I entered this year hoping that my students would form a better relationship with math. That they would see it as something that you play with and do (rather than memorize). I wanted them to see math as malleable and that there are different approaches to answers. I wanted them to “see” math, to use visuals to display their thinking (along with equations/calculations).
(Right, students working on the Marshmallow Challenge at the IBM Learning Center)
What Happened - 3 Shifts
My three biggest “shift” areas this year have been Instruction, Space, and Technology. My ALLF question ties directly into Instruction, and most of my efforts were in that area. The changes in Space and Technology happened parallel to this work, not necessarily driven by it. However, I found that my Space and Technology changes helped facilitate many positive Instruction changes.
Math Instruction
This year I attempted to change my math lessons to fit into my inquiry pursuit by making them more open-ended and visual. I used three main sources: Math in Focus lessons, self-created lessons, and Mindset Mathematics lessons.
While I continued to follow the curriculum as defined by the district and Math in Focus, and continued to use the provided program materials, I did adjust many of the lessons to better fit the criteria of my inquiry question.
Mindset Mathematics is a concept and series of books by Jo Boaler, Jen Munson, and Cathy Williams. They seek to create mathematical tasks utilizing research around how students best learn. The tasks they include in their books emphasize open-ended problems and visual thinking.
Space
In January, third grade classrooms at Grafflin received new flexible furniture (with wheels) to replace stationary desks and tables. I also was approved for a Chappaqua School Foundation grant to make my classroom library movable with wheeled shelves. These changes allowed us to change the classroom based upon the instruction. For example, it gave students more access to whiteboards to share their thinking during this fraction work.
Technology
This year our third graders had 1:1 iPads available to them (also with thanks to CSF). This affected all instruction, but especially math. Students used Seesaw to document and explain their thinking. They also used specific math-based apps to create number lines, geometric shapes, and fractions.
SCHOOL YEAR 2018-19
As I started the year, I looked at each Math in Focus unit and tried to find out how it fit with my inquiry question.
The first Math in Focus unit is Place Value. One task gives the students four digits and asks them to find the biggest number they can make from those digits. I then added to it. I asked the students what other questions we could ask about those digits. Martha asked what would be the smallest number we could make. After the students found their answer to that, I asked them again for more questions.
The class was quiet for awhile, until Michael asked, "How many numbers can we make in between the largest and the smallest?" I had them work on it. This one question led to an enormous amount of mathematical thinking. Some used individual Post-it notes for each digit. Some found ways to organize their thinking around different place values (e.g. by the thousand's digit as in the photo). Students began to generalize that, if you kept one digit the same, you could make six different numbers. They looked for the same groupings using other digits.
Finally, students began to answer that you could make twenty-four different numbers from the four digits. This led to another discussion, as Michael had said "in between". The class agreed that the correct answer would be 22, as we should not count the largest and smallest numbers.
I really liked what I saw. The students were genuinely interested in mathematical outcomes. They were coming up with appropriate ways to organize and discuss their thinking. This would become a model for me about open-ended questions as the year progressed. Could I come up with ways for the students to truly think about mathematics? Could I allow them to come up with their own questions? Could we come up with authentic inquiries?
On subsequent days, we started looking at other combinations of digits.
- What would happen if we used three digits?
- What would happen if two of the digits were the same?
- What would happen if three of the digits were the same?
- What would happen if zero was a digit? (shown in photo)
In each case, the students built upon their accumulated knowledge. They organized their inquiries, found patterns, and came up with new questions. Throughout it all, they became better acquainted with how digits and place value affect number value.
I began to look for other opportunities to adjust the Place Value curriculum to allow for more open-mindedness and authentic questions.
Later in the unit, students are asked to create number patterns of four numbers, in which a single place value changes. For example (#1 in the photo), the sequence 5833 5843 5853 5863 increases by 10 each time.
After introducing the concept of patterns, I encouraged the students to make their own. Often, this meant giving the first two or three numbers and having their partner determine the next. The class began to experiment with different ways to create patterns. We developed a class list of invented strategies, which once again helped the students develop their place value understanding.
- single place value increases by one (e.g. 1, 10, 100)
- single place value decreases by one (e.g. -1, -10, -100)
- single place value increases by > 1 (e.g. 2, 40, 300)
- single place value decreases by > 1 (e.g. -3, -20, -400)
- two (or more) place values increase or decrease
- some place values increase & some place values decrease
- numbers decreasing below four digits
- numbers increasing beyond four digits*
- numbers decreasing below 0 into negative numbers*
The strategies marked with a (*) were notable in that the students explored strategies and asked for instruction in areas that went beyond the curriculum.
