My core philosophy of math education is that:

**Achieving
a deep and enduring understanding of math requires a multi-modal approach that
begins with an engaging problem – a real reason to explore an area of math.**

Simply stated, I believe that learning math starts with a problem. It has been that way for thousands of years; small groups of mathematicians working in peer circles to solve real problems or to explore math’s curiosities. These mathematicians were engaged by the challenge of a problem and inspired to find simple and beautiful solutions.

## My Beliefs about Teaching

I believe that is what teaching is, challenging learners with an engaging problem, and letting the student set the pace of learning. The teaching role is that of the facilitator, establishing an optimal workspace, setting the lesson plan, identifying suitable problems, keeping students on task and eliciting good questions.

A classroom that is centered around problem solving is driven by two things: one, the engaging problem is such that the learner wants to know the answer, and two, the learner wants to know how to solve it in a way that avoids hard work. […]

## My Beliefs about Learning

Learners enjoy a fair struggle but are motivated to learn methods that avoid hard work. They like that feeling of reaching an answer successfully and sharing with their peers an elegant solution to a hard problem.

Real problems can open a window for introducing a new math concept. For example, I might introduce a problem involving a least common multiple (LCM) before teaching the concept. The learner might solve the problem using lesser strategies, such as using lists or number lines, to find the LCM of two small numbers. However, when working with larger numbers, or multiple numbers, the learner will start to see the need for a better strategy – a good segue to bringing up subsequent division as an approach.

I find that a handful of learners surface the higher-order methods through their own struggles, and then share their findings with their peers. My lesson plan is often taught before any recitation is needed; my lecture becomes a classroom reflection and journaling exercise of what worked and what didn’t work in the process.

Math Talk is a key component to achieving a deep understanding of math. We learn better when we can talk through our reasoning, share our understandings, and change our viewpoints upon hearing a peer’s approach. It also helps the learner and the teacher surface misunderstandings and missing precursors. […]

## My Beliefs about Learners

I believe that nobody is born into a love of math. Learners either love math or do not yet realize that they love math. Kloosterman (1994) believes that if teachers expect students to learn math, they then have the responsibility to pay attention to their beliefs and show them that they too can learn mathematics. “Beliefs have a connection to the student’s view of himself and others as learning mathematics.” (Wester 2015)

When a learner shares with me, “I don’t like math” what I hear is, “There is something that I don’t understand, and it is making me feel uncomfortable.” My role is to is to surface the learner’s misunderstandings and catch the learner back up to where they need to be; the awkwardness of math goes away.

Where this philosophy is most needed is in middle school. Our 6–8^{th} grade students have made conclusions and established an understanding of their place in math that will greatly impact their path in high school. Students are at a critical stage in their mathematical education and must be given top priority. […]

## My Beliefs about Parents and Caregivers

I believe that engaging parents and caregivers in their child’s learning of math outside of the classroom is important for achieving a good outcome for our students. I provide caregivers with what we are doing in the classroom, what they can do at home and how that is connected to their child’s outcome in math. This can include journaling our work in the classroom, describing the math concepts that we are learning, holding family math nights at school and sharing math games and books that can be played and read together at home. In the process, the caregiver may too build a deeper understanding and appreciation of math. […]

## My Beliefs about Knowledge and What is Worth Knowing

There are two important things to know about problem solving, the first is the process and, the second is the set of strategies. The process must be taught and the strategies, while standard and published, should be discovered through the learners’ own struggles. As a facilitator, I maintain a list of the strategies the learners have uncovered, and stock a toolbox full of manipulatives useful for when learners are struggling to solve a problem at an abstract level.

Students should be exposed to mental math and number sense as early as 3^{rd} grade. These powerful skills, a dexterity with numbers and the ability to work with numbers in one’s head, take a lot of the hard work and error out of problem solving.

I believe that vocabulary is just as crucial to math as it is to language arts. Math has some pretty complex terms with definitions written unnecessarily in the form of axioms. We as teachers need to spend more time covering vocabulary and using multi-modal approaches, such as those defined by Marzano and employed by Membean.

