Math Word Problems Divorced from Reality

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Introduction

Robert Albrecht and David Moursund started this IAE-pedia article at the end of October in 2011. We were talking about math word problems we had seen in recently published articles, ones that we considered to be rather ridiculous. After poking fun at and laughing about some of these word problems, we decided to begin to think about why we were seeing so many examples of such problems, and to share our thoughts with our readers.

To lighten the tone of this document, we begin with a joke.

A research mathematician and a cattle rancher were seated together on a train and began talking as the train pulled out of the station. The rancher indicated that math had always been a problem for him.
After this brief chitchat, they talked about what they were seeing in the ranch land they were passing. As they passed a herd of cattle penned together near a railroad siding, the mathematician noted, "That herd contains 144 cattle." The cattle rancher replied, "You are correct. How did you know this?" The mathematician replied, "I counted the number of legs and divided by four. To check my answer I counted the number of eyes and divided by two. I got the same answer of 144 in both cases."
The mathematician then asked, "But, how did you know I was correct?" The cattle rancher replied, "I saw that the trucks parked nearby belonged to my neighbor. Yesterday he told me he had just sold 144 yearlings and was shipping them out today."

I hope you got a chuckle out of this joke. It reminded me (Dave Moursund) of the many math problems I have seen that involve counting the legs of a mixture of 2-legged and 4-legged animals, and trying to determine how many of each type of animal were present. And, of course, it reminded me of the stereotype that mathematicians are good at arithmetic (and also are perhaps somewhat crazy). Many research mathematicians are not particularly good at arithmetic calculations, and precise "fast counting" is certainly not a requirement to be a research mathematician.

This IAE-pedia article looks at math word problems (math "story" problems) that the authors and contributors to this page think are a little "un-real world" and might benefit from a tweak or two or more. The article is designed to amuse you and to poke fun at some of the math education materials that are widely used. Evidently a number of people have enjoyed the page and told others about it. As of 3/7/2016 this page has had over 65,000 hits.

The authors add to it from time to time. Readers who have examples they would like to share should contact moursund@uoregon.edu or bob@geekclan.com.

Many of examples on this page are included in Bob Albrecht's free eBook Mathemagical Meandering.

Math Problem Solving

This section contain a brief overview of math, problem solving in math, and math word problems.

One of the key ideas in math problem solving is to check one's work and final results to "see" if they make sense. Well-designed word problems that are rooted in reality have the characteristic that one can use the reality-based roots to help check the correctness of one's results.

What is a problem?

Each academic discipline includes an emphasis on representing and solving problems. It is a challenge to write a definition of "problem" that cuts across all disciplines. Here is a definition of problem that fits reasonably well in many different disciplines.

You (personally) have a problem if the following four conditions are satisfied:

  1. You have a clearly defined given initial situation.
  2. You have a clearly defined goal (a desired end situation). Some writers talk about having multiple goals in a problem. However, such a multiple-goal situation can be broken down into a number of single-goal problems.
  3. You have a clearly defined set of resources that may be applicable in helping you move from the given initial situation to the desired goal situation. These typically include some of your time, knowledge, skills, and brain power. Resources might include money, computers, and access to the Internet. There may be specified limitations on resources, such as rules, regulations, guidelines, and timelines for what you are allowed to do in attempting to solve a particular problem.
  4. You have some ownership—you are committed to using some of your own resources, such as your knowledge, skills, time, and energy, to achieve the desired final goal.

The "resources" in the third part of the definition do not tell you how to solve a problem. Rather, they are used as aids to problem solving. In many problem-solving situations, Information and Communication Technology (including the Internet, Web, calculators, and computers) and computerized tools are resources. These resources have grown more powerful over the years. That is one reason why it is becoming more and more common to integrate the use of computers-in-problem-solving very thoroughly into the basic fabric of many academic courses.

The fourth part of the definition of a problem is particularly important. Unless you have ownership—through an appropriate combination of intrinsic and extrinsic motivation—you do not have a problem.

What is a math problem?

To answer this question, we need to combine the definition of "problem" given in the previous section with an answer to the question, "What is mathematics?"

What is an academic discipline?

Mathematics is a discipline of study. Each discipline has a set of characteristics that differentiate it from other disciplines. Each academic discipline or area of study can be defined by a combination of general things such as:

  • The types of problems, tasks, and activities it addresses.
  • Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, and so on, and its methods of preserving and passing on this accumulation to current and future generations.
  • Its history, culture, and language, including notation and special vocabulary.
  • Its methods of teaching, learning, assessment; its lower-order and higher-order knowledge and skills; and its critical thinking and understanding. What it does to preserve and sustain its work and pass it on to future generations.
  • Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
  • The knowledge and skills that separate and distinguish among: a) a novice; b) a person who has a personally useful level of competence; c) a reasonably competent person, employable in the discipline; d) an expert; and e) a world-class expert.

Notice the emphasis on solving problems, accomplishing tasks, producing products, doing performances, accumulating knowledge and skills, and sharing knowledge and skills.

Also, notice the emphasis on increasing one's level of expertise in a discipline. There has been a lot of research on how long it takes, how much effort it takes, and the roles of good teachers and coaches as a person works to be as good as he or she can be in a discipline or sub-discipline. The research-based estimates for the time it takes to "be all you can be" in a discipline tend to be about 10,000 hours or more spread out over ten years. A typical University professor who has a doctorate and has gained promotion to an Associate Professorship has likely spent more than 15,000 hours in gaining this level of expertise.

What is mathematics?

A partial answer to this question can be provided by listing math courses taught in precollege and higher education. At the precollege level students can take coursework in arithmetic, algebra, geometry, statistics and probability, pre-calculus, and calculus.

A different approach to answering this question is provided in the IAE-pedia page, What Is Mathematics?. Here is a little material quoted from that IAE-pedia webpage:

Many people have addressed the question, “What is mathematics?” See, for example, (Lewis, n.d.) and the many publications of the National Council of Teachers of Mathematics. Here are two good examples of answers to the question, “What is mathematics?”
Mathematics is an inherently social activity, in which a community of trained practitioners (mathematical scientists) engages in the science of patterns—systematic attempts, based on observation, study, and experimentation, to determine the nature or principles of regularities in systems... The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation. However, being trained in the use of these tools no more means that one thinks mathematically than knowing how to use shop tools makes one a craftsman. Learning to think mathematically means (a) developing a mathematical point of view—valuing the processes of mathematization and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade, and using those tools in the service of the goal of understanding structure—mathematical sense-making (Schoenfeld, 1992).
Notice the emphasis on thinking mathematically. One gains increased expertise in math by both learning more math and by getting better at thinking and problem solving using one’s knowledge of math.
Mathematics is built on a foundation which includes axiomatics, intuitionism, formalism, logic, application, and principles. Proof is pivotal to mathematics as reasoning whether it be applied, computational, statistical, or theoretical mathematics. The many branches of mathematics are not mutually exclusive. Oft times applied projects raise questions that form the basis for theory and result in a need for proof. Other times theory develops and later applications are formed or discovered for the theory. Hence, mathematical education should be centered on encouraging students to think for themselves: to conjecture, to analyze, to argue, to critique, to prove or disprove, and to know when an argument is valid or invalid. Perhaps the unique component of mathematics which sets it apart from other disciplines in the academy is proof—the demand for succinct argument that from a logical foundation for the veracity of a claim (McLoughlin, 2002). [Editor's note added 6/22/2014. I am not currently able to find the 2002 document. In looking through my previously published documents I found a note that in 2006 I was no longer able to access the original document online. M. Padraig M. M. McLoughlin is a prolific author, and has repeated the same general idea in a number of his papers. Learn more about him at http://faculty.kutztown.edu/mcloughl/.]
Notice the emphasis on proof or disproof.

