2017-03-28 On The Spot Ahhhaaaa

Although I have many blog posts on the tarmac ready for lift off, this entry takes special priority as I had a powerful Ahhhha moment that lasted two full periods.

Quick background, I have been teaching about trig functions in IM3 and coordinate geometry in 8th-grade math.

My IM3 students look similar to Fry from Futurama….(see below)….when I begin discussing trigonometric functions and their relationship to a right triangle and the unit circle.


Yesterday, I thought I would slow the progress down and focus on right triangle trig relationships, then move back up to unit circle and beyond. Before I even began today, the students’ lips were moistening with anticipation of the drool onset….So I changed directions, had handed out a sheet of paper to every student, and we began folding.

I didn’t know where I was going with this, but I was on an inspirational kick and was going for it. After students folded their papers, I checked each stage, I’d ask a question and have them label their own triangle, the students would confirm with their partner, and stand up when they agreed.


The 3 step process was extremely helpful in uncovering areas of stuckedness and bringing their understanding to light. My next step was to have them write each of the trig functions according to their triangle they had created.


Time quickly ran out for the students when were at this point, so tomorrow, they’ll finish the trig functions relationships. Students literally gasped when the bell ring, they were so into the lesson they couldn’t believe the period was over.

Then my 8th-grade party began, immediately after IM3, and I’m working on giving the students plenty of repetitions on problem varieties to allow them to be successful. Yesterday, the progressions began with a problem having them plot three ordered pairs, which form the vertices of a right triangle.

The students would plot the three points, then find the distance between each of the ordered pairs. Students should find that the two of the lengths are easy, just counting the distance between points, the third combination would require the use of the Pythagorean theorem. The third stage in this progression asked students to determine the slope between points forming the hypotenuse, with the hint that if stage 2 was done well, there shouldn’t be any additional requirements for calculations. The final stage asked for an explanation of how students know that the triangle formed in stage 1 was a right triangle.

Well students shared they were having a hard time with the concept of slope, so fresh of the triangle fame of IM3, I gave the same concept a try in 8th-grade.


Since this was near the end of the period, I performed a think aloud to indicate my learning. Tomorrow I’m expecting that this will be part of their work, creating the ordered pairs on a triangle, perhaps on centimeter grid paper, and do it exactly. Anyway, I felt the explanation landed with the students, several heads were nodding with that smug look of understanding….


Which showed I was on to something that may uncover more of their thinking as we explore and shed light on that which was previously in the dark.

My excitement for this simple idea and the reason for posting this moment is tied to my developing understanding of students learning and being back in the classroom for a couple of periods. Addressing students understanding in visual ways, with a manipulative and a slow pace were all pieces that helped gain students attention to make it truly impactful.

I always love these inspirational teaching moments that shed light on understanding for students, and the fact it worked in 8th through 12th yields potential for my favorite kind of tools that go K12.

What were some experience you’ve had that are similar or different? What about those experiences made them turn out the way they did?

Mission Accomplished

As I publish this reflection, I fear that this moment may turn into something along the lines of another famous moment….


I am very honored to have met #IRL (in real life) the person that inspired this learning goal for me this school year: Having students produce a weekly learnings podcast for public consumption. Joe Young is an incredible TOSA for math and education and he shared this idea a long time ago, and when I had the pleasure of teaching two courses this year, I wanted to challenge myself and use this as one way students would show what they know.

Finally, this week, I found the magic recipe to making this happen, I assigned two students, one in each class to get the job done. Friday marked the turning point for me in doing some of the things I wanted to get done and continue to build on. Friday was also our first Mystery Skype with another group of students. While the connections prevented less opportunities for conversation, it was an entertaining and fun sort of thing.

While doing the Mystery Skype, we also produced our weekly podcasts for both classes. I loved the idea of having students interact with other students around mathematics and this public display showed a proof of concept.

All in all, I was very excited to have been able to edit, publish, upload, host, and publish to iTunes. I am currently waiting on a review of my two podcasts to be approved, and if that works they’ll be in iTunes. The Mystery Skype provided the opportunities for students to strategize their thinking and be able to try to problem solve.

The podcasts are linked:  Math 8    &/or     IM3

Well looking forward to another great week. Hope your week is as magical and moving you forward to meet your new goals.

