If you don't know who Graham Fletcher is then you're really missing out on a fantastic mathematical mind. Graham has a wonderful ability to make the complicated, or overwhelming, seem simultaneously comprehensible and bite sized pieces. Having been a huge fan of his 3 Act tasks and various other ideas, like geometric number talks and visual operations on fractions I was extremely excited to meet him in real life in March for our ongoing Central Valley Networked Improvement Community and learn from him as well.
It was during this time that I thought it would be very helpful to have a single place to view all of his current work on the progressions of the operations and other information in one place. So I created a clickable PDF, Graham Fletcher's Progression Videos that allow the user to simply click on the operation or the concept they would like to learn about and Graham will expertly discuss those ideas. For example, the image below shows the clickable PDF, if the user wanted to learn about multiplication, then the user simply has to click on the multiplication symbol and watch Graham's video. I hope you enjoy and find as much learning and amazingness as I have.
"Instagram, Swiffer, and Nest had to compete with consumer habits and perceptions. Breakout products face competition from the formidable inertia powering the status quo," said Jay Samit.
Over coming inertia is hard, it is very hard to do and it is worth celebrating when we do see evidence of it being overcome. Overcoming inertia is the challenge that I am currently raging against on an apparently global scale. I love how Hamish Brewer says we need to change the conversation, and we need to be that change.
We need to change the conversation about what does effective math instruction look like in the classroom (Note, my focus is on mathematics instruction, but I believe all instructions falls under this idea). I am at this weird place in this journey as I don't know if I am crazy or if I am actually on a learning journey.
To be clear, there is nothing wrong with what we consider traditional direct instruction and students often need this guidance to introduce some concepts, skills or review some skills. The problem is when this is the only thing we do to reach students. Most people, yes students are little people, do not truly learn a concept when you lecture at them and they, at best, passively take notes. Students need to be making sense of concepts and they need to talk about their thinking. We need to provide students room to think, to make sense, to wonder, to struggle and to make mistakes. Students don’t need more testing and more worksheets.
So how do we start to change the conversation? How do we overcome that inertia to change? How do we change the conversation and invite you to be the change? How can we not be #relentless for our kids?
It's been a long, long while as this journey takes me places that I've enjoyed going and places that I have had to endure get continue on this journey. My purpose for blogging is to continue to force myself to reflect on the learning as our journey continues, and I want to establish a consistent publishing of quality writing that reflects that journey. To that end, I will figure out if once or twice a month is a frequency that is sustainable both in quality and quantity.
Currently my journey is taking me down a couple of learning paths I'll list in no particular order below:
I am looking forward to growing, reflecting, and rekindling that journey.
Recently, at #Cue16, I saw this on a grander scale. I experienced being better together with passionate, talented, and generous educators. In short, I found my tribe.
Another example of getting better together came from an amazing educator, Steve Wyborney. I am an avid fan of his work and his blog, which lead to something amazing, his work got me very curious.
Diving in, I was inspired to create this synthesis derivative of his work, combined with another math education hero, and a super amazing guy, Andrew Stadel. I had the pleasure of attending Andrew and JR's (JR is another education hero of mine) riveting discussion on Desmos Activity Builder at Cue16.
All that to say, see the #hyperdoc I made to make sense of their wisdom. Throw in a piece by Dan Meyer and Max Ray, and you may begin to see why I'm posting this now.
Moving on, both Andrew and Steve have been very helpful in enabling me to think through these pieces. So I am planning on taking their creations, to the classrooms, with two second grade rooms lined up. We'll be doing a 30 minute (or less) discussion of noticing and building math sense through the students' discoveries.
Pictures of student work and many more discussions to follow on later posts. All this to say, we grow, we learn, we inspire, and we get better together. I hope that the profoundness of this statement never ceases to amaze that young man I was, and the old man I've become.
Your turn: How are you getting better together?
