Education 6630: Critical Issues in Mathematics

Inquiry Paper: The Graphing Calculator’s Affect and Effect

Scott Walsh

Introduction

For hundreds of years mathematics could be distinguished from all other scientific school disciplines by the stability and economy of the tools used in its teaching and learning system: the lowly pencil, the one-size-fits-all textbook, the ruler, the abacus, and others. Specifically, over the last century pedagogical methods in mathematics have been in a gradual yet constant state of change. Arguably, the chief instigator of this change has been technology, specially the adaptation of scientific and more recently graphic calculators into mathematical practice. Kaput (1992) described the role of technology in mathematics education as “a newly active volcano – the mathematical mountain… changing before our eyes, with myriad forces operating on it and within it simultaneously” (p. 515). It is generally agreed upon that to teach mathematics well at any level in the K-12 education system is indeed a challenging endeavor. Moreover, teaching with a focus on investigative, student-centered, open-ended projects and problem solving routines further increases this challenge. Add to the equation the onslaught of a variety of new technologies for instructional and student’s use, and even the most refined and accomplished teachers may struggle with the demands. Today’s technology-rich environment places a unique demand on teachers at every level of the education system to implement these resources into the curriculum in an effective and efficient manner.

“Technology, and the pedagogical changes resulting from it, has a decisive impact on what is included in the mathematics curriculum” (Ellington, 2003, p. 433). Specifically, what students are taught and how they learn are influenced considerably by the technological factors at work on and within the “mathematical mountain.” The NCTM has recognized that calculators and other technologies are reshaping the mathematical landscape, and as a result our school mathematics teaching needs to reflect these changes. From being able to make and test conjectures, to working at higher levels of generalization or abstraction, research has shown that students can learn and be successful in mathematics through responsible and appropriate usage of the graphing calculator. It is interesting to note that, in 2000, the NCTM confirmed this technological choice:

Calculators are reshaping the mathematical landscape, and school mathematics should reflect those changes. Students can learn more mathematics more deeply with appropriate and responsible use of technology. They can make and test conjectures. They can work at higher levels of generalization or abstraction.

Nevertheless, the NCTM also emphasizes the necessary ‘control by teachers’ of the integration process:
Technology cannot replace the mathematics teacher, nor can it be used as a replacement for basic understandings and intuitions. The teacher must make prudent decisions about when and how to use technology and should ensure that the technology is enhancing students’ mathematical thinking.

And so it is these decisions that cause the greatest stress for teachers as they try to find the balance and integrate these tools into the curriculum. There are still many skeptics who do not perceive technology generally, calculators specifically, as being able to provide new pedagogical enhancement opportunities, but rather believe that these tools will impair students’ mathematical ability, resulting in increased mathematical illiteracy and incompetence. “They are often reluctant users of instructional technology who tend not to share the excitement associated with calculators and may consider technology as a replacement for learning skills that are the core of what constitutes mathematics” (Sturdivant et al, p. 164).

Purpose and Rationale

One particular piece of technology proven to enhance mathematical student learning is the graphing calculator. In their research, Doerr and Zangor (2000) found that graphing calculators are “effective in achieving certain instructional objectives, which are often left unchanged from traditional paper-and-pencil approaches [by] providing students with new approaches, such as the use of multiple representations, to the investigation of mathematical ideas” (p.144). However, ultimately it is the classroom teacher who decides the extent to which graphing calculators will be used in the delivery and implementation of a mathematics curriculum. Goos et al (2007) note that “to become effective users of technology in mathematics education, teachers need to make informed decisions about how and why to integrate different types of technology into their classroom practice in order to support students’ learning of mathematics” (p.75). How the teacher makes use of the graphing calculator in instruction and the reasons for using such technology in a particular way are serious underlying factors. Coupled with that is the teacher’s philosophies of math and mathematical education and how they are manifested in instructional practice. Essentially, the technology used in the classroom is only as good as the teacher allows it to be. Wilson and Krapfl (1994) suggested that there is an urgent call to better comprehend teachers’ knowledge and pedagogical strategies in their use of the graphing calculator, along with a need “to focus on the qualitative aspects of knowledge construction of students using graphing calculators” (p. 261).

In this paper I will investigate the process of learning mathematics in a technology-rich environment predominantly at a secondary school level. To do this one must consider the point of view from the two principle protagonists in the educational institution: the students and the teachers. How do students use the graphing calculator to help broaden their mathematical understanding and improve their learning capabilities? Are teacher’s ready and accepting of this new era of a becoming technologically-centered curriculum? To what extent do the teacher’s role, knowledge, and beliefs about the graphing calculator influence how they are utilized in the classroom? What are students’ responses about their teacher’s attitudes over calculator integration in their math lessons? Finally, a look into trends and standards surrounding the prevalence of and attitudes toward calculator use in our province and country will be discussed.

The Graphing Calculator – An Overview

With the availability of electronic technology since the mid 1980s mathematics education has undergone monumental changes as “educators began to study the role and impact of this tool on the teaching and learning enterprise” (Katsberg & Leatham, 2005, p. 25). According to Reznichenko (2007) “the reason for the major impact of technology on the reform in mathematics education is its effect on the cognitive processes – nature of mathematical thinking and understanding” (p. 5) From there it is possible to study the impact of the graphing calculator on mathematics education using a concept of “cognitive technologies,” defined as “media that help transcend the limitations of the mind…in thinking, learning, and problem solving activities” (Pea, 1987, p.9).

