## 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” !

## Golden’s Rings and Polyhedral “Cups”

John Golden @mathhombre just posted some very interesting GeoGebra files exploring polygons repeatedly constructed on an edge of the previous. Here’s his post.

Golden’s Rings

The sweet spots between # of sides of the polygon and the # of sides to offset for the new construction reminds me of the platonic solid and polyhedron lesson I do for Geometry.

Instead of giving students the whole net, we would explore which regular polygons can be systematically repeated around a vertex and then “bend up into 3D space” in a manner that would “hold water.”

So, considering 2 triangles… no they just fold up on top of each other. But three triangles fold up into a nice tetrahedron “cup” (missing its cap)

Four triangles? Fold them and you see a square base missing– but fold another four and you get the tetrahedron. The idea being that we should do the least complicated instructions to find these shapes. If the “instructions” are short then perhaps they are more likely to occur in nature, especially if you are lacking storage space in your DNA/RNA.

So which polygons actually form repeatable cases? The ones that can repeat around a vertex and leave a gap. Spoilers: three, four, five triangles, three squares, three pentagons. These correspond to the five platonic solids. There are some interesting differences however between the icosahedron’s repetition of its “cup” compared to the others. And consider the difference between the tetrahedron needing just one more triangle vs. the octahedron, dodecahedron, and cube needing full repetitions of their cups.

This can also be taken to extending/linking the concepts of tiling and polyhedrons. Two triangles and six triangles don’t fold up into a cup but they tile the plane instead. Are Regular Polyhedrons “closed tilings” of 3D space?  #continuummath ! 🙂

See one of my PDFs for the students here.

So… back to John Golden’s geogebra file. I wonder if there are links to different types of polyhedra from the rings he’s created. Do repetitions of this dodecagon with a triangular gap form a polyhedron? No, but it does tile. So how about some of John’s other combinations?

## 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!

### 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.

## 2015 CMC North Asilomar Wrapup – Part 1

As I was writing this, it became much longer (so it goes). I’ll be breaking this up into parts … here’s Part 1

I just got back from the California Math Council’s Northern Conference at Asilomar. Hashtag #cmcn15.

Overall, it was a very solid conference this year. It has also been interesting to seek a different product of attending the sessions from earlier years. My first few years of teaching I was hungry for any tasks or quick strategies I could implement in my classroom or honing my content knowledge. Maybe by year 5 I was feeling more confident in my content and prep so I looked to improve my pedagogy.  And recently, since I’m in a coaching/research role, I have looked into more leadership sessions. And of course, since edtech has always been interesting to me I can’t resist those no matter what… The great thing about this conference is that it has all of these types presented by math educators in a wide variety of roles: teachers of all levels, principals, coaches, policymakers, researchers, professors and more…

Here’s a brief wrapup of the sessions I went to 2015. You might notice a theme…

 Part 1: Fri 1:30 – 4:30 Eli Luberoff of Desmos Design Principles for Digital Content 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 Megan Taylor of Trellis Enacting the Gold-Standard in Teacher Education 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

(working on finding the links to people’s slides still… I swear I wrote them down somewhere 🙂

There were some tough choices in which ones to attend, but hopefully I can browse some twitter and blog recaps of my alternate universe of session choices. Onward…

## Eli Luberoff – Design Principles for Digital Content

Eli is the founder of Desmos, the online graphing calculator. But, since its inception it has grown to have a much higher potential than replacing the graphing calculators of the days of yore (the 80s). I think the team at Desmos has taken that evolution in stride: exploring what might be possible from “digital content.” Eli gave a brief history of Desmos but then quickly led us to the first demonstration:

Dan Meyer’s “Will it Hit the Hoop?” in the Desmos Calculator. We were asked to predict if the shot will hit the hoop, but with Desmos: we could type a parabola equation to help make our prediction.

