2.1

Project planning and implementation

Originally, we had planned on monthly activities to be delivered over two days in the mathematics classrooms and the Math Lab, a separate classroom being modified into a space for active learning of mathematics. Missed days due to snowstorms and other scheduling anomalies required that we limit our program to six activities in fall and spring. See Table 1 for a list of program activities.

Table 1

Illuminating Math with Light Schedule of Activities.

The optics lessons were developed in part from the Dumpster Optics lessons, a program supported by SPIE.¹ In addition, an outreach grant from SPIE awarded to St. Bernard School allowed us to purchase equipment not in the “dumpster” category, such as gas spectrum tubes and light meters. We also adapted one lesson from the Hands-On Optics program and two of the Problem-Based Learning Challenges developed by the New England Board of Higher Education.^2,3 We chose lessons and activities that illustrated mathematics concepts from the middle school curriculum. Because of the short (42 minute) class period, some activities took three days to complete.

2.2

Fractions and lenses

To illustrate a practical use of fractions, students engaged in a three-day activity based on the Hands-On Optics Terrific Telescopes lessons. On the first day, students observed a demonstration of light refracting through an acrylic block and flat lens shapes. The acrylic shapes were from a kit supplied by Arbor Scientific (Item #P2-7680) that also included a “laser ray box” providing 1 to 5 separate beams for ray tracing through the various acrylic shapes. The concept of a focus was discussed and students identified the focal point and focal length for the lens shapes. They then spent a few minutes with two different focal length “education grade” converging lenses (from Surplus Shed) projecting images of distant objects onto a wall to determine the focal length of each lens.

On the following day, the students were introduced to the thin lens equation and, guided by the teacher, performed the calculation to determine lens focal length given object and image distances. Returning to the lab, they measured object and image distances using a meter stick optical bench (Carolina Biological Supply), the lenses from the previous day, the filament of a clear light bulb for the object and a cardboard screen. Before returning to the classroom for calculations students were shown how to place the two lenses on the bench, creating a telescope for viewing through the classroom windows. Calculations were far more time consuming than expected, especially in the sixth grade classrooms. Nonetheless, the calculated focal lengths matched the ones measured on the previous day fairly well.

The third day we planned to complete the Hands-On Optics resolution activity, having students read an eye chart through a telescope from several distances in a long hallway. They were instructed to note their distance from the chart and the height of the smallest letter they could see from that distance. To expedite matters, we constructed the cardboard tube telescopes from kits purchased on EBay that were the same as those in the Hands-On Optics kit. As it turned out, the youngest students were unable to even find the eye chart. Older students were able to take some data but not enough to make meaningful conclusions. Nonetheless everyone seemed to enjoy the activity, especially looking through the 4.5” Newtonian telescope set up to look out the window. Figure 1 shows several scenes of students engaged in telescope use.

Figure 1

(top) Students experiment with different methods of supporting the cardboard telescopes while viewing an eye chart at the end of the hallway. (bottom) Looking through the small Newtonian telescope.

2.3

Structured problem solving (PBL)

For this activity we chose web-based Challenges (scenarios) developed by the PBL (Problem Based Learning) Projects of the New England Board of Higher Education.⁵ The three PBL projects, supported by the Advanced Technological Education program of the National Science Foundation, present students with “real world” problems in Photonics, Sustainable Technologies and Advanced Manufacturing. The goal of the Challenges is to teach content while at the same time allowing students to practice structured problem solving, written and oral communication and teamwork. At the heart of the method is a tool called the “Whiteboards”, four pages that walk students through the details of structured problem solving beginning with problem analysis and continuing with self directed learning (independent research), team brainstorming, and solution testing. The Challenges are presented in a series of short videos that show a problem to students in the context of the company or research university that solved the problem originally. After sharing and critiquing student solutions, comparison can be made to the organization’s solution.

Sixth and seventh graders engaged in the most straightforward of the Challenges, the Tookany-Takony Frankford Watershed Partnership problem to develop neighborhood-based solutions to the issue of storm water management. The math involved is area and volume, converting rainfall (in inches) on an area to cubic feet or gallons of water stored. For many 6th and some 7th graders this was the first real world application of mathematics Their expectation was that math should involve no more than memory and drill work.

The 7th grade classes did better working in groups than most of the 6th graders. They were able to self designate tasks for each person in the group and then put the work together as a group and present their information in front of the class using a poster. Sixth grade students had difficulty with assigning tasks so that the project work load was equally divided. In addition to the poster and presentation, students assessed their teammates and themselves on teamwork. Based on the responses, some sixth graders felt that they did more than their fair share of the work and some felt that others took over too much and their own ideas were neglected. Overall the projects went well, however, with students especially in the 6th grade taking the time to learn and understand math calculations that they may not yet have seen in math class.

