By: Kerrie Bowes
"If there is a single phenomenon of the past decade or two that can be said to be revolutionizing our lives—at home, in the work place, and in society—it is the accelerating power and ubiquitous presence of technology. Offering incredible capacities and speed for data collection, analysis, and dissemination, technology is ushering in the ‘information age.’ It is transforming our communication patterns, the work we do, and our leisure time." (Sampson, 1998) Technology, graphing calculators, computers, the Internet, the information super highway, we’ve come a long way in math since the invention of the abacus, but what does this mean for educators and what does it have to do with NCTM and Achieve standards?
" From the very first days of public education in America our nation
has held the preparation of students for productive adulthood as a basic
objective. Education is expected to be experience in knowledge building
and skill development that allows us to understand and engage with resources
and opportunities of economy and society." (Sampson,1998)
Education and schooling started out with a main goal of preparing the youth
of a society to run it and keep it going. The focus of school was job skills
and this focus is no different today than it was in the past. What has
changed, are the skills necessary to be part of today’s workforce and how
we prepare teachers to accomplish this goal.
"America’s competitiveness in the world markets of tomorrow is clearly
seen to rest on our technology competencies. The new jobs of the 21st
century require a wide variety of basic technology skills. And they will
demand an ability to continue to learn as new technologies develop, new
ways of working emerge, and new productivity is realized." (Sampson,1998)
It is important for us to realize that if the schools and education does
not focus on basic skills of technology and the ability to utilize it,
we are missing one of the main goals of education, to prepare America’s
youth for tomorrow’s future. We as educators, must also realize that once
we learn a particular skill that the learning is not over, we must continue
to learn and stay on top of new advances as well as teach this skill to
our students, the idea that learning never stops.
As more and more school districts begin to put a focus on technology
like the state of Maryland has with a goal of, "By the year 2003, Maryland
will ensure that . . . Every learner has access to and use of technology
in classrooms and schools. Technology supports achievement, enabling learners
to be independent, competent and creative thinkers, as well as effective
communicators and problem-solvers.", we will see an increased pressure
on teachers and those involved in education to implement and use technology.
(Maryland State Dept., 2001)
This is great and wonderful and nobody would disagree that it is necessary
for our students to use and understand technology, the problem is no one
has thought about how to prepare the teachers and those in education to
utilize the technology they are given. The other problem is getting the
technology into the schools.
"From using a Palm Pilot to managing a retirement portfolio, life in the 21st century is drenched in data, dominated by computers and controlled by quantitative information…Virtually all workers—custodians, secretaries, farmers, electricians, building contractors, store managers, architects, lawyers, graphic designers and others—need to deal with quantitative concepts and use reasoning skills in their jobs. Indeed, those who lack mathematical know-how face a very real threat of being left behind—as citizens, as consumers and as workers… Traditionally, it has been sufficient for only some of us to be mathematically proficient. To fully participate in the 21st century, however, all citizens must be comfortable and fluent with mathematics." (2001) Achieve continues to say that American students do well with straightforward skills tests but fail when it comes to looking at a real-life problem and choosing what math to use to solve it. I think it is here that technology can help. I would like to now turn my focus to these two sets of standards and look at where technology fits and can help to achieve these standards.
Starting with the standards for Algebra in the NCTM Principles and Standards, "In the vision of school mathematics in these Standards, middle-grades students will learn that patterns can be represented and analyzed mathematically." (2000, p.297) The Standards also state that students should be given visual models of the algebra they are studying. One way to accomplish this is with the graphing calculator. The standards go on to say that "Students should use technological tools to represent and study the behavior of polynomial, exponential, rational, and periodic functions, among others…The expanded class of functions available to high school students for mathematical modeling should provide them with a versatile and powerful means for analyzing and describing their world. With utilities for symbol manipulation, graphing, and curve fitting and with Programmable software and spreadsheets to represent interactive processes, students can model and analyze a wide range of phenomena. These mathematical tools can help students develop a deeper understanding of real-world phenomena. At the same time, working in real-world contexts may help students make sense of the underlying mathematical concepts and may foster an appreciation of those concepts." (2000, p. 297) The idea behind studying algebra and using technology then becomes a look at its real-world applications which returns us back to the main goal of education.