Another approach to answering the question, "What is mathematics?" is to discuss math content and math maturity. Math content is emphasized when we list the various components and courses in math. Math maturity focuses on understanding the content at a deep level and learning to think like a mathematician. Here is an example quoted from the Math Maturity page in the IAE-pedia.

You know that any pair of numbers can be added, subtracted, or multiplied. You know that any pair of numbers can be divided—except that it is “impossible” to divide by zero. These aspects of numbers on a number line are all big ideas. However, your mind might fixate on why the number zero has been singled out for special mention. You might ask: What is so special about zero? Why is it impossible to divide by zero? You might begin to explore what it means to say that one cannot divide by zero. You might begin to explore various aspects of the number zero that make this number distinctly different from other numbers.
One indicator of increasing math maturity is a student’s movement from rote and unquestioning memorization to learning with and for understanding. As you were reading the previous paragraph, you encountered “it is 'impossible’ to divide by zero.” What does this statement mean to you? Can you provide arguments that convince you and that might convince others that this is a correct statement? A more mathematically mature mind questions assertions such as the impossible to divide by zero assertion. It works to develop a level of understanding beyond rote learning and unquestioning acceptance of such assertions. This approach to learning with questioning and understanding is applied both in math and in other disciplines of study.
At any grade level, a teacher might encounter a student who raises such questions and who does such explorations. In the same class, there will be students who “don’t have a clue” or who “couldn’t care less” about why such questions are being asked or explored. If one student raises and explores such questions while another student “hasn’t a clue,” we are likely to take this as an indication that one student is more mathematically mature than the other. Math inquisitiveness is an aspect of math maturity.

Quoting from the same reference:

...mathematical maturity includes the ability and/or capacity to:
  • Communicate mathematics and math ideas orally and in writing using standard notation, vocabulary, and acceptable style.
  • Learn to learn math; complete the significant shift from learning by memorization to learning through understanding.
  • Generalize from a specific example to broad concept.
  • Transfer one’s math knowledge and skills into math related areas and problems in disciplines outside of mathematics.
  • Handle abstract ideas without requiring concrete referents.
  • Link a geometrical representation with an analytic representation or vice-versa.
  • Manifest mathematical intuition by abandoning naive assumptions and by readily drawing on one’s accumulated subconscious math knowledge and insights.
  • Move back and forth between the visual (e.g., graphs, geometric representations) and the analytical (e.g., equations, functions).
  • Recognize a valid mathematical or logical proof, and detect 'sloppy' thinking. Provide solid evidence (informal and formal arguments and proofs) of the correctness of one’s efforts in solving math problems and making proofs.
  • Recognize mathematical patterns.
  • Separate key ideas from the less significant.
  • Represent (model) verbal and written problem situations as mathematical problems.
  • Effectively draw upon one’s math knowledge and skills to address novel (not previously encountered) math-related problems.
Notice that the list does not contain any specific math content. Rather, it is a list of what mathematicians do. A person might have a quite high level of math maturity and not be skilled at paper-and-pencil arithmetic. An increasing level of math maturity represents progress in moving in the direction of learning to think about, understand, represent, and solve math problems and math-related problems.

What is a math word problem?

Here is a brief answer to the question, "What is a math word problem?"

The problem situation is described in a combination of natural language and the language of mathematics. There may be very little or no use of math vocabulary in describing the problem situation, and often the problem situation is "loosely" stated.
Typically the problem solver needs to have some prerequisite knowledge and understanding of the general content area of the problem in order to understand the specific problem situation and to translate this into a clearly-defined problem. Typically the narrative description of the problem situation contains sufficient information that one can use it to help check the answer or answers one believes are solutions to the problem.

Math word problems divorced from reality

Math books and other sources of math word problems often attempt to make word problems interesting to students by casting them in "semi real-world" settings. The focus of this article is on math word problems that try—but fail in important ways—to masquerade as "real-world" math word problems. The authors of this page feel that such attempts potentially do more harm than good.

  • They can lead to ridicule of the whole area of math word problems.
  • They can take away the important idea of "sense making" in checking of solutions.
  • They can help teach students completely wrong information.

Examples of Math Word Problems Divorced from Reality

Coconut-carrying swallows

The average flight velocity of an African Swallow is 17/3 that of the European Swallow. An African Swallow and two European Swallows collaborate in transporting 50 coconuts a distance of 6 KM. If the average velocity of an European Swallow is 9 KM/hour while carrying a coconut and 18 KM/hour while unburdened, how many minutes will the job require, i.e., when will the Swallow carrying the last coconut pass the 6 KM mark?

Here is a reality check on some of the supposed real-world components of this problem:

Wikipedia: the mass of a swallow is 10 to 60 grams and an unburdened swallow usually travels at 30 to 40 km/h.http://en.wikipedia.org/wiki/Swallow
Wikipedia: a "full grown" coconut weighs about 1.44 kilogram (1,440 grams). http://en.wikipedia.org/wiki/Coconut
How does a big swallow (60 grams) carry a 1,440 gram coconut?
1440g / 60g = 24. Can a swallow fly and carry 24 times its body mass?
The problem states that an unburdened swallow flies 18 km/h. But Wikipedia says that typical speed while foraging is 30 to 40 km/h.
The problem states that an unburdened European swallow flies 18 km/h and an African swallow flies 17/3 times that fast or 102 km/h. Does that sound reasonable?

Is this problem reasonable? Does it teach good science? Does it use accurate data? Should we use it with students? If we use it with students, should we discuss the accuracy of the data and the credibility of the problem with the students?

A plethora of snakes?

We sometimes tutor high school students who failed the state math test prior to their retaking the test, and will probably tutor them again when test time rolls around this school year. As Tom Lehrer once exhorted us to "be prepared", we do want to be prepared so we downloaded a sample test and spent some happy time working ON it.

We found the following problem rather interesting.

There are 6 snakes in a certain valley. The population of snakes doubles every year. In how many years will there be 96 snakes?

A. 2 B. 3 C. 4 D. 8

Here's a handy table showing the population growth of the snakes in that certain valley.

Year Snakes
0 6
1 12
2 24
3 48
4 96

The table quickly gives the answer: C. 4 years. The test taker does not need to know that this population growth can be modeled by the equation: snakes = 6 x 2n where n = number of years.

Let's keep on truckin':

Year Snakes
10 6144
20 6291456
30 6442450944

And so on. How big is that certain valley? What do all those snakes eat? Don't some snakes die every year? Aren't some snakes killed by predators such as hawks and mongeese (mongooses?)? How much territory does a snake need to survive? What is the life expectancy of a snake? Some Internet sites say the life expectancy of a snake is 10 to 40 years depending on the species.

Hmmm...the land area of the Earth is about 149 trillion square meters (149 x 1012 square meters). According to this model of population growth, in how many years will there be 1 snake per square meter of Earth's land area?

Answer: about 44.5 years.

Your Turn

According to this model (snakes = 6 x 2n), in how many years will there be 1 snake per square centimeter of Earth's land area?

