All posts by Elise Levin-Guracar

How Many Examples Do You Need to Prove Something?

December 22, 2018UncategorizedElise Levin-Guracar

By Teresa Lara-Meloy & Jennifer Knudsen | November 27, 2018

We have spent more than 10 years studying math argumentation in middle school, in part because early teenagers are in an interesting stage in their development of argumentation skills. While they often build arguments using examples only, and have difficulties in making deductive arguments, there is increasing research that highlights the use of examples in driving deductive arguments (Ozgur et al., 2017; Stylianides & Stylianides, 2009; Knuth, Zaslavky, & Ellis, 2017).

Many of us start out reasoning through examples—in life and in math. In life, we learn to generalize from few examples. From a very young age, a child learns that every time they throw or let go of something, the something falls. Children even investigate this from different heights, and love trying this out from the high chair. It is fruitless to attempt to explain gravity at this age, yet children are able to derive this key physics truth from the many examples they’ve tried. Of course, not all thinking from examples leads to truths. Sometimes, a child infers incorrectly that dolls are girls’ toys, and yet a counterexample (or more) can help dispel that faulty conclusion. We can use both of these everyday ways of drawing conclusions from examples to support middle schoolers in making arguments in mathematics class.

In math, let’s look at at a situation in which we have seen the use of examples to build up an argument. We pose a situation where students have to determine whether the distributive property applies to algebraic expressions and works for any value of the variable. Students often start by working from left to right and then get rid of parentheses in predictable but incorrect ways. This can lead to the question:

For every value of x,
does 2(x + 3) = 2x + 3?
Or
does 2(x + 3) = 2x + 6?

Many students start with examples. Let’s do the same.

Let’s say x = 4.
Then,
2(4 + 3), do what’s in the parentheses first and you get 2(7) or 2•7 = 14.
2•4 + 3, do the multiplication first and you get 8 + 3 = 11
So the first two aren’t equal. But,
2•4 + 6, do the multiplication first and you get 8 + 6 = 14
So, some of us may say, the second equality—2(x + 3) = 2x + 6— is true!

(Note: we know that students would not write it out this way. But you can help them understand the “grammar” of math by speaking in sentences such as these as you write expressions on your display.)

equations

Students may stop after trying out just one number for x, or they may try out another number just to be sure. If they are being strategic in their choice of examples, they may try a negative number (to cover both sides of zero, so to speak), or they might try a large number and a small number (to see if it works for “crazy” situations). Two cases are good enough for many of us to say that the equality is always true, no matter what x is. Even many adults would stop there (e.g. Harel & Sowder, 1998; Knuth, 2002).

But as middle school math educators, we want to press for generalization—for why the second equality is always true. Or even more generally, how the distributive property works. A good diagram can help a lot in situations like these, where you want to help students generalize in algebra. You may need to give students some direction here, but you don’t need to be completely directive. Remind them of how they represented multiplication with an area model (introduced in elementary school) and ask them if they can use the same kind of diagram to help in our investigation of the distributive property. Remember algebra is generalized arithmetic! See where they take it.

multiplication example

Technology can help because it allows students to better experience and represent what we mean when we say “for all x.” See what this looks like our examples above,—students can try out many substitutions for x, but it also connects with a more generalized form, which is “for all x”. Try out this GeoGebra interactive:

https://www.geogebra.org/geometry/att3e5r2

What other methods do you use to help students move from examples to generalizations? Tweet your answers with the hashtag #BridgingPD or #MathArgumentation.

You can find out more about this topic in our book, Mathematical Argumentation in Middle School.

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.

References

Knuth, E., Zaslavky, O., & Ellis, A. (2017). The role and use of examples in learning to prove. Journal of Mathematical Behavior (in press). Available from https://doi.org/10.1016/j.jmathb.2017.06.002

Ozgur, Z., Ellis, A. B., Vinsonhaler, R., Dogan, M. F., & Knuth, E. (2017) From examples to proof: Purposes, strategies, and affordances of example use. Journal of Mathematical Behavior (in press). Available from http://dx.doi.org/10.1016/j.jmathb.2017.03.004

Stylianides G., & Stylianides, A. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40 (3), 314-352.

Images

Graph: exploration of parabolas, generated by author on Desmos

Area model solutions: © SRI international

Using Visual Representations to Support All Students: Snippets and Resources from NCTM Presentation

December 22, 2018Visual RepresentationsElise Levin-Guracar

By Hee-Joon Kim | May 2, 2018

Last week at NCTM’s annual conference, Harriette Stevens and Hee-Joon Kim presented on using visual representations to engage all students in the mathematical practices—with a focus on argumentation. They discussed how both familiar representations (e.g. diagrams, graphs) and less familiar representations (e.g. gestures) can help students make sense of complex ideas. Also, they emphasized that making various visual tools available created opportunities for all students to be able to express their arguments without solely relying on words and symbols.

The group had fun doing math together. Participants got a chance to play with GeoGebra activities designed by the team and think about how multiple representations such as gestures, drawings and dynamic representations could be used to support students in creating and communicating arguments. One participant said the workshop made her realize she should encourage the teachers with whom she worked to use visual representations regularly in every lesson.

You can download the presentation and the handout (with QR codes to the activities) for your own use.

Download the presentation:

Download the handout:

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.

At NCTM on Friday, April 27 at 9:45 AM: Visual Representations

December 22, 2018UncategorizedElise Levin-Guracar

By Teresa Lara-Meloy | April 19, 2018

Next week, we present at NCTM on using visual representations in the math practices—mostly argumentation, considering it’s us. Here it is in the program:

Using Visual Representations

As a teaser, here are a couple of slides from the presentation:

While the practice of argumentation requires new norms, one in particular fosters students’ use of visual representations: Show how you know.

Argumentation slide

Students’ gestures are one important form of visual representation that can be overlooked. It can be particularly important for ELLs. Here’s an improv game designed to help students pay attention to each other’s gestures and more generally, to visual representations that may get used.

Magic clay directions

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.

Conjecturing for All!

December 21, 2018Mathematical ArgumentationElise Levin-Guracar

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.

Conjecturing slide 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.

How Do You Choose Tools?

December 21, 2018Mathematical ArgumentationElise Levin-Guracar

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.

Mathematical Argumentation as a 21st Century Skill

December 19, 2018Mathematical ArgumentationElise Levin-Guracar

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.