Family Math Night (1of3)

What might it look like for a learner to welcome a challenge, to see a difficult task and smile. What if we had learners that sought out challenges, learners that knew that growing their brain was the result of challenging their brain with hard tasks.

Learners that foster a growth mindset in mathematics exhibit these characteristics, and this is the message we want to communicate with our stakeholders. In our first (of three) Family Math Nights, we showcased the work of Dr. Jo Boaler, from Stanford University, and the wonderful things happening at youcubed.org.

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Families engaged in fun mathematics, guided by a wonderful group of caring teachers, where communication and visualizing mathematics were experienced. Several teachers had variations of visual mathematics, like What’s My Place? What’s My Value?, or specific visual methods  highlighted in the California Frameworks for Mathematics.

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My favorite part of all of these nights is observing parents watching their kids enjoy math and listening to their conversations. Families together engaging is mathematics in a fun way, building connections and learning. Our next event will focus on Mathematical Methods, areas that the community identify as challenging to understand, teach, and explore, which we am really looking forward to capturing.

In what ways are we fostering a growth mindset in mathematics outside events like this is an ongoing development for me, and I am wondering how to incorporate more visual mathematics across our district. What might you suggest to help us get better together? I’d love to hear if you have any similar stories?

A huge thanks to our host at Jim Maples Academy, to Dr. Boaler for sharing these transformational ideas, to our teachers for their tireless efforts to impact our learners, and to the community for sharing our night of Fostering a Growth Mindset in Mathematics.

Sequencing Transference

I am curious if there is a way for transference of skills to be acquired over time, especially for a difficult solution technique.

To satisfy my curiosity, I am trying the following scenario. In August I gave my Integrated Math 3 students a task, working with a partner, they had to sequence out the strips of paper hidden in an envelope. I made enough for each to student to work with a partner, one student would move only the words and one student would only move the equations.

The students had an entire period to work, and asked when they finished to take a picture with their device and email it to me. This first round easily took the entire period, as students became familiar with the task and had to make sense of the problem. As students worked, I circulated and asked guiding questions, but offered no direct support. A little over half way through the period, I had two groups that were close an accurate sequencing of the problem, so I prompted that students should walk around and look at what other groups were doing, develop a dialogue, and see if some fresh ideas developed.

By periods end, almost every group had some form of an accurate sequence, and I was quite impressed by their perseverance in this task. Thinking of the SMPs, we had hit three of them hard (SMP 1, SMP 3, and SMP 7), and I really think the students enjoyed it, even if “their brains hurt.”

A month later, I gave them the same envelopes, this time as a warm up problem. After 5 minutes, most groups were done, many of them could recall the flow of the solution, with minor errors; however, what stood out to me was how students were smiling once they felt they were able to do this task again, with much less effort.

A few weeks later, the students come to class they saw the familiar envelopes, they grabbed them and started sequencing the steps without being prompted. Less than three minutes later, the entire class was finished, so I took pictures of each group’s result, and projected them. As a class, I asked the students if they were similar or different than their own, this lead to a great discussion about many of the pieces that were in place, and why certain strips went where.

Following our discussion, I posted two similar problems to the one they had been sequencing to see if their understanding would transfer. The students tended to fall apart in attempting to solve these problems, their understanding of doing and undoing, i.e. the use of inverse operations and a mushy understanding of the problems became apparent, though they were clearly on the path to understanding the sequencing problem.

One thing that shocks me is that there is little connection from the sequencing of the problem to the examples shown, even after guided questioning. Meaning I’m not asking the right questions to uncover their thinking to make these connections, or I didn’t set the stage well enough for them yet, or I jumped ahead of myself and need to revisit at a future time. In any case, my curiosity is not satisfied, I do not know if students are able to transfer their understanding of one way of looking at a problem to another way, or if this task is too cognitively demanding to test this process with. In any case, there is evidence of learning in many other areas, and the collaboration over this task has been a pleasure to observe.

I am curious about your experiences with sequencing, transference, and promoting perseverance in with your students.

