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We would like to express our thanks to our colleagues, authors, reviewers, publisher, and many others who have helped us in one way or another in the preparation, editing, and production of these proceedings. Sarah M. This paper reports on the pilot and first implementation years of Generative Activities in Singapore GenSing. In the pilot year the author led a series of networked generative, function-based algebra activities with five classes of Secondary 1 students.
The functions submitted by the students for one activity were collected and analyzed. Findings were that students created a range of correct functions utilizing many important mathematical concepts.
The analysis of the functions highlighted the need for new and more powerful tools for teachers to analyze the vast quantity of data created in generative activities. In preparation for the implementation year of the grant, new analysis tools were created, this article concludes with a review of those tools and examples of the insights into student data that they allow. Keywords: algebra, generative activities, classroom networks, graphing calculators.
Generative Activities are activities where all of the data used for classroom discourse and instruction are generated by the students. Generative Activities can be created for many subject areas. Previous work by Stroup, Carmona and Davis, focused on creating a function-based algebra curriculum to be implemented in a Central Texas ninth grade years old Algebra 1 classroom. The purpose of this paper is to report on the pilot and first years of implementation of Generative Activities in Singapore GenSing.
For the pilot year of implementation school year student data is presented and analyzed. One of the major findings of the pilot implementation was that more powerful tools needed to be created for both the researcher and the teacher to make sense of all of the student generated data from the new activities Davis, For the second year of the project the current school year , examples of student data are shown to demonstrate the potential of the new analysis tools.
Schools in Singapore tend to not be air conditioned. Rooms are cooled by open windows, ceiling fans and wall mounted oscillating fans. A typical classroom will have two side walls with louvered windows starting at approximately 3 feet from the floor and going all the way to the ceiling and running their entire length.
The front wall will have a white board and a retractable projection screen. The back wall will have a bulletin board. The rooms have concrete floors, walls and ceilings.
Typical class size is 40 students and they will spend most of their day in one class with teachers rotating from room to room. With open windows, fans and concrete surfaces many teachers take hands free microphones to classes with them so as to be heard over the ambient noise.
Participants There are two different sets of participants in this study, the Sec 1 students from and the current Sec 1 students at the research site. For the study there were Secondary 1 students years old. Data was collected from five sections of both normal and express track classes. Data was collected towards the end of the first semester of school.
Students had been introduced, in the traditional curriculum, to concepts of simplifying expressions prior to the intervention. Class sizes ranged from students. In the current study there are Secondary 1 students. Generative Activities are being implemented by the classroom teacher in 7 sections of both normal and express track classes.
Data is being collected in all classes for all activities. Class sizes range from students. This system is comprised of a teacher computer running TI Navigator software, an access point, ten wireless hubs and a graphing calculator for each student.
In this study the calculator used was the TI 84 Plus. Each hub connects to four graphing calculators and transmits data to and from the teacher computer. The screen of the teacher computer is projected for the class to see. The computer displays data from all students synchronously. The Interventions In the intervention was done as a series of three, researcher lead, activities with the participating classes.
Each class was visited once every two weeks to mitigate interruption of the regular curriculum. The data reported on in this paper was collected during the second activity.
The second activity, Function Activity, has students generate a set of points according to a rule and then find multiple functions that can model the data. The activity begins with each student controlling one point in a coordinate plane via the arrow keys on their calculator. On their device each student sees only their own point. Davis as students move from coordinate to coordinate.
They are then given a rule. Students move to a location on the screen that they feel fulfills the rule and mark a point. With luck, the moving mass of points converge to form a line-like pattern. There will be points that do not fit the pattern. These are opportunities for exploration of misconceptions.
For example, there may be a point at -4, 8. What do you think the person was thinking? How do we fix it? Once the points all fit the pattern, the students are asked to find a function that goes through the data. After students have been successful finding one function to model the data they are challenged to find two more functions that also go through the data set.
For the intervention the Sec 1 teachers are leading the activities. In collaboration with the schools curriculum specialist, the Sec 1 scheme of work was reorganized to gather all of the algebraic topics covered across the year into two cohesive groupings, one lasting eight weeks and one lasting four weeks. During the first eight weeks segment, concepts of equivalence, rate and linear functions were covered. During the second four weeks segment, concepts of inequalities were explored.