According to this model (snakes = 6 x 2n), in how many years will there be 1 snake per square millimeter of Earth's land area?

Wow! Knee deep in snakes. Let's all work on our snake-charming skills!

A note on the science of NFL football: Pythagorean Theorem

We posted this information on the Oregon Council of Teachers of Mathematics (OCTM) listserv. It was later published in The Oregon Mathematics Teacher (TOMT), November/December 2011, page 31.

Date: 2011-09-25

Ahoy Mathemagicians,

As suggested by this week's MathNexus, we watched The Science of NFL: Pythagorean Theorem video.

Watch this video at http://www.nbclearn.com/portal/site/learn/nfl/cuecard/51220/. You can expand the video given in the left upper corner of the screen by dragging down and to the right on its bottom right corner.
MathNexus link: http://mathnexus.wwu.edu/Archive/websites/detail.asp?ID=237. Note added 3/14/2016. The site named is still live and is a useful site. However, the specific football link you actually want is http://www.nsf.gov/news/special_reports/football/index.jsp.

Amusing. A receiver catches a football and runs 40 yards along the side of a triangle perpendicular to the goal line. A defender is initially 30 yards to the receiver's right (at a right angle) and has to run 50 yards along the hypotenuse of the triangle in the same time in order to tackle the ball carrier.

Suppose the ball carrier is a wide receiver. Wide receivers are fast! Typical time for a wide receiver to run 40 yards is 4.3 to 4.5 seconds.

The defender who has to run 50 yards to make the tackle is probably a corner back, a safety, or a linebacker. Corner backs are fast too. Linebackers maybe not so fast. Don't know about safeties.

Assume that the defender is as fast as the ball carrier. When the ball carrier has run 40 yards along the side of the right triangle perpendicular to the goal line, the defender has run 40 yards along the hypotenuse and, alas, is still 10 yards from the ball carrier. Does he make the tackle?

The defender runs the last 10 yards along the hypotenuse, but the ball carrier is no longer there. He has run another 10 yards towards the goal line.

Suppose the ball carrier runs 40 yards in 4.3 seconds. Speed = 9.30 yd/s.

To make the tackle, the defender must run 50 yards in 4.3 seconds.

Your Turn: Calculate the defender's speed in yards per second.

How does the speed you calculated compare to Ursain Bolt's world record speed in the 100 meters? Bolt's 100-meter time is 9.58 seconds.

Your Turn. Calculate Bolt's speed in meters per second.

In order to compare Bolt's speed in meters per second to the defender's speed in yards per second, you must convert meters per second to yards per second or convert yards per second to meters per second.

1 yard = 0.9144 meter [exactly]
1 meter = 1.0936 yard [approximately]

In order to make the tackle, would the defender have to run faster than Bolt's world-record 100-meter time? If yes, how much faster? What percent faster? Et cetera, et cetera.

When the defender has run 50 yards along the hypotenuse and reached the place where the ball carrier had passed moments ago, by how much time was he late?

Suppose that the receiver was already running toward the goal line when he caught the football. The defender was probably not at that time running along the hypotenuse, but had to react to the catch and start from a standstill. It will take him a few yards to get up to speed. His average speed for the entire 50 yards will be less than you calculated above.

How did the defender know that he would have to run along the 50-yard hypotenuse of the 30-40-50 right triangle? How long did it take him to do this calculation before he started running? How far did the ball carrier run while the defender was calculating?

Well, we could go on, and you can probably think of questions that didn't occur to us.

What do you think that your students might learn by watching this video?

Cheers, Bob & George

P.S. Here are two handy Internet sites about the 40-yard and 100-meter dash times:

40-yard dash - Wikipedia http://en.wikipedia.org/wiki/40-yard_dash
100 meters - Wikipedia http://en.wikipedia.org/wiki/100_metres

The milkshake investigation: Adventures in Plopland

We sent this email to the Oregon Council of Teachers of Mathematics (OCTM) listserv in response to a posting about an interesting "real-life" algebra application.

Good Cheer Mathemagicians,

The "What does math have to do with real life?" posting was delicious, especially the milkshake plop investigation. The article "N.C. schools to bring math down to earth" quoted a math teacher describing a real-life application of algebra:

"For example, a milk shake could illustrate slope," she said. "Teachers would explain how plopping scoops of ice cream into milk makes a milk shake grow. Numbers would be assigned to the scoops of ice cream and the increasing volume of the milkshake. Then, an inclining line could be graphed. That way, when students found the slope of that line, they would see that the answer is more than just a number. It is actually the rate at which the milk shake grew."

As writers of intertwingled math and science instructional stuff, we were intrigued by the many plopifications (oops – ramifications) of this investigation.

The teacher said, "Teachers would explain." Alas, alack, and oh heck. She did not say, "Students would investigate, experiment, discover, learn, et cetera, et cetera."

Since this was given as an example of a real-life application, we assume that it was observed many times in real-life places and that many real-life experiments were run to validate the experimental method and results.

The domain, of course, is measured in scoops, which are easy to count.

Domain = {no scoop, 1 scoop, 2 scoops, 3 scoops, ..., n scoops}

The domain is a finite set of whole numbers, unless you measure tenths of a scoop, hundredths of a scoop, et cetera, et cetera

Hmmm. Do you believe that that all scoops are created equal? It is true that the volume of the total of milk plus ice cream grows as scoops are added. But, it seems to me that the person making a milk shake just keeps adding the ice cream until a certain level is reached in the container. Add too much ice cream, and the shake will overflow the container while the shake is being mixed. Add to little and the customer will feel cheated by the shake that does not seem as big as it ought to be.

The range is a measure of the volume of the milkshake. How is it measured? Cubic centimeters? Liters? Do all scoops have the same shape? Do all scoops have the same volume? Do all scoops have the same mass? More about that after we examine milkshake-making mechanics.

Milkshake-making mechanics. Is the ice cream plopped in before adding milk, or is the milk put in first and the scoops of ice cream plopped into the milk? Either way, the milk and ice cream are then mixed in a blender.

Ice cream first, then milk. If the ice cream is plopped in first, then the volume of ice cream in the milkshake container is a direct variation function of the number of scoops.

V = (Vscoop)(number of scoops) [where Vscoop is the volume per scoop of icecream]

If we identify volume with y and number of scoops of ice cream with x, the y-intercept is 0 and the slope is Vscoop measured in appropriate units.

If all scoops have exactly the same volume, say Vscoop, the range is

Range = {0 Vscoop, 1 Vscoop, 2 Vscoop, 3 Vscoop, ..., n Vscoop}

Suppose that all scoops are spheres exactly 5 centimeters in diameter. Then the volume is – well, you can work that out and restate the range in cubic centimeters.

Oops, real-life trouble! A scoop of ice cream is probably malleable. The first scoop hits the bottom of the container, compresses a bit, squeezes out a little air, and changes shape, volume, and density. (The key issue is change in volume. How compressible is a scoop of ice cream? As it warms just a little bit while the next scoop is being added, does the original scoop expand or contract due to warming?) Now plop in another scoop. The second scoop plops down on the first scoop. Both change shape, volume, and density, so the volume of ice cream in the container might not be the sum of the volumes of the two scoops before they were plopped. Changes in shape, volume, and density continue as more scoops are plopped. Soft ice cream will deform more than hard ice cream.

