High leverage apps in the mathematics classroom: Wolfram Alpha

The National Council for Teachers of Mathematics (NCTM) claims that skillful teachers

can support students in learning mathematics more deeply with the appropriate use of technology

(NCTM, 2000). This claim requires teachers to learn about new technology at a rate that can be

overwhelming considering teachers’ other responsibilities. However, teachers’ facility with the

technology is important because they need to prepare students to identify the appropriate tool to

use in particular situations and to use these tools in the solving of mathematical problems

(National Governors Association Center for Best Practices & Council of Chief State School

Officers, 2010). This article aims to describe how Wolfram Alpha, a search engine, can be used

to support classroom practice and students’ technological literacy.

Wolfram Alpha is a computational knowledge engine based on Mathematica and A New

Kind of Science that was created nearly 30 years ago (A Wolfram Research Company, 2013).

This engine provides computable knowledge based on queries that are entered by users.

Wolfram can be accessed through the use of the Internet on computers and tablets to provide

students with opportunities to explore mathematics in ways that are not well supported by hand

held calculators or when using paper and pencil. Here we describe Wolfram as a solver, a

graphing tool, and a data compilation/analysis tool.

Wolfram Alpha as a Solver

Many of the students who are familiar with Wolfram are familiar with its solver

applications. Wolfram can solve many of the mathematical problems that students encounter in

their various mathematics courses. It can solve for variables in equations, compute the sum,

difference, and product of matrices, factor expressions, evaluate functions at a particular value,

use common geometric theorems to find angle and side measures of specific triangles, find the

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area under curves, and much more. Further, Wolfram allows students to enter mathematics

problems using words, numerals, variables, or a combination of those representations. Consider

the problem:

Factor 5x2-2x+7 and sketch the related graph.

Using Wolfram students can enter this as (1) a word sentence (i.e. factor five x squared minus

two x plus seven), (2) a typical equation (i.e. factor 5x2-2x+7), and (3) a combination of words,

variables, and numerals (i.e. factor five x^2-two x+7). In each instance, the app shows input

interpretation, which shows the user what Wolfram is solving. Figure 1 shows the output for this

problem. It contains the solution and provides two graphic representations, each with a different

interval over which the graph is plotted. These multiple representations will be discussed in

greater detail in the next section.

Figure 1 shows the output for the factoring problem.

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Another feature of Wolfram Alpha is its ability to operate on matrices. The “matrices

operations calculator” supports addition, subtraction, and multiplication of up to three matrices.

Consider the problem:

Perform all of the possible operations with Matrix A=[2 3 41 −1 10 ] and Matrix B=

[6 7 52 0 11].

To begin this problem, students should type into the search bar matrices operations calculator.

This screen provides set notation of three matrices. Students can change the numbers in matrix 1

and matrix 2 and then click the equal icon (Figure 2).

Figure 2 shows how students can enter the elements of each matrix.

Then students can select addition, subtraction, and multiplication from the dropdown

menu and explain their findings. The connections to matrix dimensions and the possible

operations may support students’ abilities to explain why addition (Figure 3A) and subtraction

(Figure 3B) are feasible operations and why multiplication is not. Students can also search

matrix multiplication as a phrase to understand the results.

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Figure 3A shows the addition and Figure 3B shows the subtraction of the matrices.

Wolfram Alpha and Multiple Representations

Wolfram’s use of graphical representations is also a great feature of this app, supporting

exploration and understanding of polynomials for students at any level. To support students’

comparison of y=8x4-6x2 and y=-8x4-6x2, students may receive the following prompt.

Record hypotheses about how the graphs, y=8x4-6x2 and y=-8x4-6x2 differ. Then enter the

two graphs separately into Wolfram and record the information that you see. Compare and

describe how the information that you recorded supports or disproves your earlier hypotheses

about how the graphs differ.

When students learn to graph polynomials, a large part of what teachers share with

students is how adjustments of the coordinate window can influence the information that can be

gleaned from the graph. When graphs of equations containing local and global maxima or

minima are entered into this app, two graphs are produced. One shows the local minima or

maxima while the other global view shows the end behavior. This tool can support students’

reasoning about end behavior and local maximums and minimums in addition to developing how

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to configure an appropriate viewing window. Figure 4 shows the graphical output for y=8x4-6x2

and y=-8x4-6x2. Note, the inputs “y=8x4-6x2” and “8x4-6x2” yield the same output.

Figure 4 shows the graphs for y=8x4-6x2 and y=-8x4-6x2.

In addition to the graphical output that is produced y=8x4-6x2, the app also provides a

myriad of other information including alternate or factored forms of the equation, the domain

and range of the function, and the interval forms of the local and global maxima and minima of

the equation. This additional information can support student exploration and understanding of

many polynomial functions. Figure 6 shows the non-graphical output for y=8x4-6x2 and figure 7

shows the non-graphical output for y=-8x4-6x2.

