Learning Isn’t Linear: Desmos Final Projects

by Morika Tsujimura and Sonia von Gutfeld, Math

The last few weeks of eighth grade bring a familiar chorus every year—grumbles about the workload when summer feels so close, joyous celebrations of milestones and accomplishments and a rollercoaster of emotions as classmates reflect on their time together in anticipation of the changes high school will bring. The Desmos project is a culminating math assignment that brings out all three motifs in raucous harmony, a melding of adolescence and algebra.

Desmos is an approachable, web-based graphing program we use throughout the year. With this project, students find new, creative ways to apply this tool. First, students draw a Grace-themed design on graph paper that they then convert into equations and inequalities in order to reproduce the image on Desmos. Utilizing various types of functions learned over the year, students write equations for straight lines and curves and restrict them to the segments they need for their drawing. They also shade sections by using inequalities instead of equations. (You can view some of their projects on our Academic Excellence page.)

Anna L.’s griffin

The Desmos project is first an opportunity for the eighth graders to review the types of functions learned (linear, quadratic, and exponential). Once students start engaging with the parameters of the project, however, there are many more layers to explore. Grappling with how to generate a recognizable image on the graph often generates the “aha” moment when the abstract relationship expressed in numbers and letters suddenly makes sense in a concrete way. Manipulating slopes of lines by trial and error might be how it starts, but applying the patterns of parallel and perpendicular lines empowers students to work with greater efficiency. Though they have already studied how the terms in an equation affect the shape of its graph, sometimes it takes trying to make a curve look more like a shoulder or a piece of the Grace quatrefoil for those understandings to fully click.

Nate B.’s Grace Church School logo
Jonas C.’s Grace Church

The project goes beyond reinforcing math content and allows students to hone the skills it takes to be a good student, and specifically a math student. From time management to breaking down tasks into smaller steps, to knowing when to ask for help, to sharing new knowledge with peers, all of the benefits of project-based learning come into play. In addition, the technology that allows students to see instantaneously how a line on the graph changes according to adjustments in an equation provides a low-stakes way to take risks. This is crucial to developing problem-solving skills and the courage to try new things. Many students ended up incorporating more detail than they had first imagined possible or learned and used equations beyond the requirements and, in turn, extended their mathematical understanding.

As teachers, we initially added the requirement that the drawings relate to Grace Church School in order to streamline the decision-making process and help students make personal meaning of the assignment. It has had some surprisingly sentimental outcomes, especially at the end of this singularly stressful school year. We read reflections on how their designs revealed favorite memories of early childhood, after school routines with friends, the experience of singing in the church — special connections that made Grace home. For our students to be able to express even a fraction of those emotions and their growth using a few dozen algebraic functions might be well worth the ups and downs of those final weeks of Middle School.

IBM’s Women in Math

By Elsa Hepner, Head of Middle School

In March of 2019, Karen Uhlenbeck became the first woman to receive the Abel Prize, which is often referred to as the “Nobel Prize of Math.” With her win, Uhlenbeck further substantiated what educators have long known to be true: women have a prominent and promising role in the field of mathematics. 

Earlier this month, as part of Community Week, fifth grade students were visited by women mathematicians from IBM. Organized by fifth grade teacher Margaret Meyer and Grace parent Michelle Peluso, the event promoted the important role  of women in math. The presenters spoke of their love for mathematics and how it led them to where they are today. The passion with which they described the field was contagious. 

Rose K. ’28 remarked, “It was really cool! I loved that we heard from women specifically talking about math since you so often hear about men and what they’re doing. They made me realize that math is everywhere.”

To illustrate that patterns and numbers are all around us, the speakers led the students in a variety of games and activities. One such activity involved an example with which the students were very familiar, TikTok. The mathematicians described the elegant algorithms that work “behind the scenes,” determining what content viewers will see as they click and scroll.

Math teacher Amber Leung particularly enjoyed this activity,  in which students’ knowledge of equivalency and proportion were put to the test. “The students were social media data analysts who had to decide which videos they should advertise more heavily so that their viewers would keep watching and in turn the company could keep making more money. It was so wonderful for the students to put their fifth grade math skills in action and in cleverly relevant scenarios!”

This Year, JK-4 Math is All Fun and Games

By Leah Silver, JK-4 Math Coordinator

The vibrancy of our Early Childhood and Lower School math program can be felt both in the classroom and on Zoom screens this year. JK through Grade 4 students are questioning, constructing, noticing, playing and practicing in different ways. In a year of so much change, I’ve found it helpful to articulate guiding principles for our program this year: prioritizing the use of real materials, centering the use of games, and trusting in the resilience of our students. 

Guiding Principle 1: Prioritizing real materials 
This year presents new challenges for using materials, but we know that students learn new math while getting the opportunity to construct new understandings for themselves. While digital manipulatives exist and are very useful, when students first learn a new concept they need to hold the materials in their hands. Every student in Early Childhood and Lower School–whether learning remotely or in person–received an individual math manipulatives kit with the key materials they will use over the course of the year. Depending on the age, these kits include unifix cubes, pattern blocks, beaded number racks, base ten manipulatives, game spinners and dice. This way, we can make sure everyone has access to the same materials in a safe and sanitary way, and every kid can easily take these home, should we have to all learn remotely. 

