Gambling with Woot

MysteryBox-01

A few weeks ago, Woot.com advertised a “Mystery Box of Electronics.” One box cost $50. Woot said: “Not knowing is half the fun!” Each box would include:

  • Random consumer electronics items
  • You get 3 items per order in a brown box (nothing special about the box!)
  • Condition will be refurbished on most items. Some could be new

So, I decided, why not? And I gambled on Woot – I bought three mystery boxes of electronics! (Because not knowing really is kind of fun…)


I received my last box today! Here is what I received:

2015-09-17 13.34.58

Box 1:

I received three items:

  • A very pink “beat mixr” headset. Other listings on Amazon.com show prices for new/used (for some reason some new are cheaper than the used!), starting as low as: $110.
  • A small Acesori PowerStick. Other listings on Amazon.com show prices for new/used, starting as low as: $4.
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as:  $7.

2015-09-17 13.31.56

Box 2:

In the box, I found:

  • ifocus “Deluxe edition” educational system for kids (a set of CDs: one is an educational game and the other is a fitness program). The company that sells these says they are $199.95, but it doesn’t look like I could resell them
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as: $7.
  • Acesori Bluetooth Noise-canceling Neckband Headset with Built-in Microphone. Other listings on Amazon.com show prices for new/used, starting as low as: $30.
  • LG Electronics Gruve Bluetooth Stereo Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $30.

WP_20150922_17_20_56_Pro

Box 3:

  • BlueAnt Pump – Wireless HD Sportbuds – Black. Other listings on Amazon.com show prices for new/used, starting as low as: $26.
  • Turtle Beach Ear Force Z11 Amplified Gaming Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $24.
  • TruGuard Tempered Glass Screen Protector for iPhone 5/iPhone 5S & iPhone 5C. Other listings on Amazon.com show prices for new/used, starting as low as: $6.

So the big mathematics question is: Did I get my money’s worth?

In the Woot discussion (http://electronics.woot.com/forums/viewpost.aspx?postid=6481553&pageindex=60), many customers say, “No!” Others say, “Yes!” I’ll give several comments here, and let you decide!

  • The value is there for at least what everyone paid for when they ordered it. The problem is people discount the value simply because it’s something they don’t want or can’t use, or they deem to be useless. That’s on them, not Woot.
  • This sale rather was marked as getting mystery items that were either new or refurbished worth at least $200. So for $50 bucks, if thats the case, how could we go wrong. We should all consider that that this $200 value should not by any means be gauged on our perception of value, but rather fair market value, which is really the point of taking up a mystery box. It may be worth it to you, but it may not. If its not, you can go through the legwork of selling it off and making back your money.
  • (received my box 3 above) I am not satisfied because I don’t feel this box is worth the 50 bucks I paid. But still I will get use out of some of it. I understand that with these boxes you run the risk of getting items you don’t want or won’t use but you expect the value to still be there.
  • As to “value”, that is in the eye of the beholder. The great marketing dilemma. Do we want to talk about MSRP, street price at release, current street price, or the best deal available? Naturally, very few items ever sell at MSRP, but it is still used everywhere to convince everyone how much money they are saving at the current street price.
  • Overall – the value of the boxes I received is total crap – at least for me.
    IMHO no quality items. Protectors for outdated devices. MSRP that are years old and cannot really count against today’s value. Overall inflated MSRP assumptions of products that cost cents to make (like cases and protectors)… no value there.

So…. what do you think? Was it a good decision or bad?

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Paying a tithe (10% of earnings)

In Sunday School, I teach kids who are turning 7 this year. There is a lesson manual and a schedule to follow, but I like bringing in authentic tasks (especially when it is math-related!) whenever possible.

Today the lesson was “I can pay tithing.” We talked about the meaning of tithe (Merriam-Webster says: “to pay or give a tenth part of especially for the support of the church”).

I brought a sack of pennies (about one-year’s accumulation!) and gave each of the kids a pile. I asked them, “Find out how many pennies you have and then tell me how many pennies you’ll give for tithing.”

