**INTRODUCTION: **During my four years of working as the Montgomery College project manager for a National Science Foundation faculty development grant in the sciences, I heard a great deal about the latest research in cognitive neuroscience—i.e., what we know about how the human brain learns. During my sabbatical in the spring term of 2006, I spent many hours reading the literature in this area, and learning much more than can be contained in this brief article. However, a few notions stood out in my reading. One key discovery was that learning is actually a physiological change in the brain in which the neurons grow additional “branches” (dendrites) reaching out to other related neurons, and then strengthen their ability to transmit information more efficiently by adding layers to the main signal pathways. Another was the importance of active learning– that “the brain that does the work is the brain that learns.” (David Sousa, p. 123) A third was that our brains are naturally inclined to try to figure things out; thus, novel events (which provide unexpected stimuli to the brain) are an excellent means of providing motivation and enhancing learning.

In the fall of 2006, I taught an expanded version of an elementary algebra class, designed for students who needed extra support in mathematics. In this developmental course, many students have “seen” most of the content before, but often aren’t able to recall or use the ideas. My goal was to design a learning activity for this class that would specifically put into practice some of the “brain based” strategies I had been learning. The structure of the lesson would provide an opportunity for students to connect with prior learning, try out new ideas in a safe environment, physically manipulate models representing abstract concepts, engage the content orally and in written form, have opportunity for practice, and finally reflect on the new learning.

The specific concept of this lesson was properties of exponents, but the various educational structures in the lesson could be adapted to a variety of content areas. Some examples of adaptations in chemistry, English, economics and other disciplines are described later in this article.

The entire activity was completed in one 50-minute class period.

**PURPOSE**

- Expand student understanding of their brains as learning organs.
- Provide a positive emotional experience for students to connect with content that they often consider boring or meaningless.
- Help students learn and retain the properties of exponents, knowledge which is necessary for applications of algebra to many areas including probability, chemistry, and finance.

**SUPPLIES**

- A numbered list of equations based on the basic properties of exponents; at least half of the equations should be false. The list should be ready for display either by writing them on the board, placing them on an overhead, or projecting them via computer. (See SUPPLIES appended below for an example.)
- Index cards with a large arrow printed on them, one for each student.
- A prepared worksheet for each student containing directions for the activity and a table to record results. (See SUPPLIES below.)
- Sets of two different colored paper or plastic 5 oz cups, twenty of each color, one set for each pair of students. (I used red and blue.)

**DIRECTIONS **

**1. Self-Assessment and Class Results**

A. Display the list of six equations (Supplies A below) on the board or others means. Direct students to use their own paper to write the numbers 1 through 6 and then identify each of the six equations as True or False. They are NOT to copy the equations.

B. After students have finished, distribute the index cards with arrows on them. Tell students that an arrow pointed up represents True, an arrow down represents False. Ask them to hold the index card so that the instructor can see their selection for each statement.

C. Move through the list item by item, asking the students to show their selection for that item. Count the responses for each question and tally them next to the statement on display.

*What is happening: Students are activating prior knowledge (if any) in preparation for new learning activities. Any previously “grown” neural networks are preparing to expand and strengthen; even very weak past learning will be physiologically prepared to make new connections. Having the students use the cards to respond allows the instructor to see results without the interference of peer pressure to change answers or the fear of giving incorrect answers in public, both of which happen with a “raise your hand” type of response. Brain research reveals that the anxiety about being “wrong” can chemically block the brain’s ability to learn by preventing transmission between dendrites.*

**2. Mini-Lesson on Brain Science**

Give the following brief background on dendrite networks. Showing a basic diagram of some neurons with dendrites is helpful and can be located via the resource list below and in Chapter 1 of the * On Course* text (where you will find additional information about how the human brain learns).

“When you learn, you are using thousands of brain cells called neurons. Neurons are made up of a main cell body, a long branch called the axon, and many branchlike extensions called dendrites. Neurons communicate with each other across gaps called synapses by sending chemical signals from the dendrite of one to the axon of another. The more you learn, the more extensive your dendrite branches become and the thicker and more efficient your axon gets at receiving and using the signals. All of you have grown different kinds of neural networks for working with exponents: some of you have only developed networks for working with number exponents but not variable exponents; some of you developed your exponent networks several years ago but because you haven’t used them they have shriveled; some of you developed strong networks for exponents and practiced the ideas in a recent course so the network is healthy and efficient. The goal for this class session is to build or reinforce a strong exponent neural network so you can truly understand how to work with exponents and, as a consequence, earn a high grade in this class.”

**3. Hands-On Activity**

A. Have students select a partner (if needed, one group may have three students). Ask one student in each pair to come to the supplies location and pick up one set of cups and a worksheet for each group member.

B. Using a set of cups, tell the students that blue cups will represent variable “a” and red cups variable “b.” Ask, “What does an exponent describe?” Guide the replies if needed to the answer: “The number of times a base is used in multiplication.”

