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

example traincars

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)

1combinations only 3different 1 cars
negative traincars
7by num of pieces

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

Recursion appears as the students organized their work

This group discovers a link to Pascal's Triangle

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!

cycling shop in pascalSo 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.


Skyline Express Lesson Materials PDF – Scott Farrar 2013

NCTM – Teaching Children Mathematics – 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 doi:1

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 Eli Luberoff of Desmos 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 Megan Taylor of Trellis 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!

Types of trianglesFetter 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

function carnivalLuberoff 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”

vertical line testAdditionally, 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
Ray, M. (2013). Powerful problem solving. Heinemann.
Victor, B. (2011). Up and down the ladder of abstraction. Retrieved from

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:

hoop DesmosDan 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 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 “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.

Related links –
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.

Academic Paper Breakdown: Lampert “Teaching while students work independently”

As I’m working on my M.A., I plan on sharing my thoughts on some of the articles I read either for a class or for my own research. I hope these serve as both an accessible summary of the article for current classroom teachers and a place for me to share some of my thoughts on the article.  So here is our first one:

Lampert, M. (2001). Teaching while students work independently. In Teaching problems and the problems of teaching (pp. 121–142). New Haven: Yale University Press.

Lampert is a 5th grade teacher who is also a researcher.  During the writing of this book she was teaching part time as a math specialist for an elementary school, but she also has 8 years of full time experience.   Her background includes the philosophy of mathematics and what it truly means to know a mathematical concept. (And she believes that memorization does not constitute knowing.) She describes her teaching style as one that uses problems and therefore problem solving. “Teaching mathematics would have to engage students in doing mathematics as they were learning it.” (p5)

Chapter 6: Teaching While Students Work Independently

Lampert’s students are given an activity that is based around problems of the form (  ) groups of 2 = (  ) groups of 4.  Some of the learning targets in this lesson are for students to use multiplication to create a true statement, and to utilize problem solving strategies.

Notice a few more things here: (1) there are many ways a student can answer this.  (2) its possible but not necessary for students to bring up fractions, (3) the equals sign is not treated as a “do” symbol.  Number 1 means its more open than a normal exercise and provides opportunities for students to analyze each other’s work (great Common Core thinking 12 years early, Lampert!).  Number 2 means we can extend the problem naturally but also we have not put unnatural boundaries on the problem.  2*5 = [  ] is pure arithmetic calculation. But “x groups of 2 = [  ] groups of 4” is all of a sudden touching on algebra.  Finally number 3: the equals sign.  In a problem like “2*5 = [   ]” the equals sign is more of a symbol that the multiplication should be performed and the answer written down.  It is common for elementary students, when faced with “9 + 4 = [  ] + 5” to write, “13” in the blank.  Liping Ma, in her book Knowing and Teaching Elementary Mathematics, even notes that elementary teachers will misunderstand the equals sign: a statement like 3 + 3*4 = 12 = 15 was not flagged as incorrect by teachers.  Back to Lampert’s activity, we have the equals sign properly used as a relation, not an instruction.

Lampert describes some of the actions a teacher performs in order to keep students working on a task, while also getting the most learning out of that task.  In other words, this is not “do the worksheet then check answers at the front while I grade at my computer.”  I broke Lampert’s 11 item list (p140) down into a few categories:

  • A – Assessment – finding out what students know
  • C – Content – using content knowledge to produce help or challenges from many angles
  • S – Structure – keeping the task on track by managing student behavior or task instructions
  • P – Problem Solving – providing and modeling tools, suggestions for general problem solving strategies.

Many items had more than one category, such as “providing and maintaining appropriate use of notebooks and seating assignments” which I labelled as P and S.  Or “clarifying, inquiring, probing” as C and A: the teacher must be able to see what the student is thinking and then place themselves in that viewpoint in order to provide the next tool, hint, or question that would best support the student’s thinking.  This mental agility requires deep content knowledge.

Lampert also focused on a few case studies of her interactions with students.  One student, Varouna, had begun a problem like this:  “[ 1 ] groups of 7 = [   ] groups of 21“.  (At this point it is important to note that Varouna had completed parts (a) and (b) which were “[5] groups of 12 = 10 groups of 6” and “30 groups of 2 = [15] groups of 4”.  Note there is only one blank.)  Lampert brings up my favorite takeaway at this moment.  “She had tried an experiment and was now thinking about how to cope with its consequences.” (p123)

Lampert assessed that Varouna was unlikely to use fractions to complete the statement correctly, but to tell Varouna to erase the “1” and choose another number was to devalue Varouna’s efforts and thinking thus far.  It may appear that Varouna hasn’t done much of anything yet, but I believe that Varouna has already accomplished a lot:

  • She has applied a strategy: the other problems had only one blank, and I can’t think about this one until I fill something in
  • She (perhaps) chose a number with intent: “1” may have been a strategic choice since she assumed the problem would be easier with a smaller number.
  • She assessed her own knowledge: she stopped, realizing that she did not know the number for the second part.  But– she knows that she doesn’t know.

Lampert works a little with Varouna around how to make sense of the problem with diagrams (showing that drawing is an acceptable way to do math) and after a short bit, Varouna writes “3”: “[ 1 ] groups of 7 = [3] groups of 21”  Lampert again does not flinch to signal incorrectness:

My work here was to interpret and respond in a way that taught her something about the mathematics of multiplication and that also respected her efforts to make sense.

How often are we tempted to make the quick correction that the student is “almost right” or has the answer “backwards” ?  In doing so, we are signaling that the student can stop thinking about the problem.  Furthermore, perhaps the student has constructed the concepts in their heads in a way that is internally consistent, just not in agreement with the outside prompts. By making a quick correction on the result of their thinking, we risk destroying a correct conceptual construction.  We as teachers must take the time to get at the heart of the student’s misconception so that we can bolster their conceptual ideas.  Varouna has knowledge that 3 sevens are 21.  She has written an incorrect statment, but it is the result of that knowledge.  Telling Varouna that she is incorrect may harm her confidence in the foundational fact.

I do recommend reading the rest of the chapter for the other case studies.  Here is a link to the book, or you can find excerpts perhaps on academic journal site.