Friday, March 14, 2014

Algorithms and Flexibility

I was given a section of enriched grade nine students this semester. I decided very early on that the proper way to enrich a group of gifted students is not through speed and fractions. They came to me almost done the entire course in half the allotted time. This essentially alleviated all issues of time pressure.

The beautiful thing about this is we are able to "while" on curiosities that come up during the class (Jardine, 2008). I am not afraid to stop and smell the mathematical roses--so to speak. In a recent tweet I explained it as the ability to stop and examine pockets of wonder. This has been a blessing because our curriculum has become far less of a path to be run and more of the process of running it.

We do introductory tasks each day to get our minds thinking mathematically. We started with various estimations and have since moved into a series of numerical flexibility tasks. Here, I write a problem on the board and they are to give me the exact answer. Only two rules:
  • No calculators
  • No standard algorithms
I am careful to always write these problems horizontally, because vertical alignment is too tempting. A typical problem might look like:
5 x 38
15% of 420
The students have done remarkably well working flexibly. I have them thinking outside the standard methods they have been so successful with.
One recent flexibility task highlighted subtraction. In the midst of students explaining their strategies, two very fruitful things occurred.
One student became frustrated with the banishment of algorithms.
"How do we know if we even used an algorithm?"
I re-directed the question to the class and the response was very cool. We settled that an algorithm had three characteristics: a set structure, easy repeatability, and stability (what I called robusticity).
With these three characteristics in mind, we collected solution strategies. There were only two. First, a student explained how they separated each place value and subtracted corresponding numbers. As he got answers (3000 - 2000), he kept a running tally. In the end, the tally after the units digit was the answer.

The second student separated each digit into a subtraction problem and then stacked them on top of each other and performed the final subtraction after all four were isolated and completed. It was at this moment that I pounced and provided an alternate algorithm.

What if we just combined these numbers back into their original place value and if they were negative, we would represent it with a bar over top. I showed them the familiar addition algorithm and the response was immediate.
Many wanted clarification, and others chimed in to explain. Others wanted to know if they can use this for the rest of their assignments. I told them if we were going to use it, we better develop algorithms for counting and the basic operations. I set them into groups to decipher them.

Algorithmic language came out right away. Things like, "add a zero on the left" and "reset the far right column to 9 nought" (nought was our verbal way of reading the bar notation).

For two days we wrestled with the possible algorithms; when they are polished by the students, I will post them and link from this post.

From the outside, this task seems like a colossal waste of time. The classroom was thrown into chaos as the students worked to re-align their worlds after this major perturbation. From the inside, it showed my students the value of algorithms if they are understood. Something that these students have abandoned in favour of guaranteed right answers. Aside from the natural engagement that speaking in these numbers provides for this class, it does illustrate the fact that "knowing why" it works is a prerequisite for "knowing that" it works.


P.S. After the fact, a student used "nought" notation to give an answer in class. Another classmate noticed that the number (in this case 36) could be represented by a different nought notation number. I then had the class work to find out how many ways each number could be represented in our new notation. The results (purposely omitted here) were quite surprising.

Jardine, D. W. (2008). On the while of things. Journal of the American Association for the Advancement of Curriculum Studies, 4. 

Sunday, January 26, 2014

Road Building Task

The Pythagorean Theorem is often taught in isolation. It has connections to solving equations, but often appears in curriculum long before other equations involving radicals. It also has unique ties to both radicals as well as geometry.

Despite these connections, the theorem has developed the reputation of a surface skill. It involves the  repetition of the rule alongside numerous iterations. Something so fundamental to geometry is reduced to a droning chorus of:

" 'a' squared plus 'b' squared equals 'c' squared "

This task aims to find a middle ground between where critical problem solving and collaboration can meet the technical precision of the outcome. 

The task is originally from James Tanton's Solve This! book. In a recent Twitter conversation, a couple people were looking for good Pythagorean Theorem tasks. It was through this conversation that I began to look for extensions / alterations. 

