In the summer of 2017, all classroom teachers and learning strategists in the Oconomowoc Area School District (OASD) were trained in a new math curriculum, Bridges in Mathematics. Visual models are a key component of the Bridges curriculum and teachers at Ixonia Elementary School use them in their daily lessons.
On their website, Bridges describes visual models as, “We are not just teaching math, we are building mathematical thinkers. Seeing, touching, and sketching ideas create pictures in the mind’s eye, helping learners construct, understand, and apply mathematical ideas.” We solve math problems in our everyday lives and Ixonia Elementary teachers’ help students understand that real math does not always have an easy answer, but using visual models can help them solve those problems.
In the five months since the OASD began this new curriculum, Ixonia teachers have noticed that the integration of visual models has yielded tremendous results in how students solve problems. Using this curriculum has increased confidence and understanding in our students as mathematicians.
“Math models are tools that our students are taught starting in Kindergarten, as ways to represent their thinking. It really helps our students break down the process of mathematical thinking and gives them the structure to use words to explain that thinking,” said Ixonia Elementary School Principal, Stacy Yearling.
In Kindergarten, students use unifix cubes as a visual model to help them solve math problems. Different color cubes can represent the components of an addition or subtraction problem with a goal that students can later imagine those same cubes when trying to work out a similar math problem in their mind.
First graders use domino flap books, which allow them to visualize a math problem with the dots on a domino. They can combine different dominos to practice addition and subtraction using the dots, then use the books to write the real equation represented in their domino dots.
By second grade, students focus on breaking numbers into 1’s, 10’s, 100’s, and 1,000’s and use a splitting strategy to help them solve more complex math problems. This visual model doesn’t use the cubes or books that younger grades use, student transition to using paper and pencil, mapping out the solution to the problem in a more visual way than the standard addition or subtraction equation.
Students in third grade utilize an open number line to help solve two and three digit addition or subtraction problems. They draw the number line on paper as they work through the problem, drawing arcs, or “friendly jumps,” to help them add familiar numbers, ultimately working their way to the complex answer.
Once students reach fourth grade, they’re experienced in using all of these visuals, alternating between the models that work best for the math problem they’re trying to solve. They have a full toolkit of options to help them turn a traditional math equation into a problem that they can dissect into smaller, easier to manage parts. This experience with visual models prepares students to master fractions and they can utilize a number line to aid in that mastery.
“The beautiful part of this curriculum and these models are the real interactivity. What’s beautiful in the classroom is how much partnership and collaboration is going on as the students are working together, solving problems, using these mathematical tools,” said Yearling. “If our kids can walk away from their experience at Ixonia, having accomplished using visual models, then we have done our work in building mathematical minds,” she added.
At their OASD School Board Showcase on February 20, Ixonia Elementary students in grades kindergarten through 4, confidently demonstrated how to solve increasingly difficult addition concepts using a variety of visual models. They highlighted how these models help construct their understanding and how they apply them to solve new problems. They demonstrated how these tools strengthen their ability to make thinking visible to themselves, their classmates, and teachers, helping them talk about the process used to solve problems. Through this process, Ixonia Elementary is building mathematical thinkers who can solve real-world problems, now and in the future.
Above: Two Ixonia Elementary School students using unifix cubes as a visual model to solve a math problem.
Above: An Ixonia Elementary School student uses the number line as a visual model to solve more complex, two or three digit math equations.
Above: Ixonia students combine their visual models to master fractions in fourth grade.
Back: Stacy Yearling, Ixonia Elementary School Principal
Middle, left to right: Ixonia Elementary School Teachers, Traci Timm, Kayla Muroni, Jim Mueller, Sally Losinske, and Misty Burton.
Front, left to right: Ixonia Elementary students, Audrey, Phoebe, Ava, Dixon, and Ethan