[Dan first developed this task back in 2010 and you can get some of the initial materials and read up about it here. And I tried it out in my classes for a few years. You can see an example of my 2012 version and the handout. I switched back and forth between setting up a regression for students to fit, or setting up the y-intercept and vertex for students to fit. My students definitely enjoyed it, and I think I got more out of the vertex/yint version because I had a better idea about what I wanted students to get out of it rather than “yay parabolas”. Looking back though— so much I’d want to change. More about reflecting on my own tasks when I get to Michael Fenton’s session… ]

Ok, but in this version in Desmos, nothing is pre-made for the students. We just had the image and the standard Desmos setup. Consider the difference: if we give students the structure of a regression fit or the vertex form of a parabola– then that’s where we can take this task. However, if we give no specific conceptual tool, but rather supply the tool of the Desmos calculator itself… the students could theoretically take on multiple approaches. This is of course possible using GeoGebra as well, but in both cases we must be confident in our students abilities to be skilled users of the software. And while GeoGebra may even offer more potential conceptual paths (a higher ceiling), it’s standard interface may not have the easy entry points (low floor) of Desmos’s standard interface. That’s not to say Desmos always wins the interface comparison (try drawing a triangle) but it is a little more ready to go in common directions that a graphing calculator might go. But I digress…

We in the audience get a chance to try out some of the ways to get a predictive parabola on that shot (as people help each other out). Then Eli shows us the same task with a slightly different mode: the Desmos Activity framework. (see teacher.desmos.com) Teachers are able to create activities in Desmos that their students can run off the web. This mode of Desmos is where it is truly expanding the frontier of what is possible with tech lesson design. Eli notes that “a question still in the research phase is, ‘what is the right way to teach students with technology?'” I hope the thesis I’m writing currently can help illuminate some elements of good teaching with tech. We must not abandon all that we know about lesson design, curriculum development, and pedagogy when granted access to technology. Tech is not magic. Tech is a new kind of vehicle: we can certainly take new pathways with this vehicle, but a good teacher is still going to be the power. That’s why I like the design of the Desmos Activities so much: they keep the focus on the teacher-student interaction. The teacher is given new functionality to view and highlight student thinking. The student is given new functionality to communicate their thinking. A classroom discussion is still where a lot of learning happens– a Desmos Activity merely gives us a new angle or pathway towards that discussion.

In the Desmos Activity mode, the teachers or activity creators create a series of pages in the activity that either ask a question, provide an instance of the graphing calculator, or just provide some expository/instruction text. But each graph can be pre-set to have whichever elements the teacher desire. If we wanted to lead students towards a vertex based parabola, we could provide those tools. If we wanted to lead students towards a quadratic regression model then we could provide some starting blocks for that. But we can also string together multiple pages where each one scaffolds whichever goal– and to whatever degree– the teacher desires. Futhermore, the structure exists to prompt students to guess or explain their reasoning. And what happens to those responses and graphs? They are collected in real time and shown to the teacher interface during the activity. (The comparison of these capabilities to other environments (such as GeoGebraTube) should be the topic of a whole new blog post– I’ll have to save that for later though 🙂 One of the most important aspects of the structure of these activities is that Desmos makes no attempt to assess. That is the role of the teacher. In most activities, a “wrong” answer is accepted in exactly the same way as a “right” one. (certain activities, like Central Park, do indicate correct/incorrect but by showing the empirical result of an incorrect response– promoting experimentation). The data communicated about the students are the students own responses. There is no boiling a student down into a thin spreadsheet or checklist based upon alarmingly few datapoints. Desmos recognizes that computers are nowhere near the ability to evaluate a student’s thinking. Desmos recognizes that the role of the teacher is just the same as it always is– with or without tech.

After we played a bit with the draft (the final version will be up on teacher.desmos.com “soon”) of the Hoop Activity, Eli broke us up into groups along content level/interest and challenged us to create a task using Activity Builder. I was glad to have this time to dig in with other attendees since, while I feel like more than beginner with Desmos, I still have much to learn about its specific capabilities. I joined a group that focused on making a Triangle Centers activity. I often choose this as a sample lesson to learn a new piece of software or device, since I have done such lessons in so many ways, tech and non-tech. And with Desmos, I knew that geometry was not its strength, since the primary input is typing. (Or at least, I’m biased towards Sketchpad/Geogebra ‘s ease of geometric construction) During the next hour or so we made some headway on a triangle activity (which I promised to myself to complete soon) with some learning about specialized inputs to meet our needs– such as drawing a triangle 🙂