Recently, I was gifted the opportunity to teach a lesson in a seventh grade biology class, known for being a challenging group of learners. Although, the content for learning was heavy on the academic vocabulary, I wanted to engage the learners in a way that would bring their thinking to light, without direct input from me. Whenever vocabulary comes to mind, I immediately think of one of my favorite methods: The Frayer Model. Knowing the population I was serving, and the dependency on heavy academic language, the words of Jon Corippo came to mind, and I wanted to Seacrest the hell out of this lesson, i.e. it was a new opportunity to Iron Chef this lesson.
A little background information, I have tried with limited success to be able to produce a reasonable explanation in a finite amount of time how a group of learners facilitates this process, let alone doing in context without losing momentum with reluctant learners. In addition, I was teaching the students a new method of vocabulary usage with the Frayer model, and I was expecting to teach them the content within the 50 minutes that I had with them. I felt the cards were stacked, but I don’t mind failing and loved the idea of the challenge.
Feeling encouraged by a new understanding of the Iron Chef process, I started the lesson with a quick introduction to get the students ready for learning. Introducing the Frayer Model separately, I chose to use fast food establishments as the non-content specific example for students to get used to that idea. I read and showed mine of Taco Bell, the students then were numbered off 1 to 6, to form the groups of 4 that I needed. The students then sat in their groups of 4, each student chose a number which corresponded to a specific restaurant, and I instructed the students on where to go and what they would do. We interacted in a timed, chunked manner through this process. So the learners were writing their Frayer models while learning the structure of the Iron Chef lesson. We moved from Home groups to Expert groups, then back, it worked so beautifully, I couldn’t believe it. After less than six minutes, I had managed to explain both concepts.
Placing the words on the screen, we followed the same format, the learners were give three minutes to dive into the reading, pull out their information, before we would move. When time was up, I checked they knew where they were going, then I released them to share, if they didn’t finish their Frayer model, they could do so as they shared out. I gave 90 seconds for this, then back to their seats. Two minutes of sharing out whole group, each one writing what the other members of the group had to share, then we were off again with four new words.
Repeating this process, with less time, and the increase of tempo proved beneficial, as the structure wasn’t new anymore, they were able to get through it much quicker. The students also quickly related where they were at and how they needed to finish some pieces.
We were able to cover a large section of reading, discover new words, make associations, and synthesize information with three new structures in less than 50 minutes. Having front loaded the expectation of what was required before the leave, I was able to finish with three minutes of students creating a visual representation of how all the pieces fit together. A projected word wall of things they might consider including added a nice touch. Students surprised me with how much they pulled out of it, and I was able to instruct for less than a total of five minutes total. Not to mention this challenging class more than rose to the occasion of amazing learners, they actually were having fun.
A couple of little things that would facilitate this better next time, is a visual for where students need to go based on their number, that part I was unclear about, especially for the first few times. There is also the additional piece of differentiation for there were a few learners that already knew the academic language and the concepts, so they were done within a minute of the time to fill those pieces out and they weren’t learning as much…so adding or modifying that piece for them would also need to be taken into account…I forgive myself on this piece because I didn’t know the students at all.
Being able to climb Mt Everest in this lesson made my day, especially now that I feel I am able to successfully understand the finer parts of Iron Chef lessons, and being able to clearly communicate that piece without losing the momentum of learning, are huge wins in my book.
I wonder what is the length of time a learner can be actively engaged with a content. In this case, I am looking at a time bound for the youngest learners, Kindergarten in this case. So with the help of my wife and two 7 year-olds (as guest lecturers) I set out to determine if we could provide an engaging mathematical learning opportunity for two hours in Kinder.
The lesson format was a spiral review, though the content was a higher depth of knowledge (DOK) than the learners had previously experienced.
Inspired by a lesson from OpenMiddle, the learners were asked a question about the largest and smallest possible number given two ten frames, one of which is full and the other is not.
Originally, I wanted to just pose the problem, but having some time to think it through with the noble goal of seeing if 5 year-olds could maintain an active cognition for 2 hours, I accepted both challenges.
Now you may notice, I brought in a variety of special weapons, including having two guest lecturers/aides to assist in the engagement…and give the girls quite a unique learning experience…but mind you, I am attempting to keep kids doing math for 2 hours, after lunch, with variations in weather, and spring break around the corner. Needless to say, I needed some assistance. I also introduced a curiosity based bribery with a large box of untold goodies, they could work towards, my goal was to utilize any trick I could, to account for all possible variations.
