Sunday, April 16, 2017

Upgrade Your Card Sorts

This is the blogpost version of a talk that I gave at CMC-S 2016 and NCTM 2017 on Desmos Card Sorts. I usually post slides, but in this case I’m not sure how useful the slides are without context. Hope this post can give you a sense of the value of a card sort and ways in which a card sort can support student learning along with the benefits of using a digital card sort.

The first card sort we did in the session was Card Sort: Quadrilaterals, inspired by Lisa Bejarano. This card sort asks students to sort a set of statements about quadrilaterals according to whether they are always, sometimes, or never true.
Here’s the "Responses" view of the dashboard for the first four students in the activity (using "Anonymize").  

The green piles indicate that all of the cards in the pile have been correctly matched. Red piles tell you that one or more of the cards in the group are incorrectly matched. Gray cards haven't been grouped yet. You can also see how many cards are missing from a group. 

You can click into individual student work (above) and check the answer key to see which cards students have missed. The image above shows that Hermann only misplaced one card for the "Sometimes" true pile.

You can also use the up/down arrows on the top left to scroll through student work to get a general sense of which cards students are matching incorrectly. Better than this though, you can go back to the summary view and see right away which cards students are incorrectly matching.

From here I might pause the class and zoom in on the card that students have incorrectly matched so we can have a conversation. I might ask my students to share how they paired the card and why, and let the conversation progress from there so that I can help students build towards a correct understanding. What I love about this is the way in which it honors existing ways of thinking. Students have a chance to share what they know about rectangles, and we can use this to develop a need for a precise definition. From there, once we've agreed that a quadrilateral with four right angles is the way to define a rectangle, students can see that a square is an example of a rectangle that has perpendicular diagonals.

The second card sort we did is Card Sort: Linear Functions from the Linear Bundle, and it is an open card sort. This means that it has no answer key. After working on this card sort I showed the dashboard and we talked about different ways that the dashboard can be used to support student learning.

For an open card sort, you can see the most common groupings (above). Since there is no one right answer, the teacher can pair different groups of students together to talk about how they sorted. This is especially helpful in this open card sort where the goal is for students to deepen their understanding of the characteristics of linear functions. Instead of pairing students or groups to discuss, you might screenshot some pairings that highlight similarities and differences that you’d like the entire class to see.

Here the teacher might pose the questions “Which did you pick and why? How might the other group have decided to pair these cards together?”

This card sort had two additional screens that also provide opportunities for rich conversation and student learning. Screen 2 is below.

There are two choices a teacher can make on this screen. First, a teacher can select a set of answers that will be most beneficial for the class to hear. In many cases, sequencing these answers from less formal to more formal can help students at all levels to access the conversation and grow in their understanding of the mathematics involved. Second, a teacher can make use of a screen like this give a voice to students that don’t usually raise their hands to participate. One move I’ve seen teachers make is to let the student know ahead of time that they have an answer that will be valuable for the class to hear. 

Screen 3 (above) is similar to Screen 2 in that teachers can strategically select students to share student work. An additional benefit of a screen like this is the controversy that it can introduce. Below are some of the responses from Screen 3 from the NCTM session.

Most of us gave a response along the lines of the first three bullets. I intentionally planted three of the "Other" responses based on responses I've seen in classrooms, but there were still 5 responses from our session where participants had differing views. Being able to see these responses in the dashboard lets the teacher pair these students with other students to sort out their differences. Between this screen and the others in this open card sort, I appreciate the message that the knowledge doesn’t have to come from the teacher. I can learn from my classmates, and the can learn from me!

We ended the session with a brief tour of, along with how to find pre-made card sorts and how to build your own. For more info on this head to and Also please feel free to chime in with your card sort successes! I’d love to hear more about how people are using card sorts to support student learning.

Saturday, October 22, 2016

Mathy Activities

I was delighted to end my morning today with this:

I was even more delighted with the responses for mathy activities for my 4th grader. I'm posting links to the activities here for future reference so I don't lose them, plus maybe you can use them too!

1. Ken Ken puzzles. I hadn't seen these before and took a trip to this website. Turns out you can customize Ken Ken puzzles by operation and difficulty level, and you can also complete them online.

