Future Visions of Information Technology in Math Education:

Design Heuristics for Vignettes Embedded in the Standards

Chris Dede
Graduate School of Education
George Mason University
Fairfax, VA 22030
cdede@gmu.edu; www.virtual.gmu.edu

In implementing any set of curriculum standards, a challenge faced by teachers is developing specific learning activities from general statements about the knowledge and skills students should acquire. Inevitably, the words used in stating a standard evoke in readers a broad range of images about what constitutes exemplary practice. This ambiguity results in widely divergent materials, pedagogies, learning activities, and instructional assessments all labeled as "standards compliant."

Recognizing this problem, NCTM Standards include vignettes that portray exemplary educational situations in which a standard is applied. Writing such images of effective practice is not easy, since their goal is to depict an illustrative method for meeting a standard without implying that this is the only possible way to do so. Particularly when technology is involved, writing these scenarios is challenging because:

In a sense, this type of scenario is like a very short science fiction tale that unobtrusively briefs the reader about the future while simultaneously telling a story about goals achieved through effective action.

The new NCTM Standards will incorporate "illuminations" to aid in conveying images of effective practice. These illuminations will include written vignettes, but may take other forms as well (such as multimedia case studies). Illuminations will have the dual purpose of illustrating exemplary practice and inviting more fundamental reflections about the assumptions, beliefs, and values that underlie conventional instructional approaches. As such, they must convey the context, content, and process of mathematics education. This paper discusses design heuristics for written vignettes in the new NCTM Standards, as well as more general issues for any type of illumination.

Illustrative Vignettes Setting a Future Context for Information Technology in Education

To illustrate how a vignette can implicitly sketch a future technological context, below is a scenario I recently wrote for distribution at the 1998 Florida Educational Technology Conference (Dede, 1998).

By the Dawn’s Early Light: Distributed Learning with Technology

"Take a deep breath," Maria told her mother, "then blow it out into the balloon." Deftly, as soon as her mother had finished, Maria used a plastic clamp to pinch the neck of the special balloon, then measured its circumference. "All done, Mama!" she said, writing down the number in her notebook. Her mother sneezed, then sank back on the coach with a smile of approval. Even though her sinuses ached–and that deep breath had not helped–she enjoyed helping Maria with her daily homework. After all, participating in the allergy study project not only involved her child more deeply in school, but also subsidized the Web-TV box that provided the family access to sports and entertainment websites. Maria was navigating to the appropriate site, then logging her mother’s lung-capacity figure into the national database. Her little brother watched, fascinated by the colored visualizations displaying the complex ecological, meteorological, and pollution factors that predicted today’s likely allergic responses in Maria’s region of the city.

Maria’s teacher, Ms. Grosvenor, was also sighing out a deep breath at that moment, but not into a balloon. While eating a Ho-Ho for breakfast, she was using her home computer to access a different part of the allergy study website, a section with guidance for teachers about how to cover today’s classroom lesson on regional flora. Her preservice education a decade ago had provided some background in ecology, but–now that fifth grade students were mastering material she had not learned until the end of high school–Ms. Grosvenor frequently used the website to update her knowledge about allergenic plants. Sometimes the sophisticated multi-level model scientists and doctors were developing–made possible by the micro-regional data supplied by learners all across the country–made her head ache for reasons other than sinuses! On the other hand, at least the students were quite involved in this set of science activities. Discussions in the "Teachers’ Forum" of the website reaffirmed her own feeling that most teachers would rather have the small hassle of keeping up with new ideas than the constant struggle of trying to motivate students to learn boring lessons.

At the same time, in her elementary school’s computer Lab, Consuela was threading her way through a complex maze. Of course, the maze was not in the Lab, but in the "Narnia" MOO (a text-based shared synthetic environment developed around the stories by C.S. Lewis). Her classmates and fellow adventurers Joe and Fernando were "with" her, utilizing their Web-TV connections at their homes, as was her mentor, a small bear named Oliver (in reality, a high school senior interested in mythology who assumed a Pooh-like "avatar" in the virtual world of the MOO). Mr. Curtis, the school principal, watched bemused from the doorway. How different things were in 2003, he thought, students scattered across grade levels and dispersed across the city, yet all together in a shared, fantasy-based learning environment a full hour before school even starts! (The school building opened at the crack of dawn to enable lab-based Web use by learners like Consuela, whose family had no access at home.)

