## Treating the virus 8 ÷ 2 (2 + 2)

Recently, another instance of an arithmetic problem went viral. As math teachers, we might get asked our professional opinion and I think some of us are getting frustrated:

Or maybe many of us? (nytimes)

8 ÷ 2 (2 + 2)

People argue over the answer being 1 or 16, since it’s unclear which order to perform the operations. If truly faced with this, we’d ask what the person meant, perhaps lightly scold them for writing ambiguously.

Dan said we should be horrified. It presents an image of math as symbolic manipulation, or numerical calculation. Or worse: math as a place where there is only one correct way to do things.

I don’t disagree entirely, but we have to acknowledge the virality as evidence that this is where people are in their understanding of math. It interests them on some level, at least at the level that they are confident in their own answer enough to seek validation or comparison with others.

If we want to move people forward, like we would any student, we meet them where they are and help them take their next step.

Virality not guaranteed, but I wondered what might interest someone who has delved into the initial problem. Could we find a problem just tantalizing enough to lead them towards more interesting mathematics? (What’s this tweeter’s ZPD?)

I took a quick stab at two followup questions:

1. How many answers are there?
8 ÷ 2 ÷ 2 ÷ 2
2. What numbers can go in the blanks so that all the answers are integers?
24 ÷ [   ] ÷ [   ] ÷ [   ]

Or in image format:

Shoutout to https://www.openmiddle.com/ for the format of the second question.

The first one is intended to lead from the viral question towards a solid belief that this style of writing produces multiple answers. Well– how many?

The second question turns out to be pretty juicy. You could find some easy numbers to make it work, but can you challenge yourself to find more? How many are there? How do you know? I also think it’s interesting how the ambiguity we learned about the division operator helps us concisely communicate this problem.

We can argue about 1 or 16. We can argue that we shouldn’t have to argue about 1 and 16. But I’d rather take those infected by order-of-operations virality as an opportunity to take them someplace interesting.

## Partition Problems can Differentiate To Many Learners

Marilyn Burns (@mburnsmath) recently blogged about a problem she came across via NCTM’s Teaching Children Mathematics and Mike Flynn

This Bike Shop Lesson ended up being rich not just for the students, but also for herself. In exploring the underlying structure, you can touch on many different topics. See Marilyn’s blog, Henri Picciotto’s blog, and Simon Gregg’s tweets for a more there!

As I read these thoughts from others it reminded me of a similar lesson I did with high school Algebra 2 students. It also dealt with partitions of numbers, but exploring different constraints than the unicycles, bicycles, and tricycles. In exploring those constraints, my students found some interesting patterns including Pascal’s Triangle, powers of two, the Fibonacci sequence,

## Traincar Number Lesson

Here are three trains with length 6, but they are made up of different numbers of cars and types of cars. How many different trains are there?

my notes on negotiating “mathematical difference”

I purposely left the question vague because I wanted students to interpret it in a variety of ways. Influenced by Yackel and Cobb’s 1996 article, “Sociomathematical Norms,” I wanted students to develop their own agreements in their groups about what counted as different.

[note: I did explicitly ask my students about trains of length 5 to start them off. I may choose otherwise in the future]

### Students refine the question

I gave the students cutouts (included in PDF) to support a few aspects of problem solving. First, the scaled cutouts support non-numeric representations. Second, the easy-to-change nature means students can re-arrange the blocks quickly and are not locked to what they have first represented. Third, they must formalize thinking from the cutouts to more permanent representations. And finally, the cutouts are easily worked by more than one person so it can promote discussion in a group.

Here are some of the thoughts my students had:

1. Is 2+2+1 different than 2+1+2 ? What if they’re the same?
2. What if you don’t allow copies of a car? (so 2+2+1 is not allowed because there are two “2s”)
3. Are all “1 cars” the same?
4. Can we use negative cars? (like a car of length “-2” for example)
5. Can we use partial cars? (“1.5”)
6. Are reflected trains the same? (2+2+1 and 1+2+2 might be the same, while 2+1+2 is different)
7. what if we are limited by how many cars? (only allowed to use exactly three cars, for example)

I was very impressed by how much the students embraced the chance to question the rules and develop their own. Its something we teachers should do more often! I expected to see questions 1 and 6 from the students, but the other questions each surprised me in some way. I learned a lot about what my students could do when given the chance. They revealed a lot of thoughts they had about permutations, combinations, sorting, number, structure… I was able to grab onto this knowledge later in the year as we formally talked about those subjects.