What to do? Measure the volume of each scoop before plopping it into the container? Or somehow measure the volume of ice cream in the container? If the ice cream is soft enough to act like a liquid and conform to the shape of the container, we can calibrate the container to give the volume as a function of height. Easy if the container is a cylinder – more difficult for most milkshake containers we have seen that vary in diameter from bottom to top. Use graduated cylinders or beakers as the milkshake containers? Can we sell that to, say, Baskin-Robbins?

Milk first, then ice cream. If the milk is poured in first and then the ice cream is plopped into the milk, the volume in the container can be modeled by the equation

V = (Vscoop)(number of scoops) + Vmilk [where Vscoop is the volume per scoop of icecream]

If we identify volume with y and number of scoops with x, the y-intercept is Vmilk and the slope is Vicecream / scoop, both measured in appropriate units.

Conjecture: Ice cream plopped into milk will not deform as much as ice cream plopped into an empty container. It will be more likely to retain its original shape and volume.

Practical problems: How do we measure volume? Is the volume of milk measured before it is poured into the container? Or is it measured after it is in the container, perhaps by calibrating the container with a volume versus height scale? Is the density of ice cream greater than the density of milk so that the scoops submerge as they plop into the container? Or is the density of ice cream less that the density of milk so that part of the ice cream floats above the surface of the milk?

How about mass instead of volume? We suggest that it might be easier to measure mass than measure volume. The mass M of a scoop of ice cream doesn't change when the shape and volume change. Using mass instead of volume, the equations are

Ice cream, then milk:
Plop in ice cream: M = (Mscoop)(number of scoops) [Mscoop = mass per scoop, Mmilk = mass of milk]
Then add milk: M = (Mscoop)(number of scoops) + Mmilk [Mscoop = mass per scoop, Mmilk = mass of milk]
Milk, then ice cream:
Pour in milk: M = Mmilk [Mscoop = mass per scoop, Mmilk = mass of milk]
Then add ice cream: M = (Mscoop)(number of scoops) + Mmilk [Mscoop = mass per scoop, Mmilk = mass of milk]

Our real-life milkshake plop quest

Where is this "real-life" stuff actually used in real life? "Eureka!" we exclaimed as we ran through the streets of Newport fully clothed (too cold to run naked as did Archimedes!) and wended our way to places that make milkshakes.

Alas, no joy. In every place where people make milkshakes, not one place measured the volume or mass of the milkshake as a function of the number of scoops of ice cream plus the milk. Indeed, it was a bit scary as these real-life people backed away from our inquiry and reached for the telephone.

Rebuked by real-life people who strangely don't know about this real-life application of mathematics, we hiked many kilometers in the wonderment of Gaia, thinking about all of the above, and sat on a rock with a great view of the ocean. Suddenly there plopped into our mind an incredible eureka about ... but that's another story for another time.

We encourage you to go out in your neighborhood, find the places that measure the volume of a milkshake as a function of the number of scoops of ice cream, and post your findings.

Cheers,

Bob & George

It’s as easy as falling off a cliff, or is it?

This problem published by the NCTM is discussed in the June 16, 2011, article, Problems to Ponder section, http://www.nctm.org/Classroom-Resources/Problems/Problems-to-Ponder_-Falling-Off-a-Cliff/.

In a kingdom long ago, a king decided to let chance determine whether persons who committed major crimes would be allowed to live and stay in the kingdom or would fall to their deaths off a steep cliff. Offenders would be placed blindfolded at the edge of the cliff, and then for the rest of their lives, they would proceed to take a step forward, toward the cliff”s edge, or a step backward, away from the edge, thus saving themselves—at least for the time being.
A spinner, to be spun by a favorite of the king, would determine whether the offenders stepped forward or backward. On the first spin, a step toward the cliff would send the blindfolded criminal right over the edge. A step away from the cliff would take the offender two steps back from the edge. But then the king’s favorite would get to take another spin, randomly determining the offender’s next step. And so on…
The king, being a “merciful” ruler, wanted the criminal to have a sporting chance of .5 of not going over the cliff. So he asked the court mathematician, “How should the spinner be divided for stepping toward the cliff and stepping away from the cliff so that an offender has a .5 chance of surviving indefinitely?”

The math problem is a special case (moving only forward and backward) of random walks. The “story” attempts to make the problem more “real world” and of potential interest to students. The story is supposed to elicit ownership on the part of the problem solver. Here are some of my reactions to the problem:

  • For the rest of his or her life, the criminal stands blindfolded near the edge of a cliff. Hmm. Does the criminal get to sleep, eat, or sit down from time to time? How about bathroom breaks? How many spins per day? What if the person doing the spinning gets tired or suffers a damaged spinner finger? What if the spinner wears out?
  • I assume that the criminal is interested in remaining alive. Knowing that the cliff is straight ahead, I would assume that the criminal would take a small step when directed to step forward, and a larger step when directed to step backward. I also wondered about a variation on this observation. In my everyday life, my steps forward are longer than my steps backward, and both vary in length depending on the situation and over time. If my legs are stiff, my steps tend to be short. I wonder if the criminal’s “average” step length is measured and then used in the initial placement of the criminal near the cliff.
  • Many people have one leg slightly longer than their other leg. This can influence the length of the step of one leg versus the step of the other leg. In taking a number of steps, this can also lead to following a curved path.
  • My first image of a spinner was an inexpensive cardboard spinner with two sectors—one labeled step forward and one labeled step backward. As with fair and unfair coins, is this inexpensive spinner fair? Indeed, how difficult is it to construct a fair spinner?
  • Suppose that the person doing the spinning is absolutely consistent in spinning (making) the exact same spin each time. How does this affect the situation? I can imagine that the person doing the spinning produces an exact "five times around" spin every time. In that case, if the spinner starts in the step backward position, then every spin results in a step backwards.
  • I can imagine that the cliff is quite steep—perhaps exactly perpendicular to the ground the criminal is standing on. However, a fall off the cliff might only be a 10 centimeter fall. “Steep” is not an adequate word to describe a cliff that leads to sure death if one falls off. And, of course, there is also the possibility of a lake at the bottom of the cliff.

Water bucket conundrum

This problem was published by the NCTM in October 2011. It is discussed at the site https://mvtrinh.wordpress.com/2012/07/14/water-bucket-conundrum/.

You are staying at a rural cabin, and the only method to get water is to draw it from a well. A 4-gallon bucket and a 9-gallon bucket are the only containers for carrying water to the cabin. In one trip to the well, what whole number amount of water in gallons could you bring back to the cabin in the buckets? No other markings are on the pails, and you can’t do any estimating—you need to supply exact whole number amounts only!
What if you had a 4-gallon bucket and a 10-gallon bucket?
What if you had an n-gallon bucket and an m-gallon bucket?

Hmmm. This problem doesn't make sense to me. I can carry 4 plus 9 gallons. Surely there is more to the problem that this. Aha! I went back to the website where a solution is discussed. The problem solved at the website is how many of the integer amounts: 1 gallon, 2 gallons, 3 gallons, etc. can be measured out using a 4 gallon and a 9 gallon container? Thus, for example I can measure out 5 gallons by completely filling the 9 gallon bucket and pouring off exactly 4 gallons into the 4 gallon bucket. This leaves 5 gallons in the 9 gallon bucket.