The graphing calculator has brought reforms to mathematics education, creating a shift in “pedagogical philosophy from behavioral to constructivist, from introverted learning to cooperative, with an emphasis on conceptual understanding” (Reznichenko, p. 4). These changes in the system has resulted in the NCTM to revise it’s Principles and Standards for School Mathematics document to now deemphasize computational skills and encourage use of graphing calculators to complete routine computations in order to concentrate on conceptual understanding (NCTM, 2000). It would be interesting to determine if many of the math teachers out there are even aware of such recommended changes to the nature of the teaching and learning style. With respect to secondary school, the greatest reforms in mathematics education were found in computing graphical results in algebra, with an emphasis on functions. The graphing calculator can be used to facilitate student understanding by enabling a multiple representation approach to functions through tabular and graphical representations, symbolic expressions, and real-world modeling.

Hembree & Dessart (1986) conducted a landmark study on the effects of calculator use on student achievement and attitude in secondary mathematics. Their findings confirmed that “calculator use during instruction does not hold back computational skills and conceptual understanding; furthermore, it increases outcome results on non-calculator tests, both computational and conceptual” (Reznichenko, p. 6). A more recent meta-analysis study by Ellington (2003) helped to further open the door for a broader use of calculators in mathematics classrooms. He concluded that when graphing calculators were part of both “testing and instruction, the operational skills, computational skills, skills necessary to understand mathematical concepts, and problem-solving skills improved for participating students” (Ellington, p. 455). Dick (1992) found that graphing calculators free up time for instruction by reducing attention to algebraic manipulation, and supplying more tools for problem solving, especially useful for students with weaker algebraic skills. Another very interesting discovery in his research was that students perceive problem solving differently when they are freed from numerical and algebraic computations to concentrate on problem set up and analyzing solutions. Hence, given all of these research supports, it still is interesting to delve into the thoughts of the users (and non-users) of such technology to see how their perceptions affect the way in which math is learned. Perhaps in the ideas, data, and discussions presented we may find our own voice as an educator, and hopefully a new method of reasoning for justifying teaching practices with respect to calculator use.

Teacher Perceptions

Do we assume that teachers genuinely welcome graphing calculator technology into the classroom to provide rich and meaningful learning experiences for their students? How powerful are the teachers’ perceptions in determining the extent to which the calculator is used for teaching and learning? It is necessary to probe into the teacher’s use of graphing technology and selections of tasks to help us better comprehend their perceived usefulness of this technology in mathematics education. Simmt (1997) suggests that “all teachers have philosophies of mathematics and mathematics education and that these philosophies, even if ‘scarcely coherent,’ underlie mathematics pedagogy” (p.270). The availability of the graphing calculator now provides teachers with the opportunity to not just teach mathematical concepts differently, but also to extend the content of the curriculum far beyond the possibilities in the past. For sure, the graphing calculator expedites computations, but it also offers instructional variety, and can be used as a motivational device for all students on the mathematical spectrum, regardless of exceptionalities, abilities, or disabilities. Working with teachers who exert both eager and reluctant philosophies to technology use is fundamental to ensuring the graphing calculator is properly used in math classrooms.

In one view, some teachers perceive mathematics as a collection of procedures and rules focused on computational learning. Penglase & Arnold (1996) noticed that teachers who perceived the graphic calculator as a computational tool tended to stress content-oriented goals and viewed learning as listening. The goal then of instruction is to master formal manipulations of mathematical expressions, objects, and symbols while developing algorithmic skills. Teachers with this rule-based view present a special challenge for integrating more graphing calculator technology into the classroom. “They are often reluctant users of instructional technology who tend not to share the excitement associated with calculators and may consider technology as a replacement for learning skills that are the core of what constitutes mathematics” (Sturdivant et al, p. 164). These teachers are characterized as perceiving the graphing calculator as a crutch, preventing the learning of “real” mathematics. While this group maintains a “mastery first” attitude, an alternate view showcases a typically eager attitude to employ technology in non-routine ways to explore concepts and complex problems.

In order to justify their reluctance “teachers often appeal to a risk of social inequity (calculators are expensive), and to pedagogical difficulties linked to the diversity of calculators used by students” (Trouche, 2005, p. 18). However, Bruillard (1995) discovered that teachers still remain reluctant even when the same model of calculator is freely provided to all students, due to fears that the calculator prevents the learning of the basic elementary processes. Bernard et al (1996) found teachers believed calculators would reduce mathematics to an experimental practice that restricts the place of formal proof, quoting one teacher as claiming that “calculators deny the mathematical reflex.” Others trends in the research pointed to teachers seeing the tool as being too crude and dangerous as the graphics calculator results are sometimes approximates, resulting in particular errors and misconceptions on the part of the student.

With excitement about the graphing calculator other educators share a vision of using it to enhance instruction, especially in problem solving situations. This alternative view of mathematics “sees algorithms and symbol manipulation as components in a wider set of practices that include patterning and problem solving; here the goal for teachers is to develop students’ abilities to find and understand solutions to real problems, model phenomena, and develop conceptual understanding” (Sturdivant et al, p. 164) through the use of graphing calculators and other dynamic software. Teachers who saw it as an instructional tool had more student-centered goals and views on learning, with more interactive-driven teaching styles.