As personal learning opportunity, I wanted to involve centers for the first time. In addition, this was also a back up in case we finished early, or the kids needed something else to do….we never really needed them.
The lesson started with the learners on the carpet going over their shapes, when I was set up, we moved to our seats, after I introduced the girls as our special guests.
I had previously set up two ten frames on the floor with masking tape, this was a huge part of the lesson and meant to get the students thinking about their numbering. Playing the slides and having students come up to fill the living ten frames, asking questions and goofing around, this activity alone took almost 40 minutes. Students were so engaged with this, and laughing so much one young lady, literally, fell out of her chair on to the floor in a bout of laughter. It was magical.
When the videos played, the students had to yell out their responses, students were so invested they were literally standing up and yelling their numbers out.
When we got to the point where students were being asked to solve the OpenMiddle problem, the students used both the counters and two ten frames in front of them to build what the question was asking. With four facilitators asking questions and providing opportunities to wonder, I was very impressed at how quickly many learners were able to articulate either the largest or the smallest number. In quick succession, about half the class was able to get both, and explain, in quite vivid detail, why their numbers were the largest and the smallest….I was unduly surprised and excited.
We are 90 minutes into a very active class of learning, 5 year-olds are getting tired, one young lady even asked, if she could stop, stating her head was tired. When I asked is she would like to do something else, she flatly stated, “No.” I guess she wasn’t as tired as she thought she was.
As we came back together to finish the lesson, we were setting up the centers, when the time to begin cleaning up and getting ready for going out to the bus line started. We were just at the point of running the centers, so I will have to wait even longer to find out how these are supposed to go.
Needless to say, with a little energy, a little fun, and varying the types of involvement, even the youngest learners can be engaged in active mathematics for a lot longer than I would have guessed. For 2 hours we did mathematics, we laughed, we had fun, and everyone learned a lot.
“Tell me and I forget. Teach me and I remember. Involve me and I learn,” said Benjamin Franklin.
Learning is about involving both the teacher and the student in interactive ways, ways that provide a dance of questioning, activities, and doing. This amazing dance may occur at the seemingly most unprovoked moment, and leads to a bit of fun and a lot of learning. One such occasion occurred to me recently and it produced such an amazing bit of fun.
Setting the stage….
After a long day at work, I arrived home to the joyful glee of a precocious 7 year-old, bound with energy, and my wife making dinner. I was carrying a box of circuits from a fourth grade class, planning to return them the next day, but the physics nerd in me was tempted to play a little. When I began to play, the 7 year-old became curious, as I was thinking out loud, “What are these pieces for? How are they connected? I wonder if this will work?”
As she peered over the lip of the box, I asked her, “What do you notice?” With a tilted head she said, “I see blue color and a mess.” Side note, the box wasn’t a mess, but to a 7 year-old, with no words to describe what she was seeing, she went with what she knew.
We started pulling the materials out, when she saw that they were wires, the blue things we weren’t sure what they were for, and the light bulbs, she was both intrigued and confused. Again, I asked, “What do you notice?” She replied, “Light bulbs.” Then I asked, “What do you wonder?” She looked at me, thought a second, and said, “I think the same things you were saying earlier.” I know right?! So that happened, and we were off on our journey.
What we did….
Upon organizing the pieces, we started making connections (ha, cause it circuits...that’s funny) and we saw how the batteries connected, we placed the wires together on the board, screwed in the light bulbs, and observed what happened. Within a few minutes the light bulbs were glowing, and we were ecstatic. I didn’t expect anything to work, but it was brilliant.
So, I started asking questions and getting her to wonder about the connections, seeing if I could get her to make predictions, and then verify through the experiment. Now the nice thing about this experiment was she could immediately verify her predictions, and adjust accordingly.
She noticed that when the wire from the terminal of the battery touched certain wires one light, or both lights, would shut off. She noticed that only 1 or 2 configurations would allow for both light bulbs to be on simultaneously, and she noticed that based on how hard she pushed down on the wire, one of the bulbs was dimmer than the other at certain connectional values.