I (Dave Moursund) have long been interested in and enjoyed Water Bucket Problems. Indeed, when I first became interested in studying recursion in computer programming, I practiced what I was learning by writing a FORTRAN program that used recursion to solve water bucket problems. In addition, I have used water bucket problems in several of the books I have written.

Here are a couple of my thoughts when I read the problem given above.

  • This is a good problem area, with problems ranging from simple to quite challenging, and from concrete to abstract. I quickly developed "ownership" when I first encountered such problems. I wonder how many students find such problems interesting and develop ownership?
  • Offhand, I have trouble thinking of practical real-world problems that relate to this problem. Of course, one can make the problem quite concrete by having actual buckets and water. Students might well become engaged and have a lot of fun in that environment.
  • As a rough estimate, “a pint’s a pound the world ’round.” So, a gallon of water is about eight pounds. A nine-gallon bucket holds about 72 pounds of water. The bucket and handle have to be study enough to deal with such a weight, and so there is that additional weight to consider. I'll bet that many of the students who encounter this problem cannot carry the combined weight of the water-filled 4-gallon and 9-gallon buckets.
  • I, personally, would have considerable trouble carrying the nine-gallon bucket full of water. In addition to the weight, surely some water would slop over the edge as I carried this bucket. Aha! The problem assumes great precision in measurement and no water slop in pouring or carrying. These are unreasonable assumptions.

Email from Bob to Dave about the bucket puzzle:

See http://answers.yahoo.com/question/index?qid=20070304111813AAAVuSL. I like this statement of the 4 gallon & 9 gallon problem much better than NCTM's. I can believe Tom & Eileen, who happen to have good math maturity, discussing how to find all possible answers.

I think it is a bit silly to pose the water bucket problem as staying in a rural cabin and getting water from the well with 4-gallon and 9-gallon buckets in the manner described at the NCTM site. Not believable. If you were staying in the cabin, would you bring water from the well in this way?

9 gal + 4 gal = 13 gal. (13 gal)(8.345 lb/gal) = 108.5 lb. Too much for me to carry! Carrying one bucket containing 4 gallons (33.38 lb) is difficult because the weight is unbalanced, all on one side. Much more so for the 9-gallon bucket full of water (75 lb). I would have to stop and switch sides one or more times. Carrying 4 gallons on one side and 9 gallons on the other side is also unbalanced and difficult. I would have to stop frequently, rest, and change sides. For sure I would spill some water unless each bucket had a secure water-tight lid.

Why not just present bucket problems as puzzles to solve instead of stating the problem in the context of a silly, "real-life" situation? Or present it in a context that is more plausible, such as the Tom & Eileen scenario?

Do 4-gallon and 9-gallon buckets exist in the real world?

I searched for pails and buckets. Ulain http://www.uline.com/BL_8150/Plastic-Pails-And-Lids?keywords=5has Plastic pails: 1 gal, 2 gal, 3.5 gal, 5 gal, 6 gal, and 7 gal. [But no 4-gal nor 9-gal buckets.]

Bucket Outlet http://www.bucket-outlet.com/ has pails: 2 qt, 5 qt, 10 qt, and 16 qt. Galvanized buckets: 1.25 qt, 2 qt, 4 qt, 5 qt, 6 qt, 8 qt, 10 qt, 12 qt, and 14 qt. Buckets in gallons; 0.3125 gal, 0.5 gal, 1 gal, 1.25 gal, 1.5 gal, 2.5 gal, 3 gal, 3.5 gal. [But no 4-gal nor 9-gal buckets.]

Where did NCTM get the 4-gal and 9-gal buckets? Standard sizes carried by several sellers; 1 gal, 2 gal, 3.5 gal, 5 gal, 6 gal, 7 gal. [But no 4gal nor 9-gal buckets.]

Hey! Why not invent problems that use buckets/pails you can buy on the Internet and include links to these places. You can probably buy these standard-size buckets in a store in your town too. Use pails and buckets that exist in the real world.

A touch of humor

The following list of questions and answers has been widely circulated on the Web. See, for example, http://www.desicomments.com/forum/showthread.php?t=26651. (Note added 3/14/2016: This site continues to have silly questions and answers.) They illustrate how difficult it is to precisely state a question. An imprecise statement can result in student answers that are correct from the student's point of view but not the exam creator's point of view.

Q1. In which battle did Napoleon die?

  • His last battle.

Q2. Where was the Declaration of Independence signed?

  • At the bottom of the page.

Q3. The Willamette River flows in which state?

  • Liquid.

Q4. What is the main reason for divorce?

  • Marriage.

Q5. What is the main reason for failure in a course?

  • Tests.

Q6. What can you never eat for breakfast?

  • Lunch & dinner.

Q7. What looks like half an apple?

  • The other half.

Q8. If you throw a red stone into the blue sea what it will become?

  • It will simply become wet.

Q9. How can a man go eight days without sleeping?

  • No problem, he sleeps at night.

Q10. How can you lift an elephant with one hand?

  • You will never find an elephant that has only one hand.

Q11. If you had three apples and four oranges in one hand and four apples and three oranges in the other hand, what would you have?

  • Very large hands.

Q12. If it took eight men ten hours to build a wall, how long would it take four men to build it?

  • No time at all, the wall is already built.

Q13. How can you drop a raw egg onto a concrete floor without cracking it?

  • Any way you want, concrete floors are very hard to crack.

Unreal round-robin tournament SAT practice test question

How do you prepare to take the SAT? One way is to take online practice tests. We found several practice tests, including one at Kaplan's Test Prep site: http://216.154.212.161/KaplanQuizzes/showQuiz.jsp?TID=SATMPT1. (There are many sites that contain sample questions from various high stakes tests. This particular site was not found on 3/14/2016.)

On 2012-02-24 (Friday), Question #22 of the Kaplan test looked like this:

At a basketball tournament involving 8 teams, each team played 4 games with each of the other teams. How many games were played at this tournament?

A. 64
B. 98
C. 112
D. 128
E. 224

We selected C. 112 as our answer. Later we looked at Kaplan's answer. Kaplan said that the correct answer is D. 128.

We disagreed, so we filled out a registration form and obtained Kaplan's explanation of its answer, paraphrased here:

Difficulty: High
Strategic Advice: If each team played 4 games with each of the other teams, then each team played (8)(4) = 32 games. Use this number to determine how many total games were played.
32 games were played, and each game involved 2 teams. Therefore: 32(8/2) = 128 games were played.

Interesting. There were eight teams and, as stated in the question, each team played 4 games with each other team. [Bold added for emphasis.] Seems to us that each team played 4 games with each of seven other teams. Each team played 28 games. Apparently Kaplan thinks that each team also played itself.

We modified Kaplan's expression like so: 28(8/2) = 112 games were played.

We have never heard of a basketball tournament in which each team played each other team four times. Of course, in some sports such as baseball there are elimination rounds with pairs of teams playing short series, but that is not at all like the Kaplan problem.

We have heard of round-robin tournaments in which each team plays each other team once, and found many examples on the Internet. We also learned that there are double round-robin tournaments in which each team plays each other team twice. After much research, we are quite confident that, in the real world, there are no round-robin tournaments in which each team plays each other team four times. Kaplan's question is unrelated to the real world as – alas – are too many math test questions.

We think that a real-world round-robin tournament question in which each team plays each other team once is an excellent question. On the Net, we found such round-robin tournaments for archery, bowling, basketball, billiards (and pocket billiards, aka pool), chess, cricket, curling, darts, hockey, ping pong, poker, rugby, skeet, soccer, swimming, and tennis.