The freedom of exploration in the problem let students from multiple ability levels contribute. One student who was struggling was the one asking about negative car sizes. Its the kind of question that makes you pause; you may wonder if he’s asking about this does he understand the problem? I am very glad I let him explore it with his group. The answer, “infinite number of trains!” is much more meaningful when the student discovers it, rather than the teacher disallowing the investigation by providing the answer.

### Student work

Also, I was able to let the students experience the results of testing their own conjectures. Here are some results of their investigations: (there are some small errors in their work)

 1 3 4 7

Most groups found the pattern that with certain assumptions, the number of trains of length N is 2^N. But the exposure to other assumptions generating other patterns was a great place for a meta-discussion about mathematical practice, and the hidden structure of numbers.

Recursion appears as the students organized their work

This group discovers a link to Pascal’s Triangle

We discussed how some groups patterns fit together and how as a class we explored a lot of boundaries to the problem. We talked then about some of the boundaries we did not explore: such as limitations to only certain lengths of cars. (Notice the limitation to 1s, 2s, and 3s cars is the Cycling Shop Problem!

So when Marilyn and Henri were sharing ideas about the cycling shop problem I recalled my students work and how we were linking the various assumptions made into a larger structure. I played with thinking about the unicycles, bicycles and tricycles built from 8 wheels inside the organization of combinations and partitions.

How fun was it to come back to this problem myself and dig around for new concepts! And its not nearly done…

Limiting to only 1s and 2s has another fun result that I wont spoil here… try it yourself! try it with your students!) Another extension is to limit to prime length cars only. I concluded with sharing the Goldbach Conjecture:

Every even integer greater than 2 can be expressed as the sum of two primes

I asked my students what they thought, how they might approach it, how its connected to the traincars they’ve thought about. Then I let them know that the problem is unsolved. “You are part of the community of mathematicians… testing assumptions, making conjectures, organizing and sharing ideas.” It was one of my favorite lessons I’ve done. (Also a great way to lead into winter break… “Your HW for break is to solve Goldbach, have fun!”)

Traincar Lesson PDF – this is ‘localized’ with a playful name for my school and timing in the year (“Skyline Express”) but also I gave it as a participation quiz, so some of the introduction of the task in this pdf is specific to that classroom format.

## Differentiation to all levels

Notice aspects of these partition problems have piqued curiosity of students from elementary school, to high school, to teachers, and of course to professional mathematicians. When the learner has a chance to refine the question for themselves and explore their own ideas, the task is becomes personalized. And when the tasks are personalized then we can make them socialized: people sharing their own ideas and work.

## References

Skyline Express Lesson Materials PDF – Scott Farrar 2013

NCTM – Teaching Children Mathematics – The Cycling Shop http://www.nctm.org/Publications/Teaching-Children-Mathematics/2016/Vol23/Issue1/The-cycling-shop/ August 2016

Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in Mathematics.Journal for Research in Mathematics Education, 27(4), 458-477. doi:1. Retrieved from http://www.jstor.org/stable/749877 doi:1

## Solving real* problems with compass and straightedge

In Geometry, the unit on constructions usually begins with demonstrations and practice copying a line segment, copying an angle, bisecting a segment, bisecting an angle. These are treated as building blocks, implicitly promising more detailed constructions later. And indeed, pretty soon the unit will have constructing a parallel line through a given point and constructing a perpendicular through a point. But learning “building blocks” too often slips into disconnected procedure practice. The justification usually becomes “you’ll need it later.” Not only is this thoroughly unsatisfying to the learner, but sometimes when we get to ‘later’ we treat that topic too as disconnected procedure.

What a student stuck in these types of classes must think! The future is promised to be full of interesting problems, but the present must be slogged through.

Let us bring the interesting problems into the now. What are the problems that are solved by the use of this skill?

A few years ago I and another teacher adapted a lesson from Dan Meyer that based these problems in the statement: “the compass measures distance.” Bay Area College Map Lesson Plan (PDF) A question asks, “How far is College of Marin from SFSU?” How do you do that? We could use a ruler, measure the map distance, measure the scale, and find the proportion. Or, you could eyeball the scale, or use your thumb and finger to approximate its copies. Notice both of these have a similarity to the actual compass and straightedge construction.

If you’re measuring the scale and the map distance, you are essentially copying the length of the scale segment onto a line between the two points. This is the copy a segment construction. Don’t worry about them reaching for a ruler at first– the questions are easier without one. But note also the discussion possibilities if we ask how that ruler compares to the compass. Given 1 inch, the rest of the markings are exactly what you’d make with your compass.

