Conceptual understanding is a crucial stage in a child’s mathematical development. The Learning Principle from the NCTM Principles and Standards for School Mathematics (2000) states as one of the six principles put forward, “Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” Summit teachers engage instructional methods, including the use of manipulatives, visuals, literature, math discussions, and problem solving that provide students opportunities to make sense of the mathematical concepts.
In second grade, students engaged in counting collections. They had a variety of loose items to count, as well as larger items that they had to count without taking them out of the package. The images show how Summit students are figuring how the total number of tissues using their counting knowledge to deepen the understanding of the base 10 system.
On Curriculum night, I was asked how parents can support their child’s development of mathematics. Well, simply put: Talk to your child about math as he/she and you encounter numbers and shapes in your everyday lives. Math is all around us. I believe that students learn best when they are actively involved. Let them develop their own thinking, try not to talk ‘at’ them about… math. Christopher Danielson’s book, Talking Math With Your Kids, which provides great examples on how to talk math with 3 to 9 year olds.
www.talkingmathwithkids.com and www.tabletalkmath.com are some valuable resources with products and blogs.
…Read more about Ms. Danforth and other Summit educators here:
Our goal in creating the Math Specialist position was to provide additional instructional support for students and teachers as we further enhance our mathematics program at Summit. We’ve asked Ms. Molly Danforth, our Kindergarten-5th Grade Math Specialist, to share more of her background and approach to the role, as well as make a special announcement—the launch of Math Mondays!
I am a National Board Certified Teacher, with Bachelors and Masters Degrees in Elementary Education. I have taught at Summit since 2007 and have been in the classroom for more than twenty years. My parents were strong advocates for education and always encouraged me to continue on this path. After receiving my Masters, I decided to further my understanding of how students learn and became a National Board Certified Teacher (2004 and 2014). It was during this process that I realized that students need a balance of procedural, conceptual understanding, and application techniques in order to understand mathematics deeply. Students need to conceptually understand math concepts before learning the abstract. Throughout the year, I will be sharing information on how Summit teachers create a balance to develop mathematical thinkers and provide opportunities for parents to be involved in their child’s mathematical development. You will see what is taking place in the classroom, ideas for reinforcement at home, research studies, recommended apps and games and even a parent challenge or two!
Follow Summit’s Facebook page to see Ms. Danforth’s Math Mondays posted weekly.
Author: Erin Vosseller, Kindergarten teacher, Summit School of Ahwatukee
Published in the Ahwatukee Foothills News, October 2016
Picture by Summit staff, Micki McIntyre
Most of us grew up memorizing math facts and definitions of terms. This short term memory trick may work for test success, but does little for understanding and applying math to solve real problems.
Can you recall trying to remember which way the greater than/less than symbol should be placed? Would it have helped if you first designed your own large alligator mouth to place between real objects?
Children excel when math is taught through concrete, sensory experiences. The younger the child, the more important it is to provide hands-on, visual tools to create a deep understanding of concepts. This especially true when it comes to numbers and math symbols. Children may be able to count very high, but does the number 58 really mean anything to them?
When introducing greater than and less than symbols, kindergartners need to have already built a strong understanding of more and less. My kindergartners do a lot of work with manipulatives when learning this and other math concepts. This helps them build a visual memory of numbers so that they can find patterns and see the relationships that numbers have to each other.
There are many things we do at Summit School of Ahwatukee to help children understand symbols and other abstract ideas. A fan favorite in my class is turning greater than and less than into a very hungry alligator, which would rather eat more, something that children can relate to. Our alligators always open their mouth towards the larger group of objects or larger numbers.
Children who are still working on building a visual understanding of numeric values can use objects or drawings to represent numbers. To create a visual, put items on two sides of a table, or in two boxes, or create spaces with tape for your objects. Many stores have inexpensive buckets of cubes or animals, or you can use Lego bricks, pennies, or other favorite things you have at home.
Ask your child to look at the items and ask which they think has more, from a visual perspective. Next have them count by touching each piece to confirm their prediction. Now place the alligator mouth so he can happily eat the most! Don’t forget to have your child make an equal sign on a card to use as well.