The task:

You are building roads between four towns. The only requirement is that there is a route between every pair of towns. It doesn't have to be a direct route. What is the shortest distance of roads that needs to be built?
I originally set the towns on the vertices of a 10m by 10m square. This seems like a good place to start building familiarity with the problem. I also included a handout that provided some suggestions to test as well as spark imaginations. These starters give the task a low floor and the extensions provide a high ceiling. 

I give this task to randomly selected groups with large whiteboards. This fits into the structure of my class, but it could be given in any matter that you are comfortable. I encourage parings or groupings so students can build communal lines of thought and action. 

Student work (with original task):

As evidenced, the original task provided opportunity for iterations, but didn't provide much opportunity for branching out. Students needed to be coaxed past the original suggestions. To increase the robustness of the task, allow me to offer some suggestions: 

Begin by having students work on a grid. Whether this is with an overhead transparency or a large whiteboard, the grid will ensure they work with integers if they place towns on grid intersections. Try extending the task in the following directions:

  • Move the towns off of the vertices of the square. Recommend that the students still stick to integer coordinates--at least to begin. This variation will allow students to see the right-angle triangles that exist on the grid regardless of position. They may also get a sense for lengths of diagonals, and whether or not they are congruent. 
  • Add more towns. A simple alternative that might open up the discussion of what it means to be co-linear. Various regular polygons can be explored for patterns. 
  • Mandate direct routes. Instead of having indirect routes, make direct routes between town mandatory. This may open up conversations around number of required roads, the patterns created by regular polygons, and even--although it's a stretch--the number of distinct sections created by the roads. (an extension into inductive reasoning and counter-examples).
  • Create and place a central depot. This could possibly be my favourite extension. Have students place a point that will serve as a central depot where each town must connect to directly. Where is the best place to put it? This could possibly lead into some type of regression analysis.
The task requires students make intelligent decisions with the theorem in mind. They will begin to run iterations of the theorem as they attempt to place towns. This not only builds fluency, but blankets it in critical thinking. It is in this sense that the task meets the middle ground between fluency and critical problem solving. 


** If there are further variations or extensions from readers, please do not hesitate to comment. 

On a side note, analyzing the distances between collinear towns provides a nice visual for simplifying radicals. Use this visual to show how 2*root(2) = root(2) + root(2) = root(8)

Saturday, December 14, 2013

The Discourse Effect

This semester, I've been attempting to infuse my courses with more opportunities for students to collaborate while solving problems. This post is designed to examine the shift in student disposition throughout the process.

I have noticed an increased conceptual understanding almost across the board and this is reflected in the differing solutions on summative assessments. It is also nice to see their marks  grow on these unit tests. I do not believe that paper-and-pencil tests are the best venues for displaying conceptual understanding, but it is awesome when the two intertwine.

My unit structure

I plan my courses in units of study, and attempt to build "pods" of conceptual understanding. I am fully aware that the topics will influence one another down the road, but I prefer to have those conversations when the students make the connections.

A unit begins with a strong focus on exploration with group problem solving. Students are randomly grouped and assigned tasks to complete with large whiteboards. These problems are designed to create peer discourse and stumble upon some key features to be later formalized into the "official" mathematics. Sometimes these problems contain a foreseeable problem to be overcome (i.e. graph a function with an asymptote they don't see coming) and others require the following of a pattern (i.e. giving an example of a linear relation and asking them to find a entry out of counting range).

These classes serve as anchor moments that can be referred back to during later times in the unit. They also provide time for students to conjecture, test, analyze, and critique. These activities shift the typical role students usually possess in a math class.

During the mid-stages of a unit I purposely plan more guided discovery lessons. Often times these include a handout with prescribed exemplars that will lead them to narrowed conjectures. This narrows their thoughts, but continues to include them in the "invention" of the math. In this stage, students begin to become more exact with their solutions and notations. They often check papers with others. Here, student explanation of abstraction is key. I nominate students to explain patterns and write rules. They are well prepared for this task because of the skills built with the more open problems.