Also during this session Eli went into a little background on how Desmos, the company, works. He noted the importance of asking about the business models of edtech companies, since there are a limited set of paths a company can exist long term. If your company is supported by angel investors that support your philosophies that’s great– but not likely. If your company is sold to a larger corporation and either gets shelved or perverted in some way, that’s obviously not good either. Desmos supports itself right now via partnerships with other companies such as Pearson, CollegeBoard, Amplify, Mathalicious, College Preparatory Math, and others. This runs the gamut of size and education philosophy but perhaps because they are diversified, they can withstand being overly influenced by one side or another and keep their own identity. I was glad to hear a candid discussion of the business model. We’ve all been burned by the business of edtech: just look at how the graphing calculator industry kept its ancient calculators in the classroom via its hooks into SAT testing. In contrast, Eli promises “Desmos will be free forever for student and teachers”. Sounds great.

Eli concluded with some big takeaways:

1. hard to do tech in the classroom
2. hard to do tech curriculum well
3. feels like a research project still on addressing 1 and 2.

1 and 2 kind of go without saying, even without tech, but the good point to focus on I believe is #3. We as a community of educators are still feeling out what works best, but because Desmos is free and involves the community (activities can be created by anyone and shared), we have some good items to dig into. I still have a lot of questions to reflect on and dig into myself… Do students engage with concepts in different (better?) ways using tech based lessons? Or first, how do we even compare a tech based lesson to a non-tech counterpart? How about teachers, do the tools of teacher.desmos enable formative assessments? or enable more powerful teaching? Do the tools empirically result in more powerful teaching? What kind of training do teachers need in order to implement these kinds of lessons … or to design them? There was a lot of anxiety from audience members about how they might ever get a handle on Desmos… and Desmos is on the easier side of the spectrum. Is typing expressions/equations the best mode of input for students? How does the input format restrict or promote certain kinds of thinking?

All in all, I liked getting a “direct from the source” view of Desmos’ design principles. The session also allowed audience members to become more familiar with the activity builder in a hands-on way. I think my concerns around Desmos are around how it gets implemented in a de facto way. I also think there are some UI questions, such as the typing interface, and the choices around (lack of) menus. I’ll dig more into some of the aspects of Desmos and tech in general in future parts of this Asilomar reflection.

Bob Lochel shares some ideas about implementing the Activity Builder into his high school classroom: Activity Builder, Classroom Design Considerations

Coming up in Part 2: I’ll compare Eli’s Saturday session with Annie Fetter’s, both about tech implementation.

## Comments from other sites: Four animated gifs of the same awesome problem (dy/dan)

CFOS: Comments from other sites: I do a pretty good amount of writing online about math and math ed… just its not always here. In these posts I will link to the other site and copy over my comment with some context if needed. Mostly this is for me, so that I have a journal of what my thoughts were kept in one place.

Today’s CFOS is from Dan Meyer’s blog regarding the wonderful and somewhat famous task posed by Malcom Swan.  Dan writes,

Here is the original Malcolm Swan task, which I love:

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

paper

processing (code from Dan Anderson)

Geogebra by Scott Farrar

Desmos activity by Dan’s twitter followers

I’d love a combination of all of the above.

It appears that the Desmos activity had a drawback of no error checking. (Or, to call back your talk at CMC Asilomar, if this were a video game it would ask you to try again if your perim/area were impossible) In a customized Desmos applet, could you ask for width and height during the brainstorm rectangle phase, and then the second screen asks for perim and area? (couple more options here… you could still accept perim and area that do not match the given width, height: just flag them separately and then you can filter them out of the overlay or include them depending on where you want to take the lesson next)

It also appears that someone was investigating rectangles of a certain type, thus creating a rather straight line. Nice conversation starter there…

I think I am partial to the ability of my geogebra one to have the non-numerical entry point. On the other hand, they aren’t directly improving their skills in that arena either. A student is not hindered by calculation, but neither are they supported in it. But… I have liked giving a task to students where their interaction is manual and analog– meaning they are digging into the constraints and boundaries of a mathematical concept by physical motion and visual feedback. Does a student become immediately aware of and curious about this region that they cannot seem to get the dots into?