Additionally the topics of instruction were grouped into concepts i. Over the course of the 12 weeks of intervention students experienced ten generative activities. Data collection In the Function Activity, the students were asked to submit three functions for each rule as described above. The software allowed the teacher to save activity data in table format. At then end of each activity segment, the software was paused and the data was saved.
The implementation is building on lessons learned in the first phase of the project. These lessons made it clear that all student submitted data was of interest to the research team, not just end state data. For this reason, a major portion of the planning for the second phase of the project was devoted to the creation of a tool that would run during the activity collecting all of the data the students sent in.
Examples of data from the school year will be presented below to show exemplars of the new analysis software. Indepth analysis of the data will take place after the school year is completed. Data Analysis The pilot year data was imported into Microsoft Excel. The data was analyzed in iterative passes. Each function had to be verified and coded for correctness i. The data was sorted first by section, then by activity then by student.
This configuration allowed the viewing of all functions from one activity together and also to see the three functions submitted by one student grouped together. After coding for correctness, each function was coded for mathematical strategy and social strategies. Figure 1. Simplest Form, 4. Rational, 5. Rational 1, 6. Additive 0, 7. Multiplicative 0. Rational 1 items were rational expressions where the students used the concept that the numerator and denominator would cancel each other out and form a 1.
In items coded as Multiplicative 0, the student would put in a term or a parenthetical group of terms and then multiply it by 0 to make it disappear. Manipulations of real numbers and non-rational algebraic expressions were the most frequent mathematical strategies adopted. None of these strategies were directly taught, they immerged from the students experimentation in the private space of the calculator.
The students had previously learned that if they created two functions that made the same graph, the functions were the same. This gave them the ability to try out different combinations of terms to find things that work.
Mathematical Strategies were heavy influenced by task. The 2X task was done first. In reality, students focused on finding things the same as 6 and added X on to the end of the function. Davis Figure 2. Big Numbers, 2. Many Terms, 3. Unique Strategy, 4. Humorous, 5. The first two social strategies are ways that the students tried to get their functions to stand out in the display space. Big numbers were any that had a term with five or more digits.
Many Terms were functions that had five or more terms. The next two strategies were given to functions that were noteworthy. The act of submitting their functions to the public space, made it open for interpretation. Unique functions used mathematical elements that no one else in the section was using. Humorous may be in the eye of the beholder, but there were some functions that seemed to have been created tongue-in-cheek.
It could be argued that all of the Many Term functions, especially those with the large chains of X-X, were somewhat tongue-in-cheek, but for this analysis, those will stay coded Many Terms.
There seemed to be a quality about some functions that felt like, the student was teasing. It is in the social strategies that space creating play shines through. As in sports or other games, students explore the possibilities. They find ways to stand out, to perform. Similar to activities outside the classroom, when students see a great move, they want to copy it…or out do it. Table 1. The results from the pilot study were very exciting but they also left a number of unanswered questions.
How do noteworthy expressions in the group space affect other students? How can the mass amounts of data be meaningfully organized so the teacher can reflect on student progress outside of class? What data needs to be easily accessible so that it can be re-used in another activity or additional practice? Davis New Software for the Second Year To start the process of answering the questions surfaced by the data analysis from the first year, new software needed to be created.
For the second year of the project, a script for continuous data collection was created. In the first year of the project, only end state data was collected. This did not allow for an exploration of students immerging understanding nor of the impact of items in the group display.
Using NetLogo Wilensky, , a set of analysis tools was created to help visualize the student data. These tools are discussed below with examples of student data from the school year. The new script collects all function submissions by the students with a time stamp.
It also captures any changes to the data made by the teacher on the up front computer. This data is written out to a. One of the most ponderous parts of data analysis during the pilot year was evaluating all of the student submitted functions. Collecting just end stage data for the 5 student groups resulted in a data set of functions. While there are many softwares available that will evaluate expressions, they are not designed to import or more functions and evaluate them for equivalence.
As a first pass for an analysis tool, an interface was created that allows the user to import all of the student submitted functions from a session. A point on the desired function is then entered and the software finds all of the graphs on the screen that contain that point. The software can then add a descriptor to the spreadsheet entry for the function to identify it as being an expression in the designated set.
In this way, it saves the evaluator from having to compute the value of each function manually. With data from all of the classes composing over functions, not having to manually evaluate them is a huge benefit. The software can then exports a new. This file is now ready to be imported into the Timeline Viewer.