The final stage looks much like a "traditional" class as we move toward a summative assessment. Here, we polish up methods and notation after the time has been spent to understand where they come from. This stage is dominated by lecture, seat work, and a quiz or two.

Shift in student disposition

The focus on group problem solving through large white boards and collection of formative data using digitized exit slips has changed the ways that my students interact in class. They carry an inquisitive approach into the last stages of the unit. 

I was skeptical at first that I could cultivate an active disposition in students long term. I have had instances or lessons of great participation, but haven't tried such a large-scale change thus far in my career. Slowly, little victories convinced me it was possible. It is intimidating to give up control, and even the smallest changes take monumental efforts. Nothing about this post or blog in general is aimed at making change seem simple.

Three examples of active student disposition:

1) In-class conversations
Students have started looking past me as the sole provider of correct solutions. Many times they engage each other in conversations about differing methods. The discussions are initially centred around right and wrong answers, but quickly become arguments around efficiency. 

For instance, when given the function "2y - 6x + 12 = 0" students debated whether it was quicker to isolate the "2y" on the right hand side rather than on the left (as was the normal practice in class). In previous years, students would have just copied down the example and told themselves that it must always be on the left hand side. The opportunities for them to exercise mathematical intuition create the climate for these moments of intellectual courage. 

2) Formative data from exit slips
After group problem solving classes, I have students fill out their exit slips with things that they noticed or wondered. These snippets of formative data really help shape my next moves in class. They also provide the chance for every student to participate mathematically with me. After an introductory lesson on linear relations, students responded in the following ways:

Question: What did you notice / wonder from today's lesson?

"I noticed that numbers stand for specific things like corners and sides"
"I noticed there are multiple ways to figure out the constant and variables in the equation"
"I noticed that there are different ways of solving problems"
"I noticed that we got multiple different patterns from the same question. Depends on how you see it"
"I wonder if the inside of the square comes into it later"
"I noticed that some people subtract four corners and other add them in later"

The responses show the degree to which students were interacting with each other. The environment of student discourse allowed them to notice, debate, and wonder about varying strategies used by classmates. Having a private place to notice and wonder gives a chance for each student to discuss their individual thoughts. 

3) Creative responses on examinations
Students that have the chance to build mathematics and discuss their rationale become more comfortable "doing" and not "reproducing" mathematics. As the culture shifted, I noticed more students becoming playfully astute on exams. This is a good sign; it means they are making conjectures, verifying their validity, and taking the risk despite the pressure of percentage grades. 

On a recent exam on sequences and series, I asked the following question:

Build a sequence where every term is negative except the first term.

I got the following solutions:
"a=0 and d= any negative number. I know zero is not positive, but it is still not negative so it fits"
Here you can see the student teetering on the brink of mathematical precision. They creeped as close as possible to a non-solution; this is such a mathematically cheeky move. 

"a=1,000,000 and d=-1,000,001."
I love this solution. It shows very clearly that the student knows that it won't matter how large the first term is as long as the common difference can cover it. Shows mathematical generalization, without notation or conjecture. 

"Any a > 0 and d = -a + x."
This solution is not correct for all values of "x", but still shows the active nature of the student thought. They are willing to abstract the mathematics and take the risk well beyond the question's requirements. This solution served as a great class conversation the following class. 

On some days my classroom looks drastically different than the norm. Students divide into groups and work on problems with one another. They migrate from group to group looking for verification. The class is loud and active. Even when I bring everyone back together, they still talk over me and debate their solutions with each other. Even desk work is often interrupted as pods form to discuss methods for certain problems. 

On most days if you observed my classroom, you wouldn't find anything drastically different that the average classroom. Students are expected to follow examples and ask questions when confused. Students take notes and do seat work, but you would find students that question what is going on and have a tendency to break into conversations with one another. You would find students asking for me to go back and do examples in different ways. These mathematical activities are a direct result of building an environment of conjecturing, discussion, debate, verification, and sharing. Students' active disposition is thanks--in full--to the discourse effect. 