"The extra effort is worth it," thought Mr. Curtis. Two years into the technology initiative, student motivation was high (increased attendance, learners involved outside of school hours), and parents were impressed by the complex material and sophisticated skills their children were mastering. Even standardized test scores–which measured only a fraction of what was really happening–were rising. Most important, young girls such as Consuela were more involved with school. Because of their culture, Hispanic girls had been very reluctant to approach adult authority figures, like teachers–but the MOO had altered that by providing a "costume party" environment in which, wearing the "mask" of technology, children’s and teachers’ avatars could mingle without cultural constraints. "I wonder what this generation will be like in high school," mused Mr. Curtis...

This image of the future is intended to stimulate ideas about what uses of educational technology may be possible and beneficial in the near future. Such vignettes are not meant as blueprints or recipes for what should be, but as visions that help to undercut the unconscious assumptions we often make about what cannot change. Their goal is to aid readers in being more creative in thinking about the future.

My vignette was not discipline-specific, but Jim Kaput and Jeremy Roschelle wrote a vignette that focused on mathematics education for the Florida Educational Technology Conference "Futures" collection I edited (Dede, 1998):

A Gig in Cyberspace

Jeremy Roschelle and Jim Kaput
University of Massachusetts, Dartmouth
With ideas from Roy Pea, Gene Klotz, and Jim Spohrer

"Great day in Idaho," thought Robin Ursala Lerner, as she wandered across the room to wire into her job as Dr. Learner, Professional Development Facilator with the MathForum. "What’ll it be today?, she thought, "Perhaps helping a teacher find technology resources to build a curriculum unit on the recent Mars landing. Perhaps digging up examples of student work with graphing calculators for the State assessment team, or helping the innovative CyberBike development group find some teachers to test their bike-top learning tutors." That’s what R.U. Lerner liked about her job, always interesting. That and the mountain bike trail just outside her door.

"Is this Dr. Lerner?" An expectant face lit up R.U.’s screen. "My name is Lynn O. Vator, from Boston’s ThinkBig Middle School with some questions about the SimCalc calclets from the Educational Object Economy." R.U. recognized SimCalc as the NSF project that is developing interactive media for teaching calculus to urban middle school students - part of the national urban calculus initiative. (By the time they reach high school, these students are expected to study nonlinear dynamical systems.) And she worked daily with the Educational Object Economy, the world’s largest bank of interoperable software components for learning. Quickly, Dr. Lerner sized up the teacher on the other end of the line. Clearly Lynn is a cutting edge teacher, as she was going after big mathematical concepts like rate, limits, and approximation with students who would never have taken a calculus course in the 20th century. But that is what 21st century learning technology is all about -- democratic access to big mathematical and scientific ideas.

"Is your handheld calculator wired in?" Dr. Lerner asked. "Well actually I’m on cellular Internet on my subway ride home" responded Lynn. "Awesome. Then show me the graphs you’ve been working with." As Dr. Lerner and Lynn talked, they fluently shared mathematical notations back and forth, as if they were working at a shared whiteboard. But of course, Lynn was on a $99 handheld graphing videophone, while Dr. Lerner had a 100 gigahertz VR Pro installation. As Lynn showed some of the graphs her students made, Dr. Lerner pulled up some related lesson plans from the professional development workshops the Forum had sponsored recently. Because they could both see and manipulate the graphs and simulations, they could freely sketch ideas and share examples that gave life to the rather abstract mathematical concepts they were talking about. "What a relief! Those lesson plans will really help in the coming weeks," said Lynn. "But I have a more difficult problem. I want students to see the link between our velocity graphs and position graphs."

"Hmm" mumbled Dr. Lerner, scanning her directory of educational software developers. "I bet there is someone wired in at the Center for Innovative Learning Technologies (CILT) who could help us. Jay Coder has open office hours right now. I’ll dial him in." Soon the software developer had joined their conversation. "By syncing into the history of your conversation today, I can see just the examples you’ve been working on. And I think I have a script that does just what you want." While they waited, Dr. Lerner explained to Lynn thatwith dynamical component software, graphing calculators were no longer as static as they used to be, and Jay could in fact modify the software in her calculator while she waited. "That should do it. Do you see a new icon on your calculator?" "Why yes!" Lynn blurted out. "Give it a tap " urged Jay. "Hey that’s great, I’ve got both position and velocity graphs now... this will really help my students understand the idea of slope and rates! But there’s till a few things to change, like can we animate taking the limit?"