Boggle, boggle. Why does Kaplan confound an excellent real-world question by turning it into an unreal-world question? The number of games played in an 8-team round-robin tournament in which each team plays each team once is an excellent question. Asking how many games are played in a round-robin tournament in which each team plays each other team four times is silly, unreal-world busy work.

The number of games played in a round-robin tournament in which each team plays each other team once is a triangular number. A splendid teaching opportunity. We like to show the games played in a table.

Two teams: A and B. One game is played. Fanfare: 1 is the first triangular number.

Team A Team B
Team A 1
Team B

Three teams: A, B, and C. Three games are played. 3 is the second triangular number.

Team A Team B Team C
Team A 1 1
Team B 1
Team C

Four teams: A, B, C, and D. Six games are played. 6 is the third triangular number.

Team A Team B Team C Team D
Team A 1 1 1
Team B 1 1
Team C 1
Team D

Eight teams: A, B, C, D, E, F, G, and H. 28 games are played. 28 is the 7th triangular number.

Team A Team B Team C Team D Team E Team F Team G Team H
Team A 1 1 1 1 1 1 1
Team B 1 1 1 1 1 1
Team C 1 1 1 1 1
Team D 1 1 1 1
Team E 1 1 1
Team F 1 1
Team G 1
Team H

Let there be n teams. The number of games played is the (n-1)st triangular number.

number of games played = (n-1)(n)/2

Round-robin tournaments: Internet resources

Round-robin tournament http://en.wikipedia.org/wiki/Round-robin_tournament

"In a single round-robin schedule, each participant plays every other participant once. If each participant plays all others twice, this is frequently called a double round-robin. The term is rarely used when all participants play one another more than twice."

Round-robin draws for curling from 5 teams to 20 teams:

http://media.curling.ca/contentImages/File/Business%20of%20Curling/Operations%20Manual/Chapter%20N%20-%20Draws(1).pdf

Round-Robin Tournament Brackets http://www.printyourbrackets.com/roundrobin.html

Tangent to a circle

Consider the following test question:


Small Screen Shot Tangent.png



The line drawing is meant to depict a circle and some line segments. However (of course) in the drawing the circle and the line segments have width. What this "width problem" image does is make it seem like the line segment AB actually has a short line segment in common with the circle. For segments CD, OE, and CF, the figure shows a dot at the point where the segment intersects the circle, but the figure does not show a dot at the intersection of segment AB and the circle.

Now, suppose that a student has "learned" or "memorized" that a tangent line to a circle "touches" or "intersects" the circle at only one point. The student guesses that this definition also works for a line segment. The student looks at the diagram and decides that the answer is line segment OE.

In geometry textbooks, a tangent to a circle is defined as a line in the plane of the circle that intersects the circle at exactly one point. In our geometry book, we did not find a definition of a line segment tangent to a circle. We looked at several Internet definitions of a tangent to a circle. All of them defined a tangent to a circle as a line.

How might we define a line segment tangent to a circle so that the answer to the question is seg AB?

What say: A segment is tangent to a circle if 1) the segment is in the plane of the circle, 2) the segment intersects the circle at exactly one point, 3) the segment is perpendicular to a radius of the circle, and 4) all other points of the segment are exterior to the circle.

Whew! A mouthful. True? Hmmm . . . can we omit any of the four numbered phrases?

Probability problem

Probability Spinner.png




I wonder what this problem is designed to test? Probability is a challenging component of math. Do all students know about spinners? What about a spinner stopping on a line? Is the problem designed to test whether a student knows the notation for indicating a right angle and the notation for indicating that two angles are equal? What about the fact that there are six regions, but they do not all have the same angular measurement. Might a student guess 5/6 and get the answer correct by completely wrong thinking? Finally, how about the NOT? That adds complexity—Boolean algebra—to the problem. My opinion is that there are too many different reasons why a student might produce an incorrect answer to the problem.

Ball tossed in the air

Problem: A ball is tossed into the air. The height of the ball as a function of time can be described by the equation h = -16t2 + 72t. In this equation h is the height of the ball in feet and t is time in seconds. After how many seconds will the ball hit the ground?

A. 4 seconds B. 4.5 seconds C. 9 seconds D. 56 seconds

The statement h = -16t2 + 72t is a formula [ h(t) = -16t2 + 72t ] rather than an equation. At the time t = 0, the height of the ball is zero. Typically when I throw a ball, the ball is somewhat above the ground rather than lying on the ground. It is interesting to think about this throwing situation.

Does it make any difference whether the ball is thrown exactly perpendicular to the ground or not? (What do we assume the student knows about the physics of the situation?) And, of course, the problem neglects both air resistance and the fact that gravitational force on the ball decreases as the ball rises into the air.

If this is a physics problem, it is poorly stated. If it is a pure math problem, the problem is to solve the equation 0 = -16t2 + 72t, which is not much of a challenge.

Unreal measurements in 5th-grade homework

I (Bob) am living with step-granddaughter NW who is in a 5th grade GATE (Gifted and Talented Education) program. Much of her homework is copied from a 5th-grade math workbook. This workbook is a great source of silly "real-world" problems. Here are a few from one night's homework.

Sizes of beetles:

Japanese Beetle 1.295 cm

June Bug 2.518 cm

Firefly 1.063 cm

17. Which beetle has the shortest length?

18. Another type of beetle is 1.281 cm long. Which beetle has a length less than 1.281 cm?

19. Some beetles can jump as high as 15 cm. Suppose three beetles jumped 14.03 cm, 14.029 cm, and 14.031 cm. What is the order of heights the beetles jumped from least to greatest?

20. A Japanese Beetle grub may hibernate 29.301 cm underground. Between which two numbers is 29.301?

A 29.103 and 29.3 B 29.21 and 29.3 C 29.3 and 29.31 D 29.31 and 29.32

Questions we asked: Would a real-world entomologist measure the length of a beetle accurate to 1.295 cm? I don't think so. I found many Internet sites that gave the length of beetles as a whole number range of millimeters, such as 8 to 12 mm (1.0 - 1.2 cm). No site showed the length of a beetle accurate to 0.001 cm.

19. How would you measure the height that a beetle jumped. Especially, how would you measure it to an accuracy of 0.001 cm? Do you measure the height that each of the three beetles jumped ONCE and accept that measurement as an accurate measure of how high that beetle can jump? I don't think so. I think that a scientist would encourage the beetle to jump several times, record each height, and then calculate the mean height. But you who wrote this problem, please tell me how to measure the height of each jump accurate to 0.001 cm.

20. How would you measure the depth of hibernation accurate to 0.001 cm? Is the ground above the hibernation site perfectly flat and horizontal? Probably not. Do you measure to the top of the beetle? Where on the top? Head? Thorax? The measurement place on the top of beetle and the unevenness of the ground would probably cause an error of measurement much greater than 0.001 cm. Oops, what if the beetle is not hibernating in a horizontal position, but is tilted a tad? Do you take depth measurements at several locations on the beetle (each accurate to 0.001 cm) and calculate their mean?

I wonder who writes this silly unreal stuff. These math problems teach bad science. The student may think that all Japanese Beetles are exactly 1.295 cm long, all June Bugs are exactly 2.518 cm long, and all fireflies are exactly 1.063 cm long. Internet sites show a range of lengths. For example, Japanese Beetles: 8 to 11 mm (0.8 to 1.1 cm).