The processing one I like as a contrast to the hand-implemented geogebra. How would we systematically experiment around this idea? And when we devise a systematic approach, computers can be helpful in executing it. We can see the density of the dots is not uniform — what is the interaction between this perim v. area relationship and the way we have implemented our experiment? (I would guess the rectangle dimensions are chosen at random, thus creating a would-be-uniform display if not for the fundamental perimeter-area idea.)

The hand graph is limited in iteration, prone to calculation error, and slow. However, its requirements are all internal (save paper and pencil) rather than computer-based. Graphing by hand is perhaps the handle onto all of these other ideas. Is there research around how the concept of cartesian graphing changes if we grow up experiencing it in the computerized sense without the pencil and paper sense? I wonder!

For now I’d love to mix all of these in the classroom. When I spoke about this problem at Asilomar 2013, I used Malcom Swan’s initial prompts and provided grid paper for the attendees to grapple with the problem. Only afterwards did I move to the geogebra version (2013 version here, that has a slightly different scope)
And I would start the same way with students now. Here’s a rough timeline for a 1-2 day lesson:

1. paper rectangle in groups. Make a Perim v. Area graph on paper in your group that has your four rectangles. (this may correct some errors right away, like if kids put width and height instead of perim/area, their group may correct them)
2. Groups share on Desmos, add more rectangles. They are released from having to graph them on paper now. We are letting that go because Desmos will pick it up. Thus allowing students to step up their conceptual / abstracted approach.
3. class discussion, noticings, wonderings, address errors… blank region may become clearly curious here. Ok, now we’ve used the Desmos tool, but if we want to dig in with more rectangles, lets switch tools.
4. Geogebra sketch and/or programmed iteration. Groups come up with things to try and experiment with on the geogebra sketch, or devise a script that would generate perims and areas and plot them like Dan Anderson’s.
5. Return of the single case. Which points are on the edge? Can investigate with Geogebra sketch by hand. Or: can hand the class a mission on Desmos: “come up with rectangles that you think are on the edge”.
6. depending on the level of the class you can lead this towards some kind of proof or justification, entice students to make reasoned conjectures about what is going on.

Now we’ve taken the stepping stones from drawing a single rectangle to an aggregate set of many rectangles and their (p,a), to a set larger than what we could human-ly create. When we’re on step 4 there, we’re getting a handle on the abstraction of perimeters and areas of rectangles, thus allowing us to experiment with the abstraction just as easily as we worked with a single rectangle on paper at the beginning of class. But then we can go back and use each tool for what it helps with.

That, I believe, is the power of education technology: giving concrete handholds to abstract concepts. But tech is at its most powerful when working in concert with non-tech methods. And the possibilities of tech are so wide that no single tool was the magic bullet that enables this lesson– each one exposed a new vector to take.

Finally, remember Swan’s prompts (that did not aim to use technology) originally got at the existence of this concept by suggesting an impossible point and then suggesting that there are many others to find. His prompt defines these two regions (that we might call white and red for our purposes) with two examples: one red and one white. I wonder what the difference is about investigating the existence of white and red by imagining and reasoning on paper vs. bouncing up against their boundary on the geogebra applet.

## #ContinuumMath – join the movement!

I’m starting a new movement.  Maybe its not new– and in that case I’m grabbing a banner and rallying the movement!

All too often we talk only about special cases in the math classroom.  All of our teaching revolves around pointing out landmarks and marvels.  But how can our students appreciate such phenomena if their experience never includes the cases that are not marvelous?  If you live your whole life in Yosemite, how would you know how special it is?

That is my thought behind #ContinuumMath.  We need to shake up the special cases by spreading student experiences to include the countless non-special cases.

• Going to talk about SSS or SAS triangle congruency?  Why not talk about “SS” ?  Why are we always using 3 parts of a triangle?  I myself prefer “SASASA” congruency.
• The natural exponential function $e^{x}$ can be explored in the context of all of the other exponentials whose derivatives are merely proportional instead of equal to their output values.

