hickory dickory dock: navigating through data analysis t · pdf filethrough data analysis and...

This department features children’s hands-onand minds-on explorations in mathematics andpresents teachers with open-ended investiga-

tions to enhance mathematics instruction. The tasksinvoke problem solving and reasoning, require com-munication skills, and connect various mathematicalconcepts and principles. The ideas presented herehave been tested in classroom settings.

A mathematical investigation—

• has multidimensional content;• is open-ended, with several acceptable solutions;• is an exploration requiring a full period or longer

to complete;• is centered on a theme or event; and• is often embedded in a focus or driving question.

In addition, a mathematical investigation involvesprocesses that include—

• researching outside sources;• collecting data;• collaborating with peers; and• using multiple strategies to reach conclusions.

This department presents a scripted sequence andset of directions for a mathematical investigationfor the purpose of communicating what happenedin this classroom. NCTM’s Standards (NCTM2000) encourage teachers and students to explore

multiple approaches and representations whenengaging in mathematical activities. The investiga-tion will come alive through students’ problem-solving decisions and strategies in the readers’ ownclassrooms. As a result of their exploration, stu-dents will incorporate their reasoning and proofskills as they evaluate their strategies. The use ofmultiple approaches creates the richness that is soengaging in an investigation, and it also helps stu-dents find new ways of looking at things and under-stand different ways of thinking about a problem.

The activities described in this investigation areadapted from the “Row Your Boat” lesson featuredin the NCTM Navigations series Navigatingthrough Data Analysis and Probability inPrekindergarten–Grade 2 (Sheffield et al. 2002).The series of lessons featured here contain impor-

Hickory Dickory Dock:Navigating through

Data Analysis

Deborah Niezgoda, [email protected], is a National Board Certified kindergartenteacher at Westlawn Elementary School in Fairfax County, Virginia, and enjoys teaching lessonsthat link mathematics and language arts. She is interested in how children demonstrate andexplain their mathematical thinking. Patricia S. Moyer-Packenham, pmoyer@gmu. edu, teachesmathematics education courses and is the director of the Mathematics Education Center in theGraduate School of Education at George Mason University in Fairfax, Virginia. Her researchfocuses on uses of representation in mathematics and mathematics teacher development.

Edited by Patricia S. Moyer-Packenham, [email protected], George Mason University,Fairfax, VA 22030. This department is designed for teachers who wish to give students newinsights into familiar topics in grades K–6. This material can be reproduced by classroomteachers for use with their own students without requesting permission from the NationalCouncil of Teachers of Mathematics. Readers are encouraged to send manuscripts appropri-ate for this section to “Investigations,” Teaching Children Mathematics, 1906 Association Dr.,Reston, VA 20191-1502; or send electronic submissions to [email protected].

INVESTIGATIONS Deborah A. Niezgoda and Patr ic ia S. Moyer-Packenham

292 Teaching Children Mathematics / February 2005 Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

tant connections between literacy and numeracyskills for young children and provide multipleopportunities to investigate the frequency of wordsand letters in well-known nursery rhymes. Thekindergarten students depicted in these lessons col-lected data and represented it by using tally marks,frequency tables, and graphs. Students then sharedthese representations with the class to analyze andcommunicate their mathematical ideas.

Learning GoalsThis four-lesson investigation provides opportuni-ties for students to apply counting, number sense,data collection, and graphing skills. It integrateslanguage arts concepts such as recognizing lettersand words, identifying and sequencing events, andusing one-to-one correspondence to explore thefrequency of words and letters occurring in famil-iar nursery rhymes. These lessons were taught atthe close of the school year in a kindergarten class-room in a Title I school in Fairfax County, Virginia.

MaterialsColored pencils or markers should be available foreach lesson.

Lessons 1 and 3 require the following materials:

• Large print version of “Rain, Rain, Go Away”with tally/frequency chart (see fig. 1)

• Blackline masters of “Hickory Dickory Dock”with tally/frequency chart (see fig. 3)

Lessons 2 and 4 require the following materials:

• Completed tally/frequency chart for “Rain,Rain, Go Away” from lesson 1 or 3

• Large print graph of “Rain, Rain, Go Away”(see fig. 5)

• Completed tally/frequency chart for “HickoryDickory Dock” from lesson 1 or 3

• Blackline master graph for “Hickory DickoryDock” (see fig. 9)

Objectives of theInvestigationThe students will—

• calculate the frequency of words in a familiarnursery rhyme;

• calculate the frequency of letters in the first lineof a familiar nursery rhyme;

• represent and describe data using tally marksand frequency tables; and

• display data by constructing bar graphs.