"Well, I’ll leave you two to work out the details," said Dr. Lerner, "I’ve got an appointment now." Just outside a certain mountain slope beckoned, and with the new interactive graphing capability on her CyberBike, Dr. Lerner had some limits of her own to push.

URLS for more information

MathForum – http://mathforum.org

SimCalc – http://www.simcalc.umassd.edu

EOE – http://trp.research.apple.com

CILT – http://www.cilt.ltc.vanderbilt.edu

This image of the future captures a variety of themes in the NCTM Standards, helping the reader imagine new ways advanced information technologies may empower learning, professional development, and assessment.

Modifying Existing Vignettes in the Current NCTM Standards

Sometimes, an existing vignette in the current NCTM Standards can be modified to reflect the inclusion of sophisticated educational technology. For example, in 1992 I altered two vignettes in the NCTM Standards to illustrate how adding technology could make their portrayal of educational excellence more powerful (Dede, 1992). Even though these were written before the emergence of the last two generations of information technology (including the World-Wide Web), the forward looking depictions I created six years ago have stood the tests of time and technological evolution quite well.

This scenario incorporates advanced telecommunications into Vignette 8.2, Standards for the Evaluation of the Teaching of Mathematics, from the NCTM Professional Standards for Teaching Mathematics (1991).

A Preservice Observation Experience

Pierre Bordeaux, a master teacher, has an ongoing working relationship with the teacher education classes taught by his friend Sally Witt, a mathematics educator from Southwest State University. The preservice students periodically watch Pierre teach a lesson; he is skilled at creating classroom environments that fosters the development of each learner's mathematical power is crucial [Standard 8, Teaching]. In exchange, each of Sally's students regularly uses the network to work with a pupil in Pierre's class, providing him with valuable aid in tailoring instruction to individual needs. The two instructors jointly publish the results of research they conduct on teaching styles, using both the classroom pupils and the preservice teachers as sources of data.

Whenever the methods students are observing, both the videocameras and the liveboards in Pierre and Sally's respective classrooms are linked to provide an annotated videoconference between the two rooms. Each preservice teacher's notepad is also linked to the notepad of one of Pierre's learners, so that each of Sally's students can "look over the shoulder" of his or her particular pupil.

The day set for an observation has arrived, and the methods class is watching their liveboard and notepads. Mr. Bordeaux has informed his class that they are being observed by Dr. Witts' students and has reminds them that the preservice teacher assigned to each learner will later give some suggestions and comments to that pupil at their next virtual meeting. His class quickly loses any self-consciousness as the lesson unfolds.

Mr. Bordeaux begins the lesson by reminding his students why they are meeting for an unusual experience: beginning the class by lecture-and-discussion rather than by working as usual in small groups on real-world problems having mathematical content [Standard 1, 9-12 Curriculum]. In many of their groups, situations calling for the analysis of right triangles have arisen. Some students have been using synthetic approaches to analyzing these geometric problems; others have been applying algebraic concepts [Standards 7 and 8, 9-12 Curriculum]. Now, the time has come for all the groups to share their different representations and to integrate these understandings under the overall rubric of the Pythagorean theorem.

Mr. Bordeaux tells them that the Pythagorean theorem was known to the ancient Chinese, Greeks, and Egyptians. He adds that a file on each of their notepads contains references to books that discuss the evolution of the theorem–and of mathematics–in the three different cultures. The "knowbot" in his desktop machine has already automatically forwarded these references both to the Civilization teaching staff to see if some of these themes can be woven into their curriculum and to the Library so that the materials are ready for downloading to any students who request them. {A "knowbot" is a semi-intelligent agent that can be programmed to act on a user's preferences. Knowbots can save people time by automating fairly simple tasks such as sorting electronic mail, responding to routine messages, and filtering publications for articles of potential interest.} Several minority students in the class are interested, as they have been learning through virtual field trips to other countries how non-Western cultures contributed many seminal ideas to American society.

After some introductory formal exposition on the Theorem, Mr. Bordeaux rearranges the students into new collaborative groups. His pupils relax as the unfamiliar lecture format is replaced by the standard environment of small group interchange. Every group has one student from each team that encountered right triangles in the context of real-world problems. The university students, who have already done a similar exercise in their class, watch the transmission window on their notepads with fascination as the figures created by Pierre's learners begin to appear.