Multiple Answers

One child said, “I’ll give them all.” I responded, “Sure. You don’t have to give all of your pennies, but you can. Do you still want to?”  “Uh-huh.” “Why do you want to give all of them instead of just 10%?” “Well. I don’t need them. I could keep some.” After church was finished, he came running back to tell me “I kept some of my pennies but I put some on the ground so that someone can find it for a lucky penny!”

Another child said, “I’ll give 4 because I’ve got 43 pennies.” I responded, “Okay. How did you figure that out?” “It’s easy! Every time I count 10 pennies, I take one.”

Another child said, “Can I keep the extra pennies?” I responded, “Sure!” “Okay, I’ll give 17 because I have 39 pennies.” “Okay, how did you decide to give 17?” “Because it looks like half.” “Ok. You can decide how much you want to give. The church asks for 10% but it’s okay to give more. Do you still want to give 17?” “Uh-huh!”

I gave them each a tithing envelope, they filled out their own slip (mostly), put the pennies in the envelope, and brought them to the bishop.

Assumptions

Sometimes in textbooks a word problem is given and the question asks: “Which should you choose?” or “What’s the best choice for Rohit?” Those problems make me cringe because it is making an assumption that “optimization” means the same thing for everyone, especially when some context is given. In this example, the kids all have different reasons for wanting to keep or give away the pennies – some told me they have $20 or $30 at home to spend, so maybe they don’t value pennies much. Others were already making plans for how they would spend their pennies. Some kids are just too darn nice.

But, if I pushed them to only pay 10%, then it feels like I’m making an assumption that you should always give the minimum. At the same time, by not pushing them to pay only 10%, I feel like I’m telling them they should always pay more than the minimum. So, even for me, I’m not sure what the “right” response is in this real-world math problem.

Choosing a Bus Pass

Mathematics Problem

Students can receive discounts when they ride the bus in three ways: (1) Show your student ID and one trip is 60 cents; (2) Use your student number to buy a 10-ride pass for $6.00; or (3) Buy a semester-long pass that lasts 5 months for $50. Which choice would you prefer?

Quick answer: Choices 1 and 2 are the same value. Choice 3 will be cheaper if you ride the bus at least 84 times over the five months, so if you plan to ride the bus about 17 times per month, or about 4 times per week, then it will be cheaper to buy a semester pass, so you should prefer to buy the semester pass.

Real-world problem

“Prefer” often indicates in mathematics problems that you should optimize the situation. For this problem, optimization probably means finding the cheapest way. But in the real-world, we have to think about other constraints that might not be named in the problem. Here is where you can ask your students (if they have thought about this in their own life) to share their thinking and interpretation of the problem.

I have thought about this situation each semester in the past four years. I choose not to buy the semester pass, because I will feel pressured to always ride the bus. But I want to choose to walk or ride my bike most of the time, except when weather prevents, so that I am incorporating exercise into my daily routine. So my choice really becomes between options 1 and 2.

They seem like the cost should be the same, but in practice option 1 is much more expensive for the following reasons: First, I don’t use cash a lot so there are times I don’t have cash at all. When I do, I generally have quarters or dollar bills. If I use quarters or dollar bills, I receive a change card from the driver. Somehow because of the coins I usually have, I end up with lots of nickel change cards. I can only use one change card at a time, so (for me) the nickel change cards only result in more nickel change cards. So I throw those away. Second, I sometimes misplace my student ID and so at those times I have to pay $1.25 as the full fare.

For both of those reasons, option 1 is more expensive. And yet – I end up using it much more often than option 2. Why? Because option 2 requires me to buy the 10-ride card online, one at a time, and wait for it to arrive in the mail. So it is not convenient, even though it is cheaper.

GeoGebra – Malfatti’s Problem

In Axiomatic Geometry, I and a partner explored Malfatti’s problem (see references below). I created a GeoGebra worksheet to support the presentation we later gave. I share some details here.

GeoGebra-Malfatti-02

Malfatti originally asked this question is 1803 and the question, “How to arrange in a given triangle three non-overlapping circles of greatest area?” remained open until 1994.

