If you remember from our last blog, we were creating the ambiance for rich mathematical discussions. To continue our four-course meal metaphor we have now set the table and equipped our students with models of how to communicate productively, and sentence starters to lean on in case of emergency. It is now time to put our new social skills into action. As we set the table for the main meal, we invite you and your students to enjoy a cocktail (for you) and some tasty hors d'oeuvres (for them). Did you know that starting the meal with those little food concoctions originally served as conversation starters? “Oh, you must try these sweet potato biscuit sliders. They’re divine!”

The same strategy applies to creating great mathematical discussions! Students need opportunities to practice engaging in mathematical discourse in delicious bite-sized pieces. Below are few of our favorite mathematical “hors d’oeuvres” to get students used to talking about math.

In this classic opener, students are asked to determine which item in a collection doesn’t belong and to provide a reason why. It may help to remind students to use their sentence frames if they need support getting started.

“I think ________doesn’t belong because____________”.

2. Always, Sometimes, or Never True

Provide a mathematical statement and allow students to decide if the statement is always, sometimes, or never true.

For example:

“When multiplying two numbers the product is bigger than the factors”

“It takes 3 dots to make a triangle”

An extension/push is to have students rewrite the statement in three ways – so that it would always be true, sometimes be true, or never be true.

3.Number, Fraction, or Graph Talks

3.Number, Fraction, or Graph Talks

We love to use number/fraction/graph talks to get students used to sharing their thinking verbally and to practice their reasoning skills in a safe setting. There is a lot of research about the benefits of using number talks with any grade level, including university students! In this video Jo Boaler talks about the benefits – from building number flexibility to seeing mathematics as creative subject – and esteemed educators Cathy Humphreys and Ruth Parker describe how these activities develop students’ agency and identity as mathematicians.

Since it is always useful to see these practices in action, here are some classroom examples: 5th Grade, Middle, University

Great resources for fraction talks can be found here fractionstalks.com and this is an excellent blog post about how to facilitate them in the classroom. We especially appreciate the sample questions to ask after the initial question: “What part of the image is shaded ____ (color)”

4. Proclaim the claim

Start with a mathematical claim for your students to consider.

Mr. Leo’s Claim: Zero divided by zero is 1. Do you agree or disagree with this claim?

Provide mathematical justification for your claims.

Providing students with sentence starters and an option to ask a question about this claim are great ways to get started. Push students to see if they can come up with an argument that would be counter to their claim. i.e. if they agree with the statement ask them to consider what a counter claim might be.

To make any of the above hors d'oeuvres work well we need to think through how to roll out the activity. Some of our favorite pedagogical moves for making these activities work are below.

Seed the Discussion. Start with an elbow partner pair share. Don’t forget to remind students about good conversation skills. While students are sharing, walk around and listen to their thinking and ask specific students privately if they would be willing to share their thinking with the class. Sometimes it helps students to know ahead of time that they will be sharing and this private conversation can be used to boost their academic and social status!

Math & Mingle. Using the Which One Doesn’t Belong image, label each option on the screen as A, B, C, or D and pick 4 areas in your room that correspond to each letter: D-door, B-bookcase, etc. After having a chance to check in with their partner ask students to move to an area that corresponds to their choice and give them a chance to confer with the people in the same area. Did they have the same or different reasons for being there? Then use the practice of safe cold calling to get some ideas out on the board.

Keep Them Thinking & Talking. For the Which One Doesn’t Belong activity, a possible extension is to challenge students to think of a reason why each one doesn’t belong in the group with the others or leave the 4th option blank and allow students to fill it in themselves. Have follow up questions ready in your back pocket. For example, “Can you shade a section with twice the area? Can you shade exactly one-quarter of the shape?”

Stir the Pot. Students love to argue! Especially with their teacher. Teaching students how to hold a respectful academic debate is an important life skill and what better way to develop this skill than in the context of a math class! A strong mathematician should be able to convince a skeptic! With any of the starters above, spice things up by arguing for the wrong idea! This not only creates a culture of debate but allows students to practice constructing arguments and critiquing the reasoning of others! #MP3

Yes, and…While all of the hors d'oeuvres above are great ideas, they can fall flat if we don't attend to how we use them! Pedagogy is really the foundational key to making everything work. We found this post and videos from math educator and instructional designer David Wees compelling. What key pedagogical moves work in your class? Please share!

Brightspot: Mathematicians often collaborate to solve interesting problems and there is no age barrier to that collaboration. Over at HTH North County David Corner & Karen Underwood brought together their 11th and 3rd grade class to collaborate on the problem below. The beauty of mathematics is that there is not age requirement to wonderment! If you are interested for more context, here are the slides from the lesson.

We are wishing our MAIC community a relaxing, rejuvenating, and radical winter break! Some holiday fractal fun for you and students (or family)

Yay Math!