Through promoting careful decision making about technology use and effectively integrating it into the curriculum these teachers help students to become judicious technology users.
Discussions are held that gets students reflecting on their own mathematical thinking; that is, their metacognition. By sharing decision making about mental, pencil-and-paper, and technology approaches with the class, as well as allowing students to discuss decisions about these three approaches with one another, judicious technology use can be acquired. Ball & Stacey (2005) suggest “an instructive exercise to promote careful decision making is to have students monitor their own overuse or underuse of technology on an assignment” (p. 12). What they found in other research was that students underuse technology even when it is freely available. Secondly, mathematics teachers who are more eager technology users in class integrate calculator use through a mix of problems, some that are best done without technology and others that really demonstrate its power. “If the curriculum includes only questions that are comfortably within the range of expected paper-and-pencil skills, then permitting technology sends the message that technology is either useless or (worse) should be used on simple problems” (Ball & Stacey, p. 12). To encourage the discriminating use of technology we need to create assessments that contain some items which are most effectively done with calculator use and some without it.

Student Perceptions

The supports and connections that can be made through the implementation of the graphing calculator are too numerous to discuss here but let it be said that this piece of technology can support inductive thinking by allowing students to efficiently generate and explore a large host of mathematical examples, and then make conjectures about their patterns and relationships. With access to the graphics calculator students no longer have to be restricted to learning from and working with data sets published in textbooks or contrived by their teachers to make calculations easier. One of the greatest advantages of student use of the graphing calculator is the opportunity for the learner to visualize mathematical concepts. Since students are now able to see patterns, observe changes, and view images of geometric figures, data, and relationships, visualization through the use of the graphing calculator, it has gained more prominence as a means of learning math. As a result “visual reasoning has become more widely acknowledged as acceptable practice for mathematicians in the mathematical discovery process” (Goos et al, p. 84).

As an educator and collaborator with other mathematicians it is evident that most perceptions of the advantages of the graphing calculator appeared to be instructionally related. On the flip side, most teachers deem the disadvantages to be primarlily logistical in nature. Doerr and Zangor (2000) found that a common fault is that students do not have a meaningful strategy for the use of the calculator. They found “students attempted to use it as a ‘black box’ without attempting to form meaningful interpretations of the problem situation” (Doerr & Zangor, p. 158). In many classrooms the teachers uses the device during class discussion, as a shared tool via the TI-Smart View or overhead screen. When used in a shared, public display, the calculator fosters communication among the students, encourages student initiative, and often results in students leading the discussion. However, in its use as a private device downfalls can be created. It is a common occurrence that student’s use of their calculators as private devices leads to breakdown of group interactions within a group. Doerr and Zangor (2000) found that “when two or more students in a group tested or checked a possible conjecture or computation on their own calculators, they then continued to use the tool to explore possible situations, interpretations or refinements of their own thinking” (p. 158). Once the closed communication network is established within a group, students tend to continue to pursue their investigations as individuals, with very little talking, sharing of ideas, results, and representations. Too often the students then turn to the teacher for assistance rather than to each other. Hence, much consideration must be given from a logistical perspective on how to achieve balance with students in class so that communication is fostered and student group discussions are optimized. Collaboration needs to take place between teachers and students so that effective modeling strategies of how to work together using technology is fostered, encouraged, and productive during the required time of class.

While we have demonstrated that teacher beliefs about graphing calculators influence student access to graphing calculators, research has also been done to show how access to graphing calculator influences student performance. Harskamp, Suhre, and van Streun (1998; 2000) carried out a study that compared the performance of students given varying levels of access to graphing calculators. Students from 12 math classes were randomly assigned to one of three cohorts: those with access for one full year to graphing calculators, those with access during one unit of instruction, and those with no access. While the graphing calculator groups received additional instruction on how to use it to perform tasks such as graphing functions, finding solutions graphically, and checking algebraic solution, more advanced operations were not explored. Interestingly, none of the teachers had any prior experience in teaching with the graphing calculator.

Kastberg & Leatham (2005) reference in their research “results showed that students with the longest access to calculators used a wider range of problem-solving approaches and tended to attempt more problems and obtain higher test scores than the students who had not” (p. 27). Also, students tended to replace more common algorithmic practices and heuristic strategies with graphical approaches. Furthermore, students classified as “below average” by the researchers made more frequent use of graphical strategies and “achieved a significantly higher [score] than students in the control group” (Harskamp et al., 2000, pp. 47-48). In the end they suggested that even limited access to graphing calculators may have a positive effect on students’ attitudes, approaches, and accomplishments in mathematical problems. Something for us to consider for sure the next time we question whether it is worth the “aggravation” (a word teachers use all too often) to take the extra time in class, even if it is for just a small period, to get the calculator in the hands of the students.

In 2001 Faure & Goarin analyzed the results of a survey of 527 eleventh grade students and the relationship they established with their graphing calculators. Over the span of nearly eight years there was not a significant change in the way that students learned how to use their calculator, even though the prevalence of this tool has grown in leaps and bounds since 1992. The survey found the main process of appropriating the graphing calculator is based on individual exploration-discovery; afterwards a social dimension with friends is applied. The most startling piece of data in this figure (1-1) is that the teacher was the least involved factor in the process of calculator appropriation by math students. While ‘playing’ with the calculator in formal and informal settings is no doubt good hands-on practice for students, having a professional lead them in their in-servicing is no doubtful purposeful and necessary.

