Fun with obfuscated definitions

I am teaching College Algebra and Pre-Calculus this year out of two college textbooks, Ratti & McWaters for the former and Blitzer for the latter. I happened to be teaching conic sections in both classes on the same day. Here’s the Pre-Calculus take on “parabola”:

A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, that is not on the line.

Simple, clear, exact.

And here is College Algebra (mind you, the “easier” course meant for those not necessarily taking any higher math):

Let l be a line and F a point not on the line l. Let P be the plane determined by F and l. Then the set of all points P in the plane that are the same distance from F as they are from the line l is called a parabola. Thus, a parabola is the set of points P for which d(F,P) = d(P,l), where d(P,l) denotes the distance between P and l.

Er, what?

Outline from my talk on Lessons from Game Design for the Classroom

This post is set to go up a few hours before my talk at MEAD. It’s meant primarily for those who attended but I figured others might be interested so I made it public. It is slightly cryptic in that a.) I show the actual websites and games I use as much as possible rather than slides, so I don’t have a linear Powerpoint b.) the slides I do have tend to be understated and don’t make total sense
without the talking to go with them and c.) there’s a fair amount affected by audience participation and ad libbing.

Game Design Presentation Slides

Instructions for puzzles
Puzzles
Puzzles 2
Puzzles 3

Pre-prologue: Pronouncing Mihály Csíkszentmihályi

THE ONE WHO KNOWS HOW TO PRONOUNCE CSIKSZENTMIHALYI

Prologue: What this talk could be about and what it is actually about

Game design in the classroom could refer to
a.) Analyzing mathematical properties of games (Angry Birds parabolas, Super Mario physics, Pac-man ghost logic)

b.) Having students create games

National STEM Video Game Challenge

Lincoln Elementary School

c.) Having students play games

World of Warcraft in the classroom

Teaching Educational Psychology with Neverwinter Nights

d.) Using ideas from game design to modify the classroom environment. <– focus will be here, although we will also consider using a game design angle to improve (c.)

I. Hint tokens

Participants are in groups. Set up gamification of talk: show prizes, set up leaderboard.

Give two hint tokens to each group.

Have them work on average speed problem.

Award points after set time limit (10 minutes?) as necessary.

II. Nonlinear lesson planning

Open style problem often presented linearly but not discovered that way by students.

Example from Adventure (Nonlinear lesson design)

Example from Bronze

Example from Treasure Hunt AD&D Module

Example of average speed problem just solved; have participants try their own structure diagram

Dicussion of using structure to anticipate student questions; what help to provide

III. Gamification

Modifying “mundane reality” to be games: badges, progress bars, levels, experience points, leaderboards.

Problem of renaming without substantive content change: “Experience points”

More substantive examples: socialPsych

Khan Academy badges

Mozilla Open Badges Infrastructure

3D GameLab (aka “Quest Based Learning”)

Gamification ethics (optional based on time, may also be moved to later portion of talk): Badges etc. accused of being behavioralist. What are the ramifications? Noschese criticism (may have the “disastrous consequence” of making pupils mechanically repeat lower-level exercises to win awards, rather than formulating questions and applying concepts.)

IV. Flow

Mihály Csíkszentmihályi measuring happiness, flow being the optimal point between too much difficulty and too little

Jenova Chen: Design Flow in Games

Demonstration of flOw game, discussion of other instances in gaming (Max Payne, Parodius)

Comparison to traditional scaffolding: not only gains in difficulty but reduces in difficulty as necessary

Demonstration of “paper” version of flow (using puzzles)

Downside of flow: gaming the system (intentional losing, artificial feel when too mechanical), difficulty of paper implementation. Possibility of implementation into Khan-like software.

V. The McLeod criticism and Devlin criticism of educational games

McLeod: Do most educational games suck? Graphics-based criticism.

Devlin: No need to to present traditional symbolic-based learning (“has to arise naturally out of the natural environment and have meaning in it”)

Terminology of 1st generation / 2nd generational / 3rd generation educational games

1st generation, drill with game attached: relation to “Soup Cans” problem of compromising environmental mimesis by presenting a nonsensical task (7th Guest example)

2nd generation, integration around mathematics.