(click for animated GIF)

• I’ve previously discussed that lines are not always parallel.  Corresponding and Alternate Interior angles are not very interesting if our “desire for congruence” is always fulfilled.
• Algebra: $5x+4 = 34$.  “2 step equation”?  Heck, I can solve that equation in 7 steps!  How about this: Students… try using 5 properties of equality (“steps”) for this equation to get to x=6 — I bet you’ll use at least one additive inverse and at least one multiplicative inverse.  The inverse properties come out naturally as the most efficient moves.
• We have lots of new technology to assist us in exploring whats around the special case. Constructing equidistant points with a compass and straightedge is what can be done after we experience [GGB] how distances change.

Students who experience all of these non-special cases then see the special cases as actually special.

Every time your textbook or curriculum offers up a wonderful case– they’re just begging to be shaken up and explored in the context of the structures they depend upon.  This is the Gift of Sometimes True.  Whenever we are giving a theorem or an if-then statement, we can parameterize the condition.  Here’s what I mean:

The diagonals of a parallelogram bisect each other.

We can take that and say, the diagonals of a quadrilateral bisect each other if the quadrilateral is a parallelogram.  And now we want to consider the entire continuum of quadrilaterals– I want students to see quadrilaterals that are almost parallelograms, quadrilaterals that are nowhere near parallelograms, and everything in between!  We further get to talk, as a bonus, about all of the properties of diagonals in other quadrilaterals.  Try the geogebra applet for yourself.

In this applet I parameterized the quadrilatral, but not fully: its impossible to get a rectangle or rhombus here– a tradeoff I made for simplicity of interaction: you can only drag point C.  But as students drag, they observe the diagonals and their dissected lengths.  They can make observations and comparisons in real-time, and they can conjecture and posit as they experience and interact the structure of the geometry.  Students will find the parallelogram.  Students will find the trapezoid.  Students will also find locations where three segments are equal.  Students will find quads that contain isosceles triangles formed by their diagonals… Students will discover so much more because they are seeing so much more than the narrowness of the special case.  Embrace the Continuum.

Check out #ContinuumMath on twitter and/or leave replies here.

## Student Inquiry into the Neighborhood of the Special Case

Lets look at Triangle Centers: a fun unit for many Geometry classes.  Its a topic with some good “real-world” applications, certainly, but we need not always justify our lessons with application.  And it may hold us back from important mathematical practices.  Pure math is underrated!  Lets compare some traditional textbook style application problems with an interactive style problem.  First, Holt Geometry:

Those two problems are somewhat typical of some application motivators.  But consider a correct response from a student:

18.  Main street should be the angle bisector of the angle between Elm Street and Grove Street.
37.  Bisect the angle between the streets.  Draw the perpendicular bisector between the museum and the library.  The visitors center should be where those two lines cross.

How does this capture the imagination of a student?  If they know what an angle bisector is, they can supply an answer.  But if they do not know it (or haven’t made the connection about equidistance from sides) what kind of feedback can you give them that guides them to the answer without providing the answer?

But the book already provided the direct answer, earlier in this section: Theorem: a point is on the bisector of an angle if and only if it is equidistant from the sides.  #18 is almost explicitly asking for this theorem.  #37 asks a little more: it wants this theorem combined with the perpendicular bisector theorem, which led off this section– no prelude.  These questions are merely dressed up versions of those on Level 1 and Level 2 of Bloom’s Taxonomy.

With GeoGebra, we can provide loads and loads of information that (1) helps guide the student around the topic we want and (2) does not provide progress towards “the answer” in discrete unassailable steps.  We can be more helpful while being #lesshelpful.

Here is a picture of an applet I created to give some real-time information to students about point D and triangle ABC.  And here is the interactive applet itself, try it!

In this first static picture, what information is given to students?  Most students will be keen enough to see the comparison of the distances DA, DB, DC.  And all students will catch on after they start dragging point D.

What do we need to ask here to get students to think about perpendicular bisectors?  I say: “not much!”  My introduction to the applet is “Drag point D.  Find special locations.”  That’s not to say I’m not communicating with the students.  I’m communicating a great deal more!  I’m communicating through the boundaries and feedback programmed into the Geogebra manipulative.  Manipulatives are powerful in any setting, but computerized manipulatives enable modes of lessons not possible before.

