## Abstract

This paper describes a laboratory exercise designed to provide students with experience testing a hypothesis by systematically isolating and controlling determinant variables. The study involves an analysis of walking and is performed by the students on a subject from within their lab group. The study requires use of a motorized treadmill, tape measure, stop watch, metronome, personal cassette player, and calculator. The exercise is designed to include factors that the students are familiar with, so they can focus on the isolation of variables without being confused about the process they are investigating. However, the exercise will not turn out as the students anticipate, meaning they will be forced to reevaluate the assumptions that formed the basis of their original hypothesis. This exercise is designed for a college-level course in exercise science, physiology, or biology but could easily be managed by a high school honors class with appropriate guidance.

- walking
- hypothesis
- variable

To perform a valid test of a hypothesis it is necessary to isolate and control the variables that affect the system under consideration. Although most laboratory exercises used in college and high school classes go to great lengths to control specific variables, this strategy is often lost to the student who is confused by unfamiliar circumstances and has difficulty distinguishing between the controlled and uncontrolled variables.

One solution to this problem is to use a simple relationship where it is readily apparent which variables are being controlled. The human walk is a system familiar to all students. Although complex as a process, forward locomotion using a walk is based on a very simple relationship where the forward speed *v* (distance covered in unit time) is equal to the product of the step frequency *f*_{s} (number of steps in unit time) and step length *d*_{s} (distance covered in each step) 1 The simplicity and conceptual familiarity of this relationship provide an opportunity for the student to systematically explore the consequences of isolating and manipulating each of the three variables independently (speed *v*, step frequency *f*_{s}, and step length *d*_{s}).

What hypothesis could be tested by this manipulation? First, it is necessary for the students to be absolutely clear on what a hypothesis is and where it originates.

The hypothesis is a concept that is held dear as the cornerstone of the scientific process, yet its role in modern science is often misunderstood. The hypothesis is defined as “an idea or explanation of something that is based on known facts but has not yet been proved” (*Cambridge International Dictionary of English*). Although deceptively simple, the concept of the hypothesis holds substantial implications for the manner in which scientific investigation progresses. In practice, modern experimental science uses the hypothesis as a prediction of the outcome under specific experimental manipulations, and it functions to evaluate a proposed *understanding* of how the system works. A simple diagram can describe the common understanding of the scientific method (Fig. 1). The hypothesis is a prediction that originates as a logical consequence of an understanding of how a system operates. How can this be applied to walking?

Numerous studies published in the literature indicate that humans tend to select a consistent relation between speed, step frequency, and step length (2–6). It is sensible, then, to assume that, for each individual, there is a repeatable relationship among the three variables that define the walk (*Eq. 1*). It is usually assumed that this relationship exists because it provides the most economical option for walking. That is, for any given walking speed, the individual will choose a certain step frequency and step length because this is the most comfortable (and least metabolically costly), even though it is possible to walk at that speed using a different combination of step length and frequency.

We can predict that, if there is an optimum relationship between speed and step frequency that represents natural walking, then for each step frequency or step length the individual should select speeds that fit this relationship. The expectation is that manipulation of each of these parameters results in the selection of the same speed-frequency relationship, and that the three variables can be interchanged as the factor that determines the other two. This hypothesis can be written as where the subscript *c* indicates the variable being controlled.

## THE EXPERIMENT

The aforementioned hypothesis can be tested using a remarkably simple analysis. The objective of the experiment is to isolate and independently manipulate each of the variables in *Eq. 1* while measuring the subject’s free selection of the other two. Because *Eq. 1* must hold, when one variable is controlled it is only necessary to measure one of the nonconstrained variables; the other can be calculated directly from *Eq. 1*. The experiment consists of systematically controlling each of the variables in turn while measuring a second and calculating the third. A list of required materials and general instructions are provided in appendix a. An example handout of detailed instructions and a data sheet appropriate to be distributed to the student groups are given in appendix b. Following are descriptions of each of the constraint experiments.