Working collaboratively by sharing and discussing the representations on their notepads, Pierre's learners gradually evolve an understanding that all these real-world contextualizations of right triangles have the same underlying mathematical structure [Standard 14, 9-12 Curriculum]. During this cooperative learning exercise, several of the university students have been assigned to watch Mr. Bordeaux rather than the learners. Each observes both the figures he is drawing on his notepad as he moves among the workgroups, explaining and guiding, and his facial expressions, words, and body posture on the liveboard. At times, Mr. Bordeaux displays the figures from some notepad on the liveboard in his classroom to illustrate a particular point for the entire group. These preservice teachers will later present a report to Sally's students on Pierre's classroom management strategies.

All the university notepads are automatically archiving what is transmitted to them, as is the liveboard. In this way, the preservice teachers can focus completely on what is happening rather than diverting their attention to take notes; the technology is serving as an automated extension of human short-term memory. Sally scans everything that is happening from her desktop workstation, instructing the machine to save particular segments that she later will review with her students. {"Knowbots" are limited in their ability to understand which aspects of complex data people will find important.}

The methods students are quite impressed with the mathematical activities. They observe that the learners are busy exploring mathematics and that the teacher is supportive of their doing so. To aid the cooperative learning situation, the notepads provide a shared focus for his students to stay on task even when Pierre is not monitoring their particular group. The preservice teachers are also attending carefully to what is happening. Their notepads are sending summaries of their observations to Sally, who can later use these "cognitive audit trails" to assess their clinical skills and methods.

The methods students are particularly impressed when Mr. Bordeaux has his learners verbalize their shared understandings [Standard 2, 9-12 Curriculum]. This is consistent with the emphasis that Sally has given in the methods class, as well as the criteria for effective teaching that they had established. Pierre is also pleased by how well the observation experience is going. Having the preservice teachers watching by virtual means is much less disruptive to his learners' concentration than the university students' physical presence would have been. The university class gets more from seeing his pupils work "live" than on video, although at times his teaching is videotaped to reduce the load on the broadband network linking the two institutions.

The class concludes with Mr. Bordeaux's learners able to access on their notepads an interactive version of Dr. James Blinn's classic visualization graphics on the Pythagorean theorem. The university students watch carefully as the learners manipulate the images [which include animated versions of the still images on page 117 of the NCTM Standards for Teaching Mathematics, in the vignette from which this scenario is drawn]. Pierre is grateful that the university students will be doing the diagnostic assessment of the evaluative activity; this lightens his work load and provides fresh insights on his learners.

After the class, Mr. Bordeaux, Dr. Witt, and the university students conduct a brief videoconference to debrief the observation experience. Pierre explains that the notepads generate such a wealth of individualized evaluative information he often cannot assess learner performance on an individual basis. Instead, he focuses on those students having difficulties and on gifted students who need individualized enrichment activities. This type of assessment aids him in understanding how best to pair the learners for their cooperative activities.

Mr. Bordeaux thanks the preservice teachers for their help and offers to provide advice should they later attempt a similar lesson during student teaching. Dr. Witt is grateful for his cooperation; teaching exemplary methods for creating good learning environments is certainly improved by these types of observation experiences. A record of this professional development activity is automatically entered in both instructors' personnel files and in all the students' portfolios by their respective knowbots.�

The next vignette depicts how parental involvement in education can be leveraged through the facile communication of data over distances. This scenario incorporates sophisticated networking technology into Vignette 6.3, Standards for Teaching Mathematics, from the NCTM Professional Standards for Teaching Mathematics (1991).

A Parent-Teacher Conference

Ms. Lundgren has been trying to change her approach to teaching mathematics so that students are learning to reason and communicate about mathematics, to make sense of mathematical ideas, and to make connections. She believes that she has been successful in moving the discourse of her classroom away from a focus on the right answers and the teacher as authority [Standards 2 and 3, Teaching].

Although she finds it difficult, she has been designing better mathematical tasks. With the help of a regional network of fifth-grade mathematics teachers acting as a virtual support group for implementing the NCTM Standards, Ms. Lundgren has also come up with some different ways for keeping track of what students are learning. Today she is meeting with parents to go over their children's report cards, and she will draw on her new records for those conferences.

