To incorporate a research component into the curriculum, two aspects must be considered. First, students need a research model that is easy to understand and apply. Second, students must have an understanding of some basic statistical tools, such as mean, median, mode, and range. Rather than being taught as isolated topics, the statistical tools are used in applying the research model to real situations.
The Elementary Mathematics Research Model (Irby & Bohan, 1991) has students
move through seven steps to produce knowledge through mathematics, as shown below.
You may wish to project this model on an overhead projector, as shown on the Elementary Mathematics Research Model overhead.
In step 1 students must attempt to identify a
problem. For the students to become involved and have ownership in the project, let them think: of things that they would
like to know, of some questions that they would like to answer, or of some
problems that they have observed in the school or community. During this
brainstorming session, establish a rule that no one is to judge the thoughts of
another. Let the ideas come freely.
Step 2 is a natural outcome of step 1. One of the issues from the
brainstorming session is chosen, a problem to be solved is developed, and a
research question is stated. The following is a problem which could be formulated from a
brainstorming session with students:
The students were concerned with the amount of garbage produced in the
school cafeteria and its impact on the environment (the problem). The research
question
was, What part of the garbage in our school cafeteria is recyclable?
In Step 3 students hypothesize the expected outcome of the research. The
teacher might ask, "What do you think will be the outcome of your research or
investigation?" Students might answer, "We
believe that half of the waste is recyclable."
Step 4 will find students developing a plan for how to test the hypothesis
and answer the question. The following items will need to be considered in
developing the plan:
- permission - who will give us permission: the principal, the cafeteria
supervisor, the maintenance director, or others?
- courtesy - when can we conveniently discuss this project with the cafeteria
management?
- time - how much time can we spend on this investigation? when should we do
this project each day? how long do we think it will take to gather all the
data?
- money - will it cost anything? how can we get the money? do we need to
write a grant proposal to request the money through the principal or the PTA?
- safety - what measures must we take to ensure safety, for instance, gloves
and masks?
The students will need to develop an exact plan to address these concerns.
Note: The rest of this lesson is based upon the following plan:
We will have our study last for three weeks, giving us fifteen opportunities
to collect data. We will check the garbage every day and request that it not be
thrown out until we do so. We will request the help of our fellow students when
throwing out their garbage in the cafeteria by requesting that they separate it
into six different cans that are clearly marked - uneaten food, partially eaten
food, Styrofoam, paper, plastic, and aluminum. We will weigh the amount of each
can and keep the records each day. The number of aluminum cans will be
counted.
As the students determine how they will gather the data, they need to
determine what variables are involved in the research study. In this example,
they might determine that the weight of each individual can would be one
variable, the length of the study might be another, and so on.
Step 5 is "carry out the actual plan." During the time the data are being
collected, discuss ways in which the students might report the findings. Graphs
should certainly be discussed as a possibility, as should types of graphs best
used for various purposes. At this point, the need for
statistical measures to describe the data becomes apparent. For example, since
this study is to last fifteen days, it is not probable that the same number of
aluminum cans would be collected each day. How can the number of cans collected
daily be described without having to list fifteen numbers?
Developing Measures of Central Tendency
The Mean
To teach the concept of mean, pose a situation for students in which eighty
cans are collected one day and sixty the next. Have students use a meter stick
and adding-machine tape to represent these numbers by cutting off pieces 80 and
60 centimeters in length. This tactic gives students a physical representation
of their two-day collections. Have students attach the tapes end-to-end. Hold
up the combined tapes and ask, "What does this paper represent?" [The total
number of cans collected for two days] Students use this paper and the meter stick to
decide
the total number of cans they have collected.
Say to the students, "Suppose that on two other days you collected the total number of cans
represented by the combined tapes. However, an equal number of cans was
collected each day. Use the combined tape to decide what that number was."
Since
this paper represents two days of collecting, the combined tape can be folded
into two equal parts and compared with the meter stick to find the number. Once
the number, 70, has been determined, define this number as the mean.
Repeat the activity with different numbers of cans and days; this extension is
necessary, as otherwise some students form the misconception that we
always divide by 2 when finding the mean.
Present various situations in which students try to predict what would
happen to the mean if, on the next day, a greater or smaller number of cans was
collected. Predictions can be investigated by using adding-machine tapes. The
conceptual work done with the tape can readily be connected to the symbolic
procedure for finding the mean. Connecting the tapes represents finding the sum
of the numbers, and folding the combined tapes represents dividing the total
into equal parts. The number of parts into which the combined tape is folded is
determined by the original number of pieces of tape.
Demonstrate the need for other measures of central tendency by pointing out
the main weakness of the mean - the extent to which its value can be affected
by extreme scores. This weakness can be demonstrated within the framework of
activities discussed earlier by showing the effects that a day when no cans
were collected would have on the mean of three days of collection averaging
eighty
cans per day. The median and mode can then be presented as different measures
of central tendency that minimize the effect of extreme scores.
The Mode
To teach the meaning of the mode, have students write fifteen numbers on index cards and place them in a box. (You can use classroom data, such as grades (anonymously, of course) to make the activity more authentic.) The example below uses the following numbers: 76, 80, 84, 72, 85, 80, 74, 61, 72, 84, 76, 80, 91, 87, and 85. Have students pull a card at random from the box and place it on a chart. A second card is then extracted, and the question asked, "Is this number greater than, less than, or equal to the first number?" This second card is placed on the chart to the right, to the left, or above the first card depending on whether the number on it is, respectively, greater than, less than, or equal to the number on the first card.
The students continue to pull cards, asking the same question over and over until all cards are arranged on the chart in order, left to right, from smallest to largest. Looking at all the cards on the chart, ask, "What number appeared the greatest number of times?" After identifying the tallest column, define the number of cards in that column as the mode.
A Table of Fifteen Number Cards
The Median
To teach the concept of median, have students use the fifteen numbers they have placed on the chart. Ask, "Where have you heard the word median used before?" [The median of the highway is the part that divides the highway into two equal parts.] In mathematics the word median is used to tell us something about a set of numbers. Ask, "What do you think it tells us?" [It is the point that separates a set of numbers into two equivalent subsets.] Have students work with a partner to find the median of the set of numbers represented by the cards on the chart. One way is to begin removing cards from either end of the set simultaneously, one with each hand. This process is continued until only a single card remains. Have groups share their method with the class. Identify the number on the middle card as the median.
Ask students, "If we find the median by eliminating cards from each end, will we always
get to a point where a single card remains?" After getting a consensus that we
would not, discuss the conditions under which this outcome would or would not
occur, capitalizing on the opportunity to review the concept of even and odd
numbers. Next place an even number of cards in order as indicated previously,
and eliminate cards simultaneously from either end until only two cards remain.
Give students an opportunity to discuss how the median might be identified. Try
to get the class to agree that the best solution would be to call the point
halfway between the two remaining cards the median. Introduce situations in
which
- the median is not a whole number, as when the two remaining cards contain such consecutive numbers as 87 and 88 (median 87.5); and
- the median is a whole number but not a number on one of the cards; for
example, the two remaining cards have such whole numbers as 84 and 88 (median 86).
In either case, the median is the mean of the two remaining numbers.
Dealing with the Data
In step 6, at the end of the three weeks, analyze the data. The question to
be answered is, "Did the test support our hypothesis?" The data will be
analyzed on the basis of the statistical tools previously developed.
As they "look back" in step 7, students should ask such questions as the ones suggested in Questions for Students, as shown below.