There will be those that argue why can’t we teach this the old-fashion way with paper and pencil. And in fact, with many of the functions we can still do this fairly easily. But then it becomes a measure of the students spending time to accurately calculate a solution and create a graph of information. Instead they could be using the technology tools to help calculate the answer and provide a visual in a short matter of time, leaving the rest of the time for the students to analyze the data for its accuracy in the event and what in means. One of the expectations in the NCTM Algebra standards is that students will "judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology." (2000, p.296) As Achieve stated, one of our downfalls when compared to other nations in mathematics, is that are students are really good at straight forward computations but we lack the skills to see the problems conceptual. By using technology students can focus on why the results occur, where the information fits and where it will be useful to use, instead of just calculating an answer.
Throughout the Algebra Standards chapter of NCTM Principles and Standards, many references are made to use and benefit of using technological tools such as the graphing calculator. For example, " A major goal in the middle grades is to develop students’ facility with using patterns and functions to represent, model, and analyze a variety of phenomena and relationships in mathematical problems or in the real world. With computers and graphing calculators to produce graphical representations and perform complex calculations, students can focus on using functions to model patterns of quantitative change. " (2000, p. 227) By using the graphing calculator to do such things as create a scatterplot and find a curve or line of best fit, which becomes "more apparent" when the students have the visual aid of the calculator, students are able to test their theories and connections about the set of data more easily then if they were to do it by paper- and – pencil. (NCTM, 2000,p.228-229)
Technology can also help provide a real-life quick visual for the concept of rate of change. Using a calculator based ranger, which is a type of motion detector that connects to a graphing calculator, students can move about the room and see the effects it has when they pick up the pace, when they slow down and when they remain still. This allows them to see what happens what rate of change and a constant are. But still the underlying message at the end of the chapter is, that all this is possible when "supported by careful planning by teachers." (NCTM, 2000, p. 230)
Achieve has similar goals when it comes to the standards for Algebra, they seem to have a real focus on real-world experiences. While Achieve does not focus much on the technology aspect of achieving standards, in fact they say "Because these expectations focus on the mathematical competence to be expected of students entering high school, they rarely make any reference to the role of calculators ( or other pedagogical aids) in classroom instruction. With few exceptions, students should be able to carry out most calculations implied by these expectations without the aid of a calculator. Although it is appropriate for students to use calculators to learn about mathematics and to help solve problems, they should be able to demonstrate understanding and competence both with and without the aid of calculators." (2001, p. Math Expectations) I feel realistically many of their goals would be better achieved if they took advantage of the technology available and made it part of their document on improving middle school mathematics.
When I look at Achieve’s goals for Algebra, what first shocks me is
that they expect all students to have an understanding of quadratic and
exponential functions in the middle school years, this is a focus that
the NCTM standards have placed for the high school level Algebra students.