What was the outcome….
I prompted her thinking with guided questions, about why these observations are true. Her first responses hit on everything from the size, shape, direction, and straightness of the wires to the type of battery. Clearly this 7 year-old had no understanding of currents, Ohm’s and Kirchhoff's laws, and I was excitedly prompting for clarifications. Then an inspiration hit me….water.
So I painted a different picture, I said imagine if there was water flowing from this end of the battery, how might the water act as it flows around the circuit. Now, in case you’re wondering, this 7 year-old has limited exposure to electric circuits in a formal manner, but she has been to multiple rivers and we have had conversations on how silly water seems to act as it flows. So my inspiration was to try to tie this to things she has had a context from which to jump to this abstraction.
The next few moments were pure magic, she started experimenting with the which lights would turn off when she struck certain wires, and said that if the water flowed in a circle, then when the other light doesn’t come on, there’s a “rock” in the way. I knew she had the idea. So I asked, “So how might the correspond to when you touch this wire, and neither light turns on?”
She tilted her head, verified that when she touched the wire to that spot, both lights didn’t come on….”Oh I know!” she gasped. “It’s because there’s a rock stopping the water from reaching both lights.” Tracing out the circuit, she could see that the water wouldn’t reach the lights.
So, we traced out the circuit, played with a few more configurations, and then it was bedtime. May I tell you, we worked on the circuits for 2 hours, it felt like 15 minutes, and she was exhausted. My wife reported to me later, that she had complained her head was hurting, but she didn’t know why.
Two days later, the other 7 year-old exasperatedly proclaims, “I want to learn about the electricity thing.”
With some time to reflect on this experience I’m going to follow more of Benjamin Franklin’s advice: I am ready to provide a better stage, to involve them in their learning.
The beginning of my work with this idea came to light from an article I read from NCTM, this lesson was a handout. My lesson materials is found here, which includes the PPT and handouts I used for this 5th grade lesson.
What I liked….
The first thing that spoke to me about this lesson is the low barrier to entry, specifically, the task has students follow a pattern of crossing out numbers in a distinguishable pattern. With very little in depth thinking, once students see that pattern, the process of finding numbers is relatively easy. Connecting the students’ understanding of prime numbers to composite numbers, this process introduces “hidden” language, like factors in a natural setting as we explore this process. Specifically, the link of why content has developed a language as a necessary way to describe the things that we are interacting with. Finally, the cross curricular idea stands out as a natural progression of this lesson, from the history component of learning a little about Eratosthenes to the connection between mathematics and music.
What I wonder….
This section has less to do with the lesson, and more to do with my implementation of the lesson. Originally, I thought this lesson would take a few days, realistically though, this lesson is more like a unit, and probably would take a full week of instruction for students to really make connections. My implementation of the lesson required me to create a full sequence that spoke to “my style” of teaching, realizing that I would most likely leave behind the idea of the connection to music.
The teacher kind enough to let me use her classroom, also pointed out a really great point, that in showing videos, it would be beneficial to have questions next to video, so students would be able to refer to them, as they were not reading the questions on the handouts I created to accompany the lessons.
What the students got….
So all of the above to say, that the students got a very nice introduction to who Eratosthenes is, what he did, and some insight on how to say his name. Almost all students were able to get the process of how the sieve works, and the students were excited to go ahead with both examples. These pieces of the lesson went well, combined with the fun and many student interactions made for a success in those ways.
On the other hand, I felt I didn’t facilitate the flow of the lesson on the two days very well. The combinations of interacting with individual students, not explicitly detailing what interactions would look like, and being purposeful in the questioning, providing examples, or explicitly modeling left many students without the proper understanding.
My learning objective was to have students be able to recognize how composite and prime numbers are similar and how they are different. This was my target baseline, and I did not make that mark, though many students developed a deeper understanding, and a few were really trying to answer the questions.