#### Speed constrained.

This is usually considered “normal” walking, because we naturally use motions of our limbs to move our bodies from place to place. To constrain forward speed, a standard treadmill is used (but it must be set for zero incline). Speed is then set, and the subject uses whatever step frequency and step length feels most comfortable. A speed range from 0.5 to 5.0 mph (0.222–2.22 m/s) works well, although some smaller individuals may not be able to walk at 5 mph. A useful strategy is to identify the subject’s fastest walking speed and make a frequency measurement and then find the slowest speed at which the treadmill can maintain a consistent belt speed. With these two bounds, the range in between can be divided into five more speeds to provide seven speeds covering a large range. It is necessary for the subject to walk steadily, and it is recommended that the subject walk for at least one minute (especially at slow speeds) to ensure that they have settled into a consistent gait. With the forward speed set by the treadmill, it is an easy matter to measure step frequency from the duration of a known number of steps using a stop watch and the sound of the subject’s footfalls on the treadmill. Because speed is set by the speed of the treadmill belt and step frequency has been measured, the subject’s step length can be calculated (*v*_{c}/*d*_{s} = *f*_{s}). These data will define the relationship for walking when speed is constrained.

Technical note: when timing footfalls, many students will naturally start the stopwatch on the count of “one.” If they then stop the watch on the count of “20,” they will have timed the duration of only 19 steps. It is important to ensure that timing starts with “zero” at the first contact and stops at the 20th contact.

#### Step frequency constrained.

An individual is readily able to walk to the sound of an even beat, for instance as in walking to the beat of music. A metronome can be used to provide a reliable signal of known frequency. Walking for any duration, especially at higher frequencies, will take the individual out of earshot of mechanical or AC-powered metronomes. Battery-powered metronomes are available (e.g., KDM-1, Korg, Japan), but a low-cost alternative is taping a metronome beat and replaying it using a personal cassette player that can be carried with the subject. Because frequency is determined in this experiment, either step length or average forward speed can be measured. The easier of these to measure is average forward speed. Once again, a stopwatch can be used to measure the time it takes the individual to cover a known distance (e.g., some number of meters marked on the floor or wall). From the set frequency and measured speed, the step length used by the subject can be calculated (*v* = *f*_{sc} × *d*_{s}). Several frequencies should be used (again 7 are suggested), with the subject walking at the beat for an adequate duration to become comfortable. The widest possible frequency range should be used. We have been successful using frequencies from 0.8 beats/s (48 beats/min) to 2.9 beats/s (176 beats/min).

#### Step length constrained.

Placing tape markers on the floor can control the step length a subject uses. The subject is instructed to walk in a normal manner, placing the same part of the foot at the level of each marker (some students may try to place their feet very precisely, which may alter their gait, but high precision is not necessary). A range of seven step lengths from 0.25 to 1.0 m are recommended. Each length should be marked with at least 30 markers, to ensure that the subject becomes familiar with the spacing before the measurement is taken, and the subject can practice several times at a given spacing to become comfortable with it. Either average speed or step frequency can be measured at each step length, but step frequency seems the easier because it can be measured with a stopwatch measuring the duration of a known number of steps (e.g., 10 steps taken in the middle of the set of 30 markers). Speed can then be calculated

#### Analysis.

*Experiments A, B,* and *C* above provide data on speed, step frequency, and step length for the three independently constrained walking parameters. It is possible to compare the relationship between step frequency and speed for each of these conditions by use of regression analysis. This is a simple exercise using a variety of software packages (Excell, Kaleidagraph); it can be accomplished on a graphing calculator or can be easily plotted by hand. This comparison is the test of the original hypothesis, i.e., that the same combination of walk parameters will be used regardless of conditions.

## RESULTS

An example of results generated by a typical student group is provided in Fig. 2. The results indicate that the original hypothesis must be rejected; the expectation that the variables are interchangeable is not supported, and a single relationship for human walking under these three constraints does not exist. Instead, three distinct relationships are observed.

Some of the comparisons may show only subtle differences. This is most evident when the frequency-constrained relationship is compared with the speed-constrained one. In a sample of 12 subjects (6 male and 6 female), we found statistically verifiable significant differences between speed- and frequency-constrained relationships in 10 of the 12 (1). The step length-constrained relationship is substantially different from the other two, and we have always seen an obvious and significant difference between this and the other relationships.