Ironically, further data in their report indicate that students desired for their teachers to be involved in the process of learning about the graphing calculator, indicative of the institutional recognition of the tool. It is quite evident that in hindsight many of the students wished they did not have to do the majority of their learning by trail and error, through friends, or even with instructions for use, but rather with their teacher in class. We need to carefully reflect on such statistics and ponder the degree to which we use the precious time we have with our students to ensure their needs in mastering new technological tools like the graphing calculator are met. It is through these opportunities with them that we can ensure the real power of the tool can be showcased, the misgivings can be highlighted, and the intentions of it can be made explicit. The results are displayed in figure 1-2.

Figure 1-3 investigates the reasons why students actually use a calculator both in math class and at home. The authors note that the calculator is mainly used by students during the reinvestigation of knowledge through drill exercises and assessments, moderately during more open processes of investigations and explorations, and disturbingly low when new concepts are being presented and established in class. Faure & Goarin (2001) suggest this situation is linked to the weak integration of this tool into the classroom. Major initiatives also need to be taken in our pedagogical approaches to increase the calculator use outside of class, where more time is available for students to master the concepts through home studying.

Canadian Frameworks

In the curriculum guides for the math courses across the K-12 system in Newfoundland and Labrador the department of education (2006) outlines that the “learning environment will be one in which students and teachers make regular use of technology” since “calculators will be an integral part of many learning activities” (www.edu.gov.nl.ca). When the WNCP Common Curriculum Frameworks for Mathematics K-9 and Mathematics 10-12 (WNCP, 2006 and 2008) were adopted as the basis for the K-12 mathematics curriculum in this province, technology was one of seven key “critical components that students must encounter in a mathematics program in order to achieve the goals of mathematics education and embrace lifelong learning in mathematics” (www.edu.gov.nl.ca). It went on to say that technology and calculator use “contributes to a learning environment in which the growing curiosity of students can lead to rich mathematical discoveries at all grade levels” (www.edu.gov.nl.ca; emphasis added). This is just not an issue for intermediate and secondary teachers but one that primary and elementary educators must also make significant in their pedagogical practices. Of course, one of the biggest barriers that remain is access to the technology, getting graphing calculators to become common place in our classrooms.

A survey done by the Ministry of Education in Ontario during the 2003 school year with grade nine teachers found that fifty-one percent of these educators indicated their students had access to graphing calculators at school during the course only a few times, never or hardly ever (EQAO, 2003c). As a means of integrating handheld technologies such as the graphing calculator into the classroom, the Expert Panel on Student Success in Ontario (2004) recommended “time for teachers to observe classrooms where technology is an integral part of the learning process” (p. 57) Just the technology and just the handbooks will not be enough for educators to see how to best use these powerful tools in the classroom. Real discourse and concrete professional development in a classroom setting would be a great enhancement to learning how to best integrate the graphing calculator into the curriculum.

Conclusion

The graphing calculator is a powerful, rich, multi-dimensional tool that needs to be optimized and further integrated into the mathematics classrooms. This incorporation can improve students’ conceptual understanding, attitudes towards math, and arguably, retention beyond secondary level studies in mathematics and related fields. The benefits of the graphing calculator go a long way in advancing student understanding of the topic they are learning. Enabling hands-on activities, students are then able to collect, explore, and analyze data through the use of many applications and data collection extension devices. Teachers need to become familiar with the disadvantages as well as the advantages of using graphing calculators in their classrooms. Supplemented with that, there needs to be discussion about the limitations of graphing calculators to ensure valuable learning experiences exist not just for students but for teachers as well. As my inquiry paper has discussed students do acquire many of their skills with their calculator through trial and error, when in actual fact they seek out more involvement from their teachers. A lack of proper integration of the tool in class has resulted in an overemphasis of the calculator for simple algorithmic computations in lieu of more powerful explorations and investigations that the calculator can be very useful in supporting.

My research has also found that a prolonged exposure to the graphing calculator can result in higher test scores for students of varying abilities in their mathematics, something that surely would be interesting to carry out in today’s test obsessed mode of teaching and learning. While there still remains much reluctance on the part of many teachers to give this technology the recognition and respect it deserves, minds are changing, pedagogical practices are evolving, and mathematical arguments are transforming. For sure, finding the balance with calculator use requires careful consideration by the teacher, in-depth consultation with all levels of academia, and most of all prudent communication with the most important players, the students.

Toward the end of this research study I became increasingly curious about the role that undergraduate education programmes have on establishing particular attitudes about calculator use in the classroom and how effective they are in either establishing or dismissing a full integration of the calculator with the curriculum. Identifying the needs, adapting courses appropriately, and aptly skilling all pre-service mathematic teacher educators are among the challenges facing the system today that would be interesting to study and build upon in my inquiry project. In addition, finding antidotal evidence from teachers at the local level in terms of their attitudes, pressures, and success surrounding calculator use and misuse would be a very interesting venture to extend my work. There are so many facets to this truly fascinating subject of the graphing calculator’s affect and effect on our math education system that I truly feel the surface has been merely scratched. I look forward to continuing this investigation in my practice.

References

Ball, L., and Stacey, K. (2005). Teaching strategies for developing judicious technology use. In W.J. Masalski (Ed.), 2005. Technology-Supported Mathematics Learning Environments, Sixty-Seventh Yearbook; pp.3-16. Reston, VA: National Council of Teachers of Mathematics.

Bernard R., Faure C., Noguès M., & Trouche L. (1996). L’intégration des outils de calcul dans la formation initiale des maîtres, Rapport de recherché IUMF-MAFPEN. Montpellier: IREM, Université Montpellier II.