Prior KnowledgeBefore the lessons were introduced, the studentshad many experiences with singing and “touchreading” (Hall and Williams 2000) the nurseryrhymes used in these investigations. They also hadprevious opportunities to sequence and complete“sentence building” activities (Hall and Williams2000) on a wide variety of texts, including nurseryrhymes. Sentence-building activities involve tak-ing a piece of well-known text and cutting it upinto individual words. Students then reassemblethe words in the appropriate sequence in order tomake sense of the text. In addition, the studentshad multiple formal and informal experiences withcreating tally charts and graphing data, and theyhad achieved facility with counting to twenty.

Teaching Children Mathematics / February 2005 293

Lesson 1: Gathering Dataon Word Frequency in theRhymeThe teacher began the investigation by singing thenursery rhyme “Rain, Rain, Go Away” with theentire class. She used a large print version of therhyme on chart paper and pointed to the words ofthe rhyme as she sang it. The children sang therhyme with her. After they repeated the rhyme twotimes, the teacher asked, “How many words are inour rhyme?” Some of the children replied 17,whereas others said that there were 18 words in therhyme. Then she asked two students to count thewords in the rhyme by touching them as theycounted. The class discovered that some of the stu-dents were counting the title and others were not.The teacher asked the children if they could thinkof a way to make a number sentence that showedthis difference. One child volunteered and wrote“18 – 1 = 17” on the white board. Capitalizing onthese student discoveries is an important part ofany successful mathematics lesson.

Next the teacher asked the students to determineif any of the words repeated in the rhyme. A lot ofhands were raised as the children saw severalrepeated words. One child said that the word rainrepeated in the rhyme. The teacher gave her a pur-ple dry-erase marker to circle the word each timeshe saw it. As the class found other words thatrepeated, they used different color markers to cir-cle each group of repeated words. (For example,they used purple to circle all the occurrences of theword rain, and red to circle all the occurrences ofthe word go; see fig. 1). When students found allthe words that repeated in the rhyme, the teacherasked several questions about the data such as“What word repeats the most?” and “How manytimes does the word rain repeat?” The question“How many words repeat the least?” sparked aninteresting discussion. Some of the children saidthat the words that repeated the least were thewords that repeated two times, because this was theleast number of repeats. Other students said thatthe words that occurred only once in the rhymerepeated the least because one was less than two.The teacher encouraged the children to talk abouttheir reasons for choosing each position.

Once all the repeated words were circled, theteacher read the list of words, then modeled count-ing the number of times a word appeared in therhyme and recording the data in the tally chart. Stu-dents volunteered to count the number of times that

different words occurred in the rhyme and to indi-cate this number by writing tally marks in the tallycolumn on the group chart. After the tally chart wascomplete, the teacher asked the students how theywould know what to put in the frequency column.Their experience with recording data was evidentwhen one child said, “If it’s four tally marks, youput a four number on it.” The children took turnsconverting the tally data into a frequency table bywriting numbers on the group chart.

After completing the tally chart and frequencytable for the rhyme, the teacher revisited an earlierquestion and tied it to the newly gathered data. Sheasked, “How can we use the chart and the fre-quency table to figure out how many words are inour rhyme?” One child said, “Count the ones in thetally marks.” Another child said, “I’m trying to addwith 4 and 2 and 2 and 1 and 1 and 1 and 1 andkeep going.” Next the teacher asked the children todecide whether there were more words thatrepeated in the rhyme or more words that did notrepeat. A discussion ensued, with children count-ing and arguing their positions. One child pointedto the bottom of the chart, saying, “There are toomany down here and less up here,” referring to thenumber of ones on the bottom of the table.

294 Teaching Children Mathematics / February 2005

Figure 1Students use different colors to track

repeated words.

