By Teresa Lara-Meloy | February 8, 2019

I recently started a martial arts class. Towards the end of the first class, we began something I have never done before: the five animal poses. As I focused on following the leader, making sure I’ve got the right motions, we embodied different animals, starting with the bear. I could see the hands for these movements were always in claw-like shape. That made sense. The next animal was the tiger. Suddenly, the class started growling and roaring. Because I was totally clueless as to what was expected of me, I started to panic. Is that how we were to embody the tiger? In searching for clues, I looked around and saw that not everyone was engaging in these vocalizations, even when they were all following the stealthier, cat-like version of the poses. I felt some comfort in knowing I was not alone and continued with the class. We proceeded through the rest of the animals, ending with the deer – which is apparently a more silent animal. I sighed in relief.

What does this story have to do with math argumentation, you may ask. Think about what is usually expected of students during math class. In the past, and even many classrooms today, a math class involved the teacher presenting a lesson, then students practicing the procedures therein, and the teacher correcting students along the way. But things are changing!

Now, we ask students to participate in math conversations, and in particular, we ask them to engage in mathematical argumentation. In this new situation, students may not always know what to do, what is expected of them, and what successful participation in math class means. In fact, I suspect that their level of panic when confronted with new ways of engaging in math discussions and class is similar (or possibly worse) than what I felt in that martial arts class.

In Mathematical Argumentation in Middle School – The What, Why, and How, we propose the use of warm up games to help students learn productive norms for classroom discussion and each stage of argumentation (See our suggested norms for argumentation). These games are particularly important to play at the beginning of the year, to get the class off to the right start. But we suggest using these games throughout the year. They are a great way to open class, using a playful, non-mathematical way to remind students of how to be respectful, listen to one another, make bold conjectures, build off of other people’s ideas, and be unafraid to make mistakes. Each game has simple rules of interaction that foster spontaneity and strong collaboration among participants. The games bridge between students’ everyday experiences and the academic practice of mathematics in a lively, engaging way that is safe even for those with little confidence in their mathematical competence. Once you play a game, it is important to make explicit, through discussions, what the connection is between the norms and the game.

While there are games for each of the stages of argumentation in our book, here’s one game to help students see that building an argument in class is like telling a story together. In the game, students learn that it is important to listen to others (before you), so that you can easily make a contribution when it is your turn. Then you, as their math teacher, connect that idea to math conversations. The games are non-mathematical in nature, to make them accessible to all students, even those with less confidence in math.

See our book for more games and ideas of how to establish fun and explicit classroom norms for engaging your students in argumentation.

This blog post was originally published on Corwin Connect: http://corwin-connect.com/2018/11/establishing-norms-for-mathematical-argumentation/

Sources:

• Tiger image: https://www.publicdomainpictures.net/en/view-image.php?image=26472&picture=chinese-style-tiger-art. License: CC0 Public Domain
• How to Do Math Argumentation poster
• Mathematical Argumentation in Middle School—The What, Why, and How by Jennifer Knudsen, Harriette S. Stevens, Teresa Lara-Meloy, Hee-Joon Kim, and Nicole Shechtman. Thousand Oaks, CA: Corwin: page 130.

By Jennifer Knudsen | March 7, 2018

I recently re-tweeted Christina Cho’s (@ccho01) tweet on her second graders’ conjecture wall. So inspiring! If second graders can conjecture, so can middle schoolers, don’t you think? We’ve found it to be true.

I thought I’d share a little bit about how we think about conjecturing.

A simple, particular conjecture could be a guess at what value of x makes 3x+35 =74 true.  That would be a very particular conjecture. Students could approach it, before they know how to solve “two step equations,” by making their best guess. What if x is 10?

Would that work? Maybe x needs to be a little bigger than that. Justifications could be substituting the value for x and seeing if the equation is true. The whole process can lead to development of the standard methods for solving such equations—and why they work.

Two related general conjectures we explore in the book are about proportional relationships and their representations. We start by defining such relationships via equations and then ask students to establish properties of their graphs and tables.

Proportional relationships can be represented by equations of the form y = kx, where k is any constant number.
What are the properties of tables of a proportional relationship? Why?
What are the properties of graphs of a proportional relationship? Why?

It’s interesting to consider why, for example, you can “add” two rows in a proportional table to get a third, but not in more generalized linear ones (with non-zero intercepts)—reasoning from algebra. While often all these properties of proportional relationships are often taught as observations, they are an opportunity for argumentation.

In your own practice, we recommend keeping a balance of particular and general conjectures for students to justify.

Try out one of our activities and let us know how it goes!
Follow @jen_knudsen to keep up to date on our blog entries.

The image in this blog post can be found in our book: Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). Mathematical Argumentation in Middle School—The What, Why, and How. Thousand Oaks, CA: Corwin.