The new information regarding matching a desired function is utilized in the individual student view in that tool. Figure 3. Graphical Viewer The Timeline Viewer Figure 4 gives an overall idea of when activity is happening in the class. It also allows for the identification of patterns in submissions. Each row represents one of the students in the class designated by a unique icon. Each icon in that row represents the submission of a new expression.
The icons are color coded to give a quick idea of which function Y1-Y10 is being updated. Students who submit multiple times have many icons in there row, where other student may only submitted once toward the beginning of the session and once toward the end and will only have two icons showing in their row.
Early analysis reveals that neither strategy is better or worse. Each new idea or attempt is submitted to the group space, independent of the current correctness of the expressions. By the final submission the expressions tend to be correct.
In this way, the researchers actually have a better view into these students through process. Other students prefer to think privately until they know in their private space the calculator that they have a matching function.
These students only submit their functions when they are sure they are correct. Either strategy can yield positive results, the timeline view gives access to this type of analysis.
Davis Figure 4. Timeline Viewer, Whole Class In addition to the whole class view, the Timeline Viewer lets you filter out other data and view the submissions from just one student Figure5. Using the placement of the icon as the timestamp not the leading edge of the expression the student view below shows that this student submitted a new Y1, Y2, Y3 and Y4 at the same time. The software knows which match and which do not because of the identifying data added to each expression in the Graphical Viewer.
The student then corrected Y4 and returned to Y1 to work at finding a new expression to replace 4X. In the end this student submitted 5 correct expressions. Next Steps Analysis of the student data for the implementation is just beginning. Some next steps will be to look at how social strategies the students use influence their peers. When an expression with many terms shows up in the display, is it quickly followed by other students doing expressions with many terms?
What cues can the teacher use to elicit specific strategies? Figure 5. Timeline Viewer, Individual Student In addition to researcher style questions there are also questions about how to make the data useful to teachers to do formative analysis of the students understanding.
This formative knowledge can then be used to plan instruction for the following lessons. Because there is such a vast amount of data created by this type of activity, tools to synthsise the data and give useful reports back to the teacher are needed.
These questions and challenges are what the GenSing project will be exploring over the coming school years. Generative Activities in Singapore: A Beginning. Davis, S. GenSing Graphical Viewer. GenSing Timeline Viewer. Stroup, W. A taxonomy of generative activity design supported by next-generation classroom networks. Generative Activities and Function-Based Algebra.
The nature and future of classroom connectivity: The dialectics of mathematics in the social space. Efforts to improve such knowledge require redefining teacher education through new teaching and learning. One such action has been the development of a Foundations Unit, Scientific and Quantitative Literacy, for all first year pre-service primary teacher education students at Queensland University of Technology.
This paper will outline the implementation of the unit and compare and contrast the performance of a group of students from QUT and the Institute Perguruan Raja Melewar, Negeri Sembilan. Quantitative comparisons will include those from journal and exam marks.
Possible reasons for any differences observed will be suggested as topics for further research. In semester one, students completed prescribed studies undertaken in Malaysian Institutes. In semester 2, all first year students following the Bachelor of Education Studies program for Primary Science major, took three professional studies which consisted of Information and Communication Studies; Primary Curriculum and Pedagogy: Health and Physical Education and a no credit unit for professional practice called School Based Experience.
These students also took a content studies unit, a foundation unit, MDZ, called Scientific and Quantitative Literacy. The unit was taught in English by staff in the existing two teacher education colleges involved in the program.
At the end of four years successful students will graduate with a QUT degree. This unit consists of an integration of topics from mathematics and science. Objectives of this unit include the development of an understanding of the nature of the disciplines, the acquisition of knowledge and the basis of beliefs in the disciplines.
Beliefs about the nature of mathematics, its roles in society, and the contribution it has made to the growth of knowledge. Myths and misconceptions about mathematics and science. The scientific method and the formation of hypotheses.
There is a focus on challenging existing beliefs and justifying personal beliefs. Induction and deduction in mathematics and science. The role of patterning and generalisations. At both institutions implementation of the Unit consisted of thirteen weeks, each with a two hour lecture and two hour tutorial. Assessment of the Unit Assessment at both institutions was identical. During and after each tutorial, students were required to complete writings in a reflective journal.
