Making Math Formulas Tactile


The credit for this unusual idea goes to Math Giraffe and the Tactile Formula Project . Here’s how it played out in my 9th grade Algebra I classroom.

The challenge? Making Exponential Formulas out of concrete images

I believe the lesson is applicable with any formula and any level of student. Try it yourself… how well do you understand the formulas you teach?

The Lesson

I started by telling the students about ‘this cool idea I saw on the internet’.

The idea is to create formulas out of three dimensional objects, like candy or pennies, or legos, or cubes, or squares that suggest or seem to define the elements in a formula. I didn’t have the luxury of time to collect objects, so I told my students to use drawings of items that they planned to use to make the elements.

To get the creative juices flowing, I shared the images from the Math Giraffe blogpost, and started the conversation about what I saw in the images. The students quickly started noticing and interpreting what they saw in the images, and questioning why students used some of the images in the examples. Some students began interjecting their own ideas for images they would have used. This really primed the pump, so to speak, for the formulas we were going to interpret.

Images from Brigid’s Math Giraffe blog post.

Brigid, of Math Giraffe, used the classic slope-intercept formula,

y = mx + b

well known to my students. The task for my students was to do this with the multiple versions of the exponential formula:

I wrote all of the different forms of exponential formulas on the board, and let each student choose the formula they wanted to use for their visual interpretation. What followed was a deeply enriching -and eye-opening – mathematical and conceptual conversation among all of my students!

Note: We used exponential formulas, but any formula would work with this activity.

The challenge for my students was using pictures to relay the action contained in the formula elements vs just using a picture that looked like an item in the formula. An example of this was incorrectly using bananas for parentheses, because, as the student explained, “they looked like parentheses.”

The items here don’t exactly illustrate the action of the elements.

This student was just a little shaky on choosing an item that revealed the inclusive nature of the parentheses. Another student didn’t use any pictures at all. He wrote out his formula using a color for each letter, basically color-coding the identity of each element in the formula with an elaborate system for understanding each part:

Did they miss the point? Absolutely not! Both were much more able to explain when and how to use the formula after this task.

This task got my students thinking more deeply about the formula itself, than just trying to memorize when to use it!

Interestingly, the exponential growth element was mostly identified as time, with one child drawing a clock face whose hands illustrated the exponent variable.

Y is “get out” because it is output. B is the rate, so percentage signs. The crossed watches were used to show the x as time.
Cars, roads, and streets were the subject of this formula.

This student uses bacteria below the formula to show starting amount (3), the growth multiplier with a question mark, and the x was shown as time. It’s interesting that he used a calculator with an ‘answer’ to illustrate the y variable!
Dinner was the subject here. The ingredients to make the dinner were multiplied. Heat from the burner was the common ratio. X was the amount of time the food was cooked!

One of my favorites: chocolate and peanut butter to make candy, with the exponent increasing the amount of candy produced!

I noticed that after this exercise, students were beginning to talk about the formulas by identifying individual elements and their actions. When explaining why representational items were chosen, students were recalling examples from word and story problems from our practice work.

Research (source articles noted below) shows the value of connecting the abstract ideas of math to concrete items and has been proven to help with retaining the material. In this lesson, connecting physical objects (or in this case, drawing pictures of physical objects) allowed my students to make clear connections to what the formula was representing. More than that, this exercise functioned as a valuable informal assessment for me about what my students really understood about the elements and actions of the formulas we were learning. I look forward to using this task again!

Research that supports connecting concrete and abstract ideas: Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993);  Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O’Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003).

A little creativity goes a long way towards getting kids to do their homework!

My students do not do homework. Okay, maybe the young man from Vietnam. And he does it neatly, completely. But he is the exception.

Homework should allow students to practice what they’ve learned. They have to see a reason to really learn (aka understand, put in context, synthesize…), so this year my students will be required to videotape themselves explaining to another person (or their dog, or to me) what they have learned that day. They will need to include one new problem to be solved, with the solution. Extra points for the originality in the WAY they deliver their explanation!

Once uploaded to our class website, they can be viewed, commented upon and used in class for discussion. I also anticipate having a way for my students to see growth in their learning over time.

A side effect I anticipate is closer listening in class, and an enhanced collaborative spirit: this will force new ways for students to “share” homework. (Students are the original Creative Commons licensers!)

There are lots of great tech innovations for collecting, sharing, commenting on this type of assignment. Feel free to comment if you have a better (free) way of setting this up. I am in a BYOD school, and I want to get my students involved in texting comments during an assignment (about the actual assignment!), blogging, and more. I feel that getting multiple senses involved, and having them more involved in teaching/creating their own lessons, will generate more learning.

As a side note, I also plan to have them create other assignments on line: a magazine page on a topic; a cited problem solving journal article on one or more of our math topics: online posters illustrating concepts, a word wall, and more. This way they can incorporate music, visuals, and color. I may have to purchase a monitor for letting their creations run in a constant loop, like one of those electronic picture frames!

I will track their progress. Algebra II, here we come!


#MTBoS I am going to try this 30 day thing AGAIN! Here’s my challenge: school’s out. No kids to write about!
Let’s just see what a teacher does with summer!
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