GeoGebra-Malfatti-03

As far back as 1384, in a manuscript of Italian mathematician Gilio de Cecco da Montepulciano a problem is presented of how to construct 3 circles, inside a triangle, that are tangent to each other and tangent to 2 edges of triangle. The same problem was posed by 18th century Japanese mathematician Ajima Naonubu. In 1803, Malfatti refined the problem to include minimizing the amount of area in the triangle “wasted”, i.e., constructing the largest circles possible. In 1826, Steiner (that sly old fox) developed a construction of three circles, tangent to each other and touching 2 edges of the triangle.  This was a solution to the problems posed by Montepulciano and Naonubu, but not Malfatti.  Stiener demonstrated 3 circles in a triangle but did not minimize the area “wasted”. In 1930, Lob and Richmond showed Malfatti’s circles do not produce the largest circles (i.e., minimal waste).  Optimal circles are obtained using greedy algorithm. In 1967, Goldberg showed that the Mafatti circles will never produce greatest area and the Lob and Richmond ‘greedy algorithm’ always produces the largest total area.  In 1994, Zalgaller and Los classified all the different ways 3 circles can be arranged in a triangle and developed a formula for determining which method of arranging the circles is optimal for a given triangle.  They also proved that there are 14 combinations of circles in a triangle that will yield optimal area.

GeoGebra-Malfatti-04

Essentially, if there is room for a circle to ‘grow’ then the system is NOT rigid.  The two examples on the left are rigid, the one on the right is not, the middle circle could change radius or ‘grow’.

GeoGebra-Malfatti-05\

Zalgaller and Los proved the result systematically, reducing all possibilities to 14 combinatorially, non-identical, rigid configurations of three circles in a triangle. See cases below.

GeoGebra-Malfatti-06

Malfatti’s Marble Problem asks:

  • How can we arrange in a given triangle three non-overlapping circles of greatest area?
  • For an Isosceles triangle, the first two circles are drawn as K1 and K2. Then we choose K3a or K3b depending on the relationship between sin(alpha) and tan(beta/2).
  • Play around with the triangle and tell me –
    • Do we use K3a or K3b when sin(alpha) > tan(beta/2)?
    • Do we use K3a or K3b when sin(alpha) < tan(beta/2)? W
    • hat do we do when sin(alpha) > tan(beta/2)?
    • Also, are the three circles always symmetric for an Isosceles triangle?

GeoGebra-Malfatti-01

References

Andreatta, M, Bezdek, A, & Boronski, J (2010) The Problem of Malfatti: Two Centuries of Debate. The Mathematical Intelligencer.

Coolidge, J. L. (1923). Some Unsolved Problems in Solid Geometry. The American Mathematical Monthly, 30(4), 174-180.

Goldberg, M. (1967). On the original Malfatti problem. Mathematics Magazine, 40(5), 241-247.

Zalgaller, V. A., & Los’, G. A. (1994). The Solution Of Malfatti’s Problem. Journal Of Mathematical Sciences, 72(4), 3163-3177.

Presentation Reference

Humes, W.R. & Stehr, E.M. (2013). Malfatti’s Marble Problem. Presentation prepared for final project in D.S. Durosoy’s MTH 432: Axiomatic Geometry course.

Questioning in College Algebra

Questioning in College Algebra

College Algebra courses serve as gatekeepers to higher mathematics courses in universities, as well as to many technical and science-intensive college majors (Small, 2002).  More students enroll in College Algebra and similar courses than in all Calculus courses (Lutzer, Maxwell, & Rodi, 2007). Despite high enrollments, Small quotes a Dean of Science and Mathematics as stating, “Traditional College Algebra is a boring, archaic, torturous course that … turns off students and discourages them from seeking more mathematics learning” (Small, 2002). Small’s recommendations include a more student-centered College Algebra.

One way for teaching to be more student-centered is through focusing on types of questioning to become more intentional in using questions to support student learning. Driscoll (1999) presented a framework for questioning fostering algebraic thinking.

