Montana Council of Teachers

of Mathematics

2021 MCTM MCubed

6th Annual


MCubed & Montana Math Meet

August 10th & 11th, 2020

Butte, Montana

MCTM Karen Longhart Scholarship Awardee - Alison Lokey

NCTM Virtual 2021 Annual Meeting Take-Aways

Greetings Montana math community! My name is Alison Lokey and I have been a middle school math teacher at Target Range School in Missoula for the past 23 years. When I was awarded the Karen Longhart Scholarship last fall, I was excited that I might finally be able to attend a national math conference. Then I remembered we were in the middle of a pandemic! A silver lining of the pandemic is that we can now access great professional development ...from our couches! The NCTM Virtual 2021 Annual Meeting took place in late April, but I had access to all the session videos through June. Good thing since I finally found time in June!

It is a unique experience to search for valuable PD from a long list of workshop videos - impossible to watch them all! As I read through the list of about 75 workshops that were targeted at the 6-8 grade band, I found myself choosing videos with a similar theme. The following quote by Alexandra K. Trenfor summarizes the theme that I found throughout my conference experience: “The best teachers are those who show you where to look, but don’t tell you what to see.” The sessions I attended modeled hands-on learning experiences that went beyond tricks and rules to help students achieve deeper levels of understanding. While this is not a new idea for me, I found it insightful that I am still searching for more ways to make math meaningful to my students. As a thank you for this scholarship opportunity, I would like to share with you a few of my favorite “takeaways” by session.

Discovering What Pythagoras Knew and More Presenter: Adam Harbough

Not only are 8th graders charged with solving problems that involve the Pythagorean Theorem, they need to be able to describe a proof. This presenter showed me a way that I had not seen before that shows both the theorem and its converse. This activity requires students to have access to squares with predetermined areas and to be given the task of building triangles using one side of each square as shown below. The investigation allows students to hypothesize and search for patterns in the relationships between the areas of the three sides based on the type of triangle that they have identified it to be. Give it a try to see what you discover! Students are building their conceptual understanding before learning the formula.

The Dangers of KISSing and other Mnemonic Devices Presenters: Christy Danko Graybeal and Tricia Strickland.

Mnemonic devices are often used to help students remember certain procedures and are an evidence based practice in special education. However, the presenters urged teachers to be careful of tricks that are used for computation, as students tend to over-generalize or misuse the tricks. A trick that is risky is the “Butterfly Method'' for comparing fractions because it does not address the idea of common denominators. There are some tricks they consider to be acceptable if the conceptual understanding is built first. For example, “Keep it, switch, switch” (KISS) is a strategy for rewriting subtraction problems as addition problems. Students may appear to be getting the “right answer,” but teachers may not know whether they understand the concepts and will often see them apply the trick at inappropriate times. Number chips or number lines can be used first to establish the connection between addition and subtraction. The use of FOIL for multiplying binomials is also questionable because it is not generalizable to other types of polynomials. However, if the box method is taught first, students have a larger range of options to apply to future problems. As their conclusion, the presenters stated that they supported the use of mnemonic devices when teaching problem solving processes, conventions, definitions, or facts. For example, SOH CAH TOA is simply a restatement of the trig ratio definitions to help students with recall, but it is not a “trick” because it is not changing the way that students solve problems.

Moving Beyond Cross Multiply and Divide Presenter: Elizabeth Peyser

In a similar vein, this presenter referred to cross multiplication as an “answer getting” strategy and claims that the use of this strategy in isolation does not help students develop proportional reasoning. She emphasized the use of visual strategies such as bar models, double number lines, tables, and graphs to show how students can reason through the solution to a proportional relationship, rather than relying on cross multiplication on their calculator to automatically give them a solution. A firm understanding of diagrams and ratio tables can either be used instead of or as a precursor to cross multiplication to make sure that students are gaining conceptual knowledge needed for future mathematics topics. This proportional reasoning is taught in 6th and 7th grades, and, if done conceptually, will allow students to extend their understanding to linear functions and slope easily when in 8th grade.

In this example, the presenter recommended the “fair share” strategy for determining how many cups of sugar would be needed for 27 cups of flour. I normally have students extend the diagram until there are enough bars to represent the problem; she recommended using beads or tick marks within the bars to show how much each is representing. For example, 27 cups of flour spread out between the 3 bars means that each bar is worth 9 cups. Students can then extend this to the sugar, never having used cross multiplication.

Steps to Understanding Integers Presenters: Heidi Sbnani, Molly Vokey, Sue Looney

This presentation again was grounded in the idea that deep understanding needs to come before rules to avoid misconceptions. The presenters felt that the standard teaching of integers too quickly moves into operations, while leaving some holes in student understanding. My big “take away” from this presentation was the “beaded number line.”

A center bead is used to represent zero as shown above. Now the cool part comes in manipulating the number line to be a “folded number line.” This now can be used to teach absolute value as the magnitude of a number rather than an issue with changing signs. We all know that one of the biggest struggles with absolute value is students thinking it means to “change the sign of the number.” Students can also see a representation of a “zero pair” in how the beads line up with each other. I am on my way to Michaels to get me some beads for this rich activity!

Models and Manipulatives Presenters: Angela Godfrey and Veronica Blackham

Being a middle school teacher, I am very familiar with algebra tiles, but am not as familiar with base 10 blocks. I really enjoyed this session because of the connections made between these two sets of manipulatives. If we thoroughly teach multiplication and division of whole numbers using base ten blocks, then the jump to multiplying and factoring polynomials is an easier transition for students. The presenters tracked standards from 4th grade through high school to show the progression of conceptual understanding that helps students see what the operations actually mean. A wonderful website that they used that I have only begun to explore is I highly recommend all levels of teachers explore this site. This workshop highlighted the importance of vertical alignment of the curriculum as well as teachers being aware of how teachers of lower grades are teaching and interpreting their standards.

Going Beyond Symbols with Linear Equations Presenters: JP Han and Zandra de Araujo

The theme of vertical alignment showed up again in this session as the presenters relied on students' previous experience with bar models to create a visual representation of linear equations. Teachers can become so focused on the act of “solving” an equation using actions to “both sides” that algebra becomes a “string of symbols” with memorized rules. Visual models and drawings help students to build conceptual understanding and allows them to play with math rather than just follow rules, which leads to a more meaningful problem solving experience. Assuming that students have a good base in bar models from elementary grades, they need very little instruction from teachers to be able to make sense of the diagrams. Finding a missing value of x is more like a puzzle than a memorized set of rules.

2020 Dean Preble Award Winner

Linda Horst

The Montana Council of Teachers of Mathematics (MCTM) is proud to announce Linda Horst, as the 2020 recipient of the Dean Preble Memorial Award. Mrs. Horst received his award at the MCTM annual meeting via Zoom Thursday, October 15. Linda was nominated for this award by Jordan Graves formerly of Reed Point School.

The Dean Preble Memorial is MCTM’s most prestigious recognition and is awarded annually to a Montana educator who has made significant lifetime contributions to the teaching and learning of mathematics statewide. Preble awardees are those that have consistently assumed a leadership role among math educators. Teacher-leaders at all levels, kindergarten through university, are eligible.

This award is given in memory of longtime MCTM member Dean Preble of Forsyth, who passed away from cancer in the fall of 1998. Dean was recognized for his unfailing support for mathematics education in the state of Montana. His dedication to the mathematics teaching profession, his love of his students, his involvement in state and national mathematics organizations, and his devotion to the improvement of mathematics education for all were unparalleled.

The MCTM selection committee is honored to recognize Mrs. Horst both for her leadership role in math education and for her commitment the best possible math instruction.