Some of these required answering quantitative problems. Other responses required students to reflect on the content of the lecture. At the end of the semester, a 40 question multiple choice exam was administered. While the context of some questions was altered to suit the Malaysian environment, the two were essentially identical.
Peard, P. Roughly half come directly from high school while the others come from the workforce and other courses. Students are mostly full time, but there is a proportion of part-time students. Full time students take 4 units each semester each involving 3 hours of University attendance. The majority have part-time employment.
Those students coming directly from high school come from a variety of mathematical and scientific backgrounds. Some will have done no maths or science beyond Year 10 while others will have done advanced maths and science courses at Year There is no university accommodation for students and many travel large distances to attend.
Others reside privately nearby. They are students who have completed their form five or form six examination. They are all fresh school leavers who are twenty years old or less — young vibrant students.
The do not pay a fee for their course, instead they are given an allowance by the Malaysian Government for their food, lodging, stationary, clothes and one trip per year to and from home including flight fare for students from East Malaysia. All medical treatments are provided free in Government hospitals and clinics for students anywhere within the country even during their holidays. All students were selected through a stringent interview process and have a strong maths and science background.
Students who do not successfully complete the course will have to pay a fine according to the agreement they signed upon admission to the course.
Comparisons of Student Backgrounds The students in the Institute in Australian differ greatly from students in Malaysia in many aspects. The students in Australia are not a homegenous group. Many are adults who have other working experiences and commitments, while students in Malaysia are young school leavers eager to pursue a course in teaching. The Malaysian students do not pay a fee and are bonded by an agreement compared to their Australian counterparts who pay a fee. The Malaysian students live within the campus grounds, while Australian students find their own accommodation outside the campus grounds.
As for the academic entrance qualification, all Malaysian students have strong maths and science high school qualification, while the Australian students come from a variety of disciplines. Responses to this were voluntary. Phillip Adams in his column in the Australian Magazine March , states “We could do better if we taught our kids that kicking ideas around is at least as much fun as kicking a football around. Have you been challenged by these ideas? Do you believe your understandings of the nature of mathematics and science has changed?
Have you enjoyed the experiences of kicking ideas around? Let the teaching team know your thoughts in this discussion forum. Of the 28 responses, 22 were considered positive, 2 negative and 4 neutral.
Some responses appear in Appendix 1. One week before the forum the questions were printed and given to the students. At the forum the lecturer selected 6 leading speakers 3 male and 3 female from each of the 3 races: Malay, Chinese and Indian. After each speaker had given his or her view of one of the questions other students were called upon by the Chairperson of the forum to give their views.
The sixth speaker was first asked about what they did not find interesting about the course and then to give an overview of what they thought about kicking ideas. Every student was called upon to give a view. All 50 students gave positive responses about kicking ideas. Some responses appear in Appendix 2. Tables 2 and 3 show a comparison of the distribution of final marks and grades.
Sivasubramaniam Table 3. Nevertheless, both groups of students reported very positive results with a high degree of similarity of responses refer appendices 1 and 2. Analysis of Qualitative Data The qualitative data from each institution were examined to determine common responses and these categorised as shown in Table 4.
Table 4. This may be attributed to a number of factors. One suggestion is the diverse backgrounds of the QUT students compared with the more homogenous and generally stronger mathematical and scientific school backgrounds of the Malaysian students.
He noted that in Queensland, and in other Australian States, higher achieving students are encouraged to take the academic option while lower achieving students are encouraged to take the non- academic option.
Given these circumstances, one might expect the samples from two distinctly different populations to show considerable difference in the distribution of achievement with little overlap. However, this was not observed in the study. Furthermore, those students who had done only Year 10 or 11 mathematics achieved little differently from those who had done the non- academic subject in Year Thus, if pupil background does not correlate with achievement for the QUT group, it is unlikely to be the sole factor in the present study, though we must consider the possibility that it is a contributing factor.
Attributing the difference to different assessors can be ruled out due to the moderation of the journals and the use multiple choice items in the exam. Motivation and expectation. Malaysian students may be motived more than their Australian counterparts. The fact that the Malaysian students are given an all expenses paid for course and in addition receive free medical benefits and an allowance, relieves them of any financial burden, unlike their Australian counterparts who have to supply their own money for everything.
The Malaysian students made teaching their first choice as their career, while for many of the Australian students education was their 2 nd or 3rd choice.
