Thoughts from a first time TMCer after day 1

Day one is in the books from my very first Twitter Math Camp (TMC) at Augsburg College in beautiful Minneapolis, Minnesota.  It was a pleasure to put real, live people to the Twitter personas that I have seen over the years. 

I began my day by biking in from our rental home in Minneapolis.  I was pleased with the safe and well-kept bike paths.  The first-timers session for TMC was helpful and was well run by Glenn and Julie.  It felt good to see the abundance of other first-timers in the room - I was not alone in this new adventure.

My morning session was a continued affirmation of the friendliness of all of the TMC participants.  Chris Luzniak and Matt Baker provided a packed room with a number of useful strategies for getting our learners engaged in conversation centered around math.  Chris and Matt gave us the "My claim is...and my warrant is..."framework for student voice within the classroom.  I definitely plan on implementing this statement/justification model in the coming year. 

During the keynote address, Jose Vilson gave the whole TMC crowd a wonderful call to action to support marginalized students and to do so In a real, authentic manner.  Jose really spoke to the heart of what education should be.

During the one hour afternoon sectional, Julie Reulbach and Julia Finneyfrock did a nice job showing us all of the bells and whistles that Pear Deck has to offer.  I am intrigued by the platform.  However, I think I might need to have a bake sale or two to afford the annual costs for Pear Deck.  I really liked the way that Desmos could integrate within Pear Deck.

The half hour sectional I attended was led by Anna Hester and covered the progression and coherence of using the box (area) model for multiplication.  Anna brought a lot of energy to her presentation, especially when she brought it to a conclusion with using the box model for polynomial long division.  

The speed dating session was definitely the highlight of my day.  I had almost decided to leave before the session began, but I am very glad I did not leave.  I was able to make a number of wonderful connections during this session.  In one sense, I wish it was possible to do something like this again, where there is an organized interaction time for the participants on a quick basis.  

I suppose that is all from day 1 of TMC16.  As a newby, I felt welcomed, but I know that there are many  more people that I would like to meet.  Sounds like a challenge for tomorrow, right?

The Power of Patterns

My Number Sense (MATH119) students are taking their midterm for our course today.  A pattern problem was the very first question I asked them to think about.  I am a big fan of pattern work at all grade levels.  There is so much richness that can come out of pattern work.  Here's what I anticipate seeing from my students with this pattern.

There will be at least three different kinds of response.

• One group of students will identify a consistent pattern of addition and simply follow that pattern of sums until they get to the 50th stage.

• Another group of students will find a variable expression that connects the stage number to the number of blocks at a given stage.  These students will use their expression to find the number of blocks at the 50th stage.

• Yet another group will come up with a(n) approach(es) that I would never, ever, not in a
  hundred years had ever thought of on my own, causing me to smile, say, "That's cool!," and help me to appreciate the flexibility and beauty of math.

Pattern work allows the students to see the problem in a way that makes sense to them.  Problems like this encourage our students to see the math as more than a set of procedures to be followed, but rather as a puzzle to be solved.  Jo Boaler calls problems like this ,"low floor, high ceiling."  I really like this idea.  While there are varying levels of sophistication within the solution strategies of my students, it allows me to continue the conversation with my students about the math after the assessment occurs.  This particular pattern is quadratic in its nature.  This grounding of quadratics within a pattern allows me to have my learners connect and use representation of the mathematics in a variety of forms.  When my learners can see and make those connections on their own, that is where the real power of mathematical thinking emerges.

My new main squeeze (mathematically speaking) - Desmos Activity Builder

Over the last couple of months, I have become more and more familiar with the brilliance of Desmos Activity Builder (found at the bottom left of this linked page).  My students have become quite familiar with activity builder as well.  For the last number of years, I have been deeply appreciative of the good product put out by Desmos.  The online calculator is intuitive and really helps students to grasp the conceptual in a way that allows mathematical procedures to have a firm grounding (supporting the effective mathematics teaching practice 6 as defined within NCTM's Principles to Actions).  The relatively new feature of the activity builder has taken it to an entirely different level.

While I have used it in my college classes to support the understanding of my students, it has also been enjoyable to see my methods students develop their own activity builder lessons.  My great successes within activity builder have centered around student discovery of transformations of functions.  Yesterday, my college algebra students engaged in this activity.  

https://teacher.desmos.com/activitybuilder/custom/562e3e7f9236025b1c09a483

I had students work in groups, primarily pairs on this activity. I did this to allow my students to have discussions about their mathematical assertions.  Using collaborative groupings allows for students to have meaningful mathematical discourse as defined in effective mathematics teaching practice 4 within Principles to Actions.  