It seems to be quite a huge leap and I feel that technology would assist
greatly. Students should know how to graph some simple quadratic equations
with pencil and paper so they understand why it takes the form it does,
but when it comes to real-life applications I feel students should use
the tools of the real world. To me, it means more for a student to be able
use the graph of a quadratic equation, in the example of a ball being tossed
and see that while two roots may show up on the screen, only one is true,
because you will not have negative time or height and the person would
not toss at a level of zero. It is great that they can calculate the roots
by hand but do they understand them in real world applications ? Allowing
students to use devices like a Calculator Based Ranger to plot the travel
of a bouncing ball, and being able to understand and predict where it will
be at one time, will be more beneficial then a calculated number solved
by completing the squares or factoring on a piece of paper. If you look
at the example answers that Achieve lists to their problem sets, many are
calculator or computer generated graphs, none are hand sketched. While
Achieve says that these are not student samples of responses but rather
examples for teachers of the more complicated issues or topics in this
area, they still chose to present them using a computer or calculator sketch
rather than a hand-sketched answer, which is what we should be allowing
the students to do. (2001, p. Illustrative Problems)
Also in the final set of goals, they state that students should be able
to: " relate the solutions of a quadratic equation to the graph of the
corresponding quadratic function; and solve a quadratic equation by factoring,
finding where the graph of the function meets the x axis and completing
the square." (Achieve, 2001) Achieve also says that
it students should begin with factorable cases of quadratics and move on
to more complex ones, at the same time they should be graphing them and
thinking about their shape and relationship between the coefficients and
the zeros. (2001, Appendix D) I believe that much
of these goals could be aided by technology especially with the creation
of the graphs and through calculator software like TI-Interactive, which
allows students to manipulate one piece of a quadratic at a time and see
its effects on the graphs. I may have misinterpreted their stand on technology
but it seemed to have a hidden message that technology may be a source
of the problem for why our children are lacking and their lack of input
on technology seems to also relay this message.
The National Council of Teachers of Mathematics believes that calculators
are important tools to support the learning of mathematics and that knowing
how to use calculators intelligently is part of being prepared for an increasingly
technological world." (Straight Talk, 1999)
The National Council of Mathematics has made technology a big focus in
their newest version of standards, it even received its own principle section.
NCTM makes a big point to focus on technology as a benefit to students
pointing out that it can be a benefit to those students with special needs.
"Technology offers teachers options for adapting instruction to special
students needs."(NCTM, 2000,
p. 25) They’ve taken a position of great support for the use of calculators
and technology in the classroom saying, " Appropriate instruction that
includes calculators can extend students’ understanding of mathematics
and will allow all students access to rich problem-solving experiences."
(NCTM Position, 1998) Others have joined in saying,
"the use of technology and realistic data serves as a powerful student
motivator. With technology, students can visualize mathematical concepts
and look for creative solutions to real-world problems." (Spicer,
2001 ) Getting all student involved in more challenging mathematics
and using the data that is all around us is something that Achieve is in
great favor of and I think many of their standards could be achieved or
enhanced by the use of technology.
Looking at the Geometry standards for both Achieve and NCTM there
does not seem to be much difference and I feel here technology could be
beneficial in reaching the standards for understanding transformations
like reflection and rotation. Software like Geometer’s Sketchpad allows
students a visual to manipulate and see the effects of rotation and reflection.
While I feel basic calculations for items like area and volume should really
be calculated by hand, the problems could be made more challenging by using
some algebra. Students could be given the volume and some of the measurements
leaving one measurement to be found that might require some manipulation
of numbers on a calculator at first. "… technology can help some students
who struggle with computation move on to mathematical concepts." (Spicer,
2001) In this case the calculator could help a student who may not
be able to rearrange a volume formula to find the missing piece, while
at the same time giving them a chance to possibly discover how it might
be done. The final use of technology in the Geometry standard would be
for use with the Pythagorean theorem and square roots, allowing a more
accurate answer or understandable answer to a student. Instead of the solution
being the squareroot of5, students could calculate it to be 2.24, which
would have more meaning for the student.
The Data Standard also provides great opportunities for the use of technology. The Internet has become a great source of up to the minute data. Many websites contain real-life data that students can manipulate and analyze. Software packages and the graphing calculator also allow students to create all sorts of graphs to analyze their data with. Students can create box- and – whiskers, histograms and scatterplots. Analyzing the graphs for correlations, trends and items that don’t fit. They can predict and determine why some items do not match the rest, all of which can be valuable skills. But may favorite use of technology and one that I feel would benefit both the NCTM standards and Achieve’s standards has to do with Probability. Using a random number generator students must create their own probability test, test a situation. For example, if they wanted to test the theory that 40% of all automobile accidents occur when it is raining, they would build a test that would use 5 consecutive digits (like 0-4 or 1-5) each digit representing 20%. Because the problem they are testing states that 40% of all accidents occur when it rains they would pick two of the digits to represent that it was raining and the other three to represent that it was not. They would then program the calculator to random spit out digits that they had assigned and then analyze their results. It’s real-world and they have to think out how to test results and analyze them, all of which could be valuable skills in the work force.