C. Ask for a volunteer to help you to demonstrate the activity. Represent a^{3} by positioning three blue cups (upside down) next to each other on a desk; ask the volunteer how to represent the expression (a^{3})^{2}. Help the student if needed to position three more blue cups in a group next to the others, resulting in six cups. Ask, “How else could I describe what I have here, using exponent notation?” Obtain the answer a^{6}.

D. Next, demonstrate modeling exponent expressions in fraction form. To provide a visual “fraction,” put the “numerator” cup(s) up on a stack of books, the chalk or marker tray on a wall mounted board, or some other raised platform behind the area where you put the “denominator” cup(s). As an example, represent the fraction b/b by placing one red cup on the upper level, and another red cup on the lower level directly in front of it. Ask, “What value does the expression b/b have?” Obtain the answer “1.”

E. Ask for another volunteer to model the expression b^{6}/b^{2}. (They should have six red cups on the upper level, and two on the lower level.) Illustrate that two cups in the upper level can be matched with two cups in the lower level to make “1” as before, with four cups remaining. Ask, “What expression describes what we have left?” Obtain the answer b^{4}.

F. Direct the pairs of students to use their sets of cups to model the six problems on their “Properties of Exponents” worksheet (Supplies B below) and complete the equations in the first column of the table.

*What is happening: Students are physically modeling the abstract concept of *exponent*; this gives an opportunity for neurons in several different parts of the brain to participate in the learning process. In addition, the unexpected use of the cups meets the brain’s desire for novelty in the environment to encourage learning. In our class, we had a lot of fun playing around with the modeling, and laughter is well-documented to enhance oxygen levels in the brain as well as trigger the release of endorphins which add to the brain’s ability to focus and pay attention to a task.*

**4. Reflection and Extension**

A. When finished, have one student in each pair collect the cups and return them to the supplies area; then direct each team to join another team to form a square [If there is a trio, they can stay a trio; if there is an extra pair, identify one of the stronger students in the class and ask that student and his/her partner to work by themselves for the next activity.]

B. Tell students to INDIVIDUALLY review the statements they just wrote in column 1 of the “Properties of Exponents” handout, and complete as many of the properties in the second column as they can at this point. After a few minutes, direct students to share their results with their group and work to complete the entire column.

C. Next, display the correct answers to each of the listed properties and address any questions the students may have about the results.

D. As a closing activity, display the original True/False activity again (Supplies C below); direct students to individually identify their answers on their own paper. Ask for a volunteer to provide their entire list of answers; review for correctness with the class. Direct the students to compare their present results with their (pretest) answers from the beginning of class and welcome volunteers who are willing to share their changes.

E. Assign appropriate practice problems for homework; if time remains, have students do selected problems for class work.

*What is happening: In these steps, the student is attaching meaning to what has been discussed, reinforcing the specific neural networks that were “grown” for that content. Students have an opportunity to reflect individually on what is learned. Then, they have an opportunity for discussion; the student speaking at any given time is adding to his/her own learning network. Closure gives the student a final opportunity to practice the content, increasing the likelihood of retention of the content.*

**EXPERIENCES**

In the past, when teaching this topic, I would typically just write all the properties of exponents on the board as I talked about each one, do a handful of examples of integrating the properties, and then assign some practice problems which many of the students would do incorrectly despite my previous explanation. I have found myself bored by the lesson on exponents, so I knew my students probably felt the same. This time, my class was intrigued from the minute I walked in the room with the bags of cups in my arms! The “novelty factor” is so important to creating motivation and building memory and retention, and it doesn’t take much to create it.

During the cup activity, there was some hesitation about exactly what to do, so I ended up doing some individual demonstrations around the room to support the activity. One pair of young men was especially reluctant to pick up the cups and move them around. Once the pairs got going, however, there was enthusiasm and an occasional argument about the right way to model a problem. After the students moved into groups, I could tell from spot-checking the worksheets that the extensions to the general properties were happening. Several students volunteered their closing results on the True/False activity, with one student nearly shouting, “I didn’t know half of them at first, but now I got them all right!”

**OUTCOMES**

A couple of incidents helped me to see some direct positive impact from the day’s activity. A couple of class periods later, when we were using exponent properties in a new topic, I heard one student say to another, “You know, like the red cup things we did.” He was trying to show the other student why (x^{2})^{3} was the same as x^{6}, NOT x^{5}. So he placed his two fists together on the desk, first to the left, then in the center, then to the right, to illustrate that there were three groups of two to consider.

A more general piece of feedback on thinking about their brains came in a response to a journal prompt, “What one thing is your instructor doing that is helping you to become successful in this course?” Two different returning adult women indicated that they appreciated hearing about how their brains learned. They both said it made them feel better to realize that one reason they didn’t learn new things as quickly as some of their class members was that their neural networks had “died back” from lack of use, but that they could “regrow” those networks just as effectively with practice.

On a more concrete note, I decided to make a comparison of final exam performance on this topic between this class and my classes which included this topic in the previous semester. I found that of the twelve points assigned to assessment of exponents, in my previous semester the average earned score was 6.6 of 12. For this current semester, the average score jumped to 8.9 of 12. This improvement was impressive enough that I will definitely use the activities in that topic, and other topics as well, in my future algebra courses.

