When asked to recall the time when they were taught statistics in middle and high school, most people will not have much recollection of the subject. Since most curricula incorporate statistics into other math courses rather than dedicate an entire course to stats, few students form foundational understanding of statistics. In many middle and high school math classes, statistics comprises a small fraction of the teaching standards required for the course, and because the focus of the course is *not* statistics, teachers often spend time only glossing the key concepts or handing out a few worksheets for homework. Precious class time is certainly not spent generating and/or empirically gathering data nor on connecting statistics to everyday life. Furthermore, statistics can easily be reduced to a string of formulas, and hurried teachers are likely to encourage their students to simply memorize when to use which formula and what values are necessary to solve it.

This description of the way statistics is often taught in math classrooms epitomizes what Skemp (1978) refers to as instrumental understanding. When students have an instrumental understanding of statistics, they do not comprehend how given formulas are derived or why certain formulas should be used in a given situation. However, there is a way in which statistics can be taught relationally, so that students grasp why and how statistics models and applies to everyday life.

Graphing calculators are a common-place technology in math classrooms. In many school systems, students either have their own graphing calculator or individual schools have classroom sets. Because of their accessibility, graphing calculators are a helpful tool in illustrating concepts in statistics. Consider the following class activity in which students play a role in generating, creating graphical representations, and analyzing data.

In this activity, students will first take their pulse over a time interval of 10 seconds, recording the number of heartbeats they counted in that time. Next, they will run in place for 20 seconds, immediately after taking their pulse over a 10 second interval again and recording the second number. The teacher will then collect the before and after heartbeats of the entire class, thereby generating a set of data.

Next, students will use their graphing calculators, such as the TI-84 Plus, to interpret the data. With their calculator, students can enter the before/after values into a table and from there plot various graphical representations, including histograms and box and whisker plots. Each of these charts enable students to see different aspects of the data:

- The histogram shows the distribution of the data.
- The box and whisker plot reveals minimum and maximum values, as well as the first quartile, median, and third quartile values.

Additionally, students can use their calculators to find calculations such as the mean, sample standard deviation, and population standard deviation. Comparing these calculations to the graphs will help students make connections about the meaning of the respective values.

The above explanation only eclipses the vast applications that graphing calculators have in the realm of statistics, but it does show how technology can be used to *relationally* teach students about statistics. As stressed by Skemp (1978), the heart of teaching relationally is equipping students to think for themselves. So if students cannot remember the formula for the standard deviation of a data set in twenty years, perhaps their relational understanding of statistics will have developed critical thinking skills that will serve them just as well.

Resources

Skemp, R. (1978). Relational understanding and instrumental understanding.

*Arithmetic **Teacher*, 152-161.

Love the name of the post! My favorite part is trying to come up with a witty title! I also loved the bullet points that made it easy to understand and was organized!

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Nice title and graphic at the beginning of your blog post! I thought you did a really good job explaining the importance of relational understanding in the mathematics classroom. Your post is insightful and well organized.

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I really enjoyed your blog! Your title is very clever!

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I like that you explained the activity and the different aspects that students could explore to interpret the data.

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