Use sticky notes, cards, or dry erase boards for your child to write the number next to each group. This helps create a mental picture of the number and what it represents. Another idea is to use balance scales. Students can weigh groups of objects to help them see more, less and equal while putting the correct symbol in the middle.
Making an alligators is fun, and kids can use it to model and talk about numbers in different ways. The alligator is a visual reminder to them about what the symbol means and they soon move on to using the symbol itself.
As children are ready to progress, use larger numbers or even expressions, such as 3 +2 on one side with 5 – 1 on the other.
It is also very important for children to talk through their thinking, explaining the strategies they use. Take turns with your child reading the expression he or she made, and ask them to teach you how to determine which is greater or less and why. Explaining their thought process takes understanding to a deeper level.
In our kindergarten classroom, we ask students look at each other’s work, listen to each other describe the strategies they used, and begin to analyze someone else’s approach to see if they make sense. Learning is enhanced when math is a team sport! It should be a noisy process as children think out loud, ask questions, and listen to and talk with each other.
They should read the expressions they write, testing them to see if they make sense. With the alligator project, one child might say, “10 is less than 12” while showing classmates the visual he or she created. We teach children to listen carefully and either agree or disagree respectfully and explain why, combining a successful social skill with mathematical reasoning.
Seeing + touching + listening + explaining = visualizing and understanding! Why is this important? Think about your job. How often does your boss ask you what 28 + 52 equals? Isn’t it more likely that you are asked to analyze a problem, and find a solution that can be communicated and implemented? Let’s prepare our children for more than test grades. Let’s help them prepare to be successful in life.
Erin Vosseller enjoys using her 18 years of experience to teach kindergarten at Summit School of Ahwatukee. She holds a bachelors of arts in Elementary Education from the University of Arizona, an Early Childhood Endorsement, a full SEI endorsement, and is CLIP (Collaborative Literacy Intervention Project) certified. She also volunteers as an Assistant Director of Arizona Camp Sunrise and Sidekicks, a camp for children with cancer and their siblings, sponsored by the Southwest Kids’ Cancer Foundation.
Former Summit Student Kyle Corrette creates a 3-D model featured on NASA’s website, as his design was selected as the teen winner of the “Future Engineers 3-D Printing Star TrekReplicator Challenge.
Kyle’s brainchild was one of 405 submissions from 30 states. His design was selected by a panel of judges from NASA, the American Society of Mechanical Engineers Foundation and Made In Space, Inc.
Kyle’s success was also featured in Popular Science
In 2016, Kyle was named as semifinalist for a National Merit Scholarship. Finalist will be selected later this year.
When asked about his experiences at Summit, Kyle shared this: “One of Summit’s strengths is the ability to bring out the best within its students; kids can develop their logic skills in Mrs. Yocum’s science classes, while also cultivating their creativity as part of Chris Dorsey’s band. The school environment is geared towards building a stronger all-around person, so that no matter what discipline a student chooses to pursue, they have the ability to utilize more than one intellectual strength in their endeavors – something which I would say is absolutely necessary to be successful.
The faculty also highlights each student’s individual talents, aiding their self-confidence and personal growth. In the last few weeks, I’ve had the pleasure to meet and converse with astronauts, NASA engineers, leaders of nationwide STEM organizations, highly accomplished people within the media, and the CEO of Makerbot, one of today’s most prominent tech companies. If Summit hadn’t helped me to gain the self-assurance needed to be a competitor within this challenge, I would never have had these amazing experiences. In the future, I plan to continue to progress within the engineering field, and be a part of meaningful projects within the technological world. Everyone has to start somewhere- Summit was an excellent place to begin.” Kyle attended Summit from Kindergarten through 7th grade
This week the faculty had opportunity to dig more deeply into our goal of equipping our students with both the content knowledge and the 21st century skills they will need for the future. What are 21st century skills? They start from a foundation of robust content knowledge. Building on this base, students are challenged to develop their skills in the areas of creativity and innovation, critical thinking, communication and collaboration. We call these the 4C’s, and they will be our emphasis this year. 21st century skills also include themes such as global awareness and economic/business/entrepreneurial literacy. Information, media and technological skills are additional critical components, as are life skills such as initiative, adaptability, cross-cultural skills, productivity and leadership. Many of these themes are already integrated into Summit’s curriculum, and our goal is to accentuate and expand their emphasis from Preschool through 8th Grade.