While many QUT students are well motivated, there are those who are content simply to pass. The main employing body, Education Queensland, has up until recently not considered academic grades when selecting graduates for employment, instead relying solely on interviews of graduates for ratings.
Changes in this procedure from next year will require graduates to have a grade point average of 5 or better to be eligible for a top rating. Malaysian students are aware of scholarships for masters and Ph. D programmes conducted locally and overseas. One of the major points for the scores to award the scholarship is from the grade they obtain in their Bachelor of Education course and hence this may motivate them to strive more than their Australian counterparts.
There are some Malaysian students who would like just to pass and not attend lectures and tutorials but there are discipline procedures which have been spelt out to the students that deter them from following such a course of action.
The course guarantees the student an immediate job in Malaysia. Hence, the course is a job guaranteed course. This further provides a purpose to motivate the Malaysian students. The Physical Environment in which the two courses operated cannot be ruled out as a cause of the differences.
Living within the campus grounds is a major advantage because students have discussion sessions to kick ideas around at any time that is convenient for all in the group. In fact the Malaysian students have claimed that they enjoyed their discussions so much that it was more fun than sleeping on free afternoons. Living together within the campus grounds enables students to go out on weekends together. Sivasubramaniam their lectures and tutorial questions during such outings.
All this means that they do spend more time kicking ideas around. Other possible reasons Malaysian students have realised that their discussion session have helped them in many other ways. They claim that kicking ideas around helped them to improve their English because they had to speak in English and look up references in English. They also claim that they have become closer to each other through the interaction demanded by the course. All the Malaysian students claimed that they have discussed what they had learnt in the course with their families and or friends.
This means that they thought about their work even during their holidays. Hence, kicking ideas around has indeed instilled confidence for the Malaysian students to bravely discuss it with others well beyond the boundaries of the IPRM campus and independent of help from lecturer and friends. The main contributors for this are the content of the course and the method of delivery — students experiment and discuss their findings.
The content is related to the students real world and this draws students interest to the course. The experiments provide the concrete objects to understand the real world phenomena better. The discussion demands verbalising aptly to convey ones ideas and it also provides practice to verbalise confidently. The testing is done with their friends when they are kicking ideas around and the brave confident delivery is back home with family and friends.
The students in this course were divided into small groups of five to have discussions and to do their experiments. The small groups gave all students more opportunity to speak more often and they also felt comfortable taking risks of trying out their thinking during tutorials in the presence of the lecturer and also outside the lecture time. This method also promotes social interaction among the multi-racial Malaysian students.
These results confirm earlier research such as that reported by Brissenden who noted that the importance of language and communication, practical work and understanding expressed in the The Cockcroft Report, Mathematics Counts, is also supported by a massive body of other research evidence.
These themes demand discussion between teacher and pupils and between pupils themselves as essential features of mathematics lessons at every level. The current research reported here would support the conclusion that the implementation of the unit does in fact improve language skills, develops better understanding of concepts, and helps develop social skills. Thus the implementation of this unit is consistent with the recommendations of recent reseach in the field.
The lower achievement of the QUT students is a cause of concern and is a topic that must be investigated further by the author. Clearly, if the Malaysian students are capable of such high performance, their QUT counterparts should be capable of higher achievement. The mathematical understanding that prospective teachers bring to teacher education.
The Elementary School Journal, 90 4 , Grootenboer, P. Har Eds. Mathematics Education for a Knowledge- based Era. Peard, R. Putt, R. Faragher, M. McLean Eds. Mathematics Education for the Third Millennium. Mathematics content electives in pre-service primary teacher education in Australia. Quigley Ed. Science, Mathematics and technical education for National Development. University of Brunei: Brunei Darussalam. Relich, J. Pre-service primary teachers’ attitudes to teaching mathematics.
Southwell, B. Owens Eds. Lovell, K. University of London Press: London. Davidson, N. Ed Cooperative Learning in Mathematics. Addison-Wesley: New York. Brissenden, T. Talking About Mathematics. Basil Blackwell: Oxford, England. But, originally when I learn something and that’s that. It makes it easier for students like me, whose favourite subjects have never really included maths to get motivated and involved. I like this new way of thinking. I can’t wait till I get to use it in a classroom to excite students the same way I’ve been.