## ASSESSING LEARNING ISSUES

To evaluate the value of this exercise for a typical class, a simple assessment instrument was developed that explored the student’s perception of the experiment. This was administered to a class (30 students) consisting of junior and senior exercise science majors. These students had all taken one or more science courses, but none had been instructed specifically on walking (Table 1). The questions posed to the students were:

After completing this exercise do you feel:

*1*) you understand more or less about how an experiment should be designed?

*2*) you understand more or less about what is involved in the control of human walking?

*3*) human walking is more or less complicated than you thought?

*4*) more or less interested in the study of walking?

*5*) more or less interested in science and/or research?

The students were provided with a numerical scale of whole numbers from 0 (labeled “much less”) through 5 (labeled “no change”) to 10 (labeled “much more”). The survey was administered anonymously. The student responses are provided in Table 1.

The assessment tool also included the question: “The purpose of this lab exercise is to help you learn about (select all that apply).” The options and the percentage of the class that selected each are listed below:

how to think about the logic of an experiment | 97% |

how to conduct an experiment | 97% |

how people walk | 53% |

how to think about the logic of an experiment | 77% |

how to plot graphs | 53% |

how to evaluate a hypothesis | 77% |

how to control variables in an experiment | 93% |

why people walk as they do | 43% |

## DISCUSSION

An important feature of this laboratory exercise is the students’ faith in their own work. Despite their understanding of walking and their reasonably conceived expectation that their study should generate a single curve, they must learn to trust the results of their experiment. Undoubtedly, there will be some students who will try to fit the data to their expectations. Evidence from the accumulated class results should convince them of their error.

Hopefully, the next stage in this process is the questioning of the initial hypothesis and its underlying understanding. If the hypothesis is incorrect, then the students must question their basic understanding of human walking (Fig. 1). How could three distinct relationships be produced from the same basic movement?

When walking at a given speed, the step frequency and step length are selected that allow motion on the least energetic cost. This results in the relationship we know of as the speed-frequency relationship for human walking. However, when dealing with walking at a given step frequency or step length, the other variables are chosen such that they also minimize the energetic cost of walking. Interestingly, the minimum-energy solution for constrained step length is not the same as that for constrained step frequency, and neither of these is equivalent to the minimum-energy solution for constrained forward speed. How can this be explained to interested students?

#### Energetics of human walking.

Any given combination of step frequency and speed (and, by implication, step length) will result in a specific cost of locomotion (energy per unit distance covered). We can plot the cost against both frequency and speed to produce an energy surface (Fig. 3, here represented as the economy of walking, the inverse of cost). Due to the shape of the energy surface, the gait variables that satisfy the minimum-energy requirement for constrained step frequency or step length are not the same as those that minimize energy expenditure when velocity is constrained. This can most easily be visualized in an energy contour plot (Fig. 4), where frequency is plotted against velocity and the energy surface is indicated by contours of equivalent cost (i.e., contours at the same elevation on the surface indicate an equivalent energetic cost). Thus any combination of velocity, step frequency, and step length that lies on a contour will result in the same energetic cost to the walker.

Under a given constraint, for instance, a predetermined speed and a variety of frequencies and step lengths could be used. Graphically, these options lie on the line perpendicular to the speed axis at the location of the predetermined speed. Energetic cost will be minimized, however, only for the specific combination of step length and frequency denoted by the contour that is tangent to the perpendicular (Fig. 4, *line A*). Likewise, minimum cost for a set step frequency is a tangency between a contour and a perpendicular from the frequency axis (a horizontal line in Fig. 4). This defines a slightly different set of minimum-energy solutions (again, dependent on the shape of the minimum-energy contours) (Fig. 4, *line B*).

On the plot described above, different step lengths are indicated by lines emanating from the origin. Speed (the *x*-axis) is a linear function of frequency (*y*-axis) for any given (constant) step length. The minimum-energy solution for a range of step lengths will be indicated by the points of tangency of the minimum-energy contours and the lines emanating from the origin indicating particular step lengths. Note that the set of points that satisfy this condition differ markedly from the other two constraint conditions (Fig. 4, *line C*), just as they will in the student’s plots.