Bruillard, E. (1995). Usage des calculatrices à l’école élémentaire et au début du college, Rapport de recherche. Créteil: IUFM de Créteil.

Dick, T. (1998). Symbolic-graphical calculators: Teaching tools for mathematics. School Science and Mathematics, 92, pp.1-5.

Doerr, H. & Zangor, R. (2000) Creating Meaning For And With The Graphing Calculator. Educational Studies in Mathematics 41, pp. 143-163.

Ellington, A.J. (2003). A meta-analysis of the effects of calculators on students’ achievement and attitude levels in pre-college mathematics classes. Journal of Research in Mathematics Education, 34, pp.433-463.

Education Quality and Accountability Office. (2003c, November). Highlights of provincial achievement results: Grade 9 assessment of mathematics, 2002-2003. Retrieved December 5, 2009 from http://www.eqao.com/pdf_e/05/05P016e.pdf

Expert Panel on Student Success in Ontario: Mathematical Literacy, Grades 7-12. (2004). Leading Math Success. Retrieved December 5, 2009 from http://www.edu.gov.on.ca/eng/document/reports/numeracy/numeracyreport.pdf

Faure, C. & Goarin, M. (2001). Rapport d’enquête sur l’intégration des technologies nouvelles dans l’enseugnement des mathématiques en lycée. Montpellier: IREM, Université Montpellier II.

Goos, M. et al. (2007). Teaching Secondary School Mathematics: Research and practice for the 21st century. Allen & Unwin. Crows Nest, Austraila.

Harskamp, E., Suhre, C., & Van Streun, A. (1998). The graphing calculator in mathematics education: An experiment in the Netherlands. Hiroshima Journal of Mathematics Education, 6, 13-31.

Harskamp, E., Suhre, C., & Van Streun, A. (2000). The graphics calculator and students’ solution strategies. Mathematics Education Research Journal, 12, 37-52.

Hembree, R., and Dessart, D. (1986). Effects of handheld calculators in precollege mathematics education: a meta-analysis. Journal of Research in Mathematical Education, 17 (March), 83-89.

Kaput, J. (1992). Technology and Mathematics Education. In: D. Grouws (Ed.). A Handbook of Research on Mathematics Teaching and Learning. NY: MacMillan (pp. 515-556).

Katsberg, S., and Leatham, K. (2005). Research on graphing calculators at the secondary level: Implications for mathematics teacher education. Contemporary Issues in Technology and Teacher Education, 5(1), 25-37.

National Council is Teachers of Mathematics (NCTM). (2000). NCTM Principles and standards for school mathematics. Reston, VA: Author.

Newfoundland and Labrador Department of Education Curriculum Guides in Mathematics for K-12. Retrieved December 3, 2009 from www.edu.gov.nl.ca

Pea, R.D. (1987). Cognitive technologies for mathematics education. In A.H. Schoenfeld (Ed.), Cognitive Science and Mathematics Education. Hillsdale, NJ: Erlbaum.

Penglase, M., and Arnold, S. (1996). The graphics calculator in mathematics education: A critical review of recent research. Mathematics Education Research Journal, 8(1), 58-90.

Reznichenko, N. (2007). Learning Mathematics with Graphing Calculator: A Summary of Students’ Experiences.

Simmt, E. (1997). Graphing calculators in high school mathematics. The Journal of Computers in Mathematics and Science Teaching, 16(2-3) pp.269-289.

Sturdivant, R. et al. (2009). Preparing Mathematics Teachers for Technology-Rich Environments. PRIMUS, 19(2). pp. 161-173.

Trouche, L. (2005). Calculators in mathematics education: A rapid evolution of tools, with differential effects. In Guin, D., Ruthven, K., Trouche, L. (Eds.), The Didactial Challenges of Symbolic Calculators. New York, NY: Springer.

Western and Northern Canadian Protocol for Collaboration in Basic Education (Kindergatend to Grade 12). TheCcommon Curriculum Framework for K-9 Mathematics: Western and Northern Canadian Protocol – May 2006 and The Common Curriculum Framework for Grades 10-12- January 2008.


Wilson, M.R. and Krapfl, C.M. (1994). The impact of graphics calculators on students’ understanding of function. Journal of Computers in Mathematics and Science Teaching, 13(3), pp.252-264.

Looking To The Future

Well it's been a great run with all of you throughout our short stay discussing critical issues in mathematics. You know, I don't think there's a more critical issue out there in math classes for us to discuss than the one we have. We have discusses the misbehaviors, the low grades, the disinterested students, the textbook lesson plans, the lack of time to teach, the CRT's, the pathways program, the pressures from school district and higher personnel - the list goes on. And a common theme throughout all of our discourse, the tread that binded them all together has been how we teach and how our students learn math. The "conservative vision of education" that not only finds a home in Amber Hill but many of our schools is really having severe implciations on the future potential of so many young people. We now know unequivocally that as teachers we need to engage students, all students, and provide them with appropriate, stimulating activities in a class environment condusive to open-ended projects and tasks. We need to tune out the naysayers, we need to try something different, now. I wish all of you the very best in these endeavours we will each take back to our classrooms. I hope from time to time we can check back here with each other to update the group on any new initiatives we may be taking in class. Today and tomorrow it is our responsible, our duty to bring about the change we know we need, the change we've finally found, in all of our math classes. We too can become a Phoenix Park.

I'll see you down the road. All the best.