By Hee-Joon Kim | January 21, 2018

How do you decide which tools to allow for students to use to “do” mathematics? When we say doing mathematics, we mean students engaging in exploring mathematical ideas and concepts. Traditionally, students have been expected to use tools flexibly when they do mathematics, by drawing diagrams, using compass and straightedge to construct geometric shapes, and using calculators to work with complex numbers or graphing. Advances in technology promise to support students engaging in rich and complex mathematical ideas and even change the way they do mathematics.

However, as more tools become available, you need to consider various factors in order to decide which tools to choose for your students to use—including availability, tool features, social norms for tool use, and curriculum. Among these factors, critical is what mathematical opportunity a tool affords. Any tool has affordances and drawbacks. For example if students use a calculator to multiply fractions, students may not have an opportunity to reason about why multiplying two fractions less than 1 always results in smaller number. However, if students are to measure area involving fraction measurements, calculators can be useful for arriving at an accurate answer, even if students are less than fluent with fraction calculations.

Here are two classroom activities that utilize different tools: one using Geogebra and one, uncooked spaghetti. Both lessons aim to create opportunities for students to engage in mathematical argumentation focusing on the idea that in any triangle, the sum of the lengths of two shortest sides is always larger than the length of the third side. They offer opportunities to make a conjecture by exploring various cases, search for counterexamples, and get an insight into generalization. Yet, these activities provide different opportunities for students to engage in such argumentation. First, take the opportunity to do these activities yourself, then look at the affordances and drawback of the different tools. Understanding of affordance and challenges can help you decide which tools to use in your classrooms.

 

The GeoGebra activity (https://goo.gl/YIuz4K created with GeoGebra www.geogebra.org) allows students to explore an infinite number of cases by easily manipulating lengths of three sides. The ease with which students can generate an infinite number of cases is an affordance for creating generalized arguments. Measurements are generated automatically, which relieves students of the challenge of accurately using a ruler. Of course, access to enough technology for all students can be a problem in some school settings. And Internet access can be spotty in some places.

The uncooked spaghetti activity allows students to explore cases without using specific measurements by simply breaking a piece of spaghetti into three pieces and manipulating them to try to form a triangle. It is easy to line up the pieces to compare their lengths. However, generalizability of the idea is more difficult to achieve with the limited number of cases that students can create during a class period, or any finite number of cases they can make. Also, students need to keep track of their spaghetti cases, organizing them on a surface, which can lead to trouble. And picking up bits of spaghetti after the activity is a nuisance.

The same activity can be done with rulers. Students can explore several cases by drawing segments of specific lengths, but there is less room for manipulation of position and lengths.

The ruler, however, has the advantage of being a familiar tool for many students. This isn’t true for all students, though, and often you can end up using valuable class time teaching the use of a ruler.

Both careful design of the task and planning are key to supporting students’ productive use of tools—from spaghetti to rulers to interactive geometry apps—to explore mathematical ideas. Technology often provides access to generalizability, but the tradeoff is that these tools may be new both to you and to students, and so new norms must be in place for their use: for example, how to use tools collaboratively and how to think with tools.

Try out one—or both—of our activities and

let us know how it goes!Follow @jen_knudsen to keep up to date on our blog entries.

The PDF linked to in this blog post can be found in our book: Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). Mathematical Argumentation in Middle School—The What, Why, and How. Thousand Oaks, CA: Corwin.

By Harriette Stevens | December 12, 2017

We were recently lucky enough to present at the annual conference of the National Alliance of Black Educators. It was full of interesting sessions, which I didn’t have enough time to explore thoroughly. Our session was entitled, Advancing Students’ Engagement, Communication and Positive Identities as Mathematics Learners. One of the points we made was that mathematical argumentation is great preparation for the world of work. Here’s our slide on that:

First off, having a positive identity with regard to mathematics is a huge plus. Being seen as someone good at math is cachet enough; knowing you really are is empowering, even if your job isn’t math intensive.

Three of our points have to do with collaboration, which is considered very important in the 21st century workplace. It’s crucial to spend the time making sure everyone is on the same page—terms and concepts is how we think of it in math, but it could be a shared understanding of policy or of goals that is critical in a workplace. And decision-making discourse should be based on evidence, not just opinion—we all know of times when opinion won out, not always for the better. Finally, “innovation” is a buzz word for a reason—new ideas are used at a rapid rate, and collaborating to come up with them creates stronger ideas.

Our final two points are more mathematical, but translate into non-mathy workplaces as well. People are pattern recognizers—honing that skill is useful in many situations. And, in the current world, streams of data come at us and are used to make decisions for us. How much stronger are we if we are capable of visualizing data in a variety of ways?

We also did a math activity with the leaders who attended our workshop. It’s at the heart of our book:

 

Let us know if you try it out and how it went!
In an upcoming blog entry, we’ll show how students used the skills in our presentation in this activity. Follow @jen_knudsen to keep up to date on our blog entries.

