Math in Motion
Teaching Binomial Coefficients (or anything) with a Proficiency Mindset
You might notice something different in today's post: no lengthy note about my AI iteration process. That's because I simply used Gemini (Google) – the same tool that helped craft the Dance/Math example – and fed it this prompt: "I want to show teachers how to easily take this assignment and put it into a proficiency-based approach (using the ACTFL proficiency guide as a model) in a math class. Give me some ideas about that, including the proficiency scale from novice low to Distinguished and Can-Dos for Novice Low, middle and high. The target on this would be the equivalent of Novice High in mathematics."
I wanted to demonstrate the simplicity of moving from an "instructional" to a learning mindset when integrating AI into teaching. Additionally, if a student's answer to one of the examples in the problem set below seems too polished (perhaps done with AI's help) a teacher doesn't need to guess. A five-minute conversation could involve asking the student to talk through another problem from the set. If their explanation reveals a lack of true understanding, the teacher can then clearly differentiate between what the student does grasp (asset-based assessment) and what concepts are still fuzzy. This isn't about assigning a grade, but about providing a redirect and support until the learner achieves the desired proficiency.
—[Gemini’s response to my prompt]. This is an excellent idea! Applying ACTFL's proficiency model to a math class, especially with a creative context like dance, can make learning more engaging and meaningful. The core idea of proficiency-based learning is to focus on what students "can do" with the knowledge, rather than just what they "know."
Here's how you can adapt the math assignment on binomial coefficients into a proficiency-based approach for teachers, using the ACTFL model and targeting Novice High.
Proficiency-Based Mathematics: Binomial Coefficients in Dance
Overarching Goal: Students will be able to calculate a binomial coefficient and correctly apply the binomial coefficient to formulate and solve counting problems, demonstrating understanding through the lens of dance.
Understanding the Proficiency Scale (Adapted for Math)
The ACTFL Proficiency Guidelines describe what individuals can do with language. We'll adapt this to describe what individuals can do with mathematical concepts.
Key Idea for Math: Proficiency in math means being able to use mathematical concepts, procedures, and reasoning to solve problems and make sense of the world, moving from concrete, memorized actions to more abstract, spontaneous, and complex applications.
Proficiency Levels (Adapted from ACTFL):
Distinguished: Can skillfully and effectively apply complex mathematical concepts and models to address highly abstract and multifaceted real-world problems, integrating diverse mathematical fields and culturally relevant contexts. Can innovate and adapt mathematical solutions to novel situations.
Superior: Can confidently and fluently apply a range of mathematical concepts and procedures to solve complex, multi-step problems in various contexts. Can analyze problems, formulate strategies, and justify solutions with clear reasoning. Can generate and test hypotheses.
Advanced: Can apply mathematical concepts and procedures to solve problems in familiar and some unfamiliar contexts, using multiple representations (e.g., equations, graphs, tables). Can narrate and describe mathematical processes and outcomes. Can deal with unexpected complications in problem-solving.
Intermediate: Can create with mathematical language and concepts by combining learned material to express personal meaning and solve simple problems on familiar topics. Can ask and answer simple mathematical questions and handle straightforward situations. Produces sentence-level mathematical statements or discrete calculations.
Novice: Can communicate short mathematical messages on highly predictable, everyday topics directly affecting them, primarily through memorized facts, formulas, and isolated calculations. May be difficult to understand without significant contextual support.
Target Proficiency: Novice High (Mathematics - Binomial Coefficients)
For this specific assignment, our target is Novice High in mathematics. This means students are moving beyond just memorizing facts and can start to combine information to solve slightly more complex, but still familiar, problems.
Mathematical Focus: Calculating binomial coefficients and applying them to basic counting problems where the order does not matter.
Can-Do Statements for Binomial Coefficients in Dance (Target: Novice High)
These statements describe what students can do independently and consistently.
Novice Low (Building Blocks):
I can identify the numbers in a counting problem (e.g., "There are 5 dancers, and I need to choose 2").
I can recognize the factorial symbol (!) and its meaning (e.g., I know 3!=3×2×1).
I can calculate a simple factorial (e.g., I can calculate 4!).