Teaching through simulation, discussion of Orbiter, example forum post

3rd generation, complete integration of environment and mathematics; Physicus & Chemicus examples

My small-picture criticism on 1st generation: often only cursory gameplay, meant as excuse to drill. Can we improve an educational game as a game?

Single round of “vanilla” math basketball

The only “strategy” students can use is “get the answers right, get the shots”. Could we incorporate more strategy, that is, more choices that affect things at the game level?

(Group discussion of improvement, test of improved rules)

[Optional if time permits] Same treatment to classroom Jeopardy!

New Year type post

Some house-cleaning for the new year:

• I have done a new blogroll shuffle; three high school teachers (Sam Shah, Jason Buell, Daniel Schneider), three college-level bloggers (Robert Talbert, Peter Smith, Mike Croucher), and one “idler’s miscellany of compendious amusements” (Futility Closet). The old Mathematics links in my sidebar have now been moved to my Annotated Blogroll. Please note the disclaimer that this is only a small selection out of many good blogs.
• I’m working on a new video; feedback on my last one (Adding Logarithms with Austin Underhill, the Friendly Space Whale) would still be welcome.
• Speaking of feedback, I do plan on posting answers to the conceptual counting test at some point but only if some more brave souls attempt answers (thanks to Robert Hansen for having a go; emailing me is fine if you’re sheepish).
• The Visual Algebra book project is still alive, but I’m working on something smaller first (a novella, so to speak) as a test case that should be finished much earlier.
• While I’m going to have radio silence from now until my MEAD Conference talk, here’s a recompile of my last Queue of Planned Posts, with finished stuff removed and new stuff added. Let me know if you want something sooner rather than later.

  Ancient math history:
  What’s the earliest mathematical artifact? (part 3)
  What’s the earliest mathematical artifact? (part 4)
  On the Ancient Egyptian Value of Pi redux (Do the Pyramids encode pi?)
  On the Ancient Babylonian Value of Pi redux (more information and scans from a tablet)
  On the Ancient Phoenician (Bible) Value of Pi

  Educational futurism:
  Textbooks of the future
  Classrooms of the future

  Logic:
  On alien logic [How one might think without AND, OR, or NOT.]
  Post’s lattice
  Set theory in paraconsistent logic

  Lesson plans:
  Teaching logarithms with Stevens’ Power Law [psychology experiment]
  Doing real mathematical research with high school students

  Psychology:
  Could mnemotechnics be useful in education? (These are the memory tricks used by folks at the World Memory Championship; failed attempts to reform education with them date back to the 19th century.)

Conceptual counting test

Roughly a year ago Edmund Harriss asked me to respond to his post regarding what he called “Mathematical Scales”, that is–

I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics.

Here is a test of counting in the natural numbers. The difficulty increases as the test goes on.

1. If 3 were dropped from the counting sequence 0, 1, 2, 3, 4, 5, … what would 2 + 2 equal?

2. The Yukis were located in modern day California and used a base 4 and a base 8 system. This came about from a natural form of body counting — how?

3. Subitizing is counting without having to enumerate; for small quantities this means a “a feeling of immediately knowing how many items lie within the visual scene” and for larger quantities this becomes an estimate. For example, it is possible to note how many triangles are depicted

without having to stop and mentally think “one, two, three” but rather “three” straightaway.

Explain how subitizing and the Roman numeral system are related.

4. Describe how counting and the commutive property of addition () are related.

5. Early in a child’s development counting tends to be just “recite the numbers in order.” Later they understand that counting enumerates a set. For example, one principle is that the last number counted equals the number of items. Name three more such principles.

6. Count from 1 to 10 (where each number represents our usual base 10 understanding of 1 to 10) in base .

7. The number of sense of babies is tested by checking for essentially, level of surprise. In an experiment of Karen Wynn, one puppet was moved behind a curtain followed by a second puppet; the curtain opened to reveal either two or three puppets. The babies were more startled by three puppets than two, essentially thinking that two was the “correct” solution to 1 + 1.