The images in this blog post can be found in our book: Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). Mathematical Argumentation in Middle School—The What, Why, and How. Thousand Oaks, CA: Corwin.

By Jennifer Knudsen | November 6, 2017

Mathematical argumentation is improvisational! As you and your students create mathematical truth together, you all have to respond in the moment to what others are saying and doing. Norms of collaboration and celebrating mistakes are important. One way to set those norms is through improv warm-up games. Here’s a game from our book:

Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). Mathematical Argumentation in Middle School—The What, Why, and How. Thousand Oaks, CA: Corwin.

There are plenty more where that came from. We didn’t invent these games; most all are popular ways for improvisational theater actors to learn their craft. But we also found that the games are aligned with what research says are supportive learning environments for culturally and linguistically diverse students. Here’s how:

 

If you try Zip Zap Zop in your classroom, drop us a line and let us know how it went!

By Jennifer Knudsen | October 23, 2017

Mathematicians have been figuring out the mathematical truth for millennia. Can this possibly be relevant for today’s students? Yes!

Mathematical argumentation can make your classroom more joyful and engaging. Going beyond just rules to memorize, students are given the opportunity to make sense of those rules and to convince others of their ideas. Students get to play with mathematical ideas and take ownership of them in a way that often delights them. You’ll most likely feel a boost yourself. One teacher we worked with proclaimed that every Friday was argumentation day, and her class eagerly looked forward to it.

To engage students through argumentation, follow these steps:

1. Choose a topic on which there are likely to be differing ideas. For example, most students know that if you square a whole number greater than 1, the result is a larger number. But what happens if you square a fraction? Will the result be larger, smaller, or does it depend? The answer is not immediately obvious to most sixth graders and that makes it a good candidate for argumentation at that grade level. For other topics, you can use your state standards to come up with likely candidates for the grade you teach. Or look for places where a textbook may state some rules but there isn’t much explanation of why they work—that’s what you can have students figure out.
2. Ask students to play around with examples and make a conjecture— their best guess at what is true. For example, students may try easy fractions such as 1/3, and find that (1/3)2 is 1/9. Since a ninth of a whole is smaller than a third of the same whole, 1/9 is less than 1/3. Students may conjecture based on this that whenever you square a fraction, the result is less than the fraction you started with. Now you have something students can argue about.
3. Start by finding out which students agree or disagree by a show of hands. We don’t find out the mathematical truth by voting, but you will get a sense of whom to call on to add to the argument. Ask students who agree that the conjecture is true to go beyond the one example: How do we know it is always true, no matter what fraction we choose?
4. Recommend representations. If students aren’t sure how to go beyond giving a couple of examples, which can never be fully convincing of what’s always true, you can suggest that they try to think with other representations. Give them some individual or paired think time with the representation of their choice. Can they make a table and look for patterns? Can they draw a diagram? Make a graph? Use algebra? Try not to tell them exactly how to use a representation—let them try their own ways, which may be ways you have never thought of. Have students compare different representations they used—connecting representations is a great way to increase conceptual understanding and you’ll increase engagement as students discuss their own representations.
5. Keep the argument going with hand signals. Have students use a fist-over-fist motion to indicate they want to build on another student’s ideas, not just start a new argument. Keep asking for agreement and disagreement by a show of hands. Students can raise 1 to 5 fingers to indicate how convinced they are.
6. Be sure to call on those who disagreed with the conjecture. That’s where great counterexamples come from—students thinking of particular fractions that break the general rule that most of the class thinks is true.
7. Have students summarize the argument and the conclusion. For example: We decided that when a fraction is less than 1, if you square it, the result is less than the fraction. This is because if you take a fraction a/b, where a is less than b, and square it, you get a•a/b•b. The denominator is getting even larger than the numerator, since you are multiplying a greater number, b, by itself. But not all fractions are less than one! If you square a fraction greater than 1, such as 3/2, the result is larger, by the same argument turned “upside down.” There’s one other case: when a fraction is equal to one. When you square it, the result is still one.

Follow these steps for a topic appropriate for the grade level you teach, and support students in finding out the mathematical truth for themselves. You may find that their truth seeking expands to outside the classroom.

This blog entry first appeared on Corwin Connect on September 29, 2017.

By Jennifer Knudsen | October 22, 2017

Mathematical argumentation is the perfect opportunity for your students to develop their own mathematical authority—their sense of self as creators of mathematics, in charge of deciding the truth for themselves. We elaborate in this excerpt from our new book:

Knudsen, J., Stevens, H., Lara-Meloy, T., Kim, H., and Shechtman, N. (2017). Mathematical Argumentation in Middle School—The What, Why, and How. Thousand Oaks, CA: Corwin.