The front states "250 Milestones in the History of Mathematics" and that's exactly what it is.  It's written in chronological order all the way from circa 150 Million BCE to 2007, two years before it was published.  Every page is a different milestone with an accompanying picture.  

Not only does it discuss the famous mathematicians that we've all heard of like Pythagoras, Euler, and Fibonacci, but it also discusses people, discoveries, and inventions that I've never talked about in math class.  My favorite pages were ones that talked about different puzzles and games like Rope around the Earth Puzzle (1702), and the board game Go (548 BCE) .  So although not every page is a gigantic breakthrough that everyone has heard about, they're all important in some way, even just for entertainment.  Most of the pages discuss some history which I found really interesting because not often do we put these discoveries in context of the times.   One that I found interesting that ties in the puzzles and also history was the entry on Hex from 1942.  It discussed the game of hex and how to play and then also tells about how it was manufactured by Parker Brothers.  It's inventor, Piet Hein had to go into hiding in 1940 because of WWII.  It really makes you think about what could have been discovered and invented by some people if their society had allowed them to keep doing math.

Overall I really enjoyed the book.  It really shows how vast math is and showed how interesting it could be.  It's written at a level that is easy to read and understand.  I believe the general public would enjoy this book, but it might help if that have a slight interest in math to start with.  No deep knowledge of math is needed to read and understand this book.  For the first half of the book I read straight through in chronological order, but the second half I skipped around a bit, so if the reader decides to read in chronological order or not, either way is interesting.  My only complaint is that sometimes I wish the pages were a little more in depth instead of just introducing the topic and then moving on to the next one.  Perhaps if I had had more time to get through this book I could have looked up the ones I was really interested in online instead of immediately moving to the next one.  I definitely liked this book though and would recommend it to anyone with the slightest inclination towards math and its history.  It was an easy read and entertaining.

The Tale of the Cubic

Once upon a time there was no uniform way to solve all cubic functions--how sad.  It was a problem that puzzled mathematicians up until 1535.  What happened in 1535 you ask?  Well there was a math competition in Italy that a guy named Niccolo Fontana attended.  You might know him by the nickname Tartaglia which means "the stutterer."  Or you might not.  Anyway this guy was an engineer and amateur mathematician--because who doesn't want to do math in their free time?  So he came to this competition and SURPRISE he won it by solving a cubic function using his general formula!  Everyone was shocked because they had thought it to be impossible.  

Since it was such a coveted formula, Tartaglia wanted to keep it to himself, even hiding it by encoding it in a poem.  Here's a picture of the formula from this Vanderbilt website. Can you imagine fitting that whole thing secretly into a poem? 

So Tartaglia wanted all the glory for himself and kept it a secret until the smooth talking Gerolamo Cardano came along and got the formula from Tartaglia which he subsequently published in his book "Ars Magna" in 1545.  Needless to say Tartaglia was not pleased that Cardano broke his promise of keeping it a secret, but got his revenge by helping Cardano get arrested for heresy after Cardano made a horoscope for Jesus.  

When we talked about the cubic formula in class I attempted to use the formula to solve the following cubic.  The (-3,0), (2,0), and (5,0) are the solutions I found from graphing.  My incomplete attempt gave me imaginary numbers after taking the first square root.  I may have messed up a negative somewhere...  But take a look and then take a moment to appreciate graphing calculators, the factor theorem, and long division of polynomials that can find the solutions a whole lot faster!

It makes me wonder what kind of applications they had that needed the solution of a cubic function.  As I thought back on my schooling to try to remember an application we learned that used cubics I couldn't think of a single one.  I think we were always just given a graph and/or the equation and had to find the solutions from there without giving an adequate reason why we should even solve it.  So I went to google and found this page where several people came up with applications of cubics.  Most of them seem to be far more advanced than what we would learn about when first seeing cubic formulas.  I thought the most interesting one was the claim that the typeset letters are formed from cubic functions.  So thank you cubics for helping me type this blog!  Do you know of any other applications of cubic functions?  Any that people would have needed in 1535?

Hey look!  If you want more information about the characters in this tale click here for a more in depth story courtesy of Luke Mastin! 