Hey! Writers of this stuff, please show us how you make such accurate measurements of these insects. We would love to see a demonstration of your amazing measurement skills.

Tony buys a used car

The sample problems at Smarter Balanced are interesting. From their email summer 2012:

"Recently, Smarter Balanced released sample items and performance tasks to help teachers, administrators, and policymakers implement the Common Core State Standards (CCSS) and prepare for next-generation assessments."

You can access the sample items and performance tasks on the Smarter Balanced website at:

http://www.smarterbalanced.org/sample-items-and-performance-tasks/.

Click on Mathematics under Explore sample items and performance tasks for the various grade levels. Clicking on a task or problem causes it to be downloaded to your desktop.

We like most of the problems, but wonder if the problem about Tony buying a used car might be a tad un-real-worldly. Click here to access this problem and others online.

Tony is buying a used car. He will choose between two cars. The table below shows information about each car.

Car Cost MPG Estimated immediate repair cost
Car A $3200 18 $700
Car B $4700 24 $300

Tony wants to compare the total costs of buying and using these cars.

Tony estimates he will drive at least 200 miles per month.
The average cost of gasoline per gallon in his area is $3.70.
Tony plans on owning the car for 4 years.

Calculate and explain which car will cost Tony the least to buy and use.

We like the problem, but we wonder how many people drive only 200 miles per month (2400 miles per year).

At http://www.fhwa.dot.gov/ohim/onh00/bar8.htm we found U.S. Department of Transportation data:

AVERAGE ANNUAL MILES DRIVEN, BY AGE GROUP (MALES)

Age Miles per Year
16-19 8,206
20-34 17,976
35-54 18,858
55-64 15,859
65+ 10,304

Average for all male drivers: 16,550 miles per year

We like the problem, but we would like it more if it used typical real-world data.

We wonder: Why use such un-real-world data? Is there a reason that we don't know about? If yes -- help! -- please explain.

A related problem popped into mind: The cost of gasoline fluctuates. Is assuming that the cost of gas will be $3.70 per gallon for four years a reasonable assumption?

For given costs of gas (say, three or more different costs), at how many miles is the total cost of buying, fixing, and driving the two cars equal?

That triggered another question. The problem apparently assumes that, after the immediate repairs, neither car will require additional repairs during the four years. Is that a reasonable assumption for used cars that cost $3200 and $4700? No additional repairs in four years? Hmmm. Would it be reasonable to assume that the cost of repairs will be about equal for the two cars? After all, at the time of purchase they differed considerably.

Hey! What say we expand this into an investigation and include other costs of buying a car and using it for N years? For high school students, insurance is a big yearly cost. How does four years of insurance compare to the cost of the cars? Is it the same for both cars?

More questions are swirling about in the chaos of our minds. We will spend some happy time searching the Internet for lesson plans about how to buy a car.

We are writing this on 2013-01-01. Happy New Year! Bob & George

The water lily problem

For ten years the MathNEXUS published a Problem of the Week and other math education content. The author is now retired and has decided to take a break from this task. Quoting from the referenced site:

Important Note: The website MathNEXUS was created in August of 2005. New content has been added weekly. In many instances, the website has achieved its purpose: To enhance the teaching of mathematics at the secondary level.
But, perhaps all good things must eventually change...which is why I have decided to stop adding new content to the website. All of the previous content will remain available via the Archives...ten years of humor, problem, statistics, websites, resources, etc.! View this "break" as an opportunity for you to search the Archives section for content missed during the past ten years.
Please know that I have created two new websites that complement MathNEXUS. With a focus on the history of mathematics, a great number of varied resources are offered. See the Old Math Problems and the Teaching the History of Mathematics. Please give them your consideration...for use and enjoyment.

Here is a problem published in the Sunday 2/2/2013 issue. It is a amusing example of a math problem divorced from reality.

A water lily has a single leaf floating on the surface of a pond. The leaf doubles in size every day. After 16 days it covers the whole pond. How long will it take two such leaves to cover the whole pond?

Hmmm. If the problem talking about two water lilies each with one leaf, or one water lily with two leaves? Does this make a difference?

We think that the intended answer is 15 days, but we have a few reservations.

How does the water lily leaf double in size? Do its linear dimensions each increase by a factor of square root of two? (Hmmm. What do we mean by a leaf's linear dimensions? It seems like a significant challenge just to exactly measure the area of a leaf.) If so, does this actually exactly double the area of the leaf? That is a good math problem to ponder.)

We used the Bing search engine to find images of water lilies and found many. A water lily leaf is sort of oval with a chunk cut out of one end. The chunk is roughly triangular, but with curved sides. As a leaf grows, does it preserve its exact shape, or does its shape change over time?

Is the shape of the pond similar (ala geometry) to the shape of the water lily leaf? Is the area of the pond 2^16 times the area of the water lily leaf before it begins doubling? (Have we got that right?) Is the water lily positioned in exactly the right place in the pond before it begins doubling in size? If yes, then we can imagine the area of the water lily leaf doubling away until it covers the pond exactly.

What if the water lily is close to one side of the pond? Then, after a few doublings, part of it would be in the pond and part of it would bump up on the land near that side, and it would require more doubling to cover the pond.

But what if the shape of the pond is not similar to the shape of the water lily? Maybe it is square or rectangular or circular or perhaps irregular in shape as ponds are wont to be.

Now think about two water lilies. Are they side by side? End to end? At two random locations in the pond? And think again about the shape of the pond. Wherever the two water lilies are in the pond when they both begin doubling, might there be uncovered water between them? If their centers of mass remain in their original places, won't they overlap as they double?

Oops! What if the water lilies continue doubling after they cover the pond? How much territory will they cover in 17 days? 20 days? 100 days? In how many days will they cover the Earth?

Wow! This problem seems more like an investigation or a project. Thanks to MathNEXUS for posing such an interesting ponderable problem.

Cutting a ballot in half

The following math problem example is quoted from Annie Murphy Paul's The Brilliant Report, available at http://us2.campaign-archive1.com/?u=bc04df008d4705e4e77c2eb35&id=2362d24fc9&e=9fa2a82024. (Note added 3/13/2016. To access this specific question, see http://ww2.kqed.org/mindshift/2012/02/24/whats-your-best-guess-predicting-answers-leads-to-deeper-learning/.)

To test their theory that making predictions would facilitate learning, Lisa Anne Kasmer of Grand Valley State University and Ok-Kyeong Kim of Western Michigan University designed a lesson plan in which the teacher started off the class with a series of prediction questions. Students were asked to imagine, for example, that a boy named Alejandro was cutting a paper ballot in half, then in half again, and so on. “If Alejandro makes ten cuts, can you predict how many ballots Alejandro might have?” the teacher asked. She followed that question with another: “What is your reasoning?”

I found it a little confusing to try to understand the meaning of cutting the ballot in half, and then in half again. I assumed the meaning was that "cutting in half produced two equal-sized and shaped pieces of paper," and "cutting in half again" means to "cut each of the two equal-sized pieces of paper into equal-sized and shaped pieces of paper." This cutting process is then repeated a number of times.

