Important Note: This post is the updated version of an original manuscript that I submitted to Mathematics Teacher magazine for publication. It was rejected but I still thought its content was too valuable to not share with the wide community of mathematics educators. I am posting it here in my blog part of the MTBoS's #MTBoSBlaugust 2018 Challenge. I hope many of the blog readers will find this post useful to them in some way or fashion. Please, leave comments, suggestions, and such so that I may post additional information if need be. Thank you and here we go!
or solve the
equation x^2 + x + 41 = 0 have to be replaced
by conceptual approaches to testing students’ knowledge and understanding of
math concepts. The following examples (see Figures 1 – 3) illustrate how Wolframalpha solves such factoring
exercises and what Photomath displays when one points the camera of their iPad8
(or mobile device) at the above equation.
This paper explains how the author rewrote his mathematics
test problems so that they address challenges and opportunities that technology
brings to the arena of mathematical learning. The test problems, as shall be
illustrated, have been rewritten in such a way that it would personalize them
enough to allow the use of technology to not be shied away from or, as is the
current practice, be completely shunned from the assessment process.
With the advent of mobile devices and the
adoption of one-to-one technology programs throughout many of the nation’s
schools and districts, it has become necessary to reassess how problems in
tests are conceived and written. It would be counterproductive if educators
expected their students who were brought up in the world of selfies and Pokeman
Go1 to suddenly stop using their technology tools when they deal
with math assessment. We allow our students to use technology to explore
mathematical concepts and investigate conjectures using programs such as Desmos2,
Geogebra3, LoggerPro4, Photomath5, and
Wolframalpha6 to only inform them that they cannot use these very
same technologies during tests and quizzes.
Technology should become part of summative
assessments of our students in mathematics just as we have been allowing it to
be a tool of formative assessments and lab inquiries. It is about time for our
schools to adopt approaches to test taking with computer technology just as did
Denmark7 and other Scandinavian countries did in their school
systems. These countries’ students have outperformed our American students on
STEM tests, which strengthens the case for the need for a change in our
approach to assessment.
So, what is the solution?
In my opinion, the initial and quickest solution to the
seeming problem of technology use in tests and quizzes is to adopt an approach
to assessment that veers away from fact-based and information-laden quiz and
test items. In other words, anything that a program such as Desmos, Geogebra, Photomath,
or Wolframalpha can easily and quickly solve should no longer be considered as a
good enough assessment item for this day and age. Example questions such as
factor x^3 – 8
Figure 1
Figure 2
Figure 3
If a smart phone can provide an answer to a math problem for
a student, it means the problem is no longer useful as a genuine tool for
assessing understanding. It is merely an exercise in memorization of facts or
rehashing of algorithmic steps. In other words, such exercises would have
informed the teachers of the fact that their students became good repeaters of
what they were shown in class but certainly not whether they truly understood
such processes or not. Therefore, “problems” like those shown above should either
be modified to take advantage of such technologies or should completely be replaced
by conceptual questions. Questions that probe deeper into the students’ arsenal
of knowledge and true understanding of mathematics rather than merely elicit
the students’ useless powers of memorization and empty rote “learning” must
become the norm in math assessments. This means that the new type of problems
that we must begin to generate for our students to assess their comprehension
of mathematical concepts should take advantage of the available technology
rather than suppress it.
We owe it to our students to equip them with all possible
tools of problem solving including those that rely heavily on the use of
technology. Another solution to the lack of effective use of technology in math
classes is to emulate what many science subjects have begun promulgating, the use
of coding in science curricula. We, math educators, must start introducing our
students to mathematical problem solving through the use of coding and computer
programming. This approach has the benefit of preparing our students adequately
for the digital revolution that they are part of on one hand and equipping them
with the problem solving tool that they would employ in other fields of
learning on the other hand. In the remainder of the article, samples of
problems that emphasize conceptual understanding and encourage the appropriate and
more effective use of technology shall be given.