Ms. Byers, Stacy's mother, works in a fast food restaurant and cannot physically attend the parent-teacher conference, so she appears in a videowindow on Ms. Lundgren's desktop workstation at the time scheduled for their meeting. The corporation that runs the restaurant chain has been generous in allowing employees to use their business telecommunications infrastructure for such conferences as part of the company's commitment to improve public education. Access to such two-way, broadband telecommunications links allows teachers to share complex data-objects with parents in real-time. Less powerful networking capabilities through telephone lines have routinized the weekly, low-level exchange of information between teachers and parents, but virtual conferences such as this one are important for building an educational partnership between home and school.

Ms. Lundgren wants to show Ms. Byers how Stacy is making connections in division. Calling up her portfolio on this student, the teacher first transmits edited video segments illustrating how Stacy was able to explain a complex idea: For the fraction 28/8, 3 r 4 is effectively the same answer as 3.5 (a quotient obtained on the calculator), but the two answers differ representationally. Stacy's competence on this problem speaks well for her abilities in mathematical communication [Standard 2, 5-8 Curriculum].

Ms. Byers is a little overwhelmed by the notation, but Ms. Lundgren explains the concepts involved by transmitting a "What You See Is What I See" (WYSWIS) window with instructional materials similar to those she uses to explain these ideas to her students. {WYSIWIS has "telepointer" capabilities, so either person can move an arrow to gesture to an item on the videoscreen with both able to see what is happening.} Once she has the background to understand the material, Ms. Byers seems quite impressed with the ease with which her daughter explains the abstract concepts involved and then relates them to a real-world situation similar to those encountered in restaurants [Standard 1, 5-8 Curriculum].

Referring to her portfolio again, Ms. Lundgren then shows Stacy's mother all the ways that her daughter found to represent 8 divided by 1/2 in her journal and explains what this skill means in understanding number relationships [Standard 5, 5-8 Curriculum]. The journal entries appear in a window on the liveboard in the fast food manager's office and are large enough so that Ms. Byers can easily view them. Some of the ideas involved in the journal entries are unfamiliar to Stacy's mother, as she was not taught these types of representations when she attended school, but with Ms. Lundgren's assistance she can comprehend the progress her daughter is making.

Because she also wants to talk with Ms. Byers about Stacy's disposition toward mathematics, Ms. Lundgren refers to a chart that she is keeping on her students' mathematical attitudes. Stacy has responded well to a range of teaching strategies designed to foster learners' mathematical dispositions [Standard 6, Evaluation of Teaching]. Ms. Byers is confused by the presentation of this data in a matrix format, but finds the same ideas readily understandable when the teacher switches the representation to a pie-chart format and uses animation to illustrate changes over time.

To make these abstract ideas more concrete, Ms. Lundgren shows a short video segment illustrating Stacy's leadership in a collaborative learning experience. She is more adept than most peers at applying mathematical concepts in real-world problem solving situations; this is particularly impressive because English is Stacy's second language. Fortunately, Ms. Lundgren was able to link via the international mathematics education network to a bilingual teacher in Stacy's native country. He suggested some excellent pedagogical techniques for using shared mathematical representations to bridge the differences between English and Stacy's first language.

Ms. Byers is very impressed by the entire experience and is much more supportive of the mathematical concepts her child is learning by the end of the session than she was prior to the conference. In part because of the visual presentation of information, she sees how these abstract concepts can be used in real world situations. Ms. Byers comments that she thinks what Ms. Lundgren is doing in math is great and that she wishes her educational experience with mathematics had been similar.

Ms. Byers then spends some time discussing with Ms. Lundgren recent family events that have occurred in Stacy's life that may influence her concentration and motivation. She asks if the next conference could be scheduled late in the day; she wants her husband to come to the liveboard in the community center so that he can see first-hand Stacy's progress. They do not have sophisticated telecommunications capabilities in their home, and she despairs of explaining what she saw today in the conference without the aid of complex data-objects and Ms. Lundgren to clarify the concepts involved.

Ms. Lundgren agrees to the follow-on conference and suggests in turn some activities that Ms. Byers can do at home with Stacy to emphasize the concepts and attitudes that are being scaffolded in her classroom. Ms. Lundgren's knowbot checks that the requested privacy waiver has been signed by Stacy's parents, then sends a copy of the teleconference to the school district office to be placed in Stacy's file. This will serve to document both her progress and the school's efforts to involve her parents in the educational process.