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Figure 6 shows the additional characteristics of y=8x4-6x2

Figure 7 shows the additional characteristics of y=-8x4-6x2.

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The opportunity for students to explore the differences in the graphical representations and

compare the alternative forms, domain and range, and extrema for these two seemingly similar

functions can support a deeper understanding of function and graphing on the Cartesian plane.

Wolfram Alpha to access data

In addition to preforming mathematical calculations and providing mathematical

representations, Wolfram can be used to search for facts about almost any topic like other search

engines. The difference between Wolfram and another search engine, such as Google, is that

Wolfram will return a computed “answer” based on objective data, algorithms, and models

instead of a list of websites. To better understand Wolfram, consider a question about cell phone

use:

Is using the cell phone equally popular all over the world?

After students type in “cell phone use” to the search bar, Wolfram returns numerical data in the

form of graphs and statistics, see Figure 8. The data is not in raw form but is aggregated and

returned in many different representations. This requires students to use their understanding of

many different statistical concepts to give the data meaning.

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Figure 8 shows the output for cellphone data.

On this one query alone, statistical terminology like Median is used, a histogram is given,

a less traditional graph is presented, and more. With all this data, students can determine their

own solution, which may or may not look similar to a classmate’s solution. As teachers, we look

for tasks like this that allow for multiple solutions paths and Wolfram Alpha has this ability built

in. For example, a more traditional approach to solve the question about cell phone popularity

would be for students to look at the histogram under ‘Distributions plots’ to look at the

variability in the data. Students could discuss the range of the data, the shape of the data and

make conclusions regarding the use of cell phones around the world. Another student might look

at the ‘Mobile cellular subscriptions map’ and state that the map has many colors showing all the

differences in popularity of cell phone use around the world.

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From here, teachers can create a classroom dialog about students’ conclusions to the

question about cell phone use around the world. Students can justify their conclusions to

classmates and critique the reasoning of others. In this example, most students will conclude that

cell phones are not equally popular around the world; however, this may indicate that students

may have ignored how the different sizes of the countries factor into their solutions? For

instance, since these graphs were based on counts of subscriptions, is it fair to compare the

number of subscriptions in China to the number in Germany? This may lead to conversations

about using data correctly and how statistics can be misleading, which supports critical thinking

skills.

In addition to having students search a specific topic, Wolfram also allows for students to

search topics that are interesting to them. Instead of assigning the statistical question, you can

encourage your students to develop their own statistical questions, which is an important part of

the statistical cycle (Wild & Pfannuch, 1999). Wolfram even allows students to analyze their

own activity on Facebook allowing students to access data that is personally meaningful to them

(Lee, 2013). The output data shows type of activity, weekly distributions, and statistics about

your posts, see Figure 9. Students could analyze their own activity and compare it to a

classmate’s. To do this, students type “Facebook Report” into the search bar. Then they can

click on “Analyze my Facebook Data”. They will be prompted to sign in. If they do not already

have accounts, follow the prompts to sign up for one. Once they have signed up, students will be

able to analyze their own Facebook usage data.

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Figure 9 shows a Facebook Users activity data.

Conclusion

Wolfram Alpha has the ability to be used in any high school mathematics classroom to

enhance instruction over a variety of mathematical strands. It can be used to explore and

compare algebraic concepts in depth, or as a way to answer statistical questions about the world

around us. In guiding our students to discover this tool, we are not only giving them the

opportunity to view analysis of their own Facebook use, but also giving them a high-powered

technological tool they can take with them to college or the workforce.

References

A Wolfram Research Company. (2013). About Wolfram|Alpha. from

http://www.wolframalpha.com/about.html

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Lee, H. S. (2013).Quantitative reasoning in a digital world: Laying the pebbles for future

research frontiers. In R. L. Mayes & L. L. Hatfield (Eds.), Quantitative Reasoning in

Mathematics and Science Education: Papers from an International STEM Research

Symposium, WISDOM e Monograph #3 (pp. 65-82). Laramie, Wyoming: University of

Wyoming College of Education.National Governors Association Center for Best

Practices, & Council of Chief State School Officers. (2010). Standards for mathematical

practice. In NGACBP & CCSSO (Eds.), Common Core State Standards for Mathematics.

Washington, D.C.: National Governors Association Center for Best Practices and

Council of Chief State School Officers.

NCTM. (2000). Principles and standards for school mathematics Retrieved from

http://www.nctm.org/standards/content.aspx?id=3456

Wild, C. J., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International

Statistical Review, 67(3), 223-248.

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