In a different experiment by Xu and Carey, babies were startled by the situation depicted below

but not by this one:

Explain the ramifications for the number sense of babies.

[Diagram from The Number Sense by Stanislas Dehaene.]

8. Explain the following quote:

As Russell also helpfully pointed out, one of the reasons it took so long to discover a “definition” of the natural numbers, is that the names of the natural numbers — e.g. “3″ — are used both as adjectives and nouns, and this distinction was not sooner realized.
– From Smullyan and Fitting’s Set Theory and the Continuum Problem

9. Here is a typical (although not the only) formulation of the Peano axioms:

I. 0 is a natural number.
II. If is a natural number, so is .
III. For any natural number , does not equal 0.
IV. For any and , if then .
V. Mathematical induction: For any set , these conditions are sufficient to show contains every natural number. a.) . b.) For every natural number n, implies .

What happens if the statement “0 is a natural number” is omitted?

10. Demonstrate why “every set of natural numbers has a smallest member” can substitute for the Principle of Induction in the usual Peano axioms.

11. The (admittedly radical) philosophy of ultrafinitism denies that the infinite set of natural numbers exists. Give one possible justification for this philosophy.

12. Number can be defined without sets or classes.

What is the major downside of this formulation?

13. The now-standard formulation of natural numbers in set theory (due to John von Neumann) is , , and so on. Give an alternate formulation.

14. Design a set of axioms (without referring to previously formulated counting axioms) so it produces the sequence 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5, …

My upcoming talk at the 8th MEAD Conference

Out in Tucson, Arizona, if anyone happens to be that way:

8th Annual Mathematics Educator Appreciation Day Conference
Saturday, January 21, 2012

Session: Lessons from Game Design for the Classroom
Audience/Level: General Interest
Presenter: Jason Dyer
Description:
A trend in recent educational research has been not just incorporating games into the classroom, but methodologies from modern game design. Learn about nonlinear lesson planning, hint tokens, Csíkszentmihályi’s concept of flow, gamification, and more!

Note that this isn’t “math games for the classroom” (although a few will come up) but rather “what have we learned from game design theory that applies to educational design theory?”

I’ve posted about some of these things before; nonlinear lesson planning will be in its full-blown rather than prototype stages.

A small question on the slope formula

Why is the slope formula normally presented as

instead of

?

The latter sure seems to confuse students less.

The Educational Problem with Teaching Multiplication as Repeated Addition

The saga continues

In an article dated September 2007 entitled What is conceptual understanding? Keith Devlin mentioned that he wished teachers would stop “saying that multiplication is repeated addition”. This led to enough response email to spawn a dedicated followup article in June 2008: It Ain’t No Repeated Addition. This led to an Internet firestorm and many more articles by Mr. Devlin (one, two, three, four, five) reinforcing the contention that teaching children multiplication as repeated addition is both mathematically wrong and educationally destructive.

I chipped in marginally on the educational question (The “Multiplication is Not Repeated Addition” Research) and solidified my opinion on the mathematical end (“Multiplication is Not Repeated Addition” Revisited) but I had yet to finalize what I thought of the educational issue: does teaching multiplication as repeated addition really hurt students, or is it as Mark Chu-Carroll puts it “a choice between teaching children no intuition, and teaching them a pretty good beginners intuition”?

Since then I have taught several mathematics classes for K-8 teachers as well as helped review most of the recent “mathematics for elementary educators” books, so I’m ready for my final (?) post on the issue. Given Mr. Devlin’s “final” article has been followed by four more, I don’t know how true this will be, but nevertheless–

The short answer

It is fine to expose students to multiplication as repeated addition if great care is taken to avoid the problem of identical mental models.

The long answer

The research I’ve seen does not indicate a significant problem with multiplication-as-repeated-addition at the immediate point of learning; the papers with that goal seem to be of similar type to the paper I already wrote about of not being an honest comparison.