Tuesday, November 5, 2013

Creating Communities of Discourse: Large Whiteboards

I have talked about individual whiteboards on this blog before. My school bought me supplies and I was loving the various classroom activities. While the grouping questions facilitated good mathematical talk between peers, I was still searching for a method to encourage more collegiality where my role could diminish to interested onlooker or curious participant. 

So I had this brilliant idea. 

Why don't we get group-sized whiteboards created where students could work collaboratively on tasks?

In my mind I had just stumbled upon something uniquely genius, but soon discovered that it had been done by Frank Noschese years previous

I set out to plan tasks around this methodology. I was nervous at first, but soon discovered that the students enjoyed the shift in culture. This post is not designed to give you a bunch of whiteboarding tasks, but rather share some of my experiences/revelations from the first two months of using them consistently.

How I Set Up a Whiteboard Task (Classroom Routine)

The class begins as normal. I have some kind of cue (either projected on the board, or spoken verbally) that students do not need to unpack. This saves getting things out and then repacking immediately. 

I use the app Triptico (see David Riley for information) to group the students randomly. Triptico has all kinds of other things, but I have found their timing and grouping apps the most beneficial. Students watch anxiously as their name is highlighted and the groups are created. I allow two minutes to get settled with the new groups. 

The first few times I handed each group a board and markers, but now they just nominate a member of the group. It doesn't take too many tasks until each student has worked with every other. This only strengthens class culture. 

I then have an introduction period where each student is expected to face the front to receive the task. When that is done, they go to work on the board. 

I circulate and question. Sometimes I connect groups together if they have similar thoughts. I have even make a group "trade" if I think some different thought or a leadership change is needed. 

I allow students to take pictures of their work as they go. Often times, students record audio of the discussion as well. I haven't formalized a place to collect these archives, but that seems to be the next natural step in the process. At the end, students erase and return their boards and supplies on their own. 

I always end a task with a discussion. Students are encouraged to present their strategies. During circulation, I always prepare them for this moment. I will say something like, 

"I really like your idea here, and I think the rest of the class needs to know. I am going to ask you to explain it during the discussion if it's okay with you."

If they are really uncomfortable, I will summarize but be sure to continually reaffirm that I am not skewing their thought. Phrases like, "Correct me if I'm wrong" or "I think this is what you meant when" engage the bashful student and they often end up taking over anyway. 

My Three Biggest Revelations Thus Far
  • Tasks don't need to be complex for this to be effective
The strength is in the communication necessary to work within this framework. I have taken tasks that I consider to be "rich", but also used run-of-the-mill textbook problems. In each case it is the vibrant discourse created by the whiteboards that opens up avenues for creative and conceptual thought. The large whiteboards are what Di Teodoro et al. (2011) call an effective high-yield strategy. I do believe that good problems are often the basis of good lessons, but math teaching is not simply an archiving process of effective tasks. The structure in which a task is presented determines what type of learning is developed. Large whiteboards are an excellent way to encourage mathematical talk and cultivate deep, conceptual learning. 
  • Students enjoy to see and interpret other ideas
I try as much as possible to have students view others' work. Most of the time this is accomplished through the group discussion at the end. There are other times when students will get into informal debates after one glances at another's board. I do not discourage this, but do mediate if it gets heated (which I have affectionately termed, "math beef"). Students will naturally move groups and assume the roles of tutor or participant. Others go out of their way to check work with their friends. I encourage all of these activities. I spark them with questions like, "How is this different?" and "Is this more efficient?" Again, the peer-to-peer discourse is the strength of the strategy. 
  • It is always more effective to let a student talk about their thinking
This is the hardest for us teachers to admit, but we need to shut up a lot of the time. In the beginning, I led the summary of the strategies that I saw. I got so excited about the divergent thinking, that I lost track of the purpose of generating conceptual ideas. I now only revoice ideas after students have completed their thoughts. This has made a large difference in their stance as learners. They begin to see themselves as knowledge creators. They question what I do in class more and debate that they have easier ways to complete problems. I always tell them that communicating mathematics is often harder than the solutions themselves, but I insist that they practice it. 

