Copy of Quantitative Reasoning in a Technological World: Laying the Pebbles for Future Research Frontiers

Olive (2010) gave examples of new media

and research with these media:

• Logo and Robotics
• Online Game Environments (single and multi-user)
• Face-to-Face Collaborative Problem Solving

Let's focus on technological capabilities that

enable ways of interacting with quantities

with data and chance

Technology can be used as:

an amplifier for statistical processes, and

a cognitive reorganizer to promote conceptual understanding (Ben-Zvi, 2000)

Teachers and students should use technology to:

• Automate calculation and graphical representations
• Emphasize data exploration
• Visualize abstract concepts
• Design and conduct simulations
• Investigate real life data
• Engage in collaborative practices

(Chance et al., 2007)

Information on the web

A major problem is the ability to search for retreive and interpret information

In a study by Kingsley et al (2011), they assessed information literacy skills of first year dental students.

Nearly half of students (n = 70/160 or 43%) missed one or more question components that required finding an evidence-based citation.

A significantly higher percentage of students who provided incorrect responses (n = 53/70 or 75.7%) reported using Google as their preferred online search method (p < 0.01).

In contrast, a significantly higher percentage of students who reported using PubMed (n = 39/45 or 86.7%) were able to provide correct responses (p < 0.01).

Visualizing Complex Data

and

Visualizing Data in Complex Ways

Visualizing Data in Complex Ways

A glimpse at a recent study by Lee et al. (in press)

Ability to overlay measures on a graph and augment

a graph can create more complex representations and help transform data into useful information to support development of decisions and claims

Study of a stratified random sample of 62 teachers across 8 institutions using dynamic statistical software that can link multiple representations of data.

Sample Results

Teachers who were answering broad questions used significantly more types of augmentations (p=0.00063), and added more statistical measures to graphs (p=0.032).

Teachers whose reports indicated that dynamic linking occurred (57%) used more graphical augmentations than those teachers who did not dynamically link (p=0.013). There was no significant difference between these two groups as to their frequency of adding statistical measures to graphs.

Those that used a combination of linking, augmenting, and adding statistical measures tended to make more connections across variables and use these to support a claim about the contextual question.

Visualizing Complex Data Sets

Complex as......"Many Parts" or Multivariate

Students seem to engage easily with multivariate data and representations

Gould (2010) and Ridgeway, Nicholson, & McCuster, (2008)

Nicholson, Ridgway & McCuster (2010)

"hypothesise that working with visual representations of multivariate data at an early stage would help students to develop mental models of possible relationships between multiple variables which would give them a stronger conceptual basis for considering the formal statistical analysis they will meet...." (p. 1)

http://www.statlit.org/pdf/2010NicholsonRidgwayMcCuskerICOTS.pdf

FREE Data Visualization Tools

GapMinder

Google Public Data

Many Eyes

Tableau Public

Let's see some students making sense of

these multivariate visualizations

http://www.gapminder.org/for-teachers/

What New Mathematical Knowledge and Practices Arise With....

• Access to Data and Information
• Visualizing Complex Data
• Building Models & Simulations

Building Models and Simulating Phenomena

• Need to quantify something in the phenomena of interest---non trivial
• Need to make assumptions explicit and clear
• Need to design for encourage reasoning between data and models
• Technology can help make models explicit
• Technology can help give us different ways to examine data, in dynamic and static ways

Let's consider a task together and how technology can be used to reason about the quantities involved

Counties keep records of the number of live births that occur throughout the year. Consider the following data about live births recorded in two North Carolina counties in 2004:

A) Hyde County recorded 56 births and slightly less than 43% were male.

B) Martin County recorded 314 births and slightly less than 43% were male.

Assuming that the probability of a male birth is 50%, is event A or B more likely to occur, or are the two events equally likely? Explain.

data source: http://www.schs.state.nc.us/SCHS/data/births/bd.cfm

Task from Lee, Hollebrands, & Wilson, 2010

Current Stats for US: 51.2% Males at Birth

https://www.cia.gov/library/publications/the-world-factbook/geos/us.html

Let's imagine these events as repeatable and build a model and simulate births for each county sample size.

When you hear the name Jane,

what or whom do you think of?

Jane Fonda

fitness guru?

Lady Jane

1966 song by the Rolling Stones?

Jane Seymour

actress?

or

3rd wife of King Henry VIII?

G.I. Jane

1996 movie with Demi Moore?

Jane's Addiction

1980-90's rock band?

Question:

If you were to meet a female named Jane, she would most likely be approximately which of the following ages?

a. 21

b. 35

c. 49

d. 62

e. 75

f. 90

How did you reason about this?

How might a techno-savvy person approach this?

Google it?

The first hit for searching "Jane's age" is..

A link to a discussion about the following problem

"Jane's age is 2.5 times david's age. In 15 years jane's age will be twice of davids age.

How old was jane 8 years ago ?"

hmmm....

where might we go for access to information on how to QUANTIFY a best guess at the age of a person named Jane.

Let's use a Computational Search Engine

Wolfram Alpha

http://www.wolframalpha.com/

Let's consider access to new technologies.

What mathematical practices and knowledge

can be developed when

Using Multi-Touch & Kinetic Environments

If you are asked to describe a "blob" to a child, what would you say/do?

gestures give a window into

a person's conception of an idea.