The big issue comes later, with problems like this:

For instance, 12- to 15-year-old pupils were asked how to find the cost of 0.22 gallons of petrol if one gallon costs £1.20. (They were asked only to indicate the operation and not to perform the computation.) The most common answer was 1.20 ÷ 0.22. When the same question was asked with “easy” numbers, such as £2 for the price of a gallon and 5 gallons for the amount of petrol, the pupils answered correctly: 2 × 5. When interviewed, the pupils did not consider it incongruous for the needed operation to change when the numbers changed.

– From The Role of Implicit Models in Solving Verbal Problems in Multiplication and Division

Yes, would could represent the above operation as repeated addition by considering 0.01 to be the unit. The mathematical possibility is irrelevant in terms of the clarity of the student’s mental model; multiplication as repeated addition is inadequate to the task.

Only one of the mathematics for elementary educator books I reviewed explicitly decreed multiplication was repeated addition and left it there; the rest to greater or lesser extents emphasized that students needed to be familiar with different ways of modeling multiplication, like the area model (“the area of a rectangle with side X and Y”) or the correspondence model (“for every X, there are Y things”). What the books fail to mention is it’s quite possible in early grades to “fake” using one model with another model.

The problem of identical models

A student fixed on multiplication as repeated addition can go a long way solely with that model, because with many problems it can be mentally handled in an identical way to other models. By way of example:

You have a brownie pan divided into 3 rows containing 5 brownies each row. How many brownies do you have to serve?

The most obvious model that applies here is an area model, but as given there’s no reason a student couldn’t simply apply repeated addition anyway: 5 for the first row + 5 for the second row + 5 for the third = 15 total, and done. This can proceed for several grades until a student reaches a problem like this:

You have a brownie pan divided into 3 rows containing 5 brownies each row. Unfortunately on the bottom row the bottom half of each brownie was burned and had to be cut off. In total area, how much brownie do you have to serve?

A fairly natural repeated addition can be rescued by considering the columns instead of the rows: . However, add a burn to the right edge

and the student is now lost. Yes, one could consider 1/4 of a brownie to be a “unit” but for a 5th grader this becomes too much, and situations start to arise like the aforementioned petrol problem where whole numbers and decimals used in exactly the same problem result in students providing different operations.

The verdict

I believe that sole fixation on any mental model is a bad idea, not just multiplication as repeated addition; a student who thinks of multiplication as the area of a rectangle would solve the petrol problem with greater difficulty than a student with a correspondence model. While teachers often (or least we train them to) pay attention to the “inner life” of the mathematical brain, they do not seem to be aware of the identical model problem until it’s too late.

One of my test questions for my K-8 teachers last semester was to write a word problem where multiplication as repeated addition didn’t work naturally. It’s possible with to force students out of their “comfort zone” and make sure they understand multiple mental models. The danger of multiplication as repeated addition is it can serve as a proxy for other models in early grades; as a “default model” the correspondence model is much more flexible and the one that should have greater weight. But it isn’t necessary to avoid multiplication-as-repeated-addition to such an extreme that mulplication tables are memorized without any notion of how they are derived in the first place (as was done in the early history of mathematics education).

A resource to try

Maria Droujkova’s site Natural Math has a page dedicated to different models of multiplication, and they also sell a poster. Here is a sample:

[Thanks to Alexander Bogomolny for sleuthing out the link.]

Adding Logarithms with Austin Underhill, the Friendly Space Whale

Adding logarithms, in space!

Previously: Q*Bert Teaches the Binomial Theorem

Focus on fixing bugs rather than overwriting procedures

Oftentimes students who make mistakes have a procedure for doing a particular piece of mathematics, and they always do that procedure correctly. It is just that their procedure is wrong in some way. It is far more helpful to fix what’s broken about their procedure than to repeat the instructions over again.

[From Diagnostic models for procedural bugs in basic mathematical skills, which has a staggering 1167 cites and well worth a read if you have access.]