As students are engaged in the activity, I can monitor progress through the overview page on the teacher dashboard of activity builder.  I really appreciate this feature because I am able to monitor the ways that students are progressing through the activity.  I don't want speed to be a factor in my instruction, so if a group seems to be rushing through the activity, I know this right away.  I can check in with this group to monitor more closely their needed understanding of the concepts.  

Another of the wonderful aspects of activity builder is that I can check student work easily after the students have completed the activity in class.  At the end of this particular activity, I asked my students to find three different quadratic functions that would match to the curves on my "ugly mug" picture.  This was the creation of one group.

Desmos is also a fertile ground for students to work on appropriate vocabulary within particular contexts.  As I monitor their responses, I can step in and ask probing questions to add a certain level of mathematical specificity to their written assertions in Desmos.  

In short, activity builder has allowed me to create an environment where the sense making is placed firmly on the shoulders of my students.  They are actively engaged in the content as they explore the behavior of functions.  It is a great environment for students to take risks and revise thinking.  

Thank you, Desmos!

Emazing presentations

At our most recent School of Education retreat at the beginning of this academic year, my dean (Dr. Michael Uden) introduced us all to a presentation tool by the name of Emaze.  In some ways, it reminded me of Prezi.  However, it had a little more staying power in terms of visual appeal (for me at least).  I decided to take action on Dr. Uden's inspiration to transform one of my old Prezi's for my Number Sense course and turn it into an Emaze experience.

The reality with any presentation tool is that it can easily just be a fancy-dancy PowerPoint presentation.  This ought not to be the goal.  Active student engagement needs to be at the center of any presentation.  I will be using the Emaze presentation found below in my class tomorrow.  I plan to take significant amounts of time to pause and have the students engage each other at their tables in discussion centered around deductive reasoning and problem solving.

At the very least, Emaze is another tool for implementing the potential for an engaging learning environment.  We shall see what tomorrow holds!

Click to see Dr. Paape Math 119 Section 2.3 Emaze Presentation

Rubric to support the expression of student reasoning

Before the beginning of this school year, I spent quite a bit of time reflecting on my teaching practice as a whole.  In many ways, my regular day-to-day practice has radically changed to fall much more in line with what we know to be best practices in math education based on research.  However, my assessment strategies have lagged pretty far beyond.  Today, in my Number Sense course I am trying something new.  I am curious to find out if my adjustment increases the quality of the ways my students express their reasoning, or if I have removed some of the cognitive load for them.  Here's what I did.

1. I made most of the assessment very pattern-oriented.  For instance, I included a number of problems like the ones below.

I found the patterns above at www.visualpatterns.org/

2.  At the beginning of the quiz, I previewed the problems with the class.  I also showed them a problem solving rubric that I would be using to assess their explanations of their reasoning within their problems.  I kept this rubric visible on the front screen throughout the quiz.

I adapted this rubric from a Utah Education Network website.  I intentionally changed the rubric slightly to embed some Growth Mindset vocabulary in the "Not Yet" column of the rubric.  

3.  As students worked on the quiz, I observed them regularly looking up at the rubric.  As I consider the value of this rubric, my one concern is that I may have removed a small amount of the cognitive load for my students.  However, I think the rubric is general enough that it will likely serve more as a guide than a crutch for the students.


Here are a couple of samples of student work done within the rubric framework.

I'm pretty happy with the detailed nature of the work of my students.  As is often the case with pattern problems, it is interesting to see the variety of ways that students see the math within the problem.

I plan to continue to inspect my assessment strategies for more ways to facilitate deeper student thinking.

The Department of Representations Department

The use and connection of mathematical representations is one of the eight effective mathematics teaching practices put forward by the National Council of Teachers of Mathematics (NCTM) within their 2014 publication, Principles to actions: Ensuring mathematical success for all.  The idea of representing concepts within mathematics is nothing new for those of us in math education.  However, I think that the use of representations and emphasis of using representations in an interchangeable manner serves to further promote the understanding of concepts for students.  In this post, I am going to reflect on a Week 1 activity that I drastically changed from my old practice of teacher-centered instruction to one where student representations and discourse took center stage.

In recent months, there has been a bit of a commotion on Twitter in respect to Vertical, Non-permanent Spaces.  Take a look at Alex Overwijk's blog on the topic of VNPS.  I have also blogged on my use of large table whiteboards as well in the past.  VNPS have taken a prominent role in my teaching practice.  What follows is an example of their use within an activity designed to allow for students to compare representation usage.  

Here was the question for the class to consider.

This is a fun, little problem that allows for multiple points of interpretation by students.  I asked the additional estimation/guess question to instill in students the idea that we want to have idea of where we are going with a problem before we begin.  

Before I get into what the students did, I want to reflect a bit on how I have taught this in the past.  In years past, I would stand at the board, marker in hand, and draw the picture (maybe putting a number 1, 2, 3, and 4 in each of the squares above.  I would then write what is written below.