My biggest learning is that when you set the stage property, the students will do amazing things, this was amazing thing to do; however, I did not set the stage properly for them to be able to reach out, explore, and do those amazing things. As I write this, I am realizing what it was that was missing, I failed to set the stage properly for the students to be successful. So now I am thinking, what that might look like in a week, or so, as I will be going into another 5th grade classroom. I will modify the lesson to provide a better stage.
Inspiration has a strange way of showing up, a welcome companion when wrought of extreme desperation, which describes a recent situation I was in.
I invited myself to do a lesson in Kindergarten, my first of this school year, and I was excited. The lesson I was thinking of doing was one I had had the opportunity to practice a couple of times, so I felt confident it would go well…then I realized, this lesson is appropriate for the end of the year, but not after a month of school…then panic struck me.
Looking my manipulative chest, a plan emerged and two hours later, I had a lesson I was excited to present to 27 K and TK learners.
For the lesson, I wanted learners to have the opportunity to count, to associate numbers in various representations, to have discussions, and opportunities to ponder. Of course, with any demo lesson, a lot of the engagement will come in the presentation, so I also made sure to bring in high energy and silliness (of which, I have ample supply of).
The lesson starts with a “Which One Does Belong” slide, asking the students to choose one that didn’t belong, I asked the students to share in a Round Robin at their tables, then the students “whispered” their answers to me. Without missing a beat, the students chose the singleton unit as the one that didn’t belong, since the other three dice have a five in some representation. Interestingly, that tells me the students are numerically literate in counting and representing the number 1 and 5, this bodes well for our plans and we move on.
Not knowing what levels the learners are at this point, I wanted to build a little context with counting to five. Using a song from Sesame Street, the 3 minute video has five young adults harmonizing as they count to 5. The video went well with the learners, a pin could be heard as they watched the video. Capitalizing on this attention, I asked the students to share what the video was about, as I ready their unifix cubes.
The next slide asks students to Notice and Wonder, each student takes a little time (cued by a song) to think, then the utilize a Round Robin to share what they noticed and what they wonder. As students noticed the cubes, and I had ready the unifix cubes, the learners knew we were going to be using them.
Each student receives a strip of ten unifix cubes, and a sheet of paper. I model the first case, where I “build” the number with the unifix cube, draw it, then write the number. Practicing this, I verify each student is ready, so we begin with two, then three, and so on. Finishing with the number six, we are ready to go into the next phase, which is determining if there are five in the number six. Interestingly, when I ask the class, several students say no, that there is not five in six.
Using ten frames and red/yellow counters, I asked students to construct the same as the picture I am projecting on the screen. I was interested in how many different representations I saw, including some learners who place anywhere from seven to ten counters on the ten frame, when I asked if that was the same as the representation shown, they agreed it was. When we counted on the screen and the representation they had built, they agreed the numbers were different, but agreed that the representations were equivalent.
One student didn’t stack his as a double, like I had constructed, but made five and one more. When I asked if that was the same, he agreed, I asked him to explain, he proved it to me, by counting. Then he said, “It’s five and one” indicating his construction, and that’s 3 and 3, holding up his little fingers in two three, indicating that I should know that the two forms were equivalent. I apologized for not seeing that and thanked him for showing me.
The final part, due to time, had each student show me how five is in six, students were asked to turn over one to show me. Each student was abel to show me, and that ended the lesson.
Throughout the lesson, as we constructed the various numbers, I would ask if two or three representations were the same. For example, if the number was five, I would ask is three and two five or four and one, and we would count whole class using the unifix cubes. Doing this for each number, gave the opportunity to show the different ways to make the number, an important feature that will be exploited in number talks.
As I reflected with the teacher, I was very excited about the lesson. I felt like I reached my objective in a variety of formats. Students were being asked to draw, build, and write the various numbers. Students were asked various ways to build, we counted out loud multiple times, and students were able to talk about math as they noticed and wondered and shared which one doesn’t belong.
The pace was pretty quick, especially for young learners, but it was slow enough that almost every learner was with me. The teacher provided some very valuable feedback and had some great suggestions on extensions and moving forward. I will be very excited to head back in to continue the extensions in a short time.
A various selection of student work is shown below, the various representations and abilities speak volumes as to where the learners' accessed the content of lesson.
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