Like any strategy, it needs to be tailored to your style. Whiteboards can be brought out every day or only once a unit. I began by using them as introductions, but now find them in my room about every three days. Their strength lies in the communication conduit they open between students and within students' own mind as the access the necessary metacognition to explain their thinking. Like everything, it is a work in progress and I would love to hear of ways others use them in their daily craft. 



Di Teodoro, S., Donders, S., Kemp-Davidson, J., Robertson, P., & Schuyler, L. (2011).      Asking good questions: Promoting greater understanding of mathematics through purposeful teacher and student questioning. Canadian Journal Of Action Research12(2), 18-29.

Saturday, October 19, 2013

Digitizing Exit Slips

I've tried many forms of student written reflection in my classroom. No matter the format, I have always phased them out due to the administrative details and increased time burden. I liked the idea of having students reflecting on their learning, and believe in the benefits of writing across all curricular areas. What I needed was an easy way to orchestrate the process. 

It needed to be easy for me to access and for students to complete. Here is what I've come up with, and the results have been great:

I used my google account to set up a Google form. If you do not already use Google drive, it's time to start. 

The form is very simple. It contains two items:

(You can design the actual exit slip questions however you please. I settled on this question based on advice from @letsplaymath)

When you have your form, click "Send Form" and copy the generated URL. 

Take that URL to any QR code generator. This link was the first that came up on my Google search.

Download that QR code from the website, and name it accordingly.
(I use a separate form for each of my classes, so naming them Period 1, Period 2, etc. helps me keep things organized.)

Print off multiple copies of each code and place around the room in areas easy for students to access. I have created a couple "hot spots" in my room to avoid crowding while students herd to complete the slip. 
(Be sure to label each code with class period or title if you are doing multiples. The codes themselves are almost indistinguishable.)

It is also an option to print the QR code directly onto a course syllabus. I find that students lose these papers, so class hot spots are a safer bet.

The last piece of the puzzle is the students and their smart phones. Have them download a QR Code Scanner onto their device. There are plenty of free options available. Various apps are available across almost all devices including iPhone, iPod, Android phone, iPad etc. 

With three or four minutes left in class, have students find a "hot spot", scan the code, and fill out the form. I would recommend practicing this once before you want to collect actual data. 

All responses are automatically collected and organized into a Google Spreadsheet. Accessing the file from Google Drive makes collecting timestamped data simple. 

I have found that the students love the process and that is a victory in itself. 

A Few Notes:
  1. Save the website for your results spreadsheet in your browser bookmarks. Quick access will mean you will use it more.
  2. Take advantage of the digital format. Copy and paste the text from the Google spreadsheet into and create a visual of the student thought. 
  3. After you have read through the responses, delete the rows of data before you have students respond again. This will allow you to keep the same form and same code. If you do not delete previous responses, you may have trouble seeing which responses come from which days. All the responses are timestamped, so this is not necessary. I just don't like the file to swell too large. 
Exit slips provide students the opportunity to communicate troubles, ask specific questions, comment on class structure, summarize learning, or clarify their thinking. This digital curation system is quick to set-up and even quicker once you are accustomed to the infrastructure. 

If you have any specific questions or extensions of the process, I'd love to hear them. Tweet at me (@NatBanting) or leave a comment below. 


Sunday, August 25, 2013

What Makes a Task "Rich"?

In my short career, I have seen the death of the lesson. I remember creating 'lesson plans' to the exact standards of my college of education, and then never looking at them when I began to teach. I was never really in tune with the rigidity of the plan, but knew that there were certain learning goals I needed to get to by the end on an hour. 