DESIGNED gestures need to map well

onto intended conception

Mathematical Fidelity

“In order to function effectively as a representation of a mathematical “object,” the characteristic of a

technology-generated external representation must be faithful to the underlying mathematical properties of that object.” (Zbiek, Heid, Blume, and Dick, 2007, p. 1174)

Gestural conceptual mapping....

mapping between a gesture (physical embodiment) and the digital representation of the domain

Segal (2011)

Let's consider a few ways that gesture

is used in iPad Apps and with a Kinect

References:

• Abrahamson, D., & Cendrak, R. M. ( ) The odds of understanding the law of large numbers: A design for grounding intuitive probability in combinatorial analysis. In J. Novotná, H. Moraová, M. Krátká, N. Stehlíková (Eds.), Proceedings of the Thirtieth Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 1 – 8). Charles University, Prague, Czech Republic: PME.
• Abrahamson, D., & Wilensky, U. (2007). Learning axes and bridging tools in a technology based design for statistics. International Journal of Computers for Mathematics Learning,12(1), 23-55.
• Chance, B., Ben-Zvi, D., Garfield, J., & Medina, E. (2007). The role of technology in improving student leanring of statistics. Technology Innovations in Statistics Education 1(1).
• GOuld, R. (2010).
• Ireland, S. & Watson, J. (2009). Building a connections between experimental and theoretical aspects of probability. International Electronic Journal of Mathematics Education 4(3), 340-370.
• Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning 12(3), 217-230.
• Konold, C., & Kazak, S. (2008). Reconnecting data and chance. Technology Innovation in Statistics Education 2(1).
• Lee, H. S., & Lee, J. T. (2009). Reasoning about probabilistic phenomena: Lessons learned and applied in software design. Technology Innovations in Statistics Education 3(2).
• Lee, H. S., Hollebrands, K. F., & Wilson, P. H. (2010). Preparing to teach mathematics with technology: An integrated approach to data analysis and probability. Dubuque, IA: Kendall Hunt Publishers.
• Lee, H. S., Harper, S., Driskell, S. O., Kersaint, G., & Leatham, K. (In Press, 2012). Teachers' statistical problem solving with dynamic technology: Research results across multiple institutions. To appear in Contemporary Issues in Technology and Teacher Education.
• Nicholson, Ridgeway, McCuster (2010)
• Olive, J. (2010).
• Pratt, D. (2000). Making sense of the total of two dice. Journal of Research in Mathematics Education, 31, 602-625.
• Pratt, D. (2005). How do teachers foster students’ understanding of probability? In G. Jones Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 171-190). Kluwer Academic Publishers.
• Pratt, D., Johnston-Wilder, P., Ainley, J., & Mason, J. (2009).
• Ridgway, Nicholson, McCuster (2008)
• Rubin, A., Hammerman, J., & Konold, C. (2006). Exploring informal inference with interactive visualization software. In A. Rossman & B. Chance (Eds.), Working cooperatively in statistics education: Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.[Online: http://www.stat.auckland.ac.nz/~iase/publications/
• Sedak (2011)
• Stohl, H., & Tarr, J. E. (2002). Developing notions of inference with probability simulation tools. Journal of Mathematical Behavior 21(3), 319-337.
• Tarr, J. E., Lee, H. S., & Rider, R. (2006). When data and chance collide: Drawing inferences rom empirical data. In G. Burrill (Ed.), Thinking and reasoning with data and chance: 2006 yearbook of the NCTM (pp. 139-149). Reston: VA: National Council of Teachers of Mathematics.
• Watson, J. M., & Kelly, B. A. (2004). Statistical variation in a chance setting: A two year tudy. Educational Studies of Mathematics 57(1), 121-144.
• Zbiek, R. M., Heid, M. K., Blume, G., Dick, T. (2007).
• Zieffler, A., Garfield, J., delMas, R., & Reading, C. (2007). A framework to support research on informal inferential reasoning. Statistics Education Research Journal 7(2), 40-58.

Quantitative Reasoning in a Digital World:

Laying the Pebbles for Future Research Frontiers

Hollylynne Lee

North Carolina State University

Access to Data and Information

Many students use and create multivariate datasets every day (e.g., GeoTagging, Contact Lists on Phone, Song lists on iPod)

Students need to understand structure of data

(Gould, 2010)

Data is not the same as information. Data can be transformed into information.

Olive & Makar, 2010, p. 169

Most larger families have

either more boys or more girls

p. 18 Segal (2011)

Segal, 2011

Distribution of proportion of Males born in 2004

in each of the 100 counties in NC

Young Children Using Multi-touch to produce Many ladybugs

(though this is not my child, I have seen my son do the same thing and searched for a video rather than record my own)

I wonder ...

how/if producing many objects simultaneously helps with understanding quantities greater than one?

how/if this helps with considering one-to-one correspondence and one-to-many correspondence?

Mathination--A Multi-Touch CAS-like system on iPad

How might the various multi-touch actions create an understanding of:

equality

algebraic operations

equivalent expressions

equivalent equations

I wonder....and worry....

KinectMath

using a Kinect for Windows or Kinect for XBox to engage in gestural experiences.

I wonder how the gestures for transforming a function by stretching, shrinking, translating, etc map onto the conceptual understanding one develops using these gestures.