Investigating word frequencyindependentlyThe teacher gave the students a blackline master ofthe rhyme “Hickory Dickory Dock” and coloredpencils. Each student circled all the occurrences ofwords in the new rhyme. Independently, studentsrecorded the data they collected on the tally chartunder the rhyme. Next, students were paired as“investigation partners” to discuss the resultsrecorded on their tally charts and then fill in theirfrequency tables (see fig. 2). This gave them theopportunity to identify words they may havemissed and check the accuracy of their frequencytables. Pairs of students shared the results of theirdata collection with the larger group. Adaptationsthat were made to the blackline master to ensuresuccess for students of varying reading and mathe-matical levels included giving students a color-coded copy of the rhyme so that they could use thecolors of the words to find the repeating words inthe rhyme (see fig. 3). Five students were given acolor-coded rhyme, a color-coded tally chart andfrequency table, and the color-coded rhyme sen-tence builder so that students could physically sortand manipulate the individual words in order todetermine the frequency of repeated words (see fig.4). The teacher invited the students to explain howthey could use their tally chart and frequency tableto answer the following questions:

• How many words are in the nursery rhyme?• How many different words are in the nursery

rhyme?• Which word occurs most often?• Which words occur more than once?• Do more words repeat or do more words not

repeat?

Lesson 2: Graphing WordFrequency in the RhymeThe teacher reviewed the tally chart and the fre-quency table completed during lesson 1 for “Rain,Rain, Go Away.” She asked the students if theycould think of another way to represent the data inthe tally chart and frequency table. One studentsaid, “In a chart.” Another student pointed to theclass weather graph and said, “A graph, like that.”With the help of the students, the teacher selectedthe word away from the rhyme and one child said,“We have two [pointing to the frequency table] sowe have to put two here [pointing to the units onthe graph].” Five children were selected to convert


Figure 2Students collaborate to fill in their frequency tables.

Figure 3Color-coded copy of the rhyme

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the data from the frequency table to the graph bycoloring in units on the class graph.

After the graph was complete (see fig. 5), theteacher asked the students questions to encouragethem to analyze the data.

Teacher. Who can tell me something aboutour graph?

Student. It’s like stairs.Student. Some of them have more and some

of them have less on them.Teacher. How can you tell how many words

are in the rhyme from the graph?Student. We see the boxes.Student. Count the boxes.Teacher. Which word occurs more often in

our rhyme, the word little or the word wants?Student. They’re the same height so they’re

the same.

The teacher also asked the students whichwords occurred most, which word had fewer repe-titions—go or come, which words occurred least,how many total words were in the rhyme, howmany words repeated the most and least, how manywords repeated two times, and how the informationon the graph was the same as or different from theinformation on the frequency table. These ques-tions helped students focus on aspects of readingdata from the graph and on making numerical com-parisons among the words in the rhyme.

Creating a word frequencygraph independentlyAfter the students completed the class graph, theywere paired with their investigation partners to cre-ate a graph using the tally charts and frequencytables that they completed independently duringlesson 1. As they began their work with a partner,the teacher asked what they would use to maketheir graphs. One student replied, “Look at the tal-lies,” and another student said, “Look at the num-bers.” As students in each pair transferred their dataonto a graph, they used two different colors. (Forexample, partner A graphed the first word in red,partner B graphed the second word in green, part-ner A graphed the third word in red, and so on; seefig. 6.) Students took turns transferring the fre-quency data to the graph. Some of the children rec-ognized that the two colors created a color patternand said, “We’re making a pattern!” One studentpair used a representation on their papers to showthe words that were their responsibility to graph.


Figure 4Sorting the words

Figure 5The completed graph

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(For example, fig. 7, Minh’s paper, shows how shekept track of her words and Christian’s words.) Theteacher also created adaptations for students ofvarying levels by using color coding for the wordson the student blackline master. This made it easierfor students to identify and tally the words.

As students finished their graphs, the teacherasked questions to stimulate their thinking. Forexample, the teacher asked, “How many wordsrepeated?” and the students replied, “Seven.” Shefollowed up with the question “How do you knowit is seven?” One student responded, “Because theyhave two boxes and four boxes, and that’s the onesto count.” When teachers are discussing mathemat-ics with students, asking these “how” and “why”follow-up questions is important to encourage stu-dents to justify their answers and articulate theirmathematical thinking.