In the area of Numbers, I would have to side with Achieve that it is
best for students to know their basic facts. However the calculator can
be beneficial as quick feedback to a student who has completed their work.
If for example, the teacher were busy assisting another student, a student
who had finished their work early could double check their answers with
a calculator and receive immediate feedback to whether or not they had
done the problem right.
I feel that Achieve has chosen high standards for our students to
achieve to keep them up to the level of other foreign countries. I also
see their point that students get tired of reviewing the same material
year after year in math. I feel that it could benefit children if we challenged
them more but I feel technology should be included. Our ultimate goal as
educators is still to prepare bright hard-working individuals who will
have skills that our society will be in demand for, and currently those
skills include skills with technology and problem solving. " When technological
tools are available, students can focus on decision making, reflection,
reasoning, and problem solving." (NCTM,
2000, p. 24) All of which are the skills needed in our work force.
Achieve focus seems to be on "mathematical proficiency" and being able
to reason and use math with ease. They have stated that straightforward
calculations are not a problem for our students, so why not encourage the
use of technology and calculators?
Technology use in the schools has brought about with it much controversy.
Some parents are asking what happened to the basics? They question that
if the students are using calculators are they still doing the math? And
the answer is yes. Students still need to be able to estimate and justify
if the answer the calculator gave them was correct. Also generally students
don’t use calculators for simple calculations but rather them use them
when working with difficult numbers and more complex problems. "Graphing
calculators help students visualize problems, discover mathematical theorems
on their own, instantly check the validity of their answers, test out their
own hypotheses, and explore different ways of solving problems." (Pomerantz,
2001) Calculators and computers have been shown
to get more students involved in learning. They are great motivators, because
they can help struggling students to improve their skills. "Students who
are easily distracted may focus more intently on computer tasks, and those
who have organizational difficulties may benefit from the constraints imposed
by a computer environment. Students who have trouble with basic procedures
can demonstrate other mathematical understandings, which in turn can eventually
help them learn the procedures." (NCTM,
2000, p. 25) It is important to also realize that in the work force
and outside of schools, calculators are used everyday. Calculators are
now used on SAT and placement tests. "In the Third International Mathematics
and Science study, students with the highest scores used calculators more
frequently in mathematics instruction than students with the lowest scores."
(Straight Talk, 1999) If we are going to compare
ourselves to other countries and what their students are doing in math
then we must being willing to use the same methods that they do.
I have discussed the need for technology and how it fits into many
of the new standards arising everyday in mathematics education, now the
question becomes how can we implement it and how will we prepare teachers
to use it? As stated earlier, technology is ever changing and we need to
continually learn and stay up to date to help our students have the most
current knowledge and know how. In order to prepare our students to use
technology, we must take the time to train our educators. NCTM came up
with recommendations for the used of technology and calculators. Their
recommendations were as follows:
NCTM has also come out with a list of questions to help guide teachers, to tell whether or not it is an appropriate use of the calculator.
As of right now there are limited sources of professional development
to achieve these goals. In order to have teachers who are better prepared
we must encourage our school districts and Universities that prepare teachers
to bring more technology course in. We need to speak up as educators to
get more programs like Teachers Teaching with Technology and Corporations
like Comcast, who offer courses to school districts and now offer online
courses to teachers. We also need to encourage our local government to
come up with the funding for such programs. "Technology-related professional
development competes for attention and resources against many other professional
development concerns. This competition worsens as districts confront the
need to increase their investment in technology-related professional development
because preservice programs have, as yet, failed to incorporate technology
into teacher preparation." (McLauglin ,1998)
Again the solution becomes getting more technology courses or courses that
incorporate technology into the preservice teacher programs at many Universities.