Now, the question is, “If Alejandro makes ten cuts, can you predict how many ballots Alejandro might have?” Here my thoughts:

  1. There was only one ballot. No matter how much cutting is done, there is still only one ballot. It makes no sense to believe that the cutting process produces more and more ballots.
  2. My interpretation of cutting the ballot in half and then cutting each of the two halves in half is not what the teacher had in mind. The teacher had in mind making a stack of the two equal halves, and cutting that (two paper thick) stack in half by one cut. So, after two cuts, there are four equal-sized pieces of paper and they are to be stacked into a stack that is four paper pieces thick. My way of explaining or imagining the cutting process took three cuts to produce four pieces, while the teacher's process took two cuts. But, imaging the difficulty that arises as the stack of pieces of paper grows thicker and thicker. Eventually a child cannot cut the stack! The first cut produced 2**1 = 2 pieces of paper. The 9th cut produced 2 ** 9 = 512 pieces of paper. A ream of paper is 500 sheets. Have you every tried to cut such a stack of paper?

Along with the difficulties listed above, I have trouble imagining when a student might want to cut a ballot in half. Would this be a political statement? Were there lots of unused ballots in a recent election, so the paper was available to students to practice their cutting techniques?

Perhaps the ballots were printed on only one side of a piece of paper, and the students were making note pads from the scrap paper. Students could explore possible shapes and thicknesses of note pads that can be made in this manner.

Catching a lot of fish

Here is a math problem found on the Web:

Emily loves to fish during summer vacation. The first day, Emily catches 3 fish. The second day, Emily catches 6 fish. The third day, Emily catches 9 fish. The fourth day, Emily catches 12 fish. If this pattern continues, how many fish does Emily catch on the eleventh day? Show all of your mathematical thinking.

Hmm. Does Emily have a fishing license? Is there a daily limit on the catch? Is it realistic to think that each day she catches exactly three more fish than the previous day, day after day?

Weekly Spending on Entertainment

Here is a problem from a booklet of material prepared for sixth grade teachers.

When Susan was 11, she spent about $5 on entertainment each month. When she was 13, she spent about $15 each month. If this rate of change continues, how much will she be spending when she is 15? When she is 17? Draw a line graph that shows how her spending changes each month across these years?

Two things about this problem seem bothersome. First, is it in any sense realistic? Can one correctly infer other values of a function when the values for two points in the domain are given?

The second point is the really bothersome one. The statement "Draw a line graph..." is a clue to what the problem poser seems to have had in mind. Evidently the assumption is that the two given points are points on a (straight) line. But, the term "line graph" does not have that meaning. Even if we just stick to polynomial functions, there are an infinite number of them that will pass through the two given points.

Next example

Remember, this is a work in progress. Over time, more examples will be added. Readers are invited to send their own "divorced from reality" word problems to Dave Moursund: moursund@uoregon.edu or to Bob Albrecht: bob@geekclan.com.

Resources from IAE

An introduction to college math placement testing. Retrieved 3/13/2016 from http://i-a-e.org/iae-blog/entry/an-introduction-to-college-math-placement-testing.html.

Becoming a better math tutor. Microsoft Word file: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/220-becoming-a-better-math-tutor-2/file.html. PDF file: http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht/221-becoming-a-better-math-tutor-3.html.

Communication in the language of mathematics. Retrieved 3/13/2016 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics.

Computational thinking. Retrieved 3/13/2016 from http://iae-pedia.org/Computational_Thinking.

Folk math. Retrieved 3/13/2016 from http://iae-pedia.org/Folk_Math.

Free online math education staff development course. (This is a very detailed syllabus suitable for self-study.) Retrieved 3/13/2016 from http://i-a-e.org/iae-blog/entry/free-online-math-education-staff-development-course.html.

Good math lesson plans. Retrieved 3/13/2016 from http://iae-pedia.org/Good_Math_Lesson_Plans.

Grand challenges in math education. Retrieved 3/13/2016 from http://i-a-e.org/iae-blog/entry/grand-challenges-in-math-education.html.

Improving math education. Retrieved 3/14/2016 from http://iae-pedia.org/Improving_Math_Education.

Math assessment and math tests. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Assessment_and_Math_Tests.

Math education. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Education.

Math education digital filing cabinet. Retrieved 3/13/2016 from http://iae-pedia.org/Math_Education_Digital_Filing_Cabinet.

Math education quotations. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Education_Quotations.

Math education wars. Retrieved 3/13/2016 from http://iae-pedia.org/Math_Education_Wars.

Math maturity. Retrieved 3/13/2016 from http://iae-pedia.org/Math_Maturity.

Math methods for preservice elementary teachers. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Methods_for_Preservice_Elementary_Teachers.

Math problem-based learning. Retrieved 3/13/2016 from http://iae-pedia.org/Math_Problem-based_Learning.

Math project-based learning. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Project-based_Learning.

Math tutoring project. Retrieved 3/14/2016 from http://iae-pedia.org/Math_Tutoring_Project.

Mathemagical meanderings. PDF file: http://i-a-e.org/downloads/doc_download/238-mathemagical-meandering.html. Microsoft Word file http://i-a-e.org/downloads/doc_download/237-mathemagical-meandering.html.

Personal professional development for educators. Retrieved 3/13/2016 from http://i-a-e.org/iae-blog/entry/personal-professional-development-for-educators.html.

Play together, learn together: STEM. PDF file: http://i-a-e.org/downloads/doc_download/212-play-together-learn-together-stem.html. Microsoft Word file: http://iae-pedia.org/Good_Math_Lesson_Plans.ument from http://i-a-e.org/downloads/doc_download/213-play-together-learn-together-stem.html.

Problem solving. Retrieved 3/13/2016 from http://iae-pedia.org/Problem_Solving.

Transfer of learning. Retrieved 3/13/2016 from http://i-a-e.org/iae-blog/entry/transfer-of-learning.html.

Using brain/mind science and computers to improve elementary school math education. PDF file: http://i-a-e.org/downloads/doc_download/239-using-brainmind-science-and-computers-to-improve-elementary-school-math-education.html. Microsoft Word file: http://i-a-e.org/downloads/doc_download/232-using-brainmind-science-and-computers-to-improve-elementary-school-math-education.html.

Using math games and word problems to increase the math maturity of K-8 students. PDF file: http://i-a-e.org/downloads/doc_download/211-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html. Microsoft Word file: http://i-a-e.org/downloads/doc_download/210-using-math-games-and-word-problems-to-increase-the-math-maturity-of-k-8-students.html.

What is mathematics? Retrieved 3/13/2016 from http://iae-pedia.org/What_is_Mathematics%3F.

Word problems in math. Retrieved 3/13/2016 from http://iae-pedia.org/Word_Problems_in_Math.

Resources from Non-IAE Publications

Note to readers: Remember, this is a work in progress. Input from readers will be much appreciated. When you find a "doozy," share it with us and others. Send them to [moursund@uoregon.edu] or to [bob@geekclan.com].

We intend to locate and list good sources of good math word problems. Of course, this section will also contain more general references such as the one currently listed below.

Schoenfeld, Alan (1992.) Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. [Chapter 15, pp. 334-370, of the Handbook for Research on Mathematics Teaching and Learning, D. Grouws, ed. New York: MacMillan, 1992.] Retrieved 3/13/2016 from http://www.uio.no/studier/emner/matnat/ifi/INF4280/h08/Schoenfeld1992LearningThinkMathematically.pdf.

Authors

This Web Page was created by Robert Albrecht and David Moursund.