Fact Families

Time flies by with little regard to your desires to get things done, and I am three months behind on my blogging. I am going to start to remedy that situation with the change of the calendar.

This first example is from back in October of 2016, I will set the stage and then showcase the video below.

On this particular night, the two 3rd graders in our lives, had been working on their homework with my wife for over an hour, and everyone had exposed nerves when I showed up. The girls were frustrated and tired of the mathematics, my wife was feeling defeated at not finding a way to convey the ideas to the girls, and everyone was feeling pretty negative toward mathematics.

As I came into this situation from the perspective of we need to have some fun, and learn through exploration and their girls realizing they have the tools to make sense of the vexing problems. So we started with some work on visualizing the mathematics, utilizing the notice and wonder format to make some headway. Soon, the girls were feeling pretty good about themselves and their mathematics, and we pushed into the Fact Families, the very devil that had started this whole mess.

The girls went back and forth, each taking a turn and exploring as their understanding firmed up. The video showcases this entire process; we were using my iPad with Explain Everything to highlight the pieces. When we finished, the girls listened to pieces they created summarizing their thinking to each other, an added benefit of recording their thinking.

Love to hear your thoughts. Have you had similar experiences?

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.


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.


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.

Understanding Math Together

University of Denver mathematician, Stan Gudder, once said, “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.”

This seemingly simple statement may seem contrary to many of us who struggle at some points with our experiences in mathematics. The abstraction of thought, the seemingly endless number of steps, and hoping we remember which formula to use to solve the problem did not seem about making complicated things simple. Often our common mathematics experiences felt like a single minded journey to obtain an answer, we were finished with one problem we move onto our next answer quest.

Within this context, Burton School District is very excited to offer a series of days throughout the 2016-2017 school year that help us all to get better together. Through a partnership with Grandparents Raising Grandchildren, Burton Schools will offer courses a couple times a year where we are able to understand mathematics at a deeper level, seeing the beauty that resides within mathematics, and come to appreciate what Dr. Gudder was speaking about.

The our course, titled “Understanding Math Together,” we journey together, offering a variety of learning experiences to help us all become better together. We will also have the opportunity for three Family Math Nights, hosted at Jim Maples Academy, for the whole family to have fun doing math together.
As we conclude this exciting news, we hope through the combination of both the Understanding Math Together and the Family Math Nights, we will be able to expose, as Dr. Gudder suggests, the essence, beauty, and eloquence of mathematics, together. We hope to see you there, so we can get better together.

#makeschooldifferent Challenge: Accepted (as I understand it)

From the first time I watched the above piece by the amazing teacher, Taylor Mali, I find myself shaking my head, goosebumps rising, and, by the end, yelling, “Hell Yeah!”

This evening, I was challenge by the amazing @mathprincessC to write a blog about #makeschooldifferent, to which I am responding here.

I admit, this is a challenging piece, I am sure in my rush to respond, I am missing many things, but I have a few things on my mind, and here we go.

peter-griffin_400x400I am a total goofball (think Peter to left), so I am able to make most serious and difficult situations a little less difficult through my bad jokes, and silly antics. The limit of productivity approaches the zero, as the function of my distance to productive center simultaneously approaches zero.

I encourage people to reach for more than they are capable of, I want others to see the amazing potential they have, and the value they bring to our world each and everyday.

I try to make school different by reaching one student everyday, to make that personal connection, build relationships, and make at least one positive interaction with at least one person per day.

To poorly paraphrase Gandhi, I make school different by being a model of the change I wish I would see in the world. gandhi-21

I challenge teachers to develop and foster relationships, I ask them to care, I want them to see their growth from day to day, week to week, and month to month. Focus on growth, we are all getting better together.

Those are some of the ways that I make school different, I believe if we change the culture through positive discourse, building relationships, and foster a growth mindset, we will all be a lot better off.

I guess what Taylor Mali says, I make schools a better place, because I make a difference, what about you?

3 Things a 1st Grader Taught Me


  1. If you say you’re going to do something, you better do it.
  2. The power of observation from fresh perspectives.
  3. Curiosity and perseverance are important things for adults to model.

Although she was passionate about learning in kindergarten, her desire for academic pursuits (especially in school) have greatly diminished during her first grade year. Saddened, I wanted to combat this by engaging her in a conversation about performing some “experiments,” which garnered that familiar excitement I had loved to see a year ago.