**ADAPTATIONS FOR OTHER DISCIPLINES:**

The general flow of the lesson is one that can be applied in any content area with some adaptations.

**1. Self-Assessment and Class Results:** If you are teaching a subject which is not a review for your students, do the self assessment on previous content which is relevant to the new lesson. For example, in a chemistry class with a lesson on balancing equations, the self-assessment could be about writing the correct chemical notation for specific compounds. [Eg, sodium chloride = NaCl.]

**2. Mini-Lesson on Brain Science: ** Students in any subject area seem to love hearing about the way their brains learn. Rita Smilkstein (see Bibliography) is an English teacher who asks her students to draw their neural networks before and after they learn a new concept, to emphasize their control over the learning process. After you’ve talked about these ideas with your class (or had them read about the basics) you will find it easy to refer back to this notion on a daily or weekly basis.

**3. Hands-On Activity: ** Instead of having students use plastic cups and talk about exponents, perhaps you’ll have students do a scientific experiment, role-play a piece of literature, or create a 3-D model of the Battle of Gettysburg. Bring in a costume, move the chairs around, play a piece of music—the possibilities are endless! The key is to get all students to actively engage the learning content or skill.

**4. Reflection and Extension:** Give students a chance to process individually, in pairs or small groups, and then finally extend their understanding. For example, in an economics class, after learning about the ways that the Federal Reserve can affect inflation and output in the US Economy, students could individually draw graphs representing these two effects, then compare graphs with another student. Finally, students could be asked to identify a specific instance in US history which illustrates these effects. Each of these stages helps the neural networks to grow and enhances retention.

**SUMMARY**

While doing my research and reading, I became convinced that as educators we cannot ignore the findings of cognitive neuroscience and then wonder why our students are unsuccessful. As I work to apply the principles that I have learned (and will continue to learn), I can see that knowing what is actually occurring in my students’ brains helps me to design classroom activities that better fit their natural abilities to learn.

**BIBLIOGRAPHY**

The following is a partial list of resources I have found helpful in my quest to learn more about learning and the brain.

Bransford, John, Ann Brown and Rodney Cocking, eds. * How People Learn: Brain, Mind, Experience, and School* (Expanded Version). Washington, DC: The National Academies Press, 2000.

Brier, Georgia and Megan Hall. “From Frustrating Forgetfulness to Fabulous Forethought.” *The Science Teacher*, January 2007.

Downing, Skip. “Becoming an Active Learner.” * On Course: Strategies for Success in College and in Life*, Cengage Learning.

Jensen, Eric. *Brain Based Learning* (Revised Edition), San Diego: The Brain Store, Inc., 2000.

—. *Teaching with the Brain in Mind *(2nd Ed), Alexandria, VA: Association for Supervision and Curriculum Development, 2005.

National Research Council. *How Students Learn: History, Mathematics, and Science in The Classroom*, Washington, DC: The National Academies Press, 2005.

Smilkstein, Rita. *We’re Born to Learn: Using the Brain’s Natural Learning Process to Create Today’s Curriculum*, Thousand Oaks, CA: Corwin Press, Inc., 2001.

Sousa, David A. * How the Brain Learns* (3rd Ed), Thousand Oaks, CA: Corwin Press, Inc., 2006.

Zull, James. *The Art of Changing the Brain*, Sterling, VA: Stylus Publishing, 2002.

More sources on the Web:

http://Help4teachers.com Dr. Kathie Nunley’s Layered Curriculum Web Site for Educators (designed more for K-12, but many interesting articles)

www.pbs.org/wnet/brain Public Broadcasting System: *A Five-Part Series on the Brain*

**SUPPLIES**

**A. TRUE or FALSE (used in Step 1)**

1. **-4 ^{2 }= -16**

2. **5 ^{0} = 0**

3. **(bc) ^{4} = b^{4}c^{4}**

4. **b ^{5}/b^{8} = b^{3}**

5. **(c ^{7})^{4} = c^{11}**

6. **(3b/2c) ^{2} = 3b^{2}/2c^{2}**

**B. Properties of Exponents: Modeling Activity (used in Step 3)**

Use a blue cup to represent “a”; a red cup to represent “b”. Stack two textbooks on your desk to create a “numerator” for fractional expressions.

With your partner, use the cups to model each of the expressions in the first column, and then complete the equation. Do not go on to the second column at this time.

Example |
Property/Definition |

a |
a |

(b |
(a |

(ab) |
(ab) |

(a/b) |
(a/b) |

a |
a |

a |
a |

**C. What Did We Learn? (used in Step 4)**

Determine whether each of the following is True or False:

1. **-4 ^{2 }= -16**

2. **5 ^{0} = 0**

3. **(bc) ^{4} = b^{4}c^{4}**

4. **b ^{5}/b^{8} = b^{3}**

5. **(c ^{7})^{4} = c^{11}**

6. **(3b/2c) ^{2} = 3b^{2}/2c^{2}**

**–Deb Poese, Faculty, Mathematics and Director, School of Education, Montgomery College, MD. **