Theoretical Framework

Driscoll (1999) recommended teachers use variety in questioning to support development of algebraic habits of mind. Driscoll organized five categories of teacher intentions: (a) Managing, (b) Clarifying, (c) Orienting, (d) Prompting Mathematical Reflection, and (e) Eliciting Algebraic Thinking, as described in Table 1 below (examples are from my observations). Note that I am not including the first category (Managing) since the course I observed is lecture-based.

My research question is: How does questioning by college algebra instructors to questioning intentions recommended by Driscoll (1999) for fostering algebraic thinking in grades 6-10?

Method

I observed three 80-minutes class sessions for a small-lecture College Algebra section at Michigan State University that meets on Tuesday and Thursday evenings. The department provides class notes to instructors and this instructor emailed them to his students. Notes include definitions and examples with space provided to write additional notes and solutions. The instructor is in his final semester of a mathematics master’s program and intends to not continue teaching mathematics.

I wrote field notes focused on instructor questions and their context. I spent an hour after each session writing my impressions. I coded instructor questions were according to Driscoll’s (1999) four categories of teacher intentions, described in Table 1 with examples of questions from my observations.

Questioning-01

Results

I recorded 182 instructor questions across the three observed sessions. Table 2 below shows the results of coding each session according to four categories of Driscoll’s (1999) framework for intentions.

Questioning-02

For the most part, each of the questions asked by the instructor did fall into one of the four categories. Most of the questions that did not fall into one of these categories were of the type: “Any questions?”

Discussion

The instructor asked a variety of questions, and most questions fit into one of Driscoll’s (1999) five categories. The majority of questions fell into the Orienting category.  Almost half of the Orienting questions were “Doing” questions, that is, they ask about how to do the next step of the problem. For example, “So what’s first?” or “Okay, and we have one step left, now what do we want to do?” These questions fit in the first half of Driscoll’s description: “Intended to get students started, or to keep them thinking about the particular problem they are solving…” (Driscoll, 1999, p. 6). The remainder of the questions coded as Orienting fall into the second half of Driscoll’s description (see Table 1).  For example, [finding a radius] “Okay, and what does the square root of 40 simplify to?” or “I want you to write this formula in standard form. So what does it look like? Start with x – so what goes right there?” I felt in my coding that these types of questions were different enough that the Orienting category might be more usefully broken into smaller categories such as “Doing” as opposed to “Leading to Correct Answer.”

Questioning frameworks are useful focusing devices to help instructors reflect on their teaching. The questioning framework developed by Driscoll (1999) is intended to help secondary mathematics teachers focus on developing students’ algebraic habits of mind. I would argue that this framework would also be useful for College Algebra instructors and that the instructor I observed uses questions from this framework (whether consciously or not) to support students’ development of algebraic habits of mind. Using this framework to track questions types over time could help a College Algebra instructor become more planful about question use in support of students’ learning.

References

Driscoll (1999). Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Westport, CT: Heinemann.

Lutzer, D. J., Maxwell, J.W., and Rodi, S. B. (2007) Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States: Fall 2005 CBMS Survey. American Mathematical Society: Providence, RI. Retrieved from http://www.ams.org/cbms/cbms2005.html

Small, D. (2002) An urgent call to improve traditional college algebra programs. Focus: The newsletter of the mathematical Association of America, 12-13.

What is Algebra? (Historically)

At the beginning of my second year, I thought it would be interesting to explore definitions of Algebra in historical texts: both how the author described Algebra and also what topics were included in Algebra. Unfortunately, I have not had time to do a more comprehensive search, but I found five British and American texts that were interesting and spanned about a century. (I chose to focus on these countries because the texts are readily available in digital form and also because I am limited in my language).

Please see below for additional details, but I list their definitions here:

  • Hutton (1831) wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”
  • Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”
  • Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”
  • Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”
  • Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”

Charles Hutton

Charles Hutton had been a professor of Mathematics for the Royal Military Academy. You can see additional details about his life here: https://en.wikipedia.org/wiki/Charles_Hutton

Slide2

In his “Course of Mathematics,” he included many Algebra topics, starting from basic operations, moving through roots and powers, to arithmetic and geometric progression, to some equations, and finishing with interest formulas.

