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A Conceptual Understanding of Math

Juli Dixson at SRES

Juli Dixon, PhD, mathematician, author and consultant for DNA, is demonstrating a fraction lesson plan that she and a couple teachers developed that day. Teachers, instructional coaches, school leadership and Sand Creek Zone staff watched the demonstration at Springs Ranch Elementary School. 

Mission DNA: Developing Conceptual Understanding of Math

District 49 elementary school teachers are in the middle of a five-year math program called Mission DNA: Deploying Numeracy Achievement. The program, funded by a DoDEA grant, is designed to further extend and enhance math content knowledge for teachers and to help develop conceptual understanding that they can share with students. 

“It’s one thing to know the math, but it’s another thing to teach the math, so students understand it,” said Carolyn Merritt, Sand Creek Zone instructional coach. She is also the implementation coach for the program’s grant, which currently has 80 teacher-participants. “We are shifting how we teach. Instead of teachers telling students how to do math problems, and then they mimic the teacher, they can develop a deeper understanding of math concepts.”

The first year of the Mission DNA grant was mostly planning and an online class for about 80 teachers. In years two and three, the teachers participated in six professional development training sessions and two learning labs. (These were not the usual Pro Days the district schedules.) The teacher would attend three sessions, then in the lab they worked with an instructional consultant and a fellow teacher to come up with a math lesson. Then one teacher would present the lesson to a small group of students. After the lesson, the teachers would critique the lesson and revise. Then the second teacher would present the lesson to another group of students. 

“The cool part of the learning lab is where they put it into practice,” Merritt said. “We kept the group of students small, so teachers can focus on the intricacies of the lesson and not worry about classroom management.”


Nicole Barbera PPSEL

Nicole Barbera, kindergarten teacher at Pikes Peak School of Expeditionary Learning, participates in a learning lab, where she teaches a lesson as another teacher and instructional coach watch and critiques the lesson.


To simplify, what the consultants are teaching or coaching D49 teachers about is how to use tasks, or lessons, to develop deep understanding of math. They are helping teachers shift the way they teach to provide space for students to make sense of the math, and explain their thinking and reasoning. 

“The way we were taught is not the way we’re teaching today,” Merritt said. “Most were taught to memorize multiplication tables, but were never taught the ‘why.’ They learn the process of a procedure to get to an answer. But they don’t really have the meaning of the answer.”


For instance: 3x4. Memorized, that’s 12.

But to break it down into the ‘why’ it is 12, looks a little different.

4+4+4 is 12, or there are three groups of 4. When students understand the commutative property, they further extend understanding to 3+3+3+3 is 12 or four groups of 3. When you add context to the situation, sense-making matters. Three groups of 4 students is different from four groups of 3 students, yet in both situations there are 12 total students. 


“Kids now, through the lessons we’re teaching, are gaining meaning of this. They are making sense instead of memorizing,” Merritt said.  



Some students might struggle with multiplying 7s, but know they can break apart 7 into a 5 and 2. 

They know 5x5 is 25.

And that, 2x5 is 10

25+10 is 35 


“We were not taught flexibility with numbers,” Merritt said. “If you didn’t memorize 5x7, you had no other way of getting the answer. Students now know how to break it up into simpler problems.”

By breaking down the problem into smaller parts, they can do mental math and solve the problem. 

“We are trying to show them how to be flexible with numbers so they can be better problem solvers,” Merritt said. “So when they encounter a problem they don’t know how to solve, they have strategies. It’s helping students have a deep understanding of what the math really means. We are creating students that are flexible thinkers who can break things apart. They are encountering a problem, and learning how to strategize. The stuff people are encountering now in the world, all these new developments, it’s based on problem-solving and thinking.” 

Merritt is quick to point out that they are still teaching multiplication tables and other simple math. But, that comes after they teach the conceptual approaches. 

“We do want the efficiency and we do want our students to be fluent in math facts, but we want them to have strategies in their toolbox,” Merritt said. “Math all fits together. There is beauty in math, how it builds on itself.” 

By understanding these smaller numbers and problems, it helps students understand larger numbers as the match builds in complexity.

“When you see math competitions and people are multiplying giant numbers in their head and get it almost instantly, it’s because they can break stuff apart and work with the smaller parts,” Merritt said. “They’re not sitting there doing the algorithm in their head.” 

Teachers are learning to create tasks and use thoughtful questioning to deepen student understanding of concepts. A task may be as simple as ‘create a story problem for 26 divided by 4 where the answer is 7.’ This example task helps students make sense of the remainder when working with division. It provides space for students to gain understanding of when it is appropriate to estimate higher versus when to estimate lower.  

The consultants both work with individual teachers as well as showcase their own skills in the classroom to demonstrate how they would teach. 

“They show how to know who to call on, to build the lesson instead of letting one student tell the class the answer before it’s time,” Merritt said. “The goal is to strategically sequence our questioning to engage students in dialogue and sense making.” 

There are also sessions for the principals, assistant principals and instructional coaches. Many of whom will attend class during a lesson taught by the consultants. 

“It’s the best professional development that I’ve seen. It’s actionable. They are modeling the lessons, instead of telling them how to do it,” said Christina Tiernan, D49’s new coordinator of math. 

According to Tiernan, it is not just about the math. “We are all coachable. We all can change and improve in some facet. These are coachable moments … if you can change just one thing, even your body language that you don’t realize you are doing. It’s how to work the room, how to elicit questions from the students, so they can understand the math,” she said.  