Well it looks like another of my beliefs on the "ideal" way of teaching students has gone down the drain! Since my position as a student in high school, through my university years, and during my three years as an educator, I always though it was best to have students separated accoridng to their ability so that, as I thought, "they were learning at a learning they were capable of" and "being taught at a level they understand." I was convicned, not knowing why however, that this separation was necessary and vital to student success. Looking back I think many of my reasons as a teacher for having such a stance was to curb student misbehavior, to increase student interest, and as a result have better results. However, if we were to visit many of the general math classes today, do you think we'd find an abundance of interested, motivated, well behaved students? For that matter, would we find these behaviors in our academic classes? advanced classes? I've taught at all three levels in our high schools and in all cases there still is a lack of motivation by some, great work ethics by others, and interest (or lack thereof) in the subject matter. As we discussed in class ability grouping seems to be more of an advantage for the teacher than it does for the students in general.


Needless to say reading about Boaler's research has opened my mind up to an entirely different way of believing how we can best ensure learning takes place with our students. All of the reasons she highlighed such as working at a fixed pace, the pressure, anxiety, and restricted opportunities, as well as the setting decisions were all points I could certainly identigy with. We, as teachers, do have influence in the courses our math students enter. Can we be accurate by making a judgement based on a grade, 50% of which was decided in 2 hours on a warm day in June? Are we basing some decisions too ont he behavior of a 15 year old whose behavior is not idyllic, who doesn't fit the typical? I think we make too many assumptions about the inabaility of all our classes to achieve higher grades that we percieve them to be able to attain. Then again, our sacred textbook style may have something to do with this notion. The system we have set up in our high school sof ability grouping still pits students against each other, still makes them feel as though they are being judged. The excuse we use that "competition helps raise achievement" is relaly insignificant when you weigh that against all the implications of ability grouping.

As we noted during our discussion in class, ability grouping classes at AMber hIll did not achieve better results than Phoenix Park's mixed ability classes, despite an increased amount of time spent "working" by Amber Hill students. At this point, it really does not take much convicing to see that working is not equal to learning. Our classrooms today are full of students being busy. But, busy at what? We assign enough questions on the board to keep them working (there's that word again) for the rest of the period I don't think we ever title the time students spend a their desk as "class learning" but rather "class work", "seat work", "homework". How can we really measure how much they know through all of this work? We're trying, obviously it's not working as successful as it needs to be, especially for those with weaker math abilities. The mixed ability grouping our schools adhere to now seems to be grosely inadequate in raising the learning levels of all students.

And so as I finished reading this chaper and reflecting back on the discussions we had in class I wish now I had the opportunity to initiate a mixed ability experience with my classes. However, the textbook approach got to go if we are imlement such a trial in our classroom. Mary said in class one night that our job, as teachers, is to design effective lesson plans that can allow for multiple entry points to problems, that asks the questions in effective ways to ensure connections are made. I believe we relaly need to begin making these attempts, we really do need to step away from the status quo and develop a new plan. Phoenix Park teachers taught a wide range of students in the same class and still provided stimulating and enriching experiences for all their students. There's no reason why we can't either.

Ch 9 - XX and XY Learning Styles

I remember my grandmother often saying "It's a man's world." Now I'm sure her statement had more to do with the cultural norms her generation lived through and less to do with the shortcomings often assumed of the female gender in their education ability. But nevertheless, the notion was there and perhaps it's still there in so many ways we've yet to uncover. But if we take her simple sentence and dissect it, when who made it a 'man's world." Of course - men did! For centuries my gender has corrupted and abused the system of humanity to gain a firm grip on the upper hand, to keep our partner gender just a little (ok, a lot) more below us. Event hough it has no mathematical relevance here, check out the poem "A Work of Artifice" by Marge Piercy and surely you'll find a much more refined and interesting look at how the woman's lot in life is decided.

Thankfully, we're moving beyond this dreadful period in our existence now. However, have our classrooms? Are they still set up for male succession over the female. Are the lessons we teach, the activities we develop, the awards we present all geared more toward the male than the female student? Throughout Rosemary's presentation we learned that indeed female underachievement should never be considered a "collary of being female". From Jo Boaler's work to numerous other researchers it is clear that the teaching environment itself is what drives the majority of the confidence, beliefs, and mathematical acheievement in many of our female students. The system seems completely ignorant to the effect its structure is having on the potential for women to reach their potential.

When I recall my school days I remember there being just 1 girl ina class of 13 for advanced math in a school where there was just about 50 graduates. Then it seemed so odd, and I remember our teacher encouraging other girls in our class to move up from the academic level but they declined. Surely some of them were at comparable levels in their math to many of the boys in our advanced class. Today, looking back perhaps those girls realized they were working against a system that was setting them behind no matter how hard they worked, perhaps they were clever in their understanding of the inequities that were unknowingly being placed out in the school system. They knew the "game" they were playing int hat the time and opportunity for understanding would not be granted in a manner they preferred in math class. Whatever the reason the motivation was not there for many to participate in the top level math course available to them. Interestingly enough though, the one girl who was in the class ranked overall second in the 13 students. In some ways, apart from her brillance in academics, she was able to rise above the cloud of confusion and disillusionment many of her female colleagues were victims of.