Mental Math
Unit Two in Math in Focus had always been a difficult unit to teach. Third grade teachers often sped through it in the minimum amount of time. It focused on mental math and estimation, but appeared to require the students to follow and write down prescribed steps. In the past, I had found the unit incongruous, as it strived to teach flexible thinking through defined steps.
However, in consultation with Lisa Ultan, I began to see the real possibility in the unit. We decided to focus more exclusively on the concept of mental math, rather than on the workbook pages. Students were encouraged to come up with their own ways of adding and subtracting numbers in their heads, and exploring how to notate this thinking. This unit now became one of the most valuable and interesting to teach.
The workbook pages then became conversations instead of independent exercises. What method is #5 (see photo) using to solve 72 - 44? What other ways could we use to solve this equation mentally? What criteria should we use to evaluate different mental math strategies?
This also transitioned nicely into estimation, which is also part of the unit. Students began to think of situations in which they did not require an exact answer. They learned how to round to the nearest 10 and nearest 100, not as an isolated skill, but as another ways to be flexible with numbers.
Seesaw
Around this time, the students started using the Seesaw iPad app for math. It allowed for a variety of ways to explain their math thinking. This built in very nicely with our investigations of flexible thinking and how to notate it. Students were now able to write on whiteboards (below), type (as in the photo example), or even voice record their responses.
As the students became more familiar with the Seesaw app, they developed better ways to demonstrate their thinking. They notated photos of their work and created videos in which they explained concepts, along with arrows or circles indicating which part of the displayed whiteboard they were discussing.
(Thank you to Margaret Salmore and her class, who were especially helpful in showing us ways to fully utilize the Seesaw app.)
We used Seesaw throughout the next few unit - Addition, Subtraction, Multiplication, and Division. In each case, using the app to better demonstrate and extend our thinking.
This coincided with a heavy reliance on manipulative and visuals to work with both multiplication and division. Students created these problems with blocks or illustrations and then used Seesaw to explain their reasoning.
Furniture Arrives
Around this time, I received the new furniture in my classroom from the Chappaqua School Foundation. All of my tables, along with some of the chairs, now had wheels and could be easily moved. In addition, I now had two large portable magnetic whiteboards.
However, the biggest changes occurred as I got more used to the furniture. First, with assists from Adam Pease and Carol Bartlik, I moved my large rug to the center of the classroom, rather than in front of the SmartBoard.
In addition, I applied for and received a grant from the Chappaqua School Foundation to replace my stationary bookshelves with mobile ones. Now my entire classroom was flexible, which gave students access to the large whiteboards along my back wall. More importantly, it allowed me to tailor the room to the instruction.
iPads
We also began experimenting with different ways to use the iPads beyond Seesaw. We used math apps with base ten blocks, number lines, and fractions (shown at right).
Combined with Seesaw and the flexible furniture, this allowed the students to both expand their opportunities to explore their thinking and expand their methods of demonstrating it.
Mindset Mathematics
Finally, I began using lessons from Mindset Mathematics in the Spring of 2019. Specifically, I used lessons from the book for fractions and for perimeter (see left). These lessons coupled very nicely with using the apps on the iPads and with utilizing the new flexibility of the classroom.
These lessons were decidedly visual and exploratory. They allowed students to both build and experience the concepts. In addition, the lessons are designed to be "low floor, high ceiling", meaning that every student in my inclusion classroom could participate and that every student could find appropriate challenges.
Final Thoughts
How much will a greater focus on open-ended problems and visualization increase student ownership, understanding, and flexibility in mathematics?
In conclusion, I felt that this was my strongest year as a mathematics teacher. The multiple avenues of exploring their thinking gave the students greater ownership and understanding of the concepts. The focus on visual learning - with manipulatives, apps, whiteboards, and drawings - helped cement these concepts and contributed to their flexibility with numbers. Even the classroom furniture allowed us to better investigate concepts and discuss our findings.
I am definitely excited for next year. Not only can I use what I learned to better orchestrate math lessons geared to open-ended thinking and visualization, I will also begin the year with experience using both the iPads and flexible furniture to augment our explorations and discussions.
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(A special thank you to Lisa Ultan for her incredible support during this journey. She visited my classroom often, got to know my students and their thinking, and regularly discussed next steps with me. She was instrumental in helping me turn big ideas into real classroom progress.)