Also I think that by kicking ideas around teachers can create a really open and positive learning environment. It can really help with making friends in class. All of my previous forays into this field have seemed rigid and with no room for error. I’m looking forward to learning how to kick ideas around and approach problems logically, not just by recalling equations learnt by rote.
Being able to discuss problems with a group in an environment solely set on encouraging higher thinking is great. And yes I have had a few light bulb moments! I have been talking to family and friends about the unit and its also been valuable discussing things with students who have other tutors. Looking forward to this weeks tut! So I was a child of the rote learning method of everything.
Memorising formulas and all that jazz. On reflection I can see now why I did well in maths in some years and performed poorly in others. In the years I performed poorly I was not enjoying the content because the teachers did not convey its relevance and meaning and I could not relate anything to ‘real life’.
I enjoyed the group interaction because everyone in our group brought different strengths and skills to the table – a sharing experience that was missing from my school years. Hehe but this subject has certainly given my mind a work out. Maybe I am lazy but once I found a solution to a problem; I never bothered to ‘kick’ any further. The tutorial gave my brain a good work out. I am most definatley sic looking forward to more. Because I graduated 9 years ago, the things that I had learnt at the time always had formula or an iron fist that shoved the correct opinion in my brain.
I guess I am feeling good that my mind is being opened and I am sharing my ideas with people I live with outside uni, however there is still a lot of anxiety created within myself if I am asked to find my own way to deal with problem solving. Appendix 2: Qualitative data IP; some sample responses All positive 1.
To what extent do you believe you have ‘kicked’ ideas around in this unit so far? So, to accomplish the task we kicked our ideas around with friends, peers and lecturer as well.
By kicking ideas we had learnt and gained more ideas. We also had fun during discussion and come up with some interesting ideas. We discussed and compared to come out with the best answers. Along the process, I gained a lot of knowledge. During this unit, we kicked ideas when we faced any doubts and different views.
The questions given to us were more towards discovery approaches that helped us think beyond our limits; initiate us to do extra reading to understand it. I gained new knowledge and positive learning attitudes that useful for my future teaching. Finally I enjoyed learning this unit. Sivasubramaniam 2. When to solve the problem for the tutorial tasks I must relate each other. There was a lot of information we need to know. Besides, we not only got information from book but also other resources.
Assignments were quite difficult and challenging. We needed to cooperate with team mates to find out the answer. I found a lot of new things to be discovered when doing each of the tutorials. I had to find many materials from variety of sources such as internet, books and etc. I put a lot of effort to complete each of the tutorial and it was really challenging. My knowledge has broadened. I learnt many things that I did not know before. I understand more about the things around me and my world now.
Many new things have been learnt. More over, I came to know that there is always a logical reason for everything we have learnt. This has made mathematics and science interesting. We held a lot of group discussions. We tolerated and cooperated very well during the discussions.
The experiences were really so nice. The way our lecturer, Dr. Puma taught us, also really impressed me and I really enjoyed her lectures. Because it was more challenging. The answers are not simply right or wrong.
We need to search more information to answer one question. It makes me learn a lot of general knowledge. It will make the concept clearer. I enjoyed the experience of kicking ideas around because it lead me to gain new knowledge and the teaching methods which was used by my lecturer was very effective. All 50 students stated that they have discussed what they have learnt with family and friends. Do you find anything that is not interesting about this course? Realised maths can explain everything.
Before this I did not like maths, example probability to win a lottery. Have to find and read a lot of resources — is good to kick ideas. Thanks to Dr. This presentation describes a number of instructional strategies proven to increase understanding and achievement and discusses how the strategies apply to the use of GSP, both in hands-on activities and in whole-class presentations.
In the course of describing the strategies, the presentation surveys the breadth of mathematics that GSP activities and presentations elucidate, with attention to how these activities incorporate and enable various instructional strategies.
The survey ranges from number sense in primary classrooms through calculus and other advanced topics in secondary classrooms. The three essential elements contributing to this transformation are captured by the three descriptors at the beginning of this paragraph: interactive, dynamic, and visualization. The software is interactive. It gives students control both to create and to manipulate mathematical objects, and it gives students immediate feedback through the mathematical behaviour they observe.
The creation of mathematical objects is perhaps even more important, giving students a sense of ownership of the mathematics, a sense of their power to take control of their own mathematical universe. The software is dynamic. It allows the rapid exploration of many cases resulting from a mathematical construction, with changes in the initial conditions resulting in immediate changes to the mathematical consequences. So much of mathematics is about variation, either explicitly or implicitly.