Today, I see such a difference in the presence of female showing up in higher level math courses and in their attitudes towards their ability and beliefs. Confidence spews from math of them, and now upon revisiting my high school the advanced math class has gone compeltely the other way, with just one male student registered compared to the seven females. It was very interesting to me in that 1) girls seem to be cogniziant that they need to work that much harder to maintain an equal footing with their male counterparts and 2) the system has changed in ways that now foster the learning styles of female students in ways that encourage their participation. Whatever the case I have noticed an increase in the number of women comign to the forefront to become reputable learners and teachers in our mathematics class. All I have to do is look around our class every Thrusday night. Need I say more? :)

With respect to the females beliefs and attitudes toward math classed Boaler sums it up best when she says that "it is important not to lay blame for their disaffection on mathematics per se because the fault lies not with the instrinsic nature of mathematics, but with school mathematics as it is commonly constructed" (p. 153). Girls can achieve. Girls want to achieve. And now girls are seeing their appetite and desire for success comign true. Girls are achieving. They are not incapable but were inherently crippled in many ways because of the nature of the school mathematics they were forced to comply with. Through a more open-ended style of teaching and learning math the creativity, intuition, and experience of our bright young female students will be enhanced, will be acknowledged, and will be most importantly sustained.

Chapter 8 - Knowledge, Beliefs, and Mathematical Identities

As Sharon so effectively talked about in her blog, I too am becoming increasingly frustrated with the procedures I find myself and my students stuck in. As I read through Boaler and continue my substituting the feelings of ineffectiveness in how we're teaching and how students are learning are growing. I long now to have a full time position where I can begin to impliment some of these ideas we're discovered through Phoenix Park's approach. And like Sharon, I'm being to realize that I can't wait. Each day that goes by without bringing about change runs the risk of perhaps losing another student, turning them further away from mathematics learning, entrenching them with fictious ideas about the relevance of math in their own lives. We need to start today.

Terri-Lynn raised a very important question during her presentation: How often do you see students handing in answers in which they have no concept of whether or not they are even logical, let alone the correct response? We see it all too often in our classes where students do not (for many reasons) thinking about, analyze, and interpret the answers they have arrived at. I think our methods of extreme pencil and paper tasks without enough investigative work and explorations have often led students to a point in their learning where they haven't been taught the fundamentals of checking the reasonableness of an answer. We overhwlm them with so much classwork assigned from textbooks and extra assignment practice outside of class that it would be too time consuming to go back and check all their answers. Another thought of mine just came up here and it is this: I think we are perhaps assigning too much work for our students to complete. I think that often times our teaching strategies are too time consuming, especially when it comes to having students write off notes from the board, and as a result the majority of the practice, the time when students need to talk math between each other, is often lost in the classroom. That means the majority of the work is then left to be done in isolation usually away from class. The math is then perceived as labouors, monotonous, and routine, instead of something that should be interacitve, interesting, and fun inside class. We really can do more with less.

As we read through this book we realzie the importance of incorporating project work into our math classes. Geeno and MMAP (1998) reported that in this type of work students "develop abilities of collaborative inquiry and of using the concepts and methods of a discipline to solve problems." Given this highly significant piece of research it is quite evident that the inquiry and cleverness we expect from our students is not going ot happen by osmosis, or by us standing in front of them asking them to dictate onto their exercises what we say. We have trained our math student not to be disciplined, but rather dependent on us. While substituting for a math teacher a few weeks ago one of hte classes I had to supervise a junior high test. He had written in the notes that it was ok to provide help to the students during the test, give them hints, etc. A year ago I probably wouldn't have blinked at such a request. That day I couldn't stop blinking. We weren't five minutes into the test and hands were going up all around me. The test was constructed well and had a good range of quesitoning for the class so I didn't perceive there would be much trouble. However, the students sat there like drowning rats desperate for a push, a start in a question. Everything from what the word "prime" means to "how do I draw a numberline to show + 3 multipled by -2 was asked. It was so disappointing to see so many students suffereing through this. I wondered to myself what are we really doing to our students? Why are they failing in their learning so badly? How did they get to the point where independence was non-existent in their math? Is it too late to change the tide? I hope not.

Exploring The Differences

As I continue to work through this fall semester both in my graduate studies and in my substituting experiences in a variety of classrooms and subjects, now more than ever I feel I'm both exploring and experience the differences of our schools and students. I feel the frustrations of my students when they're subjected to note taking and textbook drill. What once fell deaf to my ears is now being hear loud and clear - our students want to be challenged, they are bored with the daily routines of pencils and paper practice. The idleness I see all around me resonates so much more now, for I am finally feeling, through our class discussions and Boaler's words, the effects of such a stagnant method of teaching and learning smothering our schools. If we really listened to our students we'd find many of the solutions to classroom management, to a lack of student engagement, and to the stress we put on all our students in over testing. By incorporating their voices, their thoughts, we can put into practice a way of doing, a means of learning as Phoenix Park has accomplished.

"The Amber Hill students believed the mathematics they encountered in school and the mathematics they met in the real world to be completely and inherently different" (p. 111). What are we doing in our classrooms to bridge this gap? Are our practices and teaching styles making the learning relevant outside our class and into the everyday lives these students live? How sad it is that our education system demands are not being linked by our students in similarities to the real world. The math is not "totally different". The methods used can be the same! Our math classrooms can be social! And most important, our classrooms should provide opportunities for students to work it out for themselves, instead of relying solely on textbooks to provide algorithm after algorithm in how to "solve" the problems. There is a need on us now, like never before, as teachers to ensure perceptions of the environments created by the real world and the mathematics classroom are no longer inherently different, but rather, the same.