For the teachers in the Mission DNA program, they are thriving under the coaching. 

“Coaching for Mission DNA has been different from any professional development and student teaching that I have taken part in,” said Tiffanie Martin, third grade teacher at Bennett Ranch Elementary School. “The frequency of the trainings, the knowledge of the trainers, the observations in the classroom, the hands-on activities that can be taken back to the classroom right away, and the learning labs really help to ensure that we are able to apply the skills we are learning to the daily classroom instruction.”

Rebecca Lentner, second grade teacher at Evans Elementary School, said, “The coaching has been a valuable program in my classroom. Mission DNA has changed how I view math.  Math has never been my favorite subject, but with Mission DNA, I have grown to enjoy teaching math.  I believe that with that change of mindset, I have become a better teacher, not just in math, but across the curriculum.”


Becky Lentner Evans

Rebecca Lentner, second grade teacher at Evans Elementary School participates in a learning lab.


“I felt like I was deepening my understanding of how to teach math and make my instruction stronger,” said Nicole Barbera, kindergarten teacher at Pikes Peak School of Expeditionary Learning. “It was very beneficial to be able to collaborate with other educators in my profession, specifically in the primary grade levels. I found this to be very powerful.”

“As a teacher, the Mission DNA coaching has changed the way I plan my instruction. It has made me look through the lens of my students’ eyes and how they perceive the skills I am teaching,” said Abbey Calderini, fourth grade teacher at Springs Ranch Elementary School.


In the classroom, teachers are taking a different approach to teaching math, and succeeding. 

Calderini, SRES, said, “I always used the  ‘I do, you do, we do’ approach for teaching math. I thought this would help students ‘practice’ their math skills and become proficient with consistent repetition of mathematical steps. Later on in my career, I made a shift to letting the students make sense of math and allow students to share multiple strategies to solving a problem. This shift led to my students understanding math skills in a deeper way and in turn my students started having higher scores. This is exactly what Mission DNA is … students talking to each other making sense of their mathematical processes. There is no one correct way to do math.”

Martin, BRES, said, “At first, the students always seemed surprised when I am not telling them what the correct answer is, but having them discuss with each other to get to the same final product. After a few weeks of teaching like this, they learn the proper ways to disagree with each other and how to defend their answers during discussion.” 

“I am more aware of allowing my students to struggle with their learning and to grow in the challenges that come with it,” Lentner, EES, said. “After going through Mission DNA I feel more freedom to allow my students to explore different strategies and use what works for them best.”  


The students are not only engaging with math and their classmates during the lessons, they are improving. Overall, the students in classrooms with teachers in the Mission DNA program are at or above where they need to be with an average of 50% on the student growth percentage.  (50% means the student is ready for the next level.)

Martin’s students are at 61%. Mark Koenig’s third grade students at Remington Elementary are at 83%. 

Looking at the Star 360 benchmarks from the Beginning of the Year (BOY) to the Middle of the Year (MOY), one can see the shift toward math proficiency. Students are improving, moving from Urgent Intervention to Intervention to On Watch to At/Above Benchmark.

Star 360 benchmarks total student

Above Graph: Total Population for D49 Elementary schools shows student improvement went from 13.3% “At or Above” benchmark to 24.7%.


Star 360 Math K-5 military


Above Graph: Military-Connected students improvement went from 14.2% “At or Above” benchmark to 27.9%. While “Urgent” dropped in both cases.


Tiernan said this improvement is a direct response to Mission DNA. 

Barbera, PPSEL, said, “My students have enjoyed math talks and sharing how they solve problems with their peers.  We spend a lot more time talking about how we solve problems and different strategies that can be used to solve a problem, than before I was a part of Math DNA.”

“Students are able to explain their thinking in a more meaningful and accurate way since using strategies from Mission DNA,” said Calderini, SRES. “The engagement level of all students has increased due to the academic social interactions taking place each day in their math lessons.”

Martin said, “They like to challenge themselves to find a different strategy to solve, so that they have an opportunity to share and discuss with the class. They also know that they are going to be held responsible to have an answer and be able to defend it, so they are all engaged in the lessons.”

Calderini added, “I have also noticed that students are able to answer much more difficult multi-step word problems and are able to think through them critically. My students are also able to find multiple solutions to a problem rather than just one way.”

“Students seem to understand math better in my classroom than in years past,” Martin, BRES, said. “They are able to solve problems in different ways, discuss and disagree with each other when solving math problems, and explain their thinking.” 

The first cohort of 80 teachers is nearing the end of its coaching. Merritt is hoping 100 teachers will be part of the cohort that will begin in the fall. 

“I think more teachers are interested in participating next year because they see the passion of the teachers that are currently involved in it and have done the training,” Merritt said. “The energy coming from the teachers has helped drive the interest of the people that want to be in the second cohort. They are talking about how amazing the training is.”

The $1.5 million STEAM grant came from the Department of Defense Education Activity (DoDEA) and has two more years. Part of the funding provides for substitute teachers to handle the classrooms while the teachers are participating in the professional coaching and learning labs.  D49 recently submitted a grant proposal to bring Mission DNA into the middle school.

For more on DoDEA grants, see the Spring 2023 edition of Discover: D49 Magazine which is out this month. Check your mailbox if you live in the district boundaries.


SRES Mission DNA observers

Carolyn Merritt, Sand Creek Zone instructional coach, along with teachers, instructional coaches, school leadership and Sand Creek Zone staff watched a lesson demonstration at Springs Ranch Elementary School.


Joel Quevillon