Rounding and a Peek-a-Boo

My Rounding Lesson

Rounding is a challenge to learn, knowing how to teach the concept is more complicated, especially to a variety of learners.

One of my tweeps, Amanda (@MsHaughs) made an excellent reference to building context around the action of rounding. Providing learners the opportunities to understand the ‘why’  supports their understanding. Amanda shares many other great ideas in a tutorial video, I highly recommend seeking out.

The goal for my 30 minute lesson is  to have learners comfortably be able to understand the general concept of covering up the digits that are not of interest with respect to the place value you are rounding too. Using a number line find your benchmark numbers, remove the peek-a-boo, and determine to which benchmark number it is closest.

One of the things that sticks with me, is finding a way to make sense of the process doesn’t equate to understanding the conceptual ideas.

The lesson starts out with two animated gifs, one showing place value, the other a silly cat being surprised by a puppet popping out of a can. The students are asked to read the question, “What might we be learning about?” One student then reads it out loud, and I ask for 10 seconds of quiet think time. A teaching tip, I included the silly cat gif because I am anticipating that students would find it both humorous and draw their attention to the screen. After the quiet think time, I ask the learners to share out at their table with their partners, as the conversations are going, I’m listening for key words, which I point out, then ask for a couple of ideas. I am assessing their reasoning, while I’m building their interest in the topic.

Moving into the content, I play a video I made through Explain Everything, and follow up with an animation I made highlighting the same steps with a different example. The first number I used was 45, rounding to the nearest tens. The idea of starting here came from a coworker of mine, she’s an amazing teacher, and now a coach in our district. She suggested that I start with something like this, keep the focus tight, and use similar examples. I used the same numbers just ten times smaller for the two examples we worked through. I also had a “secret ingredient” that allowed students ready, or wanting to challenge themselves at the end. The animations showed the case of rounding the number 78 to the nearest tens, and then students were given the choice of 3 numbers, they choose one, and use the number line to round to the nearest ten. I created a quick worksheet with the numbers, the blank number line, and the directions so they would be able to quickly apply their learning. I should mention, after the animation played twice, I paused each step, asking questions, uncovering their thinking before they began their practice.

Once students were working, I saw a variety of misunderstandings, and a variety of students able to make sense of the process. After a few minutes, I started asking students that had variety of answers to present their work on the board. From that point, I asked students to compare what they have to those examples on the board. I am trying to avoid telling the learner of their mistake, rather setting the stage for them to uncover this idea on their own.

By the second iteration, students were able to move through the process with greater fluidity, recognizing the similarity of the process with the question. We were rounding the same digit sets, for example if the first round was 82, then the second round was ten times smaller 8.2. In the former case we were rounding to the nearest ten, in the latter case we were rounding to the nearest ones. I loved that several students chose the same number, chose the same strategy, and were able to articulate that it is exactly the same, just the numbers have a different value.

Several students assumed we were learning about place value, I didn’t disagree, I only mentioned that it shows up in particular ways. Students showed evidence of this understanding by choosing to do the “same number” in the second round, rounding to ones, as in the rounding to tens because they recognized the structure of our base ten system.

When students were prompted with the “secret ingredient” I was pleasantly surprised to hear a couple of students say, “Oh, I’m taking the hard one…” Understanding that students has three numbers they could choose from, the impact of that statement was relative to what they viewed was challenging. In fact, if one student chose their version of a “hard one,” another student had chosen a different number for that same reason.

By the time the two lessons were over, I felt that the lesson was successful overall, there are a few things I think I would have done differently, like not saying, “round up/down,” I’d rather have been consistent with the idea of which benchmark number the number we were rounding is closest too. Keeping the idea of distance is important and plays out in later roles. In the second lesson, I forgot to give the story context of the importance, and I felt the closure in both cases was a little forced, with no requirement for the students to produce any reflection on the learning. A quick summary of the lesson or a 3-2-1 exit strategy, at the least, would have provided some insights into their understanding and allowed them some processing as to what we just did.