This lesson also encourages the concept that the circle drawn by the compass is the set of equidistant points from the compass’ center. Its the definition of a circle, of course, but this definition becomes actionable if we ask “are we closer to Cal or Mills right now?” We don’t have to jump to perpendicular bisector, instead we can do the slow way: where are all the points that are 10 miles from Cal AND 10 miles from Mills? Two circles get drawn. 5 miles from each? two more circles. 8 miles from each? two more circles. A pattern may start to emerge. Don’t be surprised if the students propose to draw the line between all those intersection points.

## Constructions Course Plotting

This past month I’ve observed a few classrooms doing compass and straightedge introduction. After showing how, the teacher may say to practice it some number of times. But often students papers have only imitations of the compass marks and sketches that are obviously not exact copies. This may be confusing to teachers as the whole point is to “make a copy.” But if the student isn’t doing it, they aren’t stupid, its just the task is meaningless. Literally meaningless because they do not note what the important properties of the procedures are. The important properties of the procedures have a high word count to output ratio as well– “place the center of the compass at one end of a segment and open the other end to the other endpoint” yikes.

Well lets try to pose a problem so that students need to copy segments and angles in order to complete it. What I’ve been brainstorming with is essentially “get from here to there.” Level one is shown to the left.

The rules:

1. you may only travel in full lengths of BC (given)
2. you may only turn in full angles of FDE (given)
3. you may start in any direction

From these prompts, the students need to copy segments and angles. And they are allowed to “go” in a way that enables more creativity. Informal solutions (non constructions) are also acceptable because its completely reasonable to try something informally before formalizing it.

There are multiple solutions but the points are specifically chosen so that the start and end are not a multiple of BC. Student solutions can be gauged by how close they get to the finish, providing some motivation for “better” solutions but notice that the quality of their constructions is a separate measure.

I actually began this idea with the harder version in mind: put two random points on a large piece of paper, and draw blobs in the middle. (see image) Given a single segment and a single angle, can you use copies of them to make a path from start to finish without hitting the blobs?

There were a few things that jumped out at me as I thought about this. First, students will probably copy way more angles and segments this way than you’d be comfortable assigning in a drill. (and that’s good!) Other things that I wrote as I was thinking about the implementation and potential of the lesson

1. The segment should be different than the width of the angle at the segment’s length away from the angle’s vertex. (What a mess of words — but essentially it means if the segment and angle require the compass to be almost the same opening then it can get confusing as to which measurement you have in your compass
2. copying a segment becomes pretty straightforward (ha hah) but occasionally you’ll need to extend your target line — and experiencing that need is valuable to the students since it is difficult to describe in words.
3. copying the angle requires changing the compass a lot — expect some struggle (but this is what you want them to overcome)
4. the random placement of islands may prevent a solution from existing, but discovering that is powerful. Adaptations: maybe you’re allowed to go off the paper? or… see #8
5. An easy level (like level 1 above) should probably be done first. Need to design it to require each of a segment and an angle.
6. Medium level is like I’ve pictured here, or ones in which the teacher (before class) plots a solution route first, then places islands to design the level.
7. Hard levels might be ones you let the students design for each other. These wont necessarily be hard, but just high variance of difficulty.
8. An extension: if a level is particularly challenging, you could “allow” the students to bisect one segment or bisect one angle– and use that half-sized item once. Students faced with this choice will need to evaluate which choice is best– thus potentially practicing the bisect skill a few times.
9. Elements of the parallel line construction can come out automatically, as students copy angles in the manner of corresponding angles on a transversal or alternate interior angles.
10. Speaking of which– the angles on transversals and parallel lines come out of this activity naturally as well. Students may conjecture about congruent angles on parallels lending you some fodder for discussion now or when you bring up that unit later on.
11. The underlying structure from a single segment and angle is a parallelogram grid. This can be useful to help you evaluate solutions but also can be discussed in the sense of it being an entry point into the algebra of constructible numbers. Not that you need to go into the concepts in detail, but you can lay some groundwork that

Further, I think there is ample opportunity for students to come up with creative solutions to a given level. Since the first direction is arbitrary, students are likely to have differing solutions anyway, and those can be celebrated. Students can look at each others work and notice similarities in the small issues confronted and solved (getting around an island) and also help each other with the skills without it being “the answer” to the problem at large. Students may be interested in improving their solution by doing it again with different choices. I can imagine a brilliant wall with dozens of student maps posted all over it!