I can identify when a counting problem involves "choosing" a group where order doesn't matter. (e.g., "Choosing 2 dancers for a duet" is different from "ordering 2 dancers in a line").
Novice Mid (Applying Basic Formulas):
I can write down the formula for a binomial coefficient given "n" and "k" (e.g., (kn)=k!(n−k)!n!).
I can substitute given numbers (n and k) into the binomial coefficient formula.
I can calculate a binomial coefficient with small numbers, with support or a calculator (e.g., (24)).
I can identify a simple counting problem from the dance world that uses "choosing" (e.g., "How many ways to pick 2 dancers for a trio from a group of 5?").
Novice High (Putting it Together and Applying to Dance Context):
I can independently calculate a binomial coefficient given n and k (e.g., (310)).
I can correctly identify n and k from a given dance-related counting problem (e.g., in "A choreographer has 12 unique movements and wants to create a phrase using 4 of them," I can identify n=12 and k=4).
I can set up and solve a dance-related counting problem using the binomial coefficient formula, explaining what the numbers represent in the dance context.
I can explain why the binomial coefficient formula is appropriate for a given dance problem (i.e., that the order of choosing doesn't matter).
I can formulate a simple dance-related counting problem that can be solved using a binomial coefficient.
Ideas for Implementation in a Math Class:
Contextualized Problem Sets:
Start each problem set with a dance-related scenario.
Examples:
"A ballet company has 15 dancers. How many different groups of 3 can be chosen to perform a new section of the corps de ballet?"
"A choreographer is designing a new piece and has 8 distinct costume pieces (like scarves, hats, gloves, etc.). If each dancer will wear 2 of these pieces, how many different combinations of 2 pieces are possible?"
"A dance competition has 10 finalists. How many different ways can 3 dancers be selected for gold, silver, and bronze medals?" (Initially, students might try to use binomial coefficients, leading to a discussion about when order matters vs. doesn't matter - this is a great differentiator for Novice High moving towards Intermediate).
"Choreographer's Challenge" Performance Task:
Task: Students act as "choreographers" or "costume designers" for a fictional dance piece.
They must:
Create 3-5 unique counting problems related to their dance concept (e.g., choosing dancers for specific roles, selecting costumes, forming small groups).
At least 2 of these problems must be solved using binomial coefficients.
Clearly define n and k for each problem.
Show their mathematical work (setting up the formula, calculating).
Explain their answer in the context of their dance scenario.
(For higher levels, they could also create a problem that requires permutations and explain the difference).
Assessment: Use a rubric aligned with the Can-Do statements.
Visual Aids & Manipulatives (Connecting Dance to Math):
Use actual dancers (if feasible), or paper cutouts, or even just names on slips of paper to demonstrate selection.
For example, put 5 names in a hat. "If we pull out 3 names for a small group, does it matter if I pull out Alice then Bob then Carol, or Carol then Bob then Alice?" This directly illustrates the "order doesn't matter" concept.
Self-Assessment and Goal Setting:
Provide students with the "Can-Do" statements at the beginning of the unit.
Have them reflect weekly on which statements they "can do" consistently, with help, or not yet.
Encourage them to set personal goals for moving from Novice Low to Novice High.
Differentiated Support:
Novice Low: Provide scaffolded worksheets with clear steps, worked examples, and direct questions for identifying n and k. Focus on basic factorial calculations.
Novice Mid: Provide templates for the binomial coefficient formula, encourage calculator use, and offer problems that are slightly more complex than direct substitution.
Novice High: Challenge them to formulate their own problems, justify their choices of formula, and perhaps explore simpler permutation problems to understand the distinction.
"Math in Motion" Journal Entries:
Ask students to write short reflections connecting what they learned about binomial coefficients to specific aspects of dance.
Prompts: "How could understanding combinations help a dance captain organize rehearsal groups?" or "Imagine you're choreographing a piece with 7 dancers. How many unique ways could you select 3 of them to be spotlighted in a particular moment? Explain your thinking."
By adopting this proficiency-based approach, teachers can ensure that students not only grasp the mathematical concepts but also understand their practical relevance, especially for those who see the world through the vibrant lens of dance.