You may be wondering how a 21st century classroom looks differently than those we remember from our own days in school. Here are a few examples: teachers are learning facilitators, asking challenging questions and assisting students as they work through problems. Because the learning process is as important as the product, teachers may not provide all the answers, but allow students to push through struggles to find the solutions themselves. Students often work in teams, applying what they have learned to solving real-world challenges in areas such as the environment, economics, health or global issues. Students demonstrate learning in a variety of ways– learning portfolios, multidisciplinary projects, and presenting plans and results. Summit’s focus on 21st century education will augment and extend our curriculum content as students develop their ability to apply what they have learned in creative and meaningful ways.
On behalf of the Summit staff, we are grateful to be able to serve your children and partner with you in helping them reach their full potential.
Mark Bistricky, M.Ed.
Head of School
By Melissa France, 6th grade math teacher, Summit School of Ahwatukee
Most of us grew up memorizing mathematical formulas and theorems to apply to numerical problems on a test.
Could we solve the test problems? Hopefully yes. Did we have any concept of why we memorized them, or what use the algorithms had in life? Rarely, if ever.
Fortunately, a sixth grade math class at Summit School of Ahwatukee looks and feels very differently than what we experienced. The room is not full of children quietly computing calculations. Students are engaged! T hey are talking about math with each other and the teacher. Focusing on the task at hand, they work in teams to creatively arrive at various ways to find solutions to real scenarios.
Let’s take a look at how a recent lesson, regarding a cash box and a middle school dance, unfolded in sixth grade. Students are sitting with a partner and each partnership is presented with this problem: Emma was selling tickets at the middle school dance. At the end of the night, she picked up the cash box and noticed a dollar lying on the floor next to it. She was concerned about what to do with it and wondered whether the dollar belonged inside the cash box or not. The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (buddy pricing). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not? Convince me that your answer is correct.
Students immediately started trying to solve this problem using many different strategies. The best part about my job as a teacher is watching and listening to this learning take off! Instead of standing in front of the room and lecturing the entire class period, I get to facilitate the learning that is started through student engagement and excitement!
How does this engagement and excitement happen? What am I looking for in my classroom?
I want my students to be able to be mathematically proficient in making sense of problems and to persevere in solving problems. This was apparent as the students began to analyze the given information to develop possible strategies for solving the problem. Students didn’t stop and ask me for help when they got stuck, they talked it out with each other and kept going.
I want my students to be able to be mathematically proficient in reasoning abstractly and quantitatively. Students demonstrated this to me as they began to translate the given information into a mathematical representation.
I want my students to be proficient in constructing viable arguments and critiquing the reasoning of others. Not only were the students speaking mathematically while sharing their approaches and strategies respectfully with their groups, they were recognizing and using counterexamples to refine assumptions being made and disputing or disproving the given arguments! Wow! Our math classrooms at Summit look and sound a lot different than the classrooms I experienced as a sixth grade student.
Were you able to solve the problem? Does the dollar go in the cash box or not? Can you justify your answer? Our sixth graders did so be sure to ask them to check your work…don’t forget to justify your answer!
More about Melissa France
Mathematics: Sixth Grade
Life Skills: Seventh and Eighth Grade
Student Council Advisor: Fifth – Eighth Grade
Melissa France earned a Bachelor of Science degree in Education in 1998, with a minor in Psychology from Northern State University in Aberdeen, South Dakota. She is a member of NCTM, National Council of Teachers of Mathematics. Mrs. France brought her mathematics and leadership skills to Summit in 2001. She also serves a leadership role in Summit’s math curriculum; supporting teachers in the use of research based best practices, differentiating instruction and challenging students.
Mrs. France also serves as Summit’s Student Council Advisor, guiding students in leadership roles for school wide community service projects and spirit building activities. As a result of her service and leadership, she was named Educational Mentor of the Year in 2014 by the Ahwatukee Chamber of Commerce.