One of the few tests of teachers to find a genuine correlate with teaching quality is the M.K.T. (Mathematical Knowledge for Teaching) designed by Deborah Ball.

Teaching depends on what other people think, not what you think.

– Deborah Ball

One of the key aspects of the MKT is to test if teachers understand student errors. Here are some released question samples from 2008 (link to the entire set):

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13. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students focused on particular difficulties that they are having with adding columns of numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:
Which have the same kind of error? (Mark ONE answer.)

a) I and II
b) I and III
c) II and III
d) I, II, and III

15. Mrs. Jackson is getting ready for the state assessment, and is planning mini-lessons for students around particular difficulties that they are having with subtracting from large whole numbers. To target her instruction more effectively, she wants to work with groups of students who are making the same kind of error, so she looks at a recent quiz to see what they tend to do. She sees the following three student mistakes:

Which have the same kind of error? (Mark ONE answer.)
a) I and II
b) I and III
c) II and III
d) I, II, and III

18. At the close of a lesson on reflection symmetry in polygons, Ms. White gave her students several problems to do. She collected their answers and read through them after class. For the problem below, several of her students answered that the figure has two lines of symmetry and several answered that it has four.
How many lines of symmetry does this figure have?

Which of the following is the most likely reason for these incorrect answers? (Circle ONE answer.)
a) Students were not taught the definition of reflection symmetry.
b) Students were not taught the definition of a parallelogram.
c) Students confused lines of symmetry with edges of the polygon.
d) Students confused lines of symmetry with rotating half the figure onto the other half.

(META PROBLEM: These are the “discards” from the main test, which is kept private. These questions were not considered good statistically and so are imperfect in some way. It’s hence interesting to ask: why were these questions in particular discarded?)

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With all this in mind, I’m trying to suss out what happened on a recent Algebra I test. Here are two problems, with instructions to simplify:

11. a + b + c + a + b + c + c

14. 4x + 5x + 7y – 2x

I had an enormous number of students getting 14 right but 11 wrong. If they did get 14 wrong it was because they overlooked that 2x was being subtracted rather than added, and very few made the mistake of writing what was being multiplied as an exponent. On the other hand, nearly half my students answered #11 with

instead of . I have been baffling since over what happened. I think it’s possible earlier in their education they were exposed to statements like but never the equivalent with addition. Still, it seems like that wouldn’t explain all of it; there has to be something inexorably tempting about the mistake just like my order of operations problem from a few weeks back.

ADD: This post takes on the challenge of bug-fixing students directly.

Answers to the priming questions

To follow up on this post about the psychological affect of priming on students taking math tests–

1.) What would happen if these two questions on a recent test of mine were reversed (that is, #4 was given as #5 and #5 was given as #4)?

Sam Alexander essentially got this one right in that students would be more likely to apply a difference of cubes to problem #4. The students would likely pull out a 6x to get and claim the cube root of 4 is 2. As the problem is originally written, the more common issue is to attempt difference of squares instead. It’s possible to do the factoring with difference of cubes if one does not insist the factors are rational (use ) but the flat-out mistake is more likely.

2.) The following is a multiple choice question with the image removed. If a student were to ignore the math content of the answers (that is, guess), which would be the most common answer?

The conic section depicted can be categorized as:

a.) Parabola
b.) Ellipse
c.) Cone
d.) Plane
e.) Hyperbola

This answer is nearly creepy, but the sort of thing that happens with priming all the time. Note the strong iteration of a single letter.

The Conic seCtion depiCted Can be Categorized as:

a.) Parabola
b.) Ellipse
c.) Cone
d.) Plane
e.) Hyperbola

Hence the most likely choice by randomness would be C.

EDIT: As adroitly pointed out in the comments the word “cone” is also suggested by the setup word “conic”.

Once you’re aware of priming you will start to notice its effect more often. I stay on the lookout for situations where a student gets a problem wrong not because they didn’t know the mathematics but because they were influenced by the priming in surrounding problems.