Lesson 3: Focusing onLetter Recognition in theRhymeAt the beginning of the lesson, the teacher sangthrough the nursery rhyme “Rain, Rain, Go Away”once again, pointing to the words of the rhyme asshe read them. Showing the graph and the fre-quency table from lessons 1 and 2, she asked,“Which representation was easiest to read andwhy?” One student said, “The graph because I cansee it.” Another student said, “The tally chart. Iknow the tally marks two and two makes four.” Athird child replied, “The frequency number. I canread the numbers . . . four, two, two, one, one, one,one, one, one, one, one, and one.”

Next the teacher invited the students to look atthe letters that repeated in the phrase “Rain, Rain,Go Away.” Similar to the process used in lesson 1for finding words that repeated, the students circledletters that repeated in the four-word phrase usingdifferent colored markers. They used the samecolor to circle a letter each time it repeated in thephrase. (For example, they used purple to circle allthe occurrences of the letter r and red to circle allthe occurrences of the letter a.) Once they had cir-cled all the repeated letters, the teacher read the listof letters on the poster-size tally chart and fre-quency table. She modeled counting the number oftimes a letter appeared in the first line of the rhyme,and the students knew how to record this data in thetally chart from the previous lessons. Students vol-unteered to count the number of occurrences ofeach letter in the line “Rain, Rain, Go Away,” and


Figure 6Students use different colors to transfer their data onto a graph.

Figure 7Minh keeps track of her and her partner’s

words.

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wrote the number of tally marks in the tally columnon the chart. Then the students converted the dataonto the frequency table (see fig. 8).

Throughout and following this process, theteacher questioned students to explain their think-ing to their peers. A discussion began when theteacher asked the students the following question:“Did more letters repeat or did more letters notrepeat in this first line of the rhyme?” One studentthought about all the letters in the alphabet andsaid, “More letters not repeat because there’s toomany that don’t repeat like this one, this one, thisone, this one, this one, this one,” as he pointed toall the letters of the alphabet. Another student said,“I think it’s repeat because r and a and i and n andalmost every letter in this repeats,” as he pointed tothe letters in the four-word phrase. It was interest-ing that the students could find completely differ-ent answers to the question and yet provide justifi-cations that showed both their answers werecorrect depending on the unit of comparison.

Investigating letter frequencyindependently Students worked with investigation partners to findtally and frequency data on their own. In this por-tion of the lesson they used the first line of therhyme “Hickory Dickory Dock” and identified therepeating letters using colored pencils. Studentscircled all the occurrences of repeated letters in thefirst line of the rhyme and recorded the data on thetally chart. Next they worked with their partner toreview the tally data and complete their frequencytables. Pairs of students shared the results of theirdata collection with the larger group.

The teacher made adaptations for students duringthis lesson by giving them sets of cut-up letters of thethree words so that students could physically manip-ulate the letters. When students had a difficult timetracking the print across the page from the letter to thetally chart to the frequency table, one studentrequested a blank piece of paper to place under theletter. When other students saw this, they, too,requested a piece of paper to assist them in visuallytracking their work. The lesson ended with sharingand discussion. The teacher asked, “How many lettersdid not repeat?” This prompted another discussion ofwhat constitutes “repeating.” Students discussedwhich letters they should count and a disagreementarose about whether letters that had one tally mark orthe number one written by them on the frequencytable should be counted as repeating. Gia stopped herpartner and said, “Don’t count the two. Count the oneand the zero because that thing [pointing to the letterH] is not repeating. See, H is just one and just onemeans it not repeat again.” Her partner then said, “Oh,twos mean more times, and that means repeat. Likepatterns! It needs more than one—I get it.”

Lesson 4: Graphing LetterFrequency in the RhymeThe class reviewed the tally chart and the fre-quency table for the letters in the first line of “Rain,Rain, Go Away.” The teacher asked, “How can weuse the chart and the frequency table to figure outhow many letters are repeated in our rhyme?” Astudent responded, “Look how many tally marks orlook at the number on the table.” Another studentsaid as she pointed, “The number is easiest. Thereare too many tally marks to count. See, there, there,there, they keep going and going and going.” Stu-dents created a graph to display the data collectedduring lesson 3 (see fig. 9). They experienced moredifficulty in transferring this data onto the graph


Figure 8Converting the data onto the frequency

table

because of the greater number of units on the x-axis(twenty-six items in all for the letters of the alpha-bet). The teacher encouraged student volunteers to“drive your finger over to the letter, and then turnand drive your finger up the number of boxes youneed to color” (see fig. 10). Eight children wereselected to transfer the data from the tally chart andfrequency table to the graph.