If you try out this idea or something related to it, I’d love to hear about it! Here are some related resources I’ve already received:

## Realness

Finally, the “realness”* of the problems here doesn’t rely in them being “realworld.” They are real in the sense that they can be answered by the use of the skill in question, perhaps in addition to accessing prior knowledge. This is in contrast to fake problems in which we say “practice the skill 3 times.” The main difference is that a real problem can be attacked without the skill– but the skill improves the solution. A fake problem asks directly for the skill so that it becomes the only possible solution.

What do you think? is that geogebra applet problem real or fake? Its very close, I say. A more fake version of this question would do entirely all the pre-processing for the student, telling them directly “copy EF”. Which is what I think many constructions lessons tend towards. I say the real-ness of this problem comes from the sense that I can provide a reasonable answer without using the compass and straightedge, while those tools would certainly improve my result. But– there certainly is a single right answer, and the construction is just about the only way to do it formally (if we assume pythagorean theorem to rely upon the construction). So– to make it more real we shake up the goal. The course plotting activity above is the shakeup: we have to get from start to finish using copied segments (and angles) but the students have agency in how those tools get used.

The goal here is compass and straightedge constructions. Forget “we’ll need this later” lets “need this now” !

## Evaluative Listening and Khan Academy

Dylan Wiliam writes about teachers listening to student responses,

When teachers listen to student responses, many focus more on the correctness of the answers than what they can learn about the student’s understanding (Even & Tirosh, 1995; Heid, Blume, Zbiek, & Edwards, 1999). It is easy to identify such teachers because when they get incorrect answers from students, they respond by saying things like, “Almost,” “Close,” or “Nearly; try again.” What the teacher is really saying is, “Give me the correct answer so that I can get on with the rest of my script for the lesson.” Brent Davis (1997) called such teacher behavior “evaluative listening.” Teachers who listen evaluatively to their students’ answers learn only whether their students know what they want them to know. If the students cannot answer correctly, then the teachers learn only that the students didn’t get it and that they need to teach the material again, only, presumably, better. (emphasis mine) (Chapter 4, Kindle Locations 1761-1768). Wiliam, D. (2002). Embedded formative assessment. Solution Tree Press.

Does Khan know why a response is wrong?

Consider how similar that is to Khan Academy’s current capabilities of assessment. If KA only collects the student response and evaluates it as binary correct or not– then KA only learns that the student didn’t get it and offers to reteach it (with a video) or give a hint. Furthermore, both the video and the hint are non-personalized since they do not account for what the student’s input was.

Wiliam distinguishes a separate type of listening: interpretive.

“What can I learn about the students’ thinking by attending carefully to what they say?”

Can Khan Academy do interpretive listening? Can technology in general do this? Dynamic Math Software like Geogebra or Desmos might interpret your input by attempting to incorporate it into the model it is presenting–but is that listening? Try an example:

I see it as much more valuable feedback than what KA offers, since the Geogebra feedback contains more information. A response like 6.8 now shows that what you have entered does not match the blue function. However it is still not as flexible as a human teacher, that could not only interpret the narrow numerical responses but also take an input like “why do we measure amplitude from the midline?” nimbly. KA offers a video on the subject but you’ll have to go looking for the answer to a specific question.

Of course, a teacher might be best served by a combination of Geogebra and their own human capaiblities. A geogebra applet set up ahead of time anticipating certain responses can aid the teacher in the dialogue with students. The computer can graph accurately, instantaneously, repetitively, and in parallel (multiple users at once). The teacher can use Geogebra to augment their own interpretive listening and augment the information feedback to the students. The KA question here does not augment; its listening is solely evaulative and its feedback is non-personalized and non-specific. A teacher using this KA lesson does not teach more efficiently: either the teacher abandons the student to the software’s help or the teacher supplants the software’s help by helping themselves. In either case, the teacher and KA have a substitute relationship not symbiotic relationship.

## CMC North Asilomar Wrapup Part 2! Technology: Conceptual Understanding and Intellectual Need

A continuing wrapup / reflection on the sessions I attended at the California Math Council Northern Conference, Dec 11-13.