The scene has shifted away from the harshness of a 'lesson' toward more student-action-centred words like project, problem, prompt, or task. I like these words because they accurately describe what I am trying to do as a teacher--make the students think. 

My personal favourite remains the "math task", or more desirably, the "rich math task".

The phrase "rich task" crept its way into my planning, blogging, and collegial conversations quickly. Maybe it was the way it rolled off the tongue. Maybe it was how refreshingly different it seemed when juxtaposed to 'lesson' or 'example'. "Tasks" seems free, and "rich" just seemed to fit.

The more I read, the more I realized that tasks have strong roots in the beginning of math reform. The 1991 Professional Standards for Teaching Mathematics say tasks "frame and focus students' opportunities for learning mathematics in school." (p. 24). There isn't a higher pedestal that they can be put on. They are the basis for a solid mathematical education. 

Even more confusing to me, was the fact that in the same publication, the NCTM seemed to lump a bunch of words together under the heading of "task" which included the likes of "projects, problems, constructions, applications, exercises, and so on" (p. 24). I was soon losing my long-sought distinction; I wanted "tasks" to surface as a sort of distinctive champion of effective teaching, but it seemed far more inclusive than the barriers I had set up in my mind. The word had taken on a certain amount of "semantic inflation" (Piaget, 1969). It became used by many different people to mean many different things. It was rapidly approaching buzzword territory. 

Along with the supposed interchangeability of the word "task" with many others, the adjective "rich" soon lost its lustre when I went looking. The keystone article in Rich and Engaging Math Tasks: Grades 5-9 (an NCTM publication) didn't refer to tasks as rich at all. Instead, the word "good" was used. Could such a primitive descriptor really be synonymous? It seemed like the word "rich" was only used to create a smooth title, and past that it held little unique consequence. 

In an effort to pick up the pieces, I turned to colleagues and asked them what they felt constituted a "rich task". I got several responses, and distilled the requirements into categories. What fell out was a collective--albeit primitive--definition of a "rich" task. Those characteristics mentioned most appear at the top of the list.

A "rich task" must have:

  • Multiple entry points
    • The opportunity for divergent thought was a constant theme in the responses. The availability of a low-level entry option was important for engagement of all involved.
  • Multiple solution paths
    • Much like the first attribute, the divergence of thought and method was an important feature. This leads to valuable connections and communications throughout the process.
  • A curious, captivating, or surprising element
    • This was, by far, the most vague of the requirements but several people mentioned that student intrigue played a large role. What makes a task "curious" is variable from student to student, and very hard to determine.
  • Depth
    • My favourite of the requirements, and often the hardest for classroom teachers to implement. A rich task provides natural extensions to students as they work toward solutions. 
There you have it. A muddy, crowd-sourced look at what "rich" really is. Notice that it is taken for granted that the task involves meaningful mathematics. I think the constant framework of curriculum (especially at the high school level) makes these tasks infinitely harder to find, develop, and execute. 

I'd love thoughts on tweaks, flaws, or downright blaspheme. 

Here are some of my thoughts thus far:

   1.   Rich tasks do not need a "real-world" context or consequence.

There is no question that context does help in some instances. Context can help students frame the task in familiarity. The danger in focusing on the context alone (and not the attributes listed above) is that the real-world quickly becomes skewed into what Jo Boaler calls a psuedocontext. It transports the students to math land where anything can happen no matter how trivial or abstract. Context can be helpful if it is realistic and aids in the student's mathematical modelling. 

   2.   Rich tasks should be paired with discourse.

This is where a skilled teacher separates themselves from the pack. The multiplicity of thought sets the stage for an exhibition of student learning. Rich tasks need to be taught within a rich ecology where students are sharing ideas, methods, and connections. Margaret Schwan Smith and Mary Kay Stein--who authored the keystone article mentioned earlier--have written an excellent book on teaching using productive discourse. It is a must read for any teacher hoping to use rich tasks in class. 

