Decoding the Central Limit Theorem:
Using Multiple Representations to Enhance Student Success in Statistics
by Michael Litke, Assistant Professor of Mathematics, Goodwin University
About one-third of the way through Goodwin University’s STAT 167 course, there is a massive increase in the complexity of the material. The first unit covers descriptive statistics, which students often find very familiar. Many high school math courses, as well as some college courses, cover most of the topics within descriptive statistics. I’ve noticed that as my students work to take the intellectual leap from the first unit to the second, which covers inferential statistics, many of the new concepts seem just out of reach.
I decided that I wanted to find a way to empower students to construct their own meaning of these new concepts. A new idea came to me while reviewing the Universal Design for Learning (UDL) checkpoints for the representation guidelines: I wanted to offer the students a choice in the way the material is presented to them in this new unit.
Assessing the Assessment
I analyzed my presentation of the material and noticed a number of potential barriers in representation for the students. The teaching methods, learning materials, and assessment methods could all present difficulties for the students in their existing formats. My teaching methods for lessons involved presentations that were mostly lecture-based, direct instruction, teacher centered rather than student centered, and lacking background presented prior to introducing new, more complex topics. The learning materials offered practice and variety within the questions, but there was limited real-life application of the topics. In addition, there was no background knowledge presentation and little flexibility built into the materials for the diverse learners who come through our classrooms.
The accepted practice was for all students to take the same type of exam as an assessment. This created a “one-size-fits-all” approach to education, which in recent years has proven not to be the best instrument for measuring student understanding, especially when no other options for assessment are offered. After reviewing, I decided my plan was to start making changes to the teaching methods and the learning materials.
Checking with the Experts
As I investigated solutions for these barriers, I checked to see what some of the educational experts had to say. I found dozens of articles regarding the benefits of providing student choice in the classroom as well as multiple representations for content delivery. One article from the Journal of Education for Business focused on the field of statistics education and compared teaching methods and learning materials within a quantitative business statistics class. While that field differed slightly from the course I teach, I thought the ideas presented were translatable. The authors drew many conclusions, of which one in particular stuck out: “Although it has been suggested that using a variety of techniques for evaluating student work enhances the perceived quality of online instruction, it is not clear that all types of evaluative methods are equally suitable given the challenges of learning quantitative content online… The major recommendations from these findings were that (a) faculty should use a variety of techniques for delivering content and evaluating student work… And (b) students need meaningful and timely feedback.” — Sebastianelli & Tamimi (2011)
I decided that I could incorporate the idea of using a variety of techniques for delivering content into my solution regarding the UDL representation guidelines, and save the idea of meaningful feedback for when I addressed the action and expression guidelines.
I decided to craft my UDL solution as a three-pronged attack that will target multiple UDL representation checkpoints. I plan to incorporate changes to my teaching methods, the content within the lessons, as well as the learning materials themselves. This may seem like a major change to the curriculum, but in fact the three solutions presented are highly interrelated with one another. The first set of checkpoints my solution will address falls under the Language and Symbols guideline. I plan to clarify vocabulary and symbols as well as syntax, structure, and language. I want to support the students in decoding the text, mathematical notations, and symbols within this unit, and I plan to illustrate all this using multiple types of media.
The second set of checkpoints falls under the Comprehension guideline. I want to help the students by activating and supplying background knowledge for this new content. I aim to highlight patterns, critical features, big ideas, and relationships within this new, more abstract content, and I’d like to guide the students with information processing and visualization.
Part one of my solution will introduce the central limit theorem using computer simulations. There are many online applets (a couple examples here and here) that can be used to help students simulate experimental data collection without the laborious task of real-life sample surveys or scientific experimentation. These simulations, as well as the data they produce, will allow the theorem to be derived by the students. This offers them a chance for discovery as well as ownership of the material in the unit. My aim in using the simulations is to address the checkpoints that build background knowledge, big ideas, and pattern recognition and provide a decoding of the symbols in the formulas presented within the unit.
The second part of my UDL solution will provide students with multiple media content presentations. The goal is to let the students select how the information is presented to them. Whether they choose to attend the live class demonstration (in-person or livestream), watch any of the pre-recorded videos, or complete the scaffolded self-paced activities, students will get a chance to build the background knowledge in a way they see fit.
The third part of the solution will create a glossary of new terms and symbols, so numerous in this unit that they present a huge sticking point for students. This glossary will be student created within the classroom (using teacher guidance) and allow students even greater ownership of the material. The focus is to allow the teacher to assist students with clarifying the vocabulary, symbols, syntax, and structure of the content in this unit. To summarize how all these solutions fit together: the instructor will now supply background knowledge of the central limit theorem using simulations while providing multiple options for presentation of this new, complex learning material. During this process, new vocab, symbols, and language will be aggregated into the student-created glossary of terms.
The Stage Is Set
I have always believed that a constructivist approach to math education provides students with more opportunity for success. I really want to focus on enhancing the student understanding in this unit to open more opportunities for success in the later units that build upon this content. An old analogy compares learning math to building a house: it’s very difficult to attach the roof when the foundation for the floors still hasn’t been secured. Assisting students in building a firmer foundation of the concepts within this unit will make the construction of the later concepts of the course that much easier. I also think that providing student choices makes the course more open and inviting. Providing support with the abundance of new terms in this unit will take a lot of the burden off the students so they can focus more on the abstract mathematical content itself. I’m already seeing success with the application of some of these solutions with more students this semester scoring higher and performing better on assessments within this unit compared to scores from earlier semesters. I think when I fully implement all the solutions, there will be a significant improvement in the retention of the material, enhancing the success of the students both within this course and in their future.
Goodwin University is recognized for its expertise in the application of the Universal Design for Learning framework to the design of learning experiences across a variety of learners and settings. Click here to learn more about Goodwin’s commitment to UDL and the resources that are available to educators.
Michael Litke is an assistant professor of Mathematics at Goodwin University. He received a bachelor’s and master’s degree in Math Education from Central Connecticut State University. He also received a master’s degree in Curriculum and Instruction from the University of Saint Joseph. Michael spent 15 years as a middle and high school math teacher before joining the Goodwin faculty as a mathematics and statistics instructor.