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  • Summary: A classroom activity using dried split peas exploring the reliability of a basic capture-mark-recapture method of population estimation is described using great whale conservation as a motivating example. The activity was described in C. du Feu, “Having a whale of a time,” Teaching Statistics, 31 (3) (2009), 66-71.

    Specifics: The hands-on activity uses dried split peas and involves much larger populations and has two advantages. Firstly, the split-pea populations are too large for any sensible student to contemplate counting the full population. Secondly, unlike SmartiesTM, or M&M’sTM dried split peas will not suffer loss through eating (so there is a fixed population size to be estimated). Beforehand, soak some split peas in colored food dye or simply buy both green peas and yellow peas. Students add exactly 50 of the differently colored peas to each population of unmarked split peas. We now have hundreds, if not thousands, of members in each population of which 50 were ‘captured’ and marked in the first sampling event. The sampling can be done using a teaspoon of about 5mL capacity, which gives samples of about 50 individuals. The number of marked and unmarked split peas in the spoon are counted and a population estimate is made. The peas are replaced, the populations is mixed (stirring or shaking with the lid on) and the next sample is taken. This is repeated as long as required. Once there are sufficient estimates, the sampling can be drawn to a close and discussion of the estimates can take place. 

    Supplementary materials include expository material on the motivating example and student worksheets.

    (Resource photo illustration by Barbara Cohen, 2020; this summary compiled by Bibek Aryal)

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  • A hands-on activity using the capture-recapture method to estimate the number of SmartiesTM candy pieces in a population and to study the variability in individual estimates compared to an estimate based on the mean of many estimates.  The activity was described in B. Dudley, "A practical study of the capture/recapture method of estimating population size, Teaching Statistics, 5 (3) (1983), 66-70.

    Summary: A hands-on activity to study the variability of the capture/recapture technique for estimating population sizes, demonstrated using a population of Smarties candy as an example. 

    Specifics: The capture/recapture technique is used to arrive at estimates of the size of population of mobile animals using the formula: 
    a/d = c/b, where
    a = number marked and released into the population,
    b = size of the second catch,
    c = the number recaptured in the second catch,
    d = the size of the population as a whole
    The contents of a box of smarties are poured into a saucer and all the sweets of red colour were counted (=a). After that, all the sweets are poured into a paper bag and shaken thoroughly. With an egg cup, without looking at the bag, the second sample (=b) was scooped and the number of red ones recaptured were recorded (=c). This exercise was repeated ten times and the mean was calculated. Finally, the number of Smarties in the model population were counted and compared with the estimates derived from the sampling. Students learn about the variability of individual estimates, which is quite large (remember that the mean of the estimate here is actually infinite since an observation of zero tagged items results in an infite estimate).

    (Resource photo illustration by Barbara Cohen, 2020; this summary compiled by Bibek Aryal)

     

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  • A hands-on activity using the capture-recapture method to estimate the number of M&M’sTM in a population The activity was described in G. D. Bisbee and D. M. Conway, “Studying proportions using the capture-recapture method”, Mathematics Teacher, 92 (3) (1999), 215-218.

    Summary: Scientists use the capture-recapture method as a tool to estimate population size. Animals are captured, tagged, and then released back into the population. Later, a sample is captured and a proportion used to estimate population size.

    Specifics: Let us say that we sample a beetle population of unknown size. We capture and mark ten of those beetles with a spot of India ink, then return them to the population and give them time to mix in with the population. We then recapture another sample consisting of eight beetles, one of which was previously marked. We substitute the numbers into the foregoing proportion to estimate the population size, getting 1/8 = 10/(Pop size). Solving for the Pop size gives us an estimated population of eighty beetles. Students are, predictably, less than enthusiastic about having to handle the creepy-crawly critters so this activity uses a population of M&M’s of unknown size to estimate. Each team of two to four students receives some M&M in a paper cup, which is covered on top with crumpled paper towels. The students “tag” the M&M’s from a random sample and then, after mixing them back in, sample again to estimate the number in the cup (they can later check how far off their estimates  were and compare to other teams).

    (Resource photo illustration by Barbara Cohen, 2020; this summary compiled by Bibek Aryal)

     

     

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  • A joke to use in presentations about the importance of control and replication in experimentation.  The joke was written by Larry Lesser (The University of Texas at El Paso) and Dennis Pearl (Penn State University) in March 2020.

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  • A cartoon to initiate discussions about how the correlation is a unitless number that does not change with changes in the units of the variables involved.  The cartoon was created in February 2020 by British caetoonist John Landers based on an idea by Dennis Pearl (Penn State) and Larry Lesser (Univ of Texas at El Paso). An outline of a lesson plan for the use of the cartoon is given in a 2020 Teaching Statistics article by Dennis Pearl and Larry Lesser.

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  • A poem written in 2019 by Sabrina Little, a middle school student at the Mackintosh Academy in Boulder, CO.   She entered it into the American Mathematical Society’s Math Poetry Contest contest for Colorado middle school, high school, and undergraduate students in connection with the 2020 Joint Mathematics Meetings in Denver. Sabina’s poem was judged the winner in the category for middle school students.  The poem uses imagery which can enhance a lesson on line of fit and outliers. Sabrina Little read her poem at the 2020 Joint Mathematics Meetings (see  2:42 mark of the video posted at http://www.ams.org/programs/students/math-poetry).

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  • A joke about the meaning of an inequality symbol like ≤ written in February 2020 by Larry Lesser from The University of Texas at El Paso and Dennis Pearl from Penn State University.

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  • A joke to help in discussions of the value of random assignment in experiments and in discussing pedagogical options.  The joke was written by Larry Lesser from The University of Texas at El Paso in February 2020.

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  • A joke to help discuss how random assignment is an unbiased experimental method.  The joke was written by Dennis Pearl from Penn State University in February 2020.

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  • A poem written in 2019 by Larry Lesser from The University of Texas at El Paso to discuss the normal distribution and its percentiles.  Students could first be shown a copy of the National Center for Health Statistics growth chart graph paper so they will appreciate the details of the poem. And after reading or hearing the poem, students could verify the detail that the 40th and 60th percentiles are half a standard deviation apart. The poem is part of a collection of 8 poems published with commentary in the January 2020 issue of Journal of Humanistic Mathematics.

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