2. Logic Puzzles. This was a favorite when I was a kid. Glad to have so many resources to find these and other puzzles.

3. Graphiti. As I was opening this link my son chimed in with "Oooh, I love these things!" He was also super impressed that Twitter friends were responding within minutes of my tweet. Go #mtbos!

4. This website has puzzles, games, and mini-competitions organized by grade. Can't wait to check this one out!

5. Printable number puzzles (Rosetta, Star, Subaddo).

6. The lovely Patterns of the Universe coloring book by Edmund Harriss and Alex Bellos.

7. 180 Opportunities to Notice by Steve Wyborney.

Thanks so much to everyone who contributed! What other mathy things might go on this list?

Saturday, June 4, 2016

Starting the summer with SMP 7

Back in December I attended Grace Kelemanik's closing session at CMC North in Asilomar California on unpacking the math practice standards. In her talk, she shared with us her framework on the standards for math practice.  She says that some of the standards take the lead in thinking and some do the supporting.  Standards 2(quantities and relationships), 7(structure), and 8(repetition) are the avenues of thinking.  Students with a strong foundation in these three standards will have a starting point with which to begin problem solving, and they can jump lanes if their starting strategy doesn't work.  This powerful talk has had me thinking about the SMPs ever since.

 SMP 7 is a particularly interesting standard to look at in the high school curriculum because we can find structure in expressions, diagrams, graphs and more. This year I worked closely with the Algebra teachers in my district while we implemented a new curriculum. One problem in particular from our new curriculum is below. ( From Engage NY Algebra Module 4 Lesson 8 (Student View) )

Students use the symmetry of a parabola to complete the graph (structure in the graph).  Later in the problem set students make note of the pattern in the table of values (structure in the table).

There were additional problems that made use of the structure of a parabola in order to solve.

  • Is f(4) greater than or less than -6? Explain
  • f(-4)= -13.  Predict the value for f(4) and explain your answer.

Towards the end of the lesson there was a section called Finding a Unique Quadratic Function.  Below are some of the prompts from that section:

  • Can you graph a quadratic function if you don’t know the vertex? Can you graph a quadratic function if you only know the 𝑥-intercepts?
  • Remember that we need to know at least two points to define a unique line. Can you identify a unique quadratic function with just two points? Explain.

At one of the schools that I work at we spent two days exploring the structure of parabolas and questions like the ones above using Desmos.

Day 1 Activity: Broken Parabolas

Day 2 Activity: 1-3-5-7 Parabola Challenges

In the 1-3-5-7 Parabola Challenges students explored the symmetry of parabolas by locating missing points given a set of points on a parabola. For this lesson students only looked at parabolas with an a value of 1. We didn't use that language since it was an introductory lesson, but it allowed students to use 1,3,5,7 (etc.) pattern when comparing the differences in y-values.The hope was that this informal exploration would set students up to look for and make use of structure when they graph parabolas by hand later in the unit. 

Having students reason through the problems using a pattern can also make for a great error analysis activity. Some of the student work from Activity 2 is below.  Each of these challenges started off with a set of blue points, and students were able to drag the red points to complete the parabola. They would then turn on the green function to check that it is going through all of the red and blue points. At the end there are also a couple of unique responses for challenges 1 and 2 from the 1-3-5-7 Parabola Challenges activity that would have been great to share with the class.

The original version of this activity had students entering coordinates into a table instead of dragging points. I thought this would help them focus on the patterns in a table of values as well, which could be an additional problem solving tool for them to utilize later in the chapter.  This turned out to be too challenging for the group of students that I initially worked with.  Adding points to a table of values using a pattern requires strong number sense, and is a different skill then having students use a pattern to add points to a coordinate grid.  Here is the original activity in case you'd like to take a look.

Focusing on SMP 7 in the planning of these activities shaped them in ways that led to increased reasoning and discourse.  Of course this is not the only way to increase mathematical reasoning, but definitely worthy of some focus this summer.

Footnote:  I also wanted to thank Silicon Valley Math Initiative for an inspiring end of the year session.  Their summer focus is also on SMP 7, and we launched this work in our last session by finding examples of SMP 7 in the K-12 curriculum.  I've been thinking about SMP 7 and seeing examples of it ever since.