#### Assessment results.

On the basis of the results of the simple assessment administered, it was evident that this class was readily able to distinguish the purpose of the exercise; i.e., its purpose was to increase their understanding of how to perform an experiment. High proportions of the students recognized that the exercise helped them learn about how to conduct an experiment (97%) and how to control variables in an experiment (93%). Many students also felt that the exercise helped them understand the logic behind an experiment and how to evaluate a hypothesis (both 77%). This is opposed to the responses related to the more mundane aspects of the exercise, such as how and why people walk as they do (53 and 43%). The exercise was generally considered to be worthwhile and to have increased enthusiasm for scientific inquiry; students were able to recognize that walking had more complexity than they had previously appreciated, but interest in research was increased (Table 1).

This laboratory exercise is both technically and conceptually simple enough for students to fully understand. The exercise provides an interesting activity that involves formulating an original hypothesis, conducting an experiment that isolates individual variables of a straightforward relationship, and collecting the appropriate data to test the hypothesis. The results differ from the more obvious expectation and provide the students with an opportunity to reevaluate their original understanding of the system. Learning issues include the formulation and evaluation of a hypothesis, testing via systematic control of defined variables, graphic comparison of results, and the consideration of factors influencing function in several dimensions.

## APPENDIX A

### Materials

Motorized treadmill

Personal cassette player

Tape measure

Stopwatch

Prerecorded metronome frequency tape

Colored tape

### Operation of Lab

The most time- and labor-intensive aspect of this experiment is the laying out of the tape markers for the step length-constrained portion (C). If this is done before the lab, it is well within the capabilities of student groups to accomplish the data collection and analysis with a two-hour lab period.

I have found that the lab works best if students work in groups of 4 individuals, with one of the group members acting as the subject of the study (note, this must be the same individual for each portion of the study for the analysis, i.e., comparison, to be valid). The other members of the group can serve as timer and data recorder. With a station set up for each of the three portions of the study, it is possible to have three groups of students collecting data simultaneously.

## APPENDIX B: EXAMPLE LAB DIRECTIONS

### Biomechanics of Human Movement: Lab Research Project

The objective of this project is to provide you with experience collecting and analyzing data. One of the most important aspects of this process is the isolation of the factors being manipulated from other effects that might influence the system you are investigating. You will be asked to perform your studies to isolate factors affecting human walking and will be asked to evaluate your success at achieving this at the end of the study.

### The Project

You will be evaluating the factors that determine the relationship

Although a simple relationship, it has numerous consequences when applied to the context of human walking. For instance, we know there is a limit to walking speed; which of the two, stride length or stride frequency, determines this limit? In this study, we would like to look specifically at what effect constraining one of these factors, velocity, stride length, or stride frequency, has on an individual’s selection of the other two. If the human system acts simply like a walking machine, we might expect a single solution, regardless of which variable is determined. However, this has not been determined. Your objective is to control each of these three variables in turn, and then take measurements on the other two.

### Procedure

This study will consist of three parts, in which either velocity (Sect. A), step frequency (Sect. B), or step length (Sect. C) is controlled.

For each of the controlled conditions, we would like to determine the manner in which the uncontrolled variables change as a response to the controlled variable. To identify the relationship among these features, it is necessary to determine several points over some range of possibilities. In this study, we are going to collect seven points for each analysis (i.e., 3 × 7 pts = 21 pts from each group). I suggest seven because we will want the subject’s normal walk (1 pt) and some number of points above and below that (I suggest 3 points on either side to indicate the trend).

Below are the specific instructions on how to collect each set of data.