After the class graph was complete, the teacherasked the students questions to encourage them toanalyze the data.

Teacher. Who can tell me something about ourgraph?

Student. Most boxes don’t have no colors.Student. Some of them have more and some of

them have less on them.Teacher. How can you tell how many letters are

in the first line of the rhyme from the graph?Student. We count all the boxes.Student. Yeah, a has lots of boxes colored. It’s

the most biggest.Teacher. Which letter occurs most often in our

line?Student. The a. It has lots and lots of boxes

colored.

The teacher also asked the students which let-ters had fewer repetitions, r or g, which lettersoccurred least, how many total letters were in thefirst line of the rhyme, how many letters repeatedthe most and least, how many letters repeated twotimes, and how the information on the graph wasthe same or different from the information on thefrequency table. These questions helped studentsfocus on aspects of reading data from the graph andon making numerical comparisons among the let-ters in the first line of the rhyme.

Investigating graphing letterfrequency independentlyNext the teacher paired students with their investi-gation partners and gave them a tally chart and fre-quency table for the letters in the first line of“Hickory Dickory Dock.” They took turns andtransferred their data into graph form using ablackline master of the graph. Pairs of studentsused two different colors for the graph and, again,many of them noticed that the two colors made acolor pattern. As with the group graphing, studentshad difficulty maintaining their place when trans-ferring their data from the charts to their graphs.The investigation partners shared their graphs with

the larger group and discussed their data. The teacher asked questions to help students

justify their answers and articulate their mathemat-ical thinking. For example, the teacher asked, “Is iteasier to determine how many letters repeated byreading the tally chart, the frequency table, or thegraph?” One student responded, “The tally chart.”When asked why, the student responded, “BecauseI can add it. One and one and one. That makesthree.” Another student argued, “No, the frequencytable ’cause it’s already added.” Yet another childexplained, “I can see it better on the graph. You justlook at the colored-in part.” The three students


Figure 9Graph of data from lesson 3

Figure 10Students “driving” their fingers to the letters

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decided to see who could “read the repeated lettersthe quickest.” When they read the data using eachof their preferred strategies, they were surprisedthat the student reading from the graph was thefastest at telling which letters repeated.

Teacher ReflectionThe mathematics foci of this investigation allowedstudents to collect, analyze, represent, and discussdata about information of interest to them. Animportant literacy focus on building students’ abili-ties to recognize letters and words was also present.Songs, poems, and particularly nursery rhymes areuniversally appealing to students in the early child-hood classroom and can be an excellent source ofdata for young children to collect and analyze.

Students learned how to apply data analysisconcepts to authentic mathematical situations.They collected and analyzed data, and workedcooperatively in groups and with partners. Theyhoned reading and verbal language skills as theydiscussed, justified, and solved data-analysis ques-tions. These investigations gave students an impor-

tant foundation in collecting, analyzing, and pre-senting data that they will later use in more com-plex work with data analysis and statistics.

ReferencesHall, Dorothy P., and Elaine Williams. The Teacher’s

Guide to Building Blocks. Greensboro, N.C.: Carson-Dellosa, 2000.

National Council of Teachers of Mathematics (NCTM).Principles and Standards for School Mathematics.Reston, Va.: NCTM, 2000.

Sheffield, Linda Jensen, Mary Cavanaugh, Linda Dacey,Carol R. Findell, Carole E. Greenes, and MarianSmall. Navigating through Data Analysis and Proba-bility in Prekindergarten–Grade 2. Reston, Va.:National Council of Teachers of Mathematics, 2002.

Authors’ note: The authors wish to thank LindaGarrett for her invaluable assistance in the kinder-garten classroom, and principal Kim Dockery forher support of the Professional DevelopmentSchool (PDS) partnership between George MasonUniversity (Fairfax, Virginia) and Westlawn Ele-mentary School (Falls Church, Virginia), whichmakes projects such as this possible. ▲


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