As I wrote in 1992, commenting on this exercise in repurposing the vignettes in the Standards:

Other scenarios illustrating different aspects of technology-intensive education in mathematics could certainly be conceptualized. Intelligent manipulatives; distributed simulations linked by advanced networking, data visualization, and artificial reality approaches; the personalization of education by setting mathematics exercises in virtual contexts from the local region; and the use of "cognitive audit trails" for evaluation are all intriguing possibilities for improving instruction in mathematics. (Dede, 1992)

In concluding this section, two caveats about the scenarios presented must be stated. First, these vignettes have been deliberately crafted to stress potential uses of advanced telecommunications in education that would amplify human capabilities. While realistic, situations in which the equipment doesn't work or people fail to master how to use the machines are deliberately not discussed. This paper's purpose is to describe how technologies that facilitate transmitting data over distance could leverage educational progress, rather than to assess the scope of likely obstacles and barriers.

Moreover, successfully deploying networking technology in itself does not intrinsically guarantee that mathematics education in accordance with emerging standards will occur. In fact, the current generation of integrated learning systems based on local area networking is designed with an underlying philosophy of teaching/learning directly contradictory to many ideas of the reform movement in mathematics education: Pupils are taught decontextualized concepts by rote with a stress on low-level symbolic manipulation skills. Technology can aid in reforming mathematics teaching only if a pedagogical philosophy consistent with the NCTM Standards undergirds its design.

Second, the psychosocial environment in which learning takes place is an integral part of cognitive change. Telecommunications tools change the culture of an educational organization and have the potential to empower more effe ctive instruction through peer support and access to sophisticated resources. However, their usage must be carefully monitored to preclude the inadvertent development of interpersonal dynamics that undercut learning. For example, too much use of networking could reduce learners' motivation because little person-to-person contact was occurring.

Overall, vignettes such as these accomplish several purposes. They reveal potential opportunities by violating default assumptions about what we believe is possible, yet simultaneously highlight constraints by depicting negative interactions among piece-meal innovations that seem sensible in isolation. An attractive long-range vision can both clarify short-range actions needed to create that future and indicate technology guidelines that could supplement the current NCTM Standards.

Thus, well-written vignettes embedded in curriculum standards can highlight opportunities for technology usage in exemplary instruction, as well as implicitly emphasizing the limits of what even advanced tools can accomplish.

Design Heuristic for Vignettes Embedded in the Standards

Based on my professional experience with constructing future scenarios, below are heuristics for designing and writing technology-based vignettes to embed in professional standards:

Technology-based vignettes constructed according to these guidelines are a powerful complement to generic curriculum standards, as they invite readers to imagine creative ways of reaching ambitious, evolving educational goals.

To ensure that the "illuminations" in the new NCTM Standards convey the full range of potential effective practices using technology, developing a morphology of topics to cover would be valuable. Such a morphology could include dimensions of instructional process (e.g., guided inquiry-based learning, collaborative learning, mentoring and apprenticeships), educational context (classroom, home, field settings), types of expression and communication (e.g,. asynchronous dialogue, synchronous virtual interaction, video-based anchoring in authentic situations), and technological capability (e.g., reification of abstractions, distributed cognition, modeling, situated learning). The entire set of illuminations would be crafted to ensure that all aspects of this morphology were reflected somewhere in the new Standards.

Illuminations of all types are a powerful means of providing visions that pervade the new NCTM Standards. Planning and designing these illuminations is an intriguing challenge with very high payoff in making the Standards interesting and inspirational for all stakeholders in quality mathematics education.


Dede, C., Editor. 1998. Futures: Images of educational technology in the next millennium (eight pages). Tallahassee, FL: Florida Educational Technology Conference.

Dede, C. 1992. Potential uses of telecommunications to empower implementation of the NCTM mathematics standards. In C.M. Firestone & C.H. Clark, Eds., Telecommunications as a tool for educational reform. Queenstown, MD: Aspen Institute Program on Communications and Society.

National Council of Teachers of Mathematics. 1989. Curriculum and evaluation standards for school mathematics. Reston, Virginia: NCTM.

National Council of Teachers of Mathematics. 1991. Professional standards for teaching mathematics. Reston, Virginia: NCTM.