Popular Poll: You “Probably” Don’t Remember Much about Stats

five-shocking-credit-card-debt-statistics

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.

Advertisements

Effective Classroom Technology: Should Spreadsheets Make the List?

spreadsheet

As examined in the previous post, spreadsheet software, like Excel or Numbers, has the capacity to go beyond organizing statistical data in the classroom. By making tables and graphs, spreadsheet software can be used to explore properties of sequences and series or solve algebraic equations. The question, then, is whether spreadsheets provide the most effective means by which students can gain conceptual understanding of these topics. One important consideration when answering this question is evaluating if students have existing familiarity with spreadsheet software, as this will determine the types of tasks and level of scaffolding necessary to keep students focused on the math rather than syntax and technique. However, even if students have basic skills working with spreadsheets, other available technology often proves to be more effective.

For example, consider a problem in which a student is asked to solve the following equation:

3x^2-4x-7 = x+4

We will examine two methods by which technology can be used to solve this problem.

First, we will use spreadsheet software to solve for x. In order to do this, we will set up three columns: one for the values of x, one for the left side of the equation, and one for the right side. After generating a sequence of values for x, we can then apply a corresponding formula in each of the remaining two columns, in this way quickly generating many values for the left and right sides of the equation for given x-values. Next, we can analyze the data to determine where the left and right sides of the equation are equal.

One way to do this is by creating line graphs with the coordinates from the left and right sides of the equation, as shown below. By finding approximately where the two graphs intersect, we can narrow down the interval on which the graphs are equal (as we see here, between (-2,-1) and (2,3)). Alternatively, we can examine the table of values and find the x-values at which the value of the left side of the equation becomes less than the value of the right side, and vice versa.

img_1545

img_1543

Finally, we can use the same process as before to generate x-values between (-2,-1) and (2,3) in order to find more precise x-values for which the left and right sides are equal. In this problem, the solutions (when rounded to the nearest hundredth) are

x= -1.25 and x= 2.92

Using Desmos is the second method we will examine in order to solve this problem. Using Desmos,  a student can graph the left and right sides of the equation and then tap the points of intersection to find the x-values of those points, as shown below.

img_1544

Unlike the first method, which takes at least 15 minutes and a solid knowledge of spreadsheet basics, using Desmos to find the same solution takes only a couple of minutes, not to mention that Desmos has a much more intuitive interface. Furthermore, by having to manually generate smaller and smaller increments of x in order to find the approximate solution in the spreadsheet, students are distracted from visualizing the basic concept of solving equations: the expressions are equal when they have the same y-values for given x-values. Desmos clearly demonstrates this with easy to graph and easy to find points of intersection.

As seen in this example, spreadsheets are often not the most effective technology choice for illustrating a given concept. When a teacher decides the significance of a given technology, Dick and Hollebrands (2011) propose the following question is most important: “[W]hat questions could [the teacher] ask that [he or she] could not ask before?” With regard to spreadsheets, the answer to this question is very few new questions. Overall, the limited functionality and technical requirements of spreadsheets make them less than ideal for use in the math classroom.

Resources

Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to

support reasoning and sense making. Reston, VA: National Council of Teachers of

Mathematics.

 

Spreading Out with Spreadsheets

spreadsheet

In our technologically progressive society, new software, apps, and gadgets are constantly being introduced, ever widening the range of choices from which a teacher might select a technology for his or her classroom. With this prevalence of the latest and greatest technology, it is important for teachers to keep in mind that the most effective technology for demonstrating a given concept might be a familiar one. Spreadsheet software, like Excel or Numbers, is an excellent example of this. Although spreadsheet software has been around for years now, it still has useful capabilities that merit a place in a math classroom.

When one first thinks about spreadsheets, he or she probably thinks about a means by which to organize data entries, track a budget, or tally given quantities. If used in the classroom at all, spreadsheets are often limited to tasks similar to these, and they rarely find a place in math classes outside of statistics. But what if students could utilize spreadsheets to understand broader math concepts such as sequences and series or algebra?

By employing more of the features in software like Excel and Numbers, spreadsheets can find a place in a math classroom. For example, students might use Numbers to see the difference between a sequence of odd numbers and a series of odd numbers by first generating a column of counting numbers, 1 through a positive integer n. In the next column, the student could create a sequence of odd numbers by starting with 1 in the first cell of the column, and in the remaining cells in the column enter a formula such that the value of the cell equals the value of the previous cell plus 2. In the next column, partial sums of the series of odd numbers can be found by starting with 1 in the first cell, and in the remaining cells enter a formula such that the value of the cell equals the value of the previous cell plus that of the adjacent cell (which is the next odd number in the series). For clarity, see below:

img_1539

As seen above, students can make conjectures about the explicit formula for the sequence of odd numbers, which we know to be Sn=2n-1, and test their conjectures in the spreadsheet.

Also, students can see the difference between the sequence and series of odd numbers by comparing the line graphs of each, as shown below. This might help students to recognize that the partial sum of the series of odds for a given integer n equals n^2.

img_1538

A spreadsheet could also be used to graphically see the solution to an algebra problem. Take x^2-x+3 = x+2 for example. Students can create three columns, one for x, and one  each for the left and right sides of the equation. Using formulas, a student can quickly generate a range of values for each side of the equation for given x values. A student can then create a line graph comparing the left and right sides of the equation. The solution to the equation will be where the two lines intersect, in this case, where x=1. This activity can help students graphically understand what it means for two functions to be equivalent for a given x value.

A noteworthy observation is that when using spreadsheets, there is typically multiple ways in which to achieve a given task. As a teacher, it is important to ensure that students employ correct mathematical reasoning when generating formulas and analyzing data. Further, by comparing results, students might be able to probe further into thinking about how their own solution works, and why their neighbor’s method might too. In this way, spreadsheets have the potential to satisfy Dick and Hollebrand’s (2011) stipulation that effective technology serves “to push our students’ mathematical thinking forward or to probe how students are thinking mathematically” (p. xi). However, in order to push or probe student thinking, the skills needed to manipulate the spreadsheet cannot overshadow the mathematical concepts presented (Niess, 2005). Thus, it is important for teachers to gradually build assignment complexity in order to keep students focused on the math.

Investigations like the ones described above only scratch the surface of the concepts that can be demonstrated with spreadsheets. So before you reach for the newest math app, explore the possibilities of the spreadsheet software you probably already have!

References

Dick, T. P., & Hollebrands, K. F. (2011). Focus in high school mathematics: Technology to

support reasoning and sense making. Reston, VA: National Council of Teachers of

Mathematics.

Niess, Margaret L. (2005). Scaffolding math learning with spreadsheets. Learning and

Leading with Technology, 32(5), 24-25.