Qiandu, Yangma, and the Bienao

We talked about ancient Chinese mathematics in class the other day.  Among those we discussed was Liu Hui who lived in the third century and made numerous contributions to mathematics.  Our main topic of conversation was the cube he made from three different parts that he called the qiandu, yangma and bienao.  The qiandu is half of the volume of a cube, yangma is 1/3 and bienao is 1/6, so together they make up the whole volume of a cube.  So with the finished product in front of us (but without tracing or measuring from them!), we were told to make the nets of the shapes.  A net of a three dimensional object is a flat drawing of all the faces of the shape correctly positioned so that when you cut it out you can fold it into the 3D shape.  Before I started the nets I drew the above sketch to get an idea of how they would fit together.  I decided to make my final cube 2 inches on each side which made the diagonal of each face the square root of 8 (approximately 2.8) and the diagonal of the cube the square root of 12 (approximately 3.4).






After analyzing which sides would need to fold together I made my nets.  Here's a rough sketch of each shape with the measurements.  Here's a link to Jennifer's ready made nets (with a better picture).








So with my sketches, a ruler, scissors, tape and coral paper I made my nets and then folded them up to make my three shapes!  Not the prettiest, but they still make a cube!

Left to Right: Bienao (1/6), yangma (1/3), and qiandu (1/2).

As I played around with them I decided I didn't want to use the qiandu anymore, but I wanted to make two new shapes.  I decided I didn't want the yangma and bienao to form another qiandu like in the original, so I positioned them kind of diagonally (pictured).  

Once I got started on the nets for the two new pieces, I realized they were the exact same as the yangma and bienao.  I still made them though and played around with the ways that I could put them together.  This was interesting to me because I realized I couldn't put two complete faces of the same shape together.  In other words I had to make two qiandus out of a yangma and bienao.  I couldn't make any other formation from them that would still form a complete cube.  

Overall I think it's interesting that all the way back in the third century Liu Hui was thinking about volume and how to divide it in such a way that gave a unique combination of the shapes.  (You can check out a proof and more pictures here from Sherlock Holmes in Babylon: And Other Tales of Mathematical History.)  Plus, I very much appreciate the kind of puzzle approach to putting it together.  It makes me want to make more cubes made up of more complicated pieces, although I'm not sure if I would be able to manage making shapes that have such nice divisions of the volume like he did!

Archimedes' Stomachion

Doing Math:

I love puzzles.  Love love love them.  I will spend an entire day putting together a giant jigsaw puzzles and be perfectly content.  So when I saw that one of our assignment options was trying to put together Archimedes' Stomachion I was very excited!  The Stomachion is basically a little 14 piece puzzle that you try to fit into a square.  I did this puzzle on Geogebra which you can check out here:  Stomachion on Geogebra.  Make sure your window is zoomed to show all the pieces and then you can rotate them and fit them in the square!

When I started to experiment with fitting the pieces in, I tried to be methodical and fit in the biggest pieces first.  With the big pieces in first, I could then see which smaller pieces could fit in between them.  One of the main goals was to find out which sides of the different pieces would fit together perfectly since this would make the most of the space.  At times I found it slightly difficult to spin and move the pieces exactly how I wanted them which made me wish I had physical pieces in front of me.

 This was one of my first attempts.  I tried to make bigger shapes out of the smaller pieces, like the yellow blue and green pieces at the top formed a triangle. This seemed like progress because it matched those ugly edges of the yellow piece up with others to make a shape that was pretty much the same as the triangle made from the long purple and green pieces.  Of course the two lighter blue triangles wouldn't fit, so I knew I wasn't maximizing the space as well as I needed to.

Next decent attempt.  Again I noticed certain pieces would make bigger shapes, like the rectangle in the bottom right which I kept from the first attempt.  Also the square in the top right corner.  However, I abandoned my first thoughts of putting in the biggest pieces first, which resulted in overlap from the red piece and vacant space around the small purple triangle.

This was my best and last attempt.  If there was a way to flip over that last purple triangle, it would have fit.  The right half of the square seems great.  The fact that it forms a rectangle and could be cut into multiple different triangles makes me think that it is correct.  However the left half doesn't have those same intracacies.  Unfortunately, I could not find another way to arrange it to make it fit.