Visual spatial skills, prime numbers and cryptarithms round out the knowledge needed for a student to have a successful math outcome. I also advocate for students to have exposure to other number systems (e.g. Roman Numerals and binary) as it helps us better understand our number system when we have something to contrast it with. […]

## My Philosophical Areas

I believe math is perennial, and just like an English teacher depends on Shakespeare, I include the history of Pythagoras, Euclid, Euler and Archimedes in math class, for their ways are of ageless beauty and simplicity. Thousands of years ago, math was simply prose and scrolls with steps for solving problems. It was only as recent as the 18th century that we began to use symbols and formulas and complicated definitions and terms. Over generations, we have added layers of complexity and I find it is good to go back to the origins to see what makes math so engaging and playful.

I believe in an essentialist approach to math. However, in order to develop successful students, schools need to balance and weigh the practical with the experience of the journey. And if we find a pot of gold at the end of our journey that is good, but it is the things we have discovered along the way that also matter.

My teaching philosophy, however, is progressive. Progressivism states that “The best way is to equip students with problem-solving strategies that will enable them to discover meaningful knowledge at various stages of their lives.” (Parkay, p.127) With progressivism, the learner is active. and while I am the expert, I am not the answer key. I believe the true strength in our classroom lies in the collaboration of our students. […]

## References

Ashton-Warner, S. (1963). *Teacher*. New York, NY: Simon & Schuster.

“Bloom’s Taxonomy.” *Centre for Teaching Excellence*, 4 July 2018, https://uwaterloo.ca/centre-for-teaching-excellence/teaching-resources/teaching-tips/planning-courses-and-assignments/course-design/blooms-taxonomy.

Chapin, S. H., OConnor, M. C., & Anderson, N. C. (2009). Classroom discussions: using math talk to help students learn, grades K-6. Sausalito, CA: Math Solutions.

Colgan,
L. (2018). Hey, It’s Elementary: Share-Worthy Parent Engagement Materials for
Math… No Fake News! *Gazette – Ontario Association for Mathematics*, *56*(4),
37–40. Retrieved from http://www.cse.idm.oclc.org/login?url=https://search-proquest-com.cse.idm.oclc.org/docview/2057942723?accountid=10328* *

*Ellison, Glenn. Hard Math
for Elementary School. CreateSpace, 2013. *

*G’Day Math*, 10 Nov. 2019, https://gdaymath.com/.

Kalman, Richard. *Math
Olympiad Contest Problems*. MOEMS, 2011.

Kloosterman, P., & Cougan, M. C. (1994). Students Beliefs about Learning School Mathematics. *The Elementary School Journal*, *94*(4), 375–388. doi: 10.1086/461773

“Incredibly Effective Vocabulary Learning.” *Membean
– Durable Learning*, 10 Nov. 2019, https://membean.com/.

Mills, K., & Kim, H. (2017, October 31). Teaching problem solving: Let students get ‘stuck’ and ‘unstuck’. Education Plus Development. Retrieved from brookings.edu/blog/education-plus-development/2017/10/31/teaching-problem-solving-let-students-get-stuck-and-unstuck/

“NRICH.” *NRICH Maths By University of Cambridge*, 10 Nov. 2019, https://nrich.maths.org/.

Numberphile, 10 Nov. 2019, https://www.youtube.com/user/numberphile.

Parkay, Forrest W. Becoming a Teacher. 10th ed., Pearson, 2016.

Protheroe, N., “What Does Good Math Instruction Look Like?” 10 Nov. 2019, https://www.andrews.edu/sed/leadership_dept/webinars/presentationdocuments/what_does_good_math_instruction_look_like.pdf.

Wester, R. (2015). *Matematikundervisning utifrån ett elevperspektiv: “det är klyddigt”*. Malmö: Lärarutbildningen, Malmö högskola.