Mrs. France was also a recipient of a “Golden Gator, Excellence in Teaching Award” from Xavier College Preparatory High School. This award recognizes junior high teachers who have been inspirational to Xavier’s freshmen students.
Let’s Move! Active Schools, a sub-initiative of First Lady Michelle Obama’s Let’s Move! Campaign announced its 2015 National Award honorees.
Summit School of Ahwatukee was one of 15 schools in Arizona and five hundred and twenty-five U.S schools representing 37 states, recognized for their outstanding efforts in creating an Active School environment and increasing physical activity and physical education opportunities for students.
The Let’s Move! Active Schools National Award is the top physical activity and physical education distinction for K-12 schools. The award celebrates a school’s commitment to integrating at least 60 minutes of physical activity before, during and after the school day.
“These schools are raising the bar by creating Active School environments where students are happier, healthier, and higher-performing. I am thrilled with their success through Let’s Move! Active Schools – they’re helping to create a new norm where physical activity is a fundamental aspect of a young person’s success both in the classroom and in life,” said First Lady Michelle Obama.
This is the second time Summit School of Ahwatukee has earned received this national honor. “We owe so much to our physical education teacher, Kathy Dean, who has worked tirelessly to help create a culture of health and wellness at Summit,” Head of School Patrick O’Brien proudly shares.
Studies show that Active Kids Do Better. Physical activity not only helps kids stay healthy and strong, but it can also lead to higher test scores, improved attendance, better behavior in class, and enhanced leadership and interpersonal skills.
“The Let’s Move! Active Schools National Award is the nation’s top physical activity and physical education distinction for K-12 schools,” said Charlene Burgeson, Let’s Move! Active Schools Executive Director. “We commend Summit School of Ahwatukee’s exemplary work and commitment to active learning environments. Summit’s faculty, staff and students are paving the way to a healthier, higher-performing and more successful generation of youth.”
To earn a Let’s Move! Active Schools National Award, a school must have met significant benchmarks in five areas: physical education; physical activity before and after school; physical activity during school; staff involvement; and family & community engagement.
Summit School of Ahwatukee’s programs exceed in all these areas, in large part thanks to PE teacher Kathy Dean, who was instrumental in developing Summit’s physical education curriculum focusing on physical and nutritional health. She also initiated the middle school sports program, and created a before school running club at which 50 or more students and parents voluntarily run or walk on the school’s field twice each week.
Dean began and continues to organize the annual family turkey trot, and the school’s field day, which also focuses on developing leadership in middle school students who help run the event. Last year she held a successful pedometer challenge for the school’s teachers and staff, encouraging healthy goals for daily steps.
Students benefit academically from Dean’s initiatives as she provides training to staff, sharing “brain break” activities teachers successfully use in the classroom to help young minds stay fresh and engaged.
However, from the student’s perspective the most important and best loved initiate of Dean’s is the daily morning recess / snack break enjoyed by kindergarten through eighth grade students. In addition to lunch recess, students thank her in spirit each day as they enjoy ten minutes on the playground tire jumper, shooting hoops, running on the field, or simply chatting with classmates as they enjoy a healthy snack. “We all need a break to stay fresh and motivated,” explains Dean.
Why does she do it? “The biggest reward for me is the tremendous buy in from the entire school community: teachers, staff, parents and kids, to lead healthy, active lives,” says Dean with visible warmth. “It is true that active kids are more successful academically, but they are also happier. You can actually feel the joy of students, teachers and parents on Summit’s campus. It is wonderful to experience.”
Honorees are provided with a banner, certificate and congratulatory letter from the First Lady. Recognition packages were generously sponsored by BOKS, Build Our Kids’ Success, a free before- and during-school physical activity program aimed at getting kids’ bodies moving and their brains ready to learn.
Written by: Faith Angelakis, Literacy Specialist Summit School of Ahwatukee
Faith Angelakis, the reading specialist at Summit School of Ahwatukee, has a poster hanging prominently on her office door that says: 10 Ways to Become a Better Reader: 1) Read, 2) Read….10) Read. “I often hear students chuckle as they walk by and read the poster,” says Angelakis with a smile. “It may seem simplistic, but reading to young children may be the biggest head start to building a solid literacy foundation, preparing child to read well when the time is right.” Here are just a few of the benefits that parents teach when reading aloud to children.