 Part 1: Fri 1:30 – 4:30 Design Principles for Digital Content Part 2: Sat 8:00 – 9:00 Annie Fetter of Math Forum Using Technology to Foster Conceptual Understanding 9:30 – 10:30 Steve Leinwand Mathematics Coaching: An Essential Component of Quality 11:00 – 12:00 Enacting the Gold-Standard in Teacher Education Part 2: 1:30 – 3:00 Eli Luberoff Technology and the Intellectual Need 3:30 – 5:00 Michael Fenton My Journey From Worksheets to Rich Tasks 7:30 – 10:00 Ignite! Sessions

Annie Fetter of the Math Forum gave the early morning session on Using Technology to Foster Conceptual Understanding (2014 version of presentation– very similar), while Eli Luberoff gave his second session on Technology and the Intellectual Need. Both sessions focused on how to implement technology into a classroom that is meaningful to student learning. I’m of the opinion that implementing technology in a class should be thought of in the same way as implementing a pencil: its only as what you’re going to do with it. “Implementing technology” is a phrase overloaded sometimes in education. “We must prepare our children with 21st century skills!” Does that mean that students should be using a word processor or a spreadsheet in class? Does it mean that students should be programming or scripting? Does it mean that students should be using answer clickers to say “B is the answer”? Playing computer games? Graphing equations on a calculator? Because there are so many interpretations, many of these things get accepted as “using technology” when they have vastly different levels of actual impact on learning mathematics or giving experience with “21st century skills.” While learning to word process or use other office tools is important, its not exactly the thrust of the math classroom. And while answer clickers or smart boards can make certain logistics of the classroom more efficient, they are not centered on math content. So what kinds of “tech implementation” are good vehicles for mathematics learning?  Ms. Fetter and Mr. Luberoff are here to tell us some!

Fetter begins with demonstrating a triangle activity made with Geometer’s Sketchpad. You can see the materials here, and for those without Sketchpad I made the interactive in GeoGebra here. With this activity, students are asked to drag the points of the triangles around and make observations. Fetter has developed since 2007 a method called “Noticing and Wondering” (described in detail in Max Ray’s book Powerful Problem Solving) in which she collects things that students *notice* about a math diagram or picture or situation before asking any specific questions. This means it “supports students to find as much math as they can in a scenario, not just the path to the answer.” (Ray, 2013) Furthermore, it allows communication between students as they see what others are noticing, providing ideas that they themself may not have thought of. The “wondering” part is intertwined, and works similarly: as the students drag these points around, is there something that they wonder about they observe? Perhaps “I wonder if two triangles can be the same” is a mathematically vague statement– but if that is the level of the student’s precision, then it is a teachable moment as the teacher allows the class to draw out all possible interpretations of that wondering with the class. After all, knowing the precise vocabulary probably means that the student is beyond the task, as a delicate use of terminology might be one of the last things that gets layered onto a conceptual idea in one’s mind. Why would we need* (more on this later!) a word for some property if we had never encountered such a property? In order to get to the point of mastery, a student must dig into the features and properties of the concept as their vocabulary is developing. Noticings and Wonderings help develop the class discussion.

### Tech Manipulatives

So what is the point of the technology here? The Sketchpad activity serves as a manipulative. Its not fancy, its not exactly a “21st century skill”, but it does things that other manipulatives cannot. Manipulatives open up the visual/physical communication pathways to learning about a concept, providing support for the much more difficult linguistic pathway. But manipulatives have weakness in that they cannot be as precise as a linguistic communication, or the manipulative holds some property that is not true in the abstract. However, a technological manipulative can help be both more precise and have less irrelevant properties. Fetter demonstrated this via the Algebra Tiles example. Frequently when students use Algebra tiles, they are tempted to “measure” the x tile, since those tiles must have a constant length to exist in the physical world. But its constant physical representation is at odds with the nature of what it is supposed to represent: a variable. On a computer based Algebra Tile set: the x length can change, so that there is an easy way for a student to separate the cases of “my layout is true when x = 4” and “my layout is true for all x values” by scrubbing through lengths for x. Another issue with physical Algebra Tiles is that they are necessarily 3D. We typically ignore thickness easily, but a length x and an area 1x are assigned to the same tile. The computer based Algebra Tile set Fetter opted not to fix this, but it would be possible to have lengths be represented by 1D objects, and areas by 2D objects on the screen, while this is impossible in the real world.

Sketchpad/Geogebra also allow showing and hiding things from the abstract concept impossible to access in the “real world”. Consider that JKL is always equilateral, but can have segments of any size. This is not a physical object. JKL represents an entire class of triangles. JKL = {all triangles such that JK = KL = LJ} Meaning, when we click and drag on part of it, we scrub through the infinite set it describes to display another single element. Meaning that students can investigate these abstract properties empirically. They can explore every voiced and unvoiced conjecture they have about the triangles.