I would literally say exactly what it was that I wanted my students to notice.  That is worth saying again - I would tell my students what THEY should notice.  As I reflect on that teaching practice, I actually get pretty ashamed of that kind of teaching in my past.  In my practice, I was stealing the learning moments from my students - taking away the opportunity for them to have ownership in the learning.  Ultimately, after I had "led the witness" to the point that he or she may or may not have grasped what I had hoped to be learned, I would assume that he or she could then translate this decoding skill to other representations.  For some, this was accomplished, but for the learner who struggled, this was rarely the case.  

Fast forward to a Principles to actions classroom and a new mindset on my part.  At the very forefront of my thinking, both in the preparation of my lessons and the implementation of those lessons, is the goal for my students to do the noticing.  I want them to wonder about a problem, to discuss ideas with their collaborators at the table with them and for them to have ownership of their learning.  Here are a series of representations created by my students after they were given a substantial amount of time to converse around the topic.  Four somewhat different approaches surfaced from the various groups.  

 Creating a table to help

As you can see from these two groups, they noticed the same pattern of +3 that I had regularly shown my students in the past.  The group in the top picture used a drawing to show where the 3 and the 1 in their pattern were.  The group on the bottom (a group of math majors, unlike the top picture), used a series of growing matchstick representations.  The complexity of the algebraic expression in their generalization was somewhat sophisticated.  I didn't want to discount the value of their expression, but simply asked if they realized that this expression of 4+3(n-1) would be a sticking point for some younger learners.  What does the n-1 stage mean?  How can you relate your algebraic representation to your drawing?  These students were able to explain their thinking but were open to the idea that this expression could cause confusion.  All of these interactions would never had happened, if I was at the board telling them what to notice.

60 squares of 3, plus 1 extra stick

Each of the three groups above used some version of the fact that they needed 60 total squares.  They identified that all of the squares with the exception of the very first square (with 4 matchsticks) were created by adding on three matchsticks.  

56 more squares

Each of the groups above used an adding-on approach.  Since they were given four original squares, they made the adjustment of adding on 56 additional squares with 3 matchsticks in each square.

We need 60 squares - here is what 10 is - we need 6 of those, plus adjustment

 This was the only solution of its kind.  This group actually struggled quite a bit to get to a solution.  They had initially misread the problem and had added matchsticks below the original figure, not as a continued horizontal pattern.  However, when they regrouped, reanalyzed the problem, a new solution strategy emerged.  This group noticed that they could represent the first ten matchstick boxes as drawn on their board.  They then thought that they could do six more of these groups of 10.  Initially, this was met with a problem. "We have too many sticks."  Another adjustment was made to remove the 1 overlapping matchstick.  Therefore, they knew to not continually add the 31 sticks from the first 10 boxes, but rather 30 sticks to account for the overlapping stick.  WONDERFUL!

Concluding thoughts

After each group finished their work on the whiteboards at their tables, I asked them to put their boards in the marker trays around the room.  We all took a gallery walk to make sense of the work of our classmates and to ask clarifying questions, if needed.  I encouraged the students to look for similarities and differences in methods used.  Next, each group got to verbalize to the class the work they had done and processes used in getting solutions.  

Taking this kind of time to get to an answer that I could have showed a class in much, much, much less time is an intriguing struggle.  However, I am absolutely confident that my students have grown in varying degrees in their problem solving skills as a result of the intentional kind of struggle in creating their own representations of the math.  Thanks again to the Department of Representations Department.  It may seem redundant, but I do know that the students appreciate the idea of sharing thoughts with each other.  They also appreciate seeing the varying ways that their classmates think about math.  The ways in which I validate their efforts in an activity like this go a long way in furthering the growth mindset for each of the students in my classroom.  It is a wonderful adventure we are on together to grow in our mathematical understandings.  

Group Dynamics 101

Today marks day four of a new school year.  It has been a blast to put into use all of the new ideas I've been considering throughout the summer.  One of the new implementations that has reaped immediate benefits is the use of the document below.

On the first day of my class, I had each student fill out this form.  Not only did this give me a feel for the math dispositions of my students, but I was able to use this data to form some intentional heterogeneous groupings to best promote student discourse.  I have always been a proponent for heterogeneous groupings, but never before I have been able to make these grouping choices based on student-identified markers.  I have been very pleased thus far by the kind of work my groups have been doing. In addition to my intentional grouping strategy, I have had the students create a list of what they do and don't like about group work.  

I really like the lists that my students created.  I handed out this completed copy of group work norms to the students.  We have all agreed that this is the standard to which we will hold ourselves in our group work.  In my next post, I will blog on an activity that reaped the reward of my data-driven groupings and class-originated norms for group work.