Developi loping ng Proficiency roficiency in in Gr Grad ade e 6 Common on Core re St Stat atisti istics cs By Kristen en Kent nt & Eb Ebon ony y Hitch ch Faculty ty Mentor: or: Dr. Randa dall ll Groth oth Salisbur sbury y Un Univer ersi sity ty NSF F REU EU PATHW HWAYS S 2014
Introduction Students have difficulty finding and interpreting the mean, median, and other statistical measures of center appropriately (Zawojewski & Shaughnessy, 2000). Purpos rpose: e: explore and develop students’ thinking about graphical representations of data and finding appropriate measures of center How can students’ proficiency in regard to Grade de 6 Common mmon Core e Mathem ematics tics Standa dards ds about t statis istica tical l mea easures res of cen enter er be de e devel elope ped? d?
Theoretical Framework We used the Adding it Up framework to conceptualize mathematical proficiency. It includes the following five strands: Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition
CCSSM Learning Progressions for Statistics The Common Core State Standards Writing team (2011) described key transitions and competencies in learning statistics in accordance with the Common Core State Standards in a learning progressions document. Key ideas: s: Begin with a statistical question Displaying data in dot plots Characterization of data distributions by measures of center Using their knowledge of division, fractions, and decimals in computing a new measure of center — the arithmetic mean , often simply called the mean
Additional Guiding Concepts from Literature Groth & Bargagliotti (2012) explained how to engage all students in statistical investigation using the Common Core: 1. 1. Formulati ulating ng Quest stions ions 3. An Analy lyzin zing g Data ata 2. Co Collectin cting g Data ata 4. Inter erpre reting ting Resu sults In addition to finding, using, and interpreting measures of center, we focused on helping students understand the mean’s relationship to other measures of center, such as median and mode.
Methodology Pa Partic icipants ipants Procedures cedures CCSS S Instr structi tional onal Goals: Time Frame: me: Ten weeks Understand that a set of data collected to # Of Of Pa Partic ticipa ipant nts: : Four answer a statistical question has a students & two teachers distribution which can be described by its center, spread, and overall shape Pa Participatio ticipation n Rate: : 100% Recognize that a measure of center for a Seven weekly one one-hour our numerical data set summarizes all of its sessions ns in addition to pre values with a single number and post assessment sment Display numerical data in plots on a number line, including dot plots , intervie views ws. histograms, and box plots. as well as For the privacy of the describing any overall pattern students, the following Relating the choice of measures of center pseudonyms will be used: and variability to the shape of the data distribution and the context in which the Cody dy, Flyn ynn, , Millie, lie, Giselle lle data were gathered
Methodology Data Gatheri ring & A Analysi sis Two cameras record entire hour 1. session PATHWAYS S Cycle e of Integr egrat ated d Teachin hing g Playback video & transcribe each 2. & Resear arch ch word spoken, as well as any emotions/movements Find strengths & weaknesses in 3. students’ learning in terms of the 5 Strands of Mathematical Proficiency Make data-based conjectures 4. about how to foster students’ learning These conjectures = basis for 5. developing following week’s lesson
Initial Assessment Results Overall most of the students lacked conceptual understanding when it came to finding ing typic ical al values es and when it comes to comp mpar arin ing g statistica atistical l measures sures. For an example in this problem below most students chose the median for Theater A and the mean for Theater B.
Displaying Data (Week 2, 3, 4) Le Lesson on Form rmats ts: Students generated data from rolling dice Represented data using dot plots ( conceptual understanding ) Organized & compared multiple data sets Identified middle clump ( strategic competence ) Discovered method for finding the middle data value (median) by crossing off values from each side of the graph ( procedural fluency )
Understanding Mean (Week 5, 6) Lesson on Format ats: Discussed differences in the shapes of graphs Described how each statistical measure (mean, median, mode) is affected with various data Students understood mean as a number that “evens out” or “balances” a distribution used snap cubes as data values By redistributing the snap cubes (or family members) they could easily see how the mean represented a “fair share” for the data set
Measures of Center (Week 7, 8) Lesso sson n Forma mats: ts: Presented skewed data sets to students Example: 24 Starburst candy distributed unevenly amongst four students & two teachers Asked to find the average or typical number of candies that each person received: 1, 1, 1, 1, 1, 1, 1, 1, 1, 19 During the final lesson, students analyzed a data set showing salaries of individuals in a small town:
Example EH: So with the doctor or being ng out what do you think k the typical cal income me is? Flynn: 0 Millie: Even the doctor made them fall. KK: Why do you think k 0? Giselle elle: : Because use it is the mode. e. KK: Oka kay. So do you think nk the media ian is still a good d representat sentation on becau ause se that t just t changed ged? ? Do you think nk that t is still good? Millie: Yeah because he carry’s the paper and he gets $200 for it. Giselle : The firefighters don’t get anything for it and he saves lives and houses. Dr. Groth : The stop sign guy don’t look to happy because he got 200 and you’re saying that the average is 0. He doesn’t like that.
Post Assessment Results Overall students gained conceptual understanding when it came to finding the best statistical measure to represent a typical value. Students gained procedural fluency and strategic competence in selecting and constructing data displays. These aggregated displays helped them locate the centers of data sets. Initi tial al Assess essmen ment Post t Assess essmen ment
Reflection Helped students begin to reason conceptually about measures of center, but did not have time to delve into formal measures of variability (also prescribed in the Sixth-Grade Common Core) Challen llenge ges: s: Achieving every CCSSM Standard for Grade 6 Statistics is Connecting how changing some of the data could affect the measures of center Switching back and forth between dot plots and case value bars Suggest gestion ion: Begin to develop these ideas before sixth grade
References Bremigan, E. G. (2003). Developing a Meaningful Understanding of the Mean. Mathematics Teaching in the Middle Schoo l, 9(1), 22-26. Common Core Standards Writing Team. (2011). Progression for the Common Core State Standards for Mathematics (draft), 6-8, Statistics and Probability. Retrieved from http://commoncoretools.files.wordpress.com/2011/12/ccss_prog ession_sp_68_2011_1226_bis.pdf. Groth, R. E., & Bargagliotti, A. E. (2012). GAISEing into the Common Core of Statistics. Mathematics Teaching in the Middle School, 18(1), 38-45. Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.N., & Philips, E.D. (2004). Data About Us . New York: Pearson. National Research Council. (2001) Adding it up: Helping Children Learn Mathematics . J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Zawojewski, J.S., & Shaugnessy, J.M. (2000). Mean and Median: Are they really so easy? Mathematics Teaching in the Middle School , 5(7), 436-440
Recommend
More recommend