   3.   Rich tasks are often shockingly pedestrian.

If there is one thing that the explosion of web 2.0 math resources has shown us, it's the ability to create a curious modelling context out of the most mundane of circumstances. An ounce of wonderment can go a long way. There are powerful mathematical moments waiting for students who explore the relationship between area and perimeter. Students can find it very empowering to be able to predict patterns minutes in advance. Something as simple as different arithmetic strategies can create a buzz. 

Nat Banting

Lazy references:

NCTM, Professional standards for teaching mathematics, 1991.
NCTM, Rich and engaging mathematical tasks: grades 5-9, 2012.
Piaget, Science of education and the psychology of the child, 1969.
Boaler, What's math go to do with it?, 2008.
Stein and Smith, 5 practices for orchestrating productive mathematics discussions, 2011.

Tuesday, August 6, 2013

Animating Patterns

There is a very strong emphasis on linear relations and functions in the junior maths in my province. In Grade 9, students begin by analyzing patterns and making sense of bivariate situations. The unit--which I love--concludes with writing rules to describe patterns and working with these equations to interpolate and extrapolate.

Grade 10 students continue along this path in the light of functions. There is a large degree of abstraction that occurs in a short amount of time, and droves of students abandon the conceptual background (pattern making) in favour of memorizing numerous formulas. (Slope formula, slope-point, 2-point-slope, slope-intercept, etc.)

For the record, I do not think that the transition between pattern sense making and formal function work should be made between grade levels. It is an unfortunate result of our curriculum structure.

Last semester, I worked a lot with visual patterns and group tasks. I began numerous classes with Fawn Nguyen's "Visual Patterns". We started with ten minutes, and by the end of the unit, we were making sense of several patterns within five. It served as a great warm-up or wind-down to a lesson.

We also undertook several #3Act pattern activities. The class favourite was Dan Meyer's "Toothpicks".  After they got their head around the fact that someone would "waste" all that time arranging toothpicks, there were several ingenious ways of making sense of the situation.

In fact, I noticed a distinct increase of strategies with the 3Acts approach. I started putting visual patterns into a stations format, and class participation increased. Students were creating patterns in novel ways, but still using very similar strategies to decipher and model them.

I was stuck--until a student made a comment to me during a station activity:

Me: "We know where the pattern starts, but how does it grow?"

Student: "It doesn't grow at all; it just repeats itself"

Right then I realized why it was so hard for students to describe growth, because the patterns were not growing. They were presented in stages on paper; paper--a static medium--has an incredibly difficult time representing growth.

Paper had reached its natural limitation. 

Some students just couldn't visualize the representation as a single, growing pattern. They wanted to see them as separate, and who could blame them? It makes much more sense conceptually to create a model that describes growth when you can see the growth occurring fluidly.

Here is my modification:

I began with the two problems from my textbook that I used with the inaugural relation stations (linked above).

 and used Vine to animate the growth that the textbook is trying to symbolize.

The short clips make use of colour and movement to animate the qualities of a linear relation. This is especially true for the rate of change. 

Here are three of Fawn's patterns animated with Vine:

The growth movement that these videos show may not be the pattern that the student was envisioning. This can result in larger misconceptions.

In cases where the pattern may be complex enough to entertain multiple patterns of growth, the teacher should first show a static pattern and ask students how they see the pattern growing as the stages progress. This activity holds the potential for rich classroom discourse. Students could use colour, shading, or prose to describe and illustrate their thinking. Only after the students have had time to visualize the movement, should the teacher show some animated patterns and begin working toward an equation or function to describe the growth.

In this case, several patterns may be anticipated and animated to show several avenues of thought. This takes careful anticipation on the part of the teacher. Here are two anticipated growth patterns of another visual pattern:

Great lessons meet students at their current level and provide the milieu to move them forward into deeper understanding. Animating the growth of patterns enables students to "see" the growth and make better sense of patterns when they are presented in stages, words, or in the world around them.