After discussing some possible experiments and wanting to capitalize on her recent trip to a nature preserve, we settled on making a floral arrangement from construction paper. We evolved this idea into a hike through the preserve, where we would collect samples and build replica of a flower. Side note: We are in are in a severe drought here in California, and any flowers that did bloom, occurred two months ago. Two weeks later we got all ready, off on our sample collecting adventure.

The Wife and 1st grade sister, walking way ahead of me.
The Wife and 1st grade sister, walking way ahead of me.

Before I can continue, I have to say she was with one of her older sisters (there are 20 years between the youngest and her) all day, where the 1st grader had asked and asked several times if she knew about this adventure. I hadn’t forgotten, but I thought she would. Two weeks to a first grader is a lot of time, and a lot of things have happened between then and now. Well she didn’t forget, so when she got over to our house, I was in my room recovering from a long run in 90 degree heat. She tentatively asks if we are still going, I said yes, and for the first time she hugged me, straight up long embrace, Lesson 1 learned.

Arriving at the nature preserve, she was jumping for joy.
Arriving at the nature preserve, she was jumping for joy.

Back to the narrative, so we get ready and we’re out the door in 20 minutes. She was so giddy to be going, she kept asking if we were excited too. For the next two hours we walked around, collecting samples. During this time, I learned Lesson 2, she was so observant, she saw and heard everything through fresh ears, I realized that we can stop and look around, and with a fresh perspective familiar things look new. She made me think of this hot, dry environment as if it were some amazing, new place (which it was for her) and how exciting the mundane can be when viewed from different perspectives. For example, this is important to keep in mind as a teacher, as your students will often see the mathematics you teach from fresh and alternative perspectives, we need to embrace those perspectives and view things from their eyes.

The bags of samples, classified by "families" for later use.
The bags of samples, classified by “families” for later use.

After collecting samples, we went to dinner, then off to the dollar store for supplies. Having no idea  what to get, it was a challenge to think of all we would need, but it worked out. When we got back, we were ready to start. We were trying all kinds of fun things, but I learned she was curious. I didn’t know what we were going to do for constructing a flower with our materials,  when I learned Lesson 3. When an adult wants to learn the answer to a question, by what process does she go through to answer the question. Since she would not let me off the hook, I was given a marvelous opportunity to model how one might answer such a question, and how to continue through seemingly endless possibilities to find an answer.

All in all it was a great six hours of learning and time we’ll spent. The total cost:  my time, $8, and a lot of learning. We are extending this idea and building something better from it on Wednesday. Best part is, that spark of excitement for learning is back.

My girls and the Sierra Nevada mountains.
My girls and the Sierra Nevada mountains.

Bottoms Up to Conceptually Understanding Numbers

This is just awesome! I am working my way down to the younger grades, and learning about this foundational pieces has been an eye opening experiences. I do not know if it is the length of time since I was this age, or the lack of experience in teaching these grades, but these are pieces that I just assumed we came equipped with. Learning that it is both purposeful and intentional to give students this experience plays out in so many fundamental ways, it also helps me think of the students I had in high school and college that did not experience this earlier success.

As I grow in my understanding of how students (and all peoples for that matter) learn and acquire mathematical knowledge, the more important these foundational pieces become. Funny how something as seemingly simple as an inversion to the number line creates understanding, like vertical number lines, these are difficulties we need to attend to, and we need to be careful how we accomplish these tasks.

All that to say, this is a great read and I am reblogging, more for personal reference than anything else, it acts as a reminder for that which I discussed above.

Questioning My Metacognition

The concept of 10 more/less is a beast in the primary grades.  Last week I realized that I’ve been feeding the monster that I’m continually trying to defeat. Almost every day, in every K-2 classroom across the United States students encounter this guy:

Screen Shot 2014-10-10 at 11.56.05 AM

I made a conscious effort to pop my head in every K-2 classroom in the schools I visited this week.  It was great to see that every classroom had a traditional 0-99 or hundreds chart posted like one above. 

While visiting, one of the teachers asked if I could come back today and model a lesson using a 0-99 chart because her students “just weren’t getting it.”  I gladly accepted her invitation and showed up with this guy:

Screen Shot 2014-10-10 at 11.52.19 AM

I didn’t use manipulatives for this lesson because I was specifically focusing on the rote counting process which precedes one-to-one cardinality when counting by ones OR tens.

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