### Tech in a supporting role

This is much the same with the other applets Fetter demonstrated. In each one (Runners, Galactic Exchange, Algebra Tiles) the information a textbook might confine to complicated academic language is instead reformed into an interactive, graphical format. Fetter notes that technology can generate the situation shown from diagrams, but that it further gives a place to explore and experiment, revealing information when the student “asks” for it via interaction. So the technology is this aid on the ladder of abstraction. Building conceptual knowledge with tech means to use it as a stepping stone to building the student’s mental structures. Fetter’s demonstrations also show that tech is still just an element of the classroom, not a replacement. We teachers are still having discussions with students, we are still promoting discussions between students. Tech serves to lower the burden for the entry points into a task: everyone can drag a point on a screen. Students who have the word isosceles in their vocabulary are on equal footing with those who do not: but both can notice if two sides of a triangle stay at equal lengths as the triangle’s points are dragged. And in the ensuing discussion, the class’ knowledges and experiences are combined and redistributed– all the more powerful because more students were able to engage with the properties.

### Intellectual Need

Eli Luberoff touched on tech implementation from a related view. Intellectual Need is a term used by Guershon Harel. “For students to learn what we intent to teach them, they must have a need for it.” (Harel, 2013) Where the intellectual needs can be thought of as (1) the need for certainty, (2) the need for [logical] causality, (3) the need for computation, (4) the need for communication, (5) the need for connection and structure. Frequently, mathematics is taught without considering these intellectual needs. Harel describes an example of a problem lacking need:

A student has a snow-shoveling business, and charges \$100 per customer for unlimited shoveling. However, he discounts the price by \$1 per customer for each customer over 20. What is the largest amount he can earn?

Right away, we’ve been told that there is a largest amount– when we may not have considered the possibility. A slight change to this asks a vaguer question, but also asks about a parameter, not the value:

A student has a snow-shoveling business, and charges \$100 per customer for unlimited shoveling. However, he discounts the price by \$1 per customer for each customer over 20. How many customers should he have?

Notice that the answers to these questions are the same point: the maximum of f(x) = (20+x)(100-x), which is (40,3600). But in the second version, \$3600 is the justification for the 40 customers, while the first version \$3600 is the direct answer. The second version uses f(x) as a tool to solve a need: “how many customers?” while the first version uses f(x) as the object of the problem. Note that the business model is equally ridiculous in both cases 😉 but the second version deals more directly with the situation by asking about the parameter, and letting the properties of the output value be discovered. No matter how silly the context is, it becomes “real” when the students have agency in investigating the parameter. We teacher should trust that the concepts we explore are special enough to be revealed on their own merits (the maximum point becomes interesting when digging into different customer numbers). Asking about the maximum point before we were aware of needing it results in students not understanding how or why such a point is special.

Luberoff’s modeled another example along these lines: Dan Meyer’s “Pick a Point” lesson. I personally have done this lesson in the first week of all my Geometry courses since seeing it way back in the Classical dy/dan Era. (nothing against the current era, Dan!) Actually just now in looking back at that post I loved this quote:

This math thing is easier to approach if I ask myself, what about this concept is useful, interesting, essential, or satisfying, and then work backward along that vector, rather than working toward it from a disjoint set of scattered skills. There is probably a book I should read somewhere in all of this.  – 2009 Dan

2009 Dan eventually came across Harel’s works on Intellectual Need, and of course shared the ideas with Luberoff at Desmos. The idea is that the basic principle of naming a point in Geometry should be treated as a “resolution of a problematic situation” rather than a discrete practice looked down upon as an automatic prerequisite. I feel we can really appreciate the unexpected gaps that some students present to us as indicators not that they “never learned” the concept, but rather that they were never confronted with a need to use their knowledge beyond satisfying the teacher’s requests, “Label your points!”

### Discovery and Feedback

So when learning, it is useful to experience the absence of a concept so that we feel resolution when the ideas come to solve problems for us. We might even formulate descriptions of what we need or invent, in a similar manner as historical mathematicians, methods that build upon previous ideas. “Youngsters need not repeat the history of mankind but they should not be expected either to start at the very point where the preceding generation stopped” (Freudenthal, 1981) connects to Harel’s idea that “it is useful for individuals to experience intellectual perturbations that are similar to those that resulted in the discovery of new knowledge.”

So where does technology come into this? Luberoff asserts that discoveries are born from simple questions, and that they require quick and useful feedback.