Although there are many, a particular idea Boaler raises from Lave (1996a) is that "notions of knowing should be replaced with notions of doing, arguing that the only indication that someone has knowledge is that they can use it" (p.117). This relational view is the essence of what Phoenix Park's approach to learning and teaching mathematics is all about. Like Paul and others at PP, our students too will support this logic and will fulfill it. Transmitting knowledge has been tried. It hasn't work. Why don't we try something different? That is, why don't we try holsitic means of thinking and doing with our students. Let's desconstuct the boundaries around school mathematics that currently exist, that currently cripple so many of our youth. It can be done.

One small step for math, one giant step for math minds.

Chapter 6 - What Could They Do?

I wanted to return back to the questions I asked during my lead discussion in class on Chapter 6 here within my blog. These questions were a reuslt of stunning quantitiative data Boaler presented to us in her study.

Firstly, she stated that 94% of AH and PP students correctly answered questions on the test concerning angle calculations whereas only 63% of those same students in AH and 83% of PP students correctly estimated the roof's angle in the activity. Also, the AH students in the highest sets (1 and 2) did worse that the students of AH ins ets 3 and 4. Are our teaching styles prompting inappropriate learning cues by our students? In AH and in many of our classrooms the answer is sadly yes. We are caught up in a system so focused on testing that we teach students to find clues within a question that would turn them to a particualr recipe for answering teh question correctly. Often times these cues mean the student has to do so little thinking the work becomes mechanical, thoughtless, robotic, useless. Are we truly aware of the implications such hints whether verbal or written are having on the independence of student thought and learning. We've become so good at this cueing that our students sometimes cannot succeed without it. They too have become conditioned to needing these explicit instructions in order to correclty answer many of their questions. In so many of our classes students arrive at nonsensical answers, unaware of the obvious errors in their answers. Many times it's because they take a word in the question out of context and do not have a full mastery of the outcome beng tested.

Students at both schools reported enjoying the activities immensely, particularly the AH students, many of whom asked if they could do more work of a similar nature. Are we doing a good enough job to make math class enjoyable for our students? Again, I must revert back to the system we as teachers have innocently and blindly taken as the "way" things must be done. We use the timelines from governement, the pressures from districts for improved marks, and the established doctrine of teaching to the test as all crutches for why we've taken so much of the poential fun and enjoyment out of math class. Replacing the investigations and modelling sessions we give them endless practice sheets to prepare for the summative evaluations months away. When times gets short something enjoyable is always the first to be cut because we think it's not as important as "time on task" routines of pencil and paper work. Again, it's not intentional, and some will say we don't know any better. However, Boaler is telling us better, she is now ensuring we do know better. She found only 3% of AH students added any creativity in their flat designs compared to 33% of PP students. We can create demanding cognitive takss while still having students adhere to certain rules within the class. So, let's listen to her. Let's try her approach.

Success on the GCSE exams was important for students at PP but their teachers were cavalier about exam preparation. PP provided no calculators to students needing them and the school was void of any real motivation or "gearing up" for exams. Are we giving our students enough responsibility? In many ways we, as teachers, are making our students more responsible and ready to learn. We ask them to show up to class on time, bring their supplies, complete the assigned work both in and out of class, hand in projects on time. I could go on. However, we bend ever so quickly, extend wilingly on times, and believe it or not become more stresses out over students responsibilites than they do themselves. Many of us bring the wiriting tools, the calculators, often the paper itself for our students to complete their work. Especially at test time we scurry around to bum calculators and scrap up things our students should have brought but didn't. Yes, we all forget things sometimes and there's nothing wrong in helping at times like this. However, we go to the extreme many a time. I think many of us in a desperate plea for improved scores in exams will cater to our students. I personally try to be as nice as I can when moments of "oops" come from my students. I have loaned loaned my own calculator and I have turned students away too to fend for themselves. It's a tough act to adhere to and one that we must practice every opportunity we get.

A question that raised a lot of interest in class was: How often and to what extent do we talk and stress CRT, public exams to our students, making it the focus and purpose of the course? It would be very interesting to do such a study in our classrooms, making note of the number of times we defer to "tests" as a reason for learning a concept, as a reason for paying attention, as the reson for doing well in the course. I have a fear that student hear more references to testing in their classes than any other feature of their education. From day 1 in the syllabus we highlight public exams 10 months down the road, instilling perhaps fear withint he class that everyone is here for one reason only: to be ready for that test. Again, I say our language is often deferred to test preparation because that is the system of education created and followed year after year. The top-down, hierarchial system demands a concentration on testing, on these results as a way to justify funding or lack there of, to declare success of a program or its failure. However, we as teachers are the adult, often the only adult in the classroom. We are the voice the students hear and from us they hear too much about testing, too much about getting ready for tests. Instead of putting emphais on learning for its own sake and relating it to the world around our students, we exhaust them with test prep and success. Next time, let's try to catch ourselves before we use the test as the focus of our conversations with our students. No doubt, it will be tough.


Of all the results revealed in this chapter the one that stood out to me the most was that only 9% of the AH top set of students retained the material they learned just a few months after it was assessed. PP students retained four times the amount that some AH students had.Therefore, are our tests giving a realistic picture of what our students are learning? These results show the damage, real damage that the current teaching and learning styles are having on our students long term retention and understanding. Tests really aren't all they're cracked up to be, yet we still use them to draw a line in the sand, separating those who know from those who don't. These tests determine really who succeeds, who make it, and who gets left behind. It's time to revisit the value of these test papers and their role in shaping our student's education and their futures.

Scott