Overall this was a very fun puzzle, despite being a little frustrating at times.  I always enjoy when math and puzzles connect.  Interestingly enough, after I worked on this puzzle (for a long time) I was browsing through the new book I got for class (The Math Book by Clifford Pickover) and it had a page talking about Archimedes and the Stomachion.  It even showed this solution, although it's not possible to make it this one in the Geogebra file without being able to reverse some pieces.  I like how every piece is a part of at least two bigger triangles. Plus there is a line splitting the square into two rectangles and another line on the diagonal forming two triangles.  I think it's so interesting how these strange pieces make up bigger shapes within the square.  So although I didn't find one of the "17,152 solutions" it was very fun, and interesting to examine the details of a final solution.

What is Math?

Yesterday we had a brief discussion in class about what math is.  For a bunch of math majors we came up with an explanation centered on the notion that it is a quantifiable way to explain physical phenomenon but also includes ways to predict imaginary situations.  The main thing I took away from this discussion was that our definition is probably very different from most of the general population.  I believe most people would consider math to be pure computations, glossing over its ability to apply to a vast amount of situations.  So when the follow up question "what was the first math" was asked I thought of a different question.. I started to wonder what the first people who studied math thought it was.  When did they realize this was a new subject worth studying?  I imagine early civilizations didn't think of bartering at a market and traveling long distances in terms of math like we might, so it's hard to say what they thought the first math was.

Whatever the first math was, there are points in the history of math that stand out as adding great benefit to the current study of math or even the current state of the world.  I'm personally intrigued by the mathematics that went into planning overseas journeys of discovery.  I can't fathom figuring out how many supplies it would take to last until the ship docked again, in addition to how much weight a ship would carry if it was to return with goods.  Also how to calculate times and distances of these journeys.  Without all of that careful planning that no doubt took a lot of mathematical calculations we might not be in America right now. :)

In terms of the study of mathematics, Euclid made an enormous contribution by defining the components of geometry.  Not only did his definitions give us the foundation to discuss geometry, but he gave us a way to prove concepts so every mathematician could easily communicate.

I also think the invention of a definite monetary system was a significant development.  Although it doesn't really add the to the academic side of math, it is probably the most widely used application of math and occurs every second of the day. The idea that math is only for people who actively study it is discounting it a great deal.  Even though exchanging money seems pretty basic it provides ways to talk about fractions, decimals, operations, even exponents for interest rates.  And because money plays a crucial role in people's lives I think it's an important point when talking about math.

History is so rich that I find it very interesting to pull out the details concerning mathematics.  Then analyzing it to see how those experiences tied into the minds and cultures of the past to influence the concept of what math was at that time intrigues me greatly.  Knowing that math has changed a lot since its conception makes me wonder where it could go from here!

Algebra vs. Guess-and-check

In  class, we've started talking about "algebrafying" problems which is something I really enjoy, and lots of others hate.  It's the process of taking a problem and expressing it in variables to find exact answers by solving for the variables.  Coincidentally I had an experience with a student who I interviewed in which I thought he would use algebra to solve a problem, but he was more comfortable guessing and checking.  You can see his work in the picture below.

In both of these problems he tried to solve them by guessing dimensions and then checking whether they matched the picture (like 4 would visually be a smaller side than 5).  Unfortunately he erased the dimensions that were wrong, so all that's left is the answer that he thought was correct.  But I know from sitting with him as he worked through the problems his method was to guess dimensions, like he guessed 10x18 for the bottom picture, and then to check whether it was correct.  So for the bottom one he determined 10x18 was wrong because he couldn't divide 18 (along the vertical side) into four equal integers, which represented the long side for the small rectangles.  

I think this is interesting because I have learned that the most efficient way to solve a problem like this is to use algebra.  So I would label the long side of each small rectangle as x and each shorter side as y and have equations like (x+y)(4x)=180 and 5y=4x, then solve for x and y to get x=5, y=4.  This eliminates the possibility of having multiple guesses and having only my eyes to approximate the distances drawn.  So while the student was looking at the lengths to see if his answers could make sense, his guesses would've been completely thrown off if the distances weren't drawn to scale.  It could have dramatically changed his answer, whereas my answer using algebra would've remained the same and I would still be confident in my answer once I checked whether my math was correct when solving for the variables.  