I love doing triangle center lessons via oragami (or patty) paper.  I love doing triangle center lessons via compass and straightedge.  But the boundaries of those manipulatives do not guide the students.  If you make an incorrect fold or line, the paper doesn’t tell you so.  If you want to test distances with your compass or ruler, you must do them one at a time.  If you want to pursue a question other than equidistance (e.g.) then there may be other complicated procedures to follow.

Consider with GeoGebra:  by restricting the interaction to only dragging D, the students may no longer make unproductive moves.  Every move they make produces feedback via the distance “bar graph” and the color shading of D itself.  That feedback isn’t telling the student “incorrect”.  It is telling the student “here are the distances your input asked for.”  The student is then left to parse that information, and respond with more input.  This type of interaction between student and computer occurs within a fraction of a second.  And this type of interaction is repeated tens, hundreds, or thousands of times.  The student can then use the information to start digging into the prompt: what are the special locations for D?  What makes a location special anyway?  These are questions that can be asked and answered by students without teacher interruption.

But wait, there’s more!  Lets say a student has decided upon finding the place where DA=DB=DC.  Computerized Manipulatives now allow a task between this decision and the answer.  They must spend time actually getting point D to the correct spot.  In this Geogebra applet, they have the feedback from color and the bar graph to help them, but here is where the teacher can wait for a student to ask, “is there a better way?”  Boom!  We have arrived at the motivation for the circumcenter construction.  This motivation was driven by “pure” math ideas like equidistance… only we didn’t have to say it.  Proof in geometry should be introduced as a way to perfect our conjectures and hypotheses.  If the students haven’t made a conjecture, how are they going to care about its truth?

In this applet I decided to include an extension: another triangle center.  But this is not the usual 2nd center introduced.  Your students will find the blue point and notice what is special. Try it yourself!  Think about what is also done with this wordless separation of the two points in contrast to a lecture-based introduction: the students will have their own vocabulary to parse the difference that they have played with already, instead of the students having to parse vocabulary that describes a difference that they may not have been aware of.

In this applet I included two “answer” checkboxes.  Depending on where you want to take your class next, it they may steal some thunder from the lesson, but I figured I’d include them to help illustrate the point… and points. (hah)

GeoGebra and other computerized manipulatives enable us to think about Geometry and math in a way unfamiliar from static text or lecture.  Similar to The Lines Are Not Always Parallel, I’ve created a way for students to observe the neighborhood around the special case, and constructed silent helpful barriers and footholds that students can grab onto as they discover what is special for themselves.

Let me know what you think.

Earlier today I was in a discussion with @mathhombre on Twitter about what is needed in an College Algebra course (roughly equivalent to Alg2+Precalc for high school).  I came to the (perhaps too radical) idea that lines are simulateously one of the most applicable concepts to a person’s “real life” but also one of the most useful tools for accessing higher math.  There are entire areas of study about linearization as a tool to simplify complicated cases not just in pure math, but economics, physics, engineering, etc.

Any that is just the prelude.  If we are going to talk about lines, what happens to the darling of any high school level algebra course?The quadratic function / The parabola.  Consider f(x) = x^2 – 2x – 15.  Factoring, we obtain f(x) = (x+3)(x-5).  A subtle idea that might be skipped over: f(x) is the product of two lines.  Literally lines y = x + 3 and y = x + 5.  Do our strudents face the idea that a linear factor and a line are the same?  Consider the product of the values of the lines — point by point.  Try it with Geogebra.  The f(3) = (line1 at 3)*(line2 at 3) = (6)(-2) = -12.

We also know the Fundamental Theorem of Algebra will guarantee n roots for a degree n polynomial.  But those n roots may have non-real parts, for example g(x)=x^2 + 4 does not factor over the reals  g(x) = (x+2i)(x-2i)

So can our “line*line” idea survive these new types of models?  And where can we see these complex lines?  The product relationship still holds.  g(3) = (3+2i)(3-2i) because the imaginary terms will cancel since complex roots always come in conjugate pairs.  g(3) = 3*3 + 3*2i + 3*-2i + – 2i*2i = 9 + 4 = 13.  But wait, lets slow down.  3 + 2i is a point.  A point on the line x+2i.  It is a line hidden to our view because we lack the dimensions on our plane to see it.  We have only the 1D real line as our inputs.