Problem Solving for the New Millennium
Sample Problem 0: [Use of Technology
Example 1; Redesigning the Equation Exercise]
(a) Assessment Item 1: [Photomath Time!] Point your mobile device with
the Photomath open at the equation x^2 + x + 41 = 0 and report what
the app shows as a solution and as a process.
(b) Assessment Item
2: [Wolframalpha Time!] Go to the website http://www.wolframalpha.com, enter the
equation x^2 + x + 41 = 0 and report what
the program shows as a solution and as a process.
(c) Assessment Item
3: [Compare, Contrast, Analyze, & Reflect Time!] Compare and contrast
the two solutions that are given by the two applications, analyze them using
what you learned in class about quadratic equations, and then conclude by
writing a deep personal reflection regarding this whole problem solving
experience. Recall that part of reflecting is to probe deeper and ask new
questions that enable you to expand your knowledge of the concepts at hand.
Comments: In this sample problem, the use of the
two chosen technologies becomes part of the problem solving process itself. This
reframing of the test problem encourages students to use their mathematical
skills (including rote learning type) and true understanding of the math they
learned about quadratic equations to make sense of the given results and the
different approaches taken by the two technologies. As such, the technology
becomes part of the mathematical learning process itself rather than worrying
about whether the students would cheat by using external technology sources or
not. With this new approach, we are seamlessly using technology to enhance how
we assess our students rather than let them fall into the trap of plug and chug
and hunt for answers haphazardly without much genuine understanding of the
underlying concepts.
Sample Problem 1: [Use of Technology
Example 2]
(a) Assessment Item 1: [Predictions Time!] Discuss whether log(-e) has a value or not while making sure that you refer to
specific sections from our eTextbook/Textbook that deal with the concept of
logarithms. Note: Refrain from using outside sources for now. Thank you
(b) Assessment Item
2: [Investigation Time!] Once
you answer part (a) and only after you answer it, go to the website http://www.wolframalpha.com, enter in
the textbox therein log(-e), and then analyze and reflect upon what the program
shows in the “Exact result:” section.
(c) Assessment Item
3: [Analysis & Reflections Time!] Summarize the findings, analyze them
for mathematical accuracy and validity, and then reflect upon what you learned
from this problem solving process.
Comments: In this sample problem, the use of
technology becomes part of the problem solving process itself. It forces the
students to wrestle with notation use (log vs. ln) as well as the
interpretation of Wolfram’s results that involve complex numbers and having x values that are not part of the domain
of the logarithmic function. The students’ true understanding of the math they
learned about logarithms is tested by the technology they now have to employ
during this assessment activity. As such, the technology becomes part of the
mathematical learning process and the problem solving skills and critical
thinking we would like our students to engage in while interacting with
mathematical ideas.
In the subsequent samples, a different approach to testing is taken
while devising math problems for tests and quizzes. Conceptual understanding still
reigns supreme and is always the focus of assessment. In this case, problems
are written in such a way that each student would create their own problem by
using personal data that would only be theirs and would make the solution
personal as well. This supplants and replaces the old way of having multiple
test versions (A, B, C, etc.) and reduces cases of cheating and may open the
door up for clever educational opportunities. An example would be to ask two
students to compare and contrast their personal problems along with their responses
and even go as far as having them critique each other’s work and assess it as
well.
Sample Problem 2: [Conceptual
Personalized Open-Ended Problem Example 1]
Write your own quadratic equation whose
left hand side coefficients are all prime numbers that are close to the
numbers in your birth day, birth month, and two digits in your birth year
respectively. The number on the right hand side has to also be
a prime number close to one of your pet’s ages. Use the discriminant to
predict what kind of solution(s), if any, your equation will have. Solve your
own personal equation to verify your prediction. And finally, use Desmos and
Woframalpha to further corroborate the validity of your results. Show all the
work, share your screenshots of the used program with your instructor, and
reflect on your findings.
Comments: In this sample problem, the problem is
personalized for each student because of the fact that none of the coefficients
of the equation are given beforehand. In addition, the reflection portion and
the variety in program usage add another layer of personal touch to the
students’ responses. Once again, technology is used part of the problem solving
process.