I also think this was eye-opening because he's in Algebra 1 and he has done problems out of the book for months now that use variables and solving systems and creating systems based on story problems.  However, he was unable to apply that knowledge to these interview problems.  So has he really learned how to use variables?  I would lean towards no, he has not mastered the concept.  He can do problems that solve for variables, but he cannot apply that to new problems which have no suggestion for how to solve them.  However, to be fair it could be that he preferred to guess and check just because that's more comfortable for him and perhaps since I did not tell him how to solve it he didn't even bother with thinking about variables.  Either way I think this is a great set of problems to check students understanding of variables or to even introduce the topic of variables.  I knew that it wasn't fun for him to try all those different dimensions so it could be a great way to force the topic of algebra so that he could solve this problem once using variables and be done.

Origami and Math

 This blogpost is inspired by the Ted talk by Robert Lang called The Math and Magic of Origami.

Origami is the ancient art form of folding a single sheet of paper to create objects.  You can make animals, hearts, boxes, the list never ends. In fact, the amount of objects that could be made with origami exploded once people incorporated math.  So why not incorporate origami in the math class?
Not only would students learn about an ancient culture, but it will teach them some properties about geometry and it will be super engaging because students will be able to make their own creation.  Also, origami is great because if you mess up you just unfold it and try again.  And students could probably choose their own figure to make, so it would allow for different levels of skill.  In fact I could see making a project that would include a part about learning about the culture surrounding origami, then making an origami figure and explaining the math behind their figure.  For example, Robert Lang talks about numbering the angles around the center circle and seeing that after folding, the odd numbered angles would add to be a line and the even numbered angles would add to be a line.  I think this is super interesting and it would be interesting to ask students to find angle measures based on how many folds there are.

You could also have the students examine the crease patterns (the flat unfolded page that was once folded) and ask if there are any congruent shapes and why.  Are there any similar shapes?  You could also have them take a ruler to the paper and ask them area of different shapes in the crease pattern as well as perimeter.  I think there are a lot of skills that could be practiced by doing this origami.

Robert Lang also discusses applications of origami in the real world.  Some examples he discusses are heart stints and telescopes.  I think it would be a great idea to have students work in pairs and present on a real application of origami that uses math properties that we had talked about in class.

I think this would be a really fun and engaging way to talk about shapes and geometry while learning a skill that students can be proud of.  It also provides a way to learn about another culture which will help students become better citizens of the world.  With the added component of researching a real life application of origami, we are also extending what we have learned and seeing how these things really matter, which is something a lot of math classes don't address.  Overall, I think it's a very interesting application of math and creates such beautiful objects.

Happy folding!

Decimal Pickle

In our last class we played a game called Decimal Pickle.  All you need is a deck of cards with 10, Q, and K removed, and a pencil and paper!  It's very simple to set up-- draw a path of ten steps, you could use circles, arrows (like I did), or any shape you want.  The goal of the game is to create sort of a number line where you can place decimals in order on the path from 0 to 1.  The cards you flip over on your turn are the numbers you get to arrange to form a decimal to place on the path.  Red cards mean choose another card, up to three cards.  Black card means stop.  So for example I flip over a red Jack which represents 0, and a black 8 so I stop.  Now I have two possibilities for decimals: .08 or .80.  Since there are no blank arrows between .032 and .11 I cannot place .08 on my path.  However, .80 is greater than .789 and less than .938, therefore I can place .80 on my last blank arrow, and I win!  

A couple things to keep in mind, if you draw a decimal that is a repeat, say I draw a black 5 again, I cannot fill in two spaces with the same number.  Also if there's no space for the decimal I get I must pass on that turn, for instance if I draw a red 1, and a black 2, the only possibilities are .12 or .21 which would not fit between any existing arrows, so I pass.  

I think this is a great game to develop an understanding of decimal quantity.  I think it especially emphasizes quantity in terms of the decimal places.  For instance, it helps students learn that there's a big difference between .37 and .037 because each new decimal will have a context--it will have numbers less than and greater than the number.  I also think this could be a nice intuitive introduction to start learning inequalities and their symbols, or possibly review them if they had learned about them a bit in elementary.  

There is also a benefit in being a pair game because students can make mistakes in placements and the whole class won't notice, only their partner might.  So there's an aspect of not only knowing your game, but also checking to make sure your partner is playing correctly.  It provides lots of examples and nonexamples of appropriate placement of decimals and can involve all levels of understanding.  