• Reading fluency: Hearing an adult provides an excellent example of smooth, properly phrased, and expressive oral reading. When it is time for your child to read, the lessons modeled will be remembered and emulated.
• Story language and vocabulary: When an adult reads, the child hears language and vocabulary that aren’t always used in everyday conversations. Reading builds vocabulary and your child’s ability to understand and communicate both orally and with descriptive written words.
• Directionality and 1:1 correspondence are important. Point at the words as your read to your child; this teachers that English is read from left to right, that spaces around the print indicate where a word starts and ends, and that each spoken word matches one on the page.
• Developing a love of reading may be the most important of all. Think of all the literature, textbooks, news, or professional reading we do in a lifetime. It shouldn’t be a chore. Make reading time with your child part of the daily routine. Your child will associate reading with positive experiences, developing an intrinsic motivation to continue.
• Talk about the book: Simply recounting what was read may help memory, but what does your child think about the book? What other setting could the story be in? What other choice could the characters make or what would your child do? Show how changing a word or voice can change the meaning.
Let the racers race. Making reading part of your family’s routine will do more than earning some reading “prize”; it will lay the foundation to a successful journey to reading and learning.
About the Author:
Faith Angelakis holds a Master’s Degree in Elementary Education, a Reading Specialist endorsement and is CLIP certified, (Collaborative Literacy Intervention Project).
Through collaborative efforts with grade level teachers, she leads the school’s preschool and elementary literacy team. Ms. Angelakis works with teachers, to help them continue to develop professionally, by modeling lessons or team-teaching units. She also helps teachers plan literacy instruction for the year, and provides professional reading materials about the most recent teaching techniques.
Additionally, Ms. Angelakis works directly with students in kindergarten through third grade classrooms, teaching reading to small groups of students, at their instructional reading level.
For the past three years, Summit has had a 1:1 iPad program, asking students in sixth, seventh and eighth grades to obtain an iPad for use in classes. The goal is to expand the use of technology by students, and to provide opportunities to enhance collaboration, communication, critical thinking, and creative skills.
During this time we have seen technology change rapidly. Now tablets of all kinds are available, and families often own multiple personal computers of differing operating systems, in addition to tablets and cell phones. Students will need to have technological skills that can translate to any platform. Therefore, our technology model needs to shift to meet this need.
Summit School of Ahwatukee is proud to announce that beginning in 2015 – 2016, the middle school will shift its 1:1 technology model to Bring Your Own Device (BYOD). Families will now have several options when deciding what form of personal technology each student should bring. This is manageable for us since each student will have access to Microsoft Office 365, which means that everyone – regardless of device or platform – will use Word, PowerPoint, Excel, and other Microsoft products to produce work. Cloud-based storage will provide a means for students and teachers to share work back and forth.
An Android, iOS, or Windows tablet will be acceptable, as will a Windows or Apple laptop. Devices will need browsing capability but will not need data plan agreements as students will use our Wi-Fi connections while at school. Students must use the network while at school to ensure compliance with the Children’s Internet Protection Act.
Cell phones will not be allowed as devices used in the classroom for educational purposes. Use of any device is subject to the Acceptable Use Agreement and its addendums, found in our Middle School Handbook, and is also subject to the guidelines established by the school and teacher for classroom management. Any inappropriate use will fall under the school’s discipline policy. Similar to other personally-owned items, the school is not liable for the loss, damage, misuse or theft of personally-owned devices. It is the responsibility of each family to provide adequate insurance for devices and to protect them with cases. In addition, Summit cannot provide technical support or install software on personally-owned devices.
In the fall, we will offer both student and parent “boot camps” to help your family understand how this device will be used in the classroom. Summit is excited to transition to the next stage of technology use in the middle school, and we are pleased to offer this opportunity for students to experience learning using the same devices they use to engage with their multimedia world, and to inspire learning while creating independent, critical thinkers.
Patrick O’Brien, Head of School
Summit School of Ahwatukee