### A. Speed Constrained

The object of this analysis is to constrain the speed the subject is walking and determine the step frequency and step length that feels most comfortable under these conditions. Speed is constrained by using a treadmill. It is then easy to determine step length by measuring step frequency (note again that step length × step frequency must equal speed). Note that you should not trust the instrumentation supplied with the treadmill (i.e., regarding speedometer or odometer). The measurements you make will depend on the accuracy of the speed determination. Belt speed can be determined from the rate of belt cycling (i.e., a 3-m belt moving at 3 revolutions in 3.3 s has a belt speed of

#### Data record.

For each analysis point (i.e., treadmill belt speed) you need to record:

Belt speed, step frequency, and step length. To measure step frequency, I recommend you measure the time it takes for the subject to take 20 steps. Using a large number of steps will decrease the error in stopwatch handling and in any variation in steps the subject takes. Note: be sure to start the stopwatch on the count of zero, and include the time it takes to get to the first step, i.e., do not start the stopwatch on the count of 1 or you will only measure the time it takes for the subject to take 19 steps.

Once you have an accurate measure of belt speed (m/s) and step frequency (steps/s), you can calculate step length as

Collect data for the individual’s most comfortable speed and three speeds below and three speeds above. Therefore, you should complete this section with seven sets of data.

### B. Step Frequency Constrained

In this section, you will control step frequency. You will do this using a metronome, which generates a sound pulse at specific intervals. It is fairly easy to follow a consistent beat while walking. While frequency is constrained, the individual will be free to choose the step length and forward speed that feels most comfortable. Speed will be measured by timing the subject as he/ she covers a known distance.

The subject will be supplied with a personal cassette recorder with a tape recording of the metronome beat at specified frequencies. For each frequency, the subject should walk the halls matching his/her step to the taped beat. The group will then time the subject as he/she passes a premarked distance.

### C. Step Length Constrained

In this portion of the study, the objective is to control step length and measure speed and step frequency. The main challenges are to constrain step length without interfering with normal walking style and to make accurate measurements of frequency and calculation of speed.

We will use tape markers on the floor of the hallways to control step length. Make sure the distances between markers are accurately measured. Use a sufficient number of markers. I recommend a runway at least 15 m in length. Make your measurements over the middle third, and use the first and last thirds to make sure the subject is moving consistently (i.e., place markers at the appropriate spacing over the entire 15-m runway).

Measurement of step frequency will be determined using a stopwatch by determining the time it takes the subject to take 10 steps. From the known step length and measured step frequency, average speed can be calculated. At the end of this data collection, you should have seven sets of data, each with an average forward speed, step length, and step frequency. The step lengths used should be what is normal for the individual plus three shorter and three longer than normal.

At the end of the data collection, please check and make sure that all of the measurement values on the data sheet are entered correctly and completely (i.e., check your units of measure and enter them). At the completion of each of these sections, you should have collected data on *1*) the frequency, *2*) the subject’s velocity, and *3*) the subject’s step length (Table 2).

## Data sheet

Controlled | Measured | Calculated | ||
---|---|---|---|---|

Speed (m/s) | Time (s) (20 steps) | Step Frequency (steps/s) | Step length (m) | |

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

7 |

Controlled | Measured | Calculated | ||
---|---|---|---|---|

Frequency (step/s) | Time (known dist) | Speed (m/s) | Step length (m) | |

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

7 |

Controlled | Measured | Calculated | ||
---|---|---|---|---|

Step length (m) | Time (10 steps) | Step frequency | Speed (m/s) | |

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 | ||||

6 | ||||

7 |

## Analysis

On the **same** graph, plot speed (*x*-axis) against step frequency (*y*-axis) for each of the three walking conditions.

## Acknowledgments

Dr. Anindya Chatterjee, Indian Institute of Science, wrote the analysis program that generated Fig. 3. Dr. Andy Ruina, Cornell University, first conceptualized Fig. 4. Dr. Catherine Loudon, Division of Biological Sciences, University of Kansas, helped independently test this exercise and offered valuable advice for the finalized version.

Address for reprint requests and other correspondence: J. E. A. Bertram, Dept. of Nutrition, Food, and Exercise Sciences, 436 Sandels Bldg., Florida State University, Tallahassee, FL 32306 (E-mail: jbertram{at}garnet.acns.fsu.edu).

- © 2002 American Physiological Society