For us college students it really helped try to break our habit of saying "point zero six seven" which really has no mathematical meaning and instead practice saying "sixty seven thousandths" which can help set the context of decimals as fractions of tenths, hundredths, or thousandths.  Pronouncing decimals as fractions can help students with the fluidity of switching between the different representations.  If a student was given the fraction 7/10 they would have a much easier time creating the decimal .7 if they were used to hearing .7 as seven tenths instead of point seven.  

There are some variations that can make the game more fun.  My partner and I decided to increase our paths to 15 spots so it would not only take longer to fill in, but it would also make it more difficult to know how to space the decimals.  Instead of having the strategy of putting .5 in the middle, you now no longer have an exact middle.  So in your first couple turns when you can place your decimals in any of the blank spaces, you have to think harder about what numbers could eventually go in the spaces you leave blank.

I love strategic games, and I think this is the perfect combination of a simple game concept that can really aid learning while still having enough strategy and variations to make it fun and exciting for all types of learners.

Levels of Understanding: Relational Vs. Instrumental

My post today is a reflection regarding the article called Relational Understanding and Instrumental Understanding by Richard Skemp.  First, a quick summary.  In this article Skemp discusses the two types of understanding, one could even call them levels of understanding.  Relational understanding he defines as "knowing what to do and why," with an emphasis on the why aspect.  Alternatively, Instrumental understanding is knowing a rule and when to use it, but not why.  He calls it "rules without reason."

In the article he goes on to share a few different analogies.  One was quite helpful in which he drew a comparison to music.  He said instrumental understanding was like knowing the notes and the staffs on paper, but not having those figures attached to sounds.  Relational understanding was like learning the notes in conjunction with the sounds.  Consequently when asked to write a melody the students taught relationally, would have a far easier time writing a melody.  I fully agreed with this analogy; of course reinforcing the meaning of notes with auditory information would increase the fluency and skill of students writing music.  Music class would in intensely boring if nobody heard any music.

The analogy that I didn't agree with was when he compared understanding to two different ways of navigating a new city.  Instrumental understanding was compared to just getting directions to and from essential places, which can lead you to be incredibly lost when you make a wrong turn.  Relational understanding was like exploring the city on your own and creating a mental map so you can recognize multiple ways of getting from A to B.  While I appreciate his analogy on some level, I still think it's problematic.  His description of exploring the city to find your own routes relies on doing so without help from an expert.  It's purely self discovery.  However, I believe he wants teachers to be teaching the relationships and alternate routes to get the answer.  I'm not sure that I agree with the explicit teaching of these relationships though.  I think it's sort of intriguing when I don't learn every detail about a rule and then I find it out on my own when I continue learning about the subject.  It s almost like an incentive to learn more so you can get the satisfaction of learning why a rule works by yourself.  However, I am extremely intrinsically motivated, so I understand why not all students would have the motivation to look for connections by themselves.  But I also think it's a good life lesson to not spoon feed every connection to students.  They need to be able to recognize connections on their own.

As teachers perhaps we could set up questions that will inevitably lead to discovering connections, but we should leave it for them to fully explore.  For example when discussing slope, many kids mix up the slope formula and divide the difference of x coordinates by the difference of y coordinates (like x/y), instead of the correct y/x.  To aid the relational understanding of this topic, I could ask my students to compare slope formula to the well known trick "rise over run."  Hopefully they would make the connection that the difference in y coordinates is the numerator because it corresponds to the rise.  Similarly the x corresponds to the denominator because it is the run between the two x values.  So this quick examination of the slope formula and what it corresponds to on a graph is a simple way to build a connection that will help students remember the slope formula.  But I think it's important to not answer the question for the students, but rather pose the question and leave them to find the relationship.  I think the self discovery will create independence and hopefully build some intrinsic motivation to find the relationships in upcoming lessons.

Another thing that bothers me about Skemp's outright preference for relational understanding is it's inefficiency.  Even if there are a million ways to do something, I think there is something to be said about knowing how to do it with ease.  I think math can be explained in so many ways, but since it values efficiency, we should teach with a preference to the efficient way to answer questions.  If nobody valued efficiency, no formulas would be created.  Formulas are mountains of work that are consolidated so that we can use them quickly and accurately.  I think that idea is discounted in the relational understanding model, but it shouldn't be.  Our students don't need to reinvent the wheel, just generally learn the concept of it so it can help them later on.  