Ms. Malik leads our remote JK students through a pattern activity. Students use their Unifix Cubes to construct their own two-color patterns.
Our remote Kindergarten students explore different combinations of the number ‘5’ with Ms. Moller and Ms. Silver using their Five Frames and Unifix Cubes.

Guiding Principle 2: Games are at the Center
Games have always been at the center of our math program, and this year is no different (in that regard!). With our ongoing adoption of the Bridges in Mathematics program, we have access to incredible digital versions of the games our students love to play. These games are a crucial piece of our math program, encouraging strategy development, logical thinking, and further building of math concepts. 

Grade 3 students in Mr. Schneider’s class play ‘Carrot Grab’ in a breakout room. One student shares their screen so they can both use the same game board. This game encourages students to hop to a ‘friendly number’ when adding. You can play Carrot Grab here!

Guiding Principle 3: Trust in the resilience and mathematical capabilities of our students, and keep moving forward.
While we had to make curricular adaptations to accommodate our remote learning schedule last spring, our work at the beginning of the year with students confirmed what we knew to be true: our students learned a ton of math last year and were ready to hit the ground running with their current grade level’s curriculum. We took guidance from the Bridges program not to rewind to the previous year’s content, even though some lessons may have been missed or altered. Instead, we assess as we go and identify any areas we need to re-engage with our students in real time. 

Grade 4 students in Mr. Wanyoike’s class work on building the ‘Wall of Base Ten’ to visualize numbers up to 10,000.

The Three Most Important Math Questions for EC and LS Classrooms

By Leah Silver, Math Specialist (JK-4)

While so much has changed in our teaching in the last few months, the essence of the math classroom has remained the same. Planning, teaching and visiting math classes JK through Grade 5 in recent weeks has helped me sharpen my answers to the following questions: What can our digital tools do really well? And what can our teachers do that our digital tools never can? Our digital tools like DreamBox give our students immediate and targeted feedback on their operational and problem-solving work, and we’re learning how to let these tools help us make the best use of our teaching time. 

Our teachers do incredible work every day that our digital tools cannot—and will not—ever be able to do. Our teachers work masterfully to guide our students to better articulate their mathematical thinking, learn to ask excellent questions, make keen observations and get comfortable in solving complex problems, even when the answer isn’t immediately clear. We know that the ability to work collaboratively on complex problems and express ideas clearly will be two of the most valued qualities as this group of students continues to grow up. We also know that our students’ ability to make sense of data presented to them and reach logical conclusions is essential in a world where we receive information from an infinite number of sources. Our teachers prioritize these and many other concepts in their work with students each day — no matter the distance. 

While there are so many different strategies teachers use to help students come to new understandings, one is asking the right question at the right time. Whether in a kindergarten Number Corner session or a fourth grade multiplication strategy talk, these three key questions help our students build their computational fluency: solve problems accurately, use efficient strategies and show flexibility.

Students in Mrs. Patton and Ms. Ferdinand’s Kindergarten class (the Blue Room) proudly show their completed work after an exciting math game.

Question 1: What do you notice?

Teachers methodically ask this throughout their math classes. Asking younger students what they notice helps them to make connections and to sort and categorize — an essential early childhood skill. We ask our youngest students what they notice about a group of objects, about how many days we’ve been in school, about a graph or a chart that collects student answers to the daily question. And we ask our oldest students what they notice about a set of numbers or a particular strategy to help them formulate and articulate math strategies. Asking students what they notice encourages them to put together all the math and logic learning that is in their heads to try to put some order and sense into what’s in front of them. 

Mr. Haltom based his (beautiful!) calendar creation on the one from the Bridges curriculum, adding in important days for the Grace community. He asked our youngest learners in Junior Kindergarten what they notice about the emerging pattern, and what they thought might come next. What do you notice?

Question 2: What do you wonder?

Encouraging curiosity is one of the highest priorities in our math classes. Curiosity breeds hunger for more learning and also breeds an eagerness to make sense of what’s around us. Our second grade teachers recently led an activity which gave students the opportunity to come up with questions usually left to curriculum writers. With a given amount of information, what questions could you ask? For example, if you know that there are 124 red legos and half as many blue, what questions could you come up with? 

Our students might ask:

  • How many blue legos are there?
  • How many legos are there in total?
  • How many more red legos are there than blue legos?

When we ask students to generate questions, we dismantle the notion that questions only come from those who are ‘in charge’ or older. Giving children the opportunity to come up with their own questions is one way we can show that we value their thinking encourage their developing curiosity.

Question 3: How do you know?

Verbalizing thinking is useful both in and out of the math classroom, and our students spend a lot of time writing and thinking not only about the numbers of math but about the words, too. Teachers are often encouraging students to move beyond a common initial student response, “I just know,” and instead give a reason or strategy. “I know that is ten because I started with five and added two more, which got me to seven. And I know that seven and three more equals ten.” We want to teach students that intuition does have a place in the math classroom, and that there is value to articulating what and how you are thinking.

Mr. Wanyoike guides fourth graders through a problem set from Bridges. These problems lend themselves to certain strategies that the class has been focusing on, encouraging them to flexibly solve problems and justify their thinking along the way.
In a recent first grade class, associate teacher Ms. Alonso poses a task for students. There are 18 dots all together, and some are covered up. How many are under the ‘Splat’? How do you know?

I am really proud of all of the math learning and teaching at Grace in the last few months, and I can’t wait for the day we get to safely be together, hearing our students excitedly call out everything they notice and wonder.