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.

Talking math with your kids #tmwyk

One of the wonderful privileges of being a parent is that we get to train our children in so many ways. One of my ways to nuture my children is through talking to them about mathematics. Sadly, for many who have had bad experiences with mathematics, this idea of talking math with children can cause anxiety.  This does not have to be the case when we see math as the flexible, usable, adaptable, enjoyable beauty that it is.  

On Twitter there is a hashtag, #tmwyk or talking math with your kids. This hashtag was created by Christopher Danielson. Christopher is very active on Twitter and quite responsive to individuals who post with the #tmwyk hashtag.  You can check out Christopher's website here.   

Over the last year or so, I have posted a number of interactions with my oldest daughter to the #tmwyk hashtag.  

 

Using the Chromecast to #tmwyk pic.twitter.com/9yGWvLc9hu

— Adam Paape (@Adapaa) March 21, 2015

My kiddo and I were playing, "Which one doesn't belong?" in this tweet.  Christopher Danielson is responsible for the "Which one doesn't belong?" game.

Fun work with multiples of three based on this glass of milk. 5 year old loves it. #tmwyk pic.twitter.com/FszNjPMtZP

— Adam Paape (@Adapaa) November 2, 2014

Here we had a wonderful conversation about multiples that she started by noticing the threes.


This morning my daughter and I spent some time playing a card game. I found this card game here. The basic idea is that you lay down 7 cards and then the kiddo tries to find two cards that add up to 10.

The child continues to find pairs of 10s until there are no more pairs.  When there are no more pairs to be made, the parent puts down seven more cards and the child tries to find pairs of 10s again.

In our first attempt at this game, my daughter was clearly excited. We had a really nice conversation about an Ace (1) plus 9.  We discussed whether or not 1+9 was the same as 9 + 1.  Talk about a great way to think about the commutative property of addition. My daughter said, "Yes Daddy, they are the same."  I said, "Why are they the same?"  At first she said,  "Well, just because they are." But then she gave me a counterexample.   "Dad, it isn't 9+2."  I said, "That's really good kiddo. Thank you for your thought and thank you for making me think."

 After a little while we put down 7 more cards.  An interesting thing happened with this sequence.  Take a look.

 I asked her, "What do you notice." I was expecting that she would identify the patterns of the Aces and the 9s.   However, she had a very different thought in mind.  She told me, "Well Dad, these two are the only two that are black."  I said, "Wow! You noticed something I didn't even see." She looked at me with a huge smile.  She was so proud to have noticed something that Dad didn't see.

I love activities like this where there are many possibilities for different perspectives on the math.  On this particular sequence of seven cards, I thought that she would just pick up all the Aces + 9 to get her 10s, but she didn't do that.  She remembered that there was another 5 that was hidden under one of the 9s, so she worked to get an Ace plus 9 so that she could get to a 5 + 5 because, as she said, "I know 5 + 5 is 10." Wonderful, this is why we talk math with ours kids.  They will show us things that we didn't even think about.   Kids will create strategies that are all their own and it is one of our jobs to get them to justify their reasoning behind these wonderful, kid-created strategies and thoughts.  So, let's keep the #tmwyk movement going - the payoff is huge!

Growth mindset in the classroom

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A number of weeks back I had the pleasure of listening to Jo Boaler speak at the Wisconsin Mathematics Council annual conference. She spoke to us about many aspects of best practices in mathematics education. One of her main points was that we need to make a transition within our students from a fixed mindset to a growth mindset. Boaler referenced Carol Dweck's research on mindset. Dweck makes the point that students need to realize that their brains literally grow when they are challenged in the classroom. When a child is given a complex problem, the teacher needs to encourage the students that they can do this and that this challenge will grow their brains- the challenge is good. Dweck highlights the idea of "not yet" as an important way for teachers to think about assessment of students. She mentions that students who have a growth mindset really do see learning as a valuable endeavor.

 See Dweck's Ted Talk on "The Power of Yet" below.  

As I think about this growth mindset and the idea of not labeling students as smart or dumb, I feel like this is something we ought to be doing already. However, it really isn't the case.  And, I'm sure that I am guilty of unknowingly labeling my students in the various ways that I treat them.  Many students see themselves as smart or as dumb (fixed mindset).  Teachers need to be intentional in articulating to students that the lessons teacher's design are meant to be challenging and that hard work does pay off through perseverance and persistence. Within the Common Core Standards for Mathematical Practice, there is a standard (SMP1) that highlights the benefits of perseverance in problem solving. Any teacher of mathematics ought to realize that persistence in problem solving is a valuable trait, but often times we really struggle to get this trait to be manifest within our students.  Therefore, this intentional focus on perseverance is something that a teacher needs to purposefully cultivate in his or her classroom practice.

I have spent quite a bit of time with my 6-year old, helping her to see the value of hard work both in her reading and in her work with mathematics. The other night we were working on her leveled readers and she was working hard and struggling, but I told her, "Sweetie, you can do this. It is hard work, but hard work pays off.  Let's keep trying."   Eventually, after working really, really hard on a lot of words, she said, "Dad, I can do this!  I can read!" She was so proud of her hard work.  She experienced "not yet."

The other day we received a document from her school of a self-analysis that she did within a book called, "If I lived in a castle."  You can see the image below.  She identified herself as "Lady [her name] the Smart."  I'm not too concerned about this, but I do find it interesting that she is already labeling herself "the smart."

So, take another look at Carol Dweck's video and think about the idea of a fixed mindset versus a growth mindset.  As an educator, my goal for my students and my children is that they have growth mindsets. It is work, but it is worthwhile work that pays huge dividends in learning and brain development.