Slide3

Because his definition of Algebra depended on his definition of Mathematics, I include both here. He wrote, “Algebra is the science of investigation by means of symbols. It is sometimes also called Analysis; and is a general kind of arithmetic, or universal way of computation.”

Slide4

Frances H. Smith, A.M.

Again, find additional details at https://en.wikipedia.org/wiki/Francis_Henney_Smith

Slide5

Smith (1848) defined Algebra, writing “By Algebra, which is a branch of mathematics, in which the quantities considered are represented by letters.” He continued, “Algebra dates a more recent origin than either Arithmetic or Geometry. It was not until Arithmetic had advanced to a considerable degree of perfection, and mathematicians had commenced to feel the necessity for abridging, as well as generalizing, its operations, that Algebra was introduced. In the early part of the 13th century, Leonardo, a merchant of Pisa, having made repeated visits to Arabia, returned to Italy with a knowledge of Algebra. A manuscript of his is quoted as far back as 1202.”

Slide6

Charles Davies, LL.D.

Find additional information about Charles Davies at the MAA website.

Slide7

Davies (1861) wrote that “Algebra is that branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. These letters and signs are called symbols.”

Slide8

J. M. Taylor, A.M.

James Morford Taylor served 57 years at Colgate University as a professor of Mathematics.

Slide9

Taylor (1889) wrote that “Algebra is the science of algebraic number and the equation. It differs from arithmetic (i.) in its number. The number of algebra has quality as well as arithmetical value. The double series of numbers in algebra gives a wider range to operations. Thus, in algebra to subtract a great quantity from a less is as natural as the reverse, while in arithmetic it is impossible. (ii.) In its symbols of number. Arithmetical symbols of number represent particular values; while in algebra any value may in general be attributed to the letters employed. Thus, arithmetic is confined to operations upon particular numbers, while algebra is adapted to the investigation of general principles. Again, in arithmetic all the different numbers which enter a problem are blended. (iii.) In its method of solving problems. In arithmetic we use identities, but very seldom equations; and a problem is solved by analyzing it. In algebra the characteristic instrument is the equation. To solve a problem, we translate its conditions into equations and solve those equations. The algebraic method renders easy the solution of many problems of which the arithmetical solution would be very difficult, or impossible.”

Slide10

George A. Wentworth

Slide11Wentworth (1902) wrote that “In elementary Algebra we consider all quantities as expressed numerically in terms of some unit, and the symbols represent only the purely numerical parts of such quantities. In other words, the symbols denote what are called in Arithmetic abstract numbers.”Slide12

Why study the history of Mathematics Education? The case of multiplication.

I had the privilege to take a “History of the K-16 Mathematics Course” from Dr. Sharon Senk at Michigan State University. I loved the opportunities I had to dig into old mathematics textbooks – it’s amazing how many are available digitally through Google Scholar or other resources.

I argue that looking back at old textbooks helps me in thinking about mathematics education by broadening my perspective on what “traditional” means, how often the goals of textbook authors have changed, and giving me some additional support when I talk to parents and preservice teachers about why they need to learn “new-fangled” methods for understanding basic arithmetic operations.

I like to pull out some images I found in Swetz (1995), showing that multiple methods of multiplication (yes, even the “new-fangled” methods like the lattice method) were included in textbooks as early as the 13th century:

Multiplication

I like looking in old textbooks because they also help by allowing me to “rediscover” old methods or strategies that fell into disuse, and yet could be used to help students choose from additional methods, find methods that make more sense to them, and accept that mathematics isn’t a set of rigid procedures invented by a group of old math teachers somewhere, but that creativity can enter the mix – students can come up with their own strategies!

After looking at these methods, I also found this site that gives many other strategies for mulitplying:

https://threesixty360.wordpress.com/25-ways-to-multiply/

Reference

Swetz, F. (1995). To know and to teach: mathematical pedagogy from a historical context. Educational Studies in Mathematics, 29(1), 73-88.