The other issue then becomes how to reach veteran teachers. "Convincing
teacher educators and preservice program leaders to focus on technology
has been difficult. Most teacher educators join preservice programs after
serving for a decade or more as K—12 classroom teachers. This means that
few teacher educators come from schools where advanced educational technologies
were available, let alone used instructionally or administratively. These
educators have probably not seen technology enhance student learning, may
not know how to use technology effectively, or are not convinced that technology
warrants all the attention it has received in the media as a potentially
major contributor to student learning." (McLauglin
,1998) One solution that has risen as a result is the idea of "professional
development schools (PDSs), which comprise one or more schools involved
in education reform paired with a teacher education program." (McLauglin,
1998) In this program you take trained teachers and get them involved
in helping out to train others in their schools. The cost is better than
the expense that would occur from repeatedly having to host profession
development where you higher others to come in and train your staff.
While Achieve does not offer much in the way of professional development
in technology they do state, that they will provide professional development
to teachers who have an elementary background in mathematics education
with the necessary skills and knowledge they will need to teach the fundamentals
of data analysis, geometry, and algebra. It would be encouraging to see
them add technology to the list.
Technology has become a crucial part of our everyday world and has
influenced the way we do many things, including how we teach mathematics.
After researching the pros and cons of technology, I have found that the
biggest help is that of a voice. In order to get more support and funding
for technology in the schools, educators must speak up with real evidence
of how technology is helping in education and what benefits it offers.
It is also important for educators to speak up about the fact that technology
is not replace basic computational skills that were once done with pencil
and paper, but rather it is enhancing the learning process and making math
and other subjects more accessible to all students. Students must be able
to experience the skills that they will be doing later in life not merely
calculating the answers to a set of practice problems. In the "real-world",
students with have access to computers, calculators, and texts to help
them reach the solution to a problem. No one must really memorize all the
math formulas that are used, but they must be able to understand and manipulate
them. Many groups have come up with what they feel are the standards and
goals that should be achieved by students in middle school mathematics
and they have come up with a series of assessments and evaluations to test
them. I see many of these organizations setting up these goals but providing
little resources of help and professional development for teachers to attain
these goals. Instead of constantly changing goals, standards and curriculum,
we need to focus on what we want our children to know, how we can get there
and what training needs to be provided to educators. No longer should the
issue be what to teach, how to stay up with other children of the world,
or a debate over whether or not to use technology. Technology is here to
stay and has been shown to help even the most unmotivated student to achieve.
Technology also provides skills that are necessary in our work force and
if they don’t receive the know how and uses of technology in the schools,
where will they get it and are schools still meeting their goal of preparing
America’s youth if they don’t include technology? We should focus on what
we need our children to be able to do to keep this country moving ahead
industrially, economically and technologically.
Achieve Incorporated, (2001). Foundations
for Success: Mathematics for the Middle Grades. Washington, D.C.:
Author.
McLauglin, R.T., (1998). Infusing Technology into Teacher Education. Hands On!, 21(2).
Maryland State Department of Education, (2001). Technology in Maryland Education. Retrieved from http://www.msde.state.md.us/divisions/doisd/instructtech/webdesig.html
National Council of Teachers of Mathematics, (1998). Calculators and the Education of Youth. NCTM Positions. Retrieved from http://www.nctm.org/about/position_statements/position_statement_01.htm
National Council of Teachers of Mathematics, (2000). Principles and Standards for School Mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics, (2001). Calculators-One of Many Tools to Enhance Students’ Learning. Straight Talk about Issues in Mathematics Education. Retrieved from http://www.nctm.org/news/speaksout/spksoutcalc.pdf
Pomerantz, H., (2001). The Role of Education in Mathematics Education. Teachers Teaching With Technology. Retrieved from http://www.t3ww.org/t3/therole.htm
Sampson, B.C., (1998) Technology for Education…Why Bother? Hands On!. 21(2).
Spicer, J. (2001) Why Use Calculators(and other
Technologies) in the High School Math Classroom? Eisenhower National
Clearinghouse. Retrieved from http://www.enc.org/topics/edtech/learning/document.shtm?input=FOC-000702-index