Because of their more advanced level of math knowledge and general maturity, 8th and 9th graders were presented with the Laser Safety Challenge, developed with the assistance of the International Laser Display Association (ILDA). The problem involves a legal case concerning the targeting of an aircraft with a laser pointer and students need to decide the innocence or guilt of the perpetrator. Students had no previous knowledge of laser safety and most groups enthusiastically engaged in research on both aircraft and laser specifications. When issues such as units of light measurement arose the instructor provided focused instruction on an as-needed basis. On the second day, students were given laser pointers and allowed to measure beam spreading in a long corridor. They then used ratios to determine beam area at the distance of the airplane. On the third and final day, students informally presented their solutions to the class. All teams agreed on the solution even though they had been arrived at independently.

2.4

Light, Shadows and similar triangles

All classes did the same activity for this lesson, which was developed as part of SPIE’s Dumpster Optics program. The lesson is based on a tutorial developed at the University of WA.⁶ On the first day, students used LED finger lights to trace and observe the path of light from the bulb through a 1 cm triangular hole to a screen. At each step, they first predicted what they expected to see and then made the observation. If prediction and observation did not agree they needed to figure out why before proceeding. In the final step, separate LED lights were replaced by a continuous light source, a clear globe linear filament bulb. Students observed a low resolution pinhole image of the filament on the screen. The large triangle was replaced by a smaller one to produce a clearer image of the filament. Most students found the production of an image by a small hole to be interesting.

On the second day, students were introduced to the pinhole camera and pinhole images formed, for example, under leafy trees. We had intended to allow each student to make their own camera but a shortage of materials prevented that. Instead, students made pinholes out of pieces of soft drink cans and inspected and measured them with a USB microscope. We provided home made oatmeal box pinhole cameras loaded with black and white film paper and took a 30 minute photo of each classroom while students were working. (See Figure 2.)

Figure 2

Roughly 30 minute oatmeal box pinhole camera exposure of classroom (left), inverted version (center) and oatmeal box camera (right).

The final day we had planned to set up a long filament bulb, pinhole and screen and ask students to predict and then measure image size using object and screen distance and similar triangles. However it was an exceptionally warm November day and we decided instead to take the classes outside and use the same mathematics to measure the height of a tree. Most, if not all, students were happy with this decision.

2.5

Spreading light

This simple experiment was easily accomplished by students in all grades. Given a small, bare light bulb, a square centimeter hole and a printed centimeter grid they were asked to determine how the illuminated area on the grid depended on distance from the bulb. (See Figure 3.) After some issues with keeping bulb, mask and screen aligned and figuring out how to handle illumination of partial squares, students in all grades quickly noticed the quadratic relationship.

Figure 3

Observing how light spreads from a “point” source. The light source was a MagLite^® flashlight in candle mode. Binder clips were used to hold the mask and screen vertical.

On the second day, students were asked to graph their data and observe and discuss the shape of the graph. Younger students needed quite a bit of assistance in setting up the graph and may not have grasped the full significance of it not being a straight line. Nonetheless, they were able to predict how many squares would have been illuminated at distances they did not measure. For the 8th and 9th graders this was a good introduction to the graphs of functions. Graphical Analysis software (www.vernier.com) was used to do curve fitting on student data and some time was spent exploring graphs of other functions.

2.6

Interpolation and light spectra

This final activity was undertaken in April with the class of 8th and 9th grade students only. We began by showing students videos of high speed optical food sorting machines⁷ and asking how they thought they might work. Next we gave each student a diffraction grating slide and showed them several different light sources-incandescent, fluorescent, gas tubes, lasers and LEDs. A discussion followed on using light spectra to identify materials and on the importance of the hydrogen spectrum to astronomy. Students were then challenged to determine the wavelength of the H-alpha line by comparing the hydrogen emission spectrum to the full visible spectrum and dividing the full spectrum in successive halves until they could estimate the value of H-alpha. This was done as a class with the spectra projected on the whiteboard. (See Figure 4.) Despite the approximate nature of the spectra provided, students were within a few nm of the correct value.

Figure 4

Graphic used to find the wavelength of the H-alpha line through interpolation.

The second day was spent looking at sources of light around the classroom using the RGB function on a Vernier Go Direct^® Light and Color Sensor (Item #GDX-LC). Examination of the RGB output of smartphones and laptops led to a discussion of the possible health issues related to blue light exposure at night. Students were interested to see the difference in RGB output of a phone screen with and without night blue light filtering applied.

2.7

IDL Photo contest

We decided to end the program with a photo contest to celebrate the International Day of Light (May 16). The theme of the contest was “Light in out Life” and in addition to a photo with light as the central subject students were required to submit a short statement on how their submission related to the theme of the contest. This was a voluntary activity and open to the entire student body. Approximately 20 entries were received in two divisions, middle school and high school. Entries were judged by members of the SBS staff based on artistic impression, originality and how well the photo expressed the theme Light in our Life. Three winners were awarded in each division. Prizes consisting of IDL coffee mugs and gift certificates were awarded at the annual MathCon, an annual one-day event in May when members of the Mu Alpha Theta mathematics honor society present projects on math and its applications. The authors also attended the event with optics demonstrations and the winning photographs on display. (See Figure 5.)

Figure 5

Two of the winning “Light in our Life” photo contest entries.