So lets consider the imaginary part of our domain.  Click the picture to see a larger view:

That’s g(x) = x^2 + 4.  The red axis is the standard x-axis.  The green axis is the standard y-axis.  But the blue axis represents the imaginary part of our domain.  A normal classroom might be used to plotting 3+2i on a plane, but do we often make note that its a nominally different plane than the one we graph functions upon?  3+2i will be on the plane that passes through the red and blue axes.  I chose to view it this way to keep y outputs as close as possible to our usual view.

Ok so lets see 3+2i and 3-2i and their output product 13.  I included two views here since it is difficult to get a grasp of the 3D situation.

These lines y = x+ 2i and y = x – 2i are kind of abuses of notation.  They should be specified that they are lines in space, namely y=x limited to the level planes Im = 2 and Im = -2.

There’s certainly more to do with exploring this idea, especially making the presentation more robust.

What really appeals to me is the symmetry of a quadratic’s complex conjugate roots is similar to the symmetry of a separate quadratic’s real factors around the axis of symmetry.  Even more, the idea of the imaginary number i as a 90 degree rotation fits puzzle pieces together.  The complex roots are 90 degree rotations around the axis of symmetry of a reflection of the parabola (reflect across a horizontal line passing through the vertex).  This is the part that holds some promise, but that I haven’t quite explained in my own head yet.

I invite you to play around with the Geogebra applet I used to explore this.  It is set up to start with x^2 – 8x +18.  Find the complex roots algebraically first, then see if it meshes with the visuals in the applet. Thoughts?

## The Lines are Not Always Parallel! A geogebra approach to Alternate Interior Angles

Students are frequently confused by us harping on the importance of things that seem obvious: because we hardly ever show them the cases where the theorems are false.  One of the areas this comes up is Angles in Parallel Lines.  (The whole course of Geometry may feel like this, actually).  We teachers might feel like, “how do kids get confused here? its so easy and obvious!”  That’s right… so easy and obvious that we teachers have lost sight that we only show them these narrow cases.

We spend time proving that certain pairs of angles on parallel lines are congruent or supplementary.  Over and over and over… but always parallel lines!!  Maybe maybe we make an offhand comment about “if these lines weren’t parallel then the angles wouldn’t be congruent!”.  But those words hardly paint the picture for student.

But think of the infinite number of line that are not parallel!!

So, let us turn to Geogebra.  My philosophy in making this applet (and most of my applets) is to loosen some restrictions in our normal presentation so that more of the “sample space” can be explored.  Lets consider all lines cut by a transversal, not just parallel lines.  Then how many students will not only (1) appreciate parallel’s special nature and (2) make conjectures about congruent angles all by themselves?   This is much in the same vein as Michael Serra‘s Discovering Geometry book: set up a situation so that the students actions will lead them to make those conjectures.

The applet here lets students drag the line to all positions, and illustrates the red and blue angles, giving no measurements.  We want intuition here, not measurement.  Students might already have some idea “yeah they look close enough”.

The check box “Compare Angles” allows them to update their intuition when faced with a little more precision.  This kid who wasn’t able to articulate anything before might now realize in what realm we are trying to explore.

The check box “Compare Lines” allows them to make a connection to the properties of the pairs of lines.  Again, this is some non-verbal feedback and prodding to a student.  The student must assimilate this and then can provide the verbalization themselves.

Finally, I added in a little game at the end.  It will spit out a score based on how close to congruent the angles are.  (The formula is arbitrary, but spikes up very high when you’re close).  Let me know how this plays out with your students!  Do they get invested in beating each other’s high scores?  If so, notice that in trying to beat one-another, they must know that they want to make the lines “more parallel”.  Here we can embrace the difficulty of being precise when doing this with a trackpad or touchscreen.  Will your students say “hey if I could make them parallel, that should be an unbeatable perfect score!”  (I actually think Geogebra will spit out “infinity” score if it gets close enough– a happy consequence of the data structures they’re using)