While a computer might be good at quick, it is not necessarily useful feedback. A Khan Academy “interactive” lesson can only supply a single bit of information back to you (true= your answer matches ours, false= your answer does not match ours). While it gives this feedback quickly, it is not that useful since it is based only upon the few characters you inputted yourself. Luberoff also showed slides from the SBACC released questions, which also came up quite short on the feedback. (They also had some interesting input design failures which I wrote about here, and which Steve Rasmussen wrote in greater detail here)

Ok, tech doesn’t give us good feedback automatically, but it is possible to get good feedback from tech. Luberoff referenced linerider in which users (players?) get information from the path that the rider follows, based upon their inputs of drawing a line or curve. This feedback is useful and quick. The sled is directly and immediately affected by user input. (p.s. some of the linerider creations get quite crazy!) I noticed that we are also redefining what we may think of as feedback here. It is not necessarily lingual, it is not necessarily evaluative (i.e. not an assessment), and it is not necessarily constructed with the purpose of a singular idea. As the rider of linerider falls, the user sees an empirical result of their inputs that reveal clues along many ideas, including but not limited to acceleration, slopes, curvature, maxima.

This ties back to Annie Fetter’s presentation: the student dragging a triangle receives instantaneous and continual feedback in the form of triangle JKL’s changing and unchanging properties. These programs are set up to display a phenomenon as completely as we can, and as teachers we have to trust that there is more information conveyed by an interactive phenomenon than we can convey with lingual communication. Because, if I tell you “a circle’s radius can be wrapped around the circumference about 6.28 times” that is just categorically less information than having you play with my geogebra sketch on the same topic.

### Case Study: Function Carnival

Luberoff then shows us some of the Desmos Activities that are designed with the intent of promoting intellectual needs for their related concepts, and designed to give continual useful feedback. (related: my thoughts of “Continuum Style” in lesson design) One of the lessons is Function Carnival. Students are prompted to draw graphs of small situations, like the height of a man fired out of a cannon. When they draw their graph and press play, they are shown their graph in comparison to the “actual” height in a new animation. The feedback takes their input and shows the result. It is not declaring right or wrong, it is showing, along every time value, the height you ascribed to it. This means students can modify and tune their graphs using the information they saw. “Oh my cannondude was too low in the beginning, I’ll change that part of the graph”

Additionally, analytical properties of functions arise as the students play. The formal definition of a function is wordy abstract thing, and how can a student appreciate why are we disallow more than one output per a single input if they have never seen what goes wrong in that case? Well, consider the student doodling in the function carnival and all of a sudden they will be halfway ready to explain to you what the problem with multiple outputs is. When you supply, “great! that means we should only have one output, actually when we limit one output for each input that has the special name, function” it is a resolution to our issue of multiple cannonpeople: the abstract mathematical structure of limitations on the types of graphs we can draw arises as a subthread from the simple task of graphing with quick, useful feedback. Luberoff showed David Cox‘s classroom interacting with this activity in a timelapse video. If you haven’t seen it, watch it now! Its an amazing view of how the students adapt to the structure of the lesson’s design and feedback style to create increasingly precise graphs. That video was really my favorite part of the day: it illustrates how the students tried, got frustrated, kept playing, and eventually took on more and more challenges. The Desmos Activity structure enabled Mr. Cox to talk with his students about so many ideas because they were all engaged directly with those ideas. And this was just the first slide of the activity!

### What else?

Luberoff demod a few more of Desmos Activities including a then unreleased one based upon the LineRider theme: Marbleslides.

Luberoff then concluded with reiterating “discoveries are deeply satisfying”. When students use desmos, there is a low floor but a high ceiling: easy to get started, but suggests many open ended themes allow students to explore deeply and without fear.

I think Desmos activities have some great educational design to them, but we can see from Fetter’s presentation that sometimes a technological manipulative can be quite simple. In both presentations, the focus of the technology was the students experiencing phenomena under the guidance of the teacher. The technology provided some boundaries, such as students being unable to make a triangle un-isosceles, and removed others, such as allowing a non-function to be interpreted. I highly recommend going to see both Fetter and Luberoff if you have the chance.

Harel, G. (2013). Intellectual Need. In K. R. Leatham (Ed.), Vital Directions for Mathematics Education Research (pp. 119–151). Springer New York. Retrieved from http://link.springer.com/chapter/10.1007/978-1-4614-6977-3_6
Ray, M. (2013). Powerful problem solving. Heinemann.

Coming up in Part 3: a shorter (I hope?) recap of the remaining sessions.

## iPhone

What a piece of work is iPhone, how noble in reason, how infinite in faculties, in form and moving how express and admirable, in action how like an angel, in apprehension how like a god.

Hamlet was one of the first people to get an iPhone but I was only 6 months behind! Scientifically speaking, it rocks.