Overall I'm not completely convinced that explicitly teaching relational understanding is the best method.  I think instrumental instructions has its merits as well.  Relational understanding could be beneficial when students are completely struggling with a concept, or as an extra independent enrichment, but I don't think teaching instrumental tricks to help kids remember certain concepts is terrible.  I think there's a time and place for both levels of understanding.

Fractions

To introduce the dreaded concept of fractions, our class decided to do exercises that would push the subject and necessitate the learning of fractions.  One such problem that isn't easily explained without the use of fractions was the Mixture Blues problems, some of which can be found in the image above.  Basically, in each container there is either blue dye or water and you have to find out which combination is more blue, A or B.
This problem sparked quite the discussion about how to go about solving these problems.  Of course, as college students we knew how to turn each set into fractions with common denominators, but when trying to reason through it without using fractions, we all struggled a bit.
Some of the methods we attempted was comparing how blue they were in terms of how many they would need to be all blue.  So in problem B1, A would need 1 more blue and B would need 3 more blue, so A is more blue.  We also attempted to "cancel out" similar containers in each set.  So for B4, two blue of each set would cancel out as well as two clear, so the top would be left with 1 blue and the bottom would be left with nothing.  So we argued that set A with the one blue would be more blue than set B which has nothing.  This method, although seeming intuitive to some of us, caused lots of discussion amongst the class.  Especially in the following problem,

We canceled out three blues and two clears from each set, but then were left with nothing in A and a blue and a clear in B.  From our logic, B would probably be more blue since a blue and a clear could mix to be a half blue which is more than set A which has nothing.  However, when you compute the fractions, A is actually .03 more blue.  Our intuitive method of canceling-out failed.  But why?
Let's take problem B5 and see why canceling out doesn't fit with fractions.  A= 3/5, B=4/7.  Cancel out two blues so now A=1/3, B=2/5.  If we just stop here we can already see that we've violated the rules of fractions since 3/5 does not equal 1/3 and 4/7 does not equal 2/5!  Really, the only way we can solve this problem is by introducing fractions and being able to compare them with common denominators or by turning them into decimals.  So once we reveal problems that cannot be solved without fractions, hopefully students will be more motivated to understand fractions.  Without the context of fractions representing parts of wholes, it is difficult for students to comprehend the use of fractions.  And without the frustration of failing to consistently solve a real world problem, students will have little motivation to learn the sometimes baffling, but important, concept of fractions.

As I was scrolling through twitter the other day, I noticed this image with the caption "Public Education."  As a future public educator, this really disappointed me because this is not how I view myself.  Yes, it is common to resort to teaching the way you were taught and to teach the way you think about things, but I believe classrooms should foster creativity and innovative thinking.  This can be especially hard in math because everything depends on logical reasoning which tends to be pretty straightforward.  However, when there are opportunities to find creative solutions to certain problems, teachers should embrace it.  This creativity should also apply to finding new teaching methods to solve problems that will help students' understanding.

Recently in MTH 329 we have been discussing using number lines to add and subtract.  This is a method that I had never used before because lining up numbers vertically always worked for me.  As we placed the numbered post-its on the ground and started doing some calculations by walking back and forth on our number line, I really saw the advantages of using this method.  Of course, I prefer the vertical method for it's speed, but using a number line really enforced the basic concepts of adding and subtracting, especially when negative numbers are thrown into the equation.  It was also great for the classroom involvement it provided.  Students won't be falling asleep if they're moving around the classroom, acting out math problems.  

Not only was it fun in my classroom, but it really helped a boy that I tutor.  He was having trouble figuring out problems such as -4-6.  I asked him if he had ever used a number line to help him solve these statements and he said none of his teachers have ever used one before.  So I taught him how by drawing the number line on a page, labeling a few points, and showing him where to start and when to turn around and "walk" the other direction.  I had him start at -4 facing the positive side, then since there was a subtraction (or negative sign) we turned the little arrow I had drawn around to face the negative side.  Finally he walked 6 steps forward and landed at -10.  He really seemed to like that he could see the numbers in front of him instead of just imagining their quantities in his head.  As he worked through his homework he used his number line repeatedly and had great success with it.  A week later, he even said he drew a number line on his test and it really helped!  So although there's an inclination to teach how you've been taught, and to keep your thinking in a little box, it can really pay off to expand your own mind with creative new techniques that could greatly improve student learning.