Sample Problem 3: [Conceptual
Personalized Open-Ended Problem Example 2]
Al-Khawarizmi stated to Thaabit
Ibn Qurra that the age of their niece Alya is the solution to the equation “The
square root of x plus four is equal to four!” To which Thaabit replied,
"No way! That means that Alya would have been born this very instant!"
Who do you agree with and why? Justify your response using concepts from this
specific chapter, and reflect upon your findings and conclusions. Make sure you
also verify your findings using any technology tools of your choice. Make sure
you share screenshots with the instructor and if time allows, compare and
contrast the outcomes of the two different technology tools that you opted to
employ.
Comments: The open-endedness of this sample problem
allows for great latitude in the various responses the students may come up
with, and as such personalizes student answers. The reflections would also add
another layer of personalization to the problem solving process.
Sample Problem 4: [Conceptual
Personalized Open-Ended Problem Example 3]
What astute mathematical questions
does the following diagram conjure up in your mind? Your devised questions must
incorporate concepts you have learned in this chapter. What are the
corresponding answers to your own questions and what are your solution processes?
Reflect! [Note: Challenging and creative questions that would also use technology tools
of your choice would be rewarded more than easy trivial responses that are
technology-shy ones.]
Comments: This open-ended question was devised for
a test on irrational numbers, operations with irrational numbers, and functions
and equations involving radical expressions. The nature of the question allows
for the students to be creative, broad-minded, and intentional as far as the
concepts they would choose to tackle from within the chapter on irrational
numbers.
Sample Problem 5: [Conceptual
Personalized Open-Ended Problem Example 4]
Kewl Jane Toboggan claims that if a/b = c/d is true, then (a + b)/(c + d) = b/d has to be true also. Cool Bob Sled, on the other hand,
claims that if a/b = c/d is true, then (d + c)/(b + a) = c/a instead would have to be true. Using what you have
learned in the sections of this chapter, state who you agree with and why? Your
arguments must be fully supported and are clearly logical. In addition, make sure you reflect and verify your findings
using any technology tools of your choice while making sure that you share screenshots
with the instructor and if time allows, compare and contrast the outcomes of
two different technology tools that you have used.
Comments: This open-ended question was devised for
a test on rational expressions, operations on them, and functions and equations
involving rational expressions. The tendency of most students is to use a
specific limited set of numbers in their attempt to prove or disprove the
validity of the given “equalities”. It is instructive that students learn to
prove such identities rather than rely on limited numerical verifications. It
would also be informative to figure out what technologies students would end up
using to try to “prove” such expressions and in what clever ways would they do
so.
Comments and Closing Thoughts
Technology can and should become a tool to enhance the
learning process of math students. This enhancement does not have to stop when
students are being assessed/tested. A smooth transition from learning toward
assessment has to occur so that students’ positive uses of technology would
naturally carry over to assessment situations.
The examples provided earlier in this article are but
droplets in the sea of wonderful configurations of assessment items that
instructors, content providers, and textbook authors may consider while
generating new math problems for technology-friendly and concept-laden assessment
purposes.
Dedication: I
dedicate this and everything I do to the one educator who helped me
appreciate the value of learning in the grandest of fashions, my late father
Ibrahim Nadji (RhA!) Thank you Didi and may
Allah (SWT) reward you for your dedication to raising educated citizens!
Acknowledgement: I would
like to thank the anonymous referees of Mathematics Teacher magazine who
provided me with valuable feedback that enabled me to smooth out numerous rough
edges of the original text of my manuscript.
- Pokeman Go is an Augmented Reality app developed by Nintendo company.
- https://www.desmos.com
- https://www.geogebra.org
- LoggerPro is an interface and modeling software developed by Vernier company, http://www.vernier.com
- https://photomath.net/en/
- http://www.wolframalpha.com
- http://news.bbc.co.uk/2/hi/uk_news/education/8341886.stm
- iPad is a tablet (mobile device) developed by Apple company.