Using elementary set theory, I will describe precisely every way in which it rocks, by articulating the complement.

{How the iPhone rocks}c

1. No Copy/Paste. Well, no text highlighting in general. These are essential tools, and I feel handicapped without them.
2. No GPS. This doesn’t sound so bad to most people, (“why do I need to know exactly where I am?”) but once you start playing around with the contacts and google maps then you realize you’d want some sort of functionality of “how do I get from HERE to there?” Basically, putting in GPS would have made the iPhone the closest thing to a URAT* modern society has to offer.
3. Needs a search filter on Contacts. As fun as it is to flick my contact list to scroll through them all, I would like some sort of shortcut to get to a specific person.
4. EDGE is slightly slow. But frankly, it works just fine for most of my needs. Even downloading stuff from YouTube isn’t bad with it. I sometimes even turn off Wi-Fi to go to Edge when the network signal is weak.
5. Calendar only syncs with Outlook 2003 or 2007 on Windows. I would much prefer it to sync with Google Calendar directly. You can access Google Calendar through the web, of course, but I want to use the iPhone calendar “without all that tedious mucking about in hyperspace.”*
6. Headphone jack is recessed. This is so the curved edge of the iPhone is still curved, and it actually protects the structure of the headphone jack, the internal card, and the plug. But it also means all standard headphones wont work without an adapter… \$25.
7. No 3rd party apps — yet. Sure I can go through Safari to do web-based apps, but that’s an inherently bulky way of doing things, and what did I just say about mucking about…
8. No direct interaction with files — yet. I can’t see the files I have on there (“the files are in the computer*) or do anything with them, such as saving a picture off the internet, deleting songs, editing tags, editing documents.

That’s it. And essentially, most everything will be great when Apple puts out their Software Development Kit (SDK) and lets Google work their magic. Google people are SMRT*.

## Symmetry!

What do these words have in common?
dollop, suns, noon
Ok, try turning the screen upside down and reading them… (for ‘noon’, try reading it from behind your screen… oh wait… doh!)

I was over at Tom’s the other day. We were talking about Blu-ray and HDDVD and how movies will inevitably be re-released forcing consumers to either cope with the fact that they do not have the pinnacle releases of their favorite movies, or to take cash against a sea of discs, and by conceding, buy them. Sidenote: how much clearer can a movie like the Simpsons get?

Whilst I was making fun of Tom for having multiple copies of some movies in his collection already (Princess Bride, Star Wars… ) he mentioned that there was yet another release of Princess Bride. I was ready to add another \$40 onto the running tally (yeah, I keep a tally of close friends’ leisure expenditures in my head, you wanna fight about it?) he mentioned how the cover of this latest release was quite interesting.

If you rotate the cover 180 degrees, the title reads the same way. Thus they have created an rotational Ambigram. Just like the words suns and dollop, it is symmetric under a rotation on the z axis. (z positive is coming out at you, z negative is looking directly into the screen).

The Princess Bride, however, is not a natural ambigram like dollop and suns. It took some nice artistic work for that one. Perhaps there is a job for math majors. “Oh yes I graduated with a B.S. in Math and an emphasis in ambigram synthesis.”

In reading the Wikipedia article it saw examples of other ambigrams. noon, for example, is also a 180 degree rotational ambigram, but you rotate it around the y axis (between the two o’s)

Then there’s the rather impressive x-axis symmetry of
HICKOK DIED DEC 3 1883 — DOC BEECH DECIDED HE CHOKED. Imagine the 1 has no little doohickeys on it. I actually saw this at Asilomar during Harold Jacob’s presentation.

Ambigrams have actually become more popular recently with Dan Brown’s cover for Angels and Demons. Dan Browns books are nice for inciting a little pop-math fever, they aren’t exactly rigorous… but they did lead to a re-release of John Langdon’s (he being the basis for the Da Vinci Code’s protagonist) book Wordplay.

Anyway, this got me thinking… is this a key to more hidden elements in Princess Bride? The 180 degree flips of Wesley and Inigo’s dominant hands during their duel suddenly seem more mathematically significant. This scene is actually called “swordplay/wordplay” on DVD chapter selections. Dun dun dunnnnnnnnnn….

## Pop culture data representations

Many people have seen, or even made, drawings or illustrations depicting song lyrics either sincerely or sarcastically. But how many have made accurate data representations of pop culture? I will share with you some examples.

I like the cleverly inactive “Touch this”

Layers upon layers.

Biggie never said anything about it being linear though… he merely asserted that for the function f:money->problems, f'(x) > 0 for all x.