Often students think of a variable as a mystery number, as a symbol representing an unknown and unchanging constant. Using this software, students come to realize that variables really vary, whether those variables are algebraic the value of x in an equation or geometric the position of a point in the Euclidean plane.
And the software is visual. The ability to identify mathematical invariants, and to observe the behaviour of and constraints on mathematical variables, is crucial to learning and doing mathematics. One purpose of this presentation is to examine a number of instructional strategies in relation to the educational use of this software. This observation is almost certainly true of education not just in the US, but around the world.
Marzano and colleagues have surveyed a large number of studies in order to identify those instructional strategies that result in increased student achievement. Many of these strategies are well-suited to, and enabled by, the use of interactive dynamic visualization software. This presentation provides specific examples of how student activities can take advantage of research-based instructional strategies. Another purpose of this presentation is to survey the breadth of mathematics elucidated by activities and presentations designed for interactive dynamic visualization software.
For this purpose, each instructional strategy will be illustrated by one or two Sketchpad activities. Each section below considers one instructional strategy, describing one or two examples of its application in the course of specific student activities. They can easily pick out the similarities as being the x—intercepts and the differences as being the curvature and opening direction of the graph.
They can further relate the invariants in the image to the invariant parameters r1 and r2, and the changes in the image to particular values of a. On the right is an image resulting from varying r1. Students can again pick out the similarities and differences, and relate these features to the parameters. By enabling students to identify the similarities and differences in graphs, the software focuses their attention on the relationship of the different parameters to the characteristics of the graph, and encourages students to go on to the next step: explaining why a particular parameter has the effect that it does.
The object is to get students to go beyond appearance to investigate properties. In a dynamic environment the properties of an object are shown through its behaviour rather than through its static appearance at one instant in time. In the first construction, on the left, students drag each of the four vertices, and only when they drag vertex D do they find that the construction does not truly determine a square.
This motivates students to discuss the properties of quadrilateral ABCD as revealed by dragging the figure. Is ABCD a “real” square?
Which polygons are real squares? By manipulating the figures and identifying similarities and differences, students pay more attention to the properties and come to a deeper understanding of how a square is different from other quadrilaterals.
In the context of a Sketchpad activity, the act of summarizing requires students to decide which of the behaviours they have observed are important and which are not. For instance, in an activity comparing the geometric transformations of translation sliding , rotation, and reflection, students summarize first by examining all three transformations applied to the same preimage object.
In the process, they deepen their understanding of the transformations they have been studying. Although virtual manipulatives such as dynamic geometry constructions were not specifically part of the studies summarized by Marzano et al. In the United States, there is an ongoing effort to reform the teaching of mathematics, exemplified in the publication Principles and Standards for School Mathematics National Council of Teachers of Mathematics, NCTM , p.
These representations can help students form compelling mental images of the mathematics they are studying, enabling deeper understanding and better retention.
Students begin with a point between 0 and 10 on the number line, and estimate its value. In this instance, they might say it appears to be halfway between 6 and 7. Students then press a button to zoom the portion of the number line containing the point in order to see it more clearly.
First the part of the number line containing the point drops down, as on the left, and then it expands to show more detail. Students again estimate the number. At the conclusion of the activity, they can ask to see the actual location of the point expressed numerically.
The representation entailed in zooming the number line relates to and reinforces the meaning of the representation as a string of digits. In another activity, students can explore the connection between the three-dimensional view of a cube and the two-dimensional representation of its net. Their task in the diagram on the left is to draw segments on the cube to show the pattern that should appear on each face based on the patterns shown on the net. Once they finish, they can press a button to wrap the net around the cube, resulting in the partially-completed animation shown on the right.
Many teachers prefer to have two or three students work together on an activity, finding that the small group promotes mathematical discourse and helps students learn from each other and develop a sense of self-reliance.
Two key conclusions from the research are these: 1. Organizing groups based on ability levels should be done sparingly. Cooperative groups should be kept rather small in size. Marzano et al. The value of mixed- ability groups quickly becomes apparent as all the students benefit from the process of teaching and learning from each other.
The value of small groups is clear as students have more opportunity to interact directly with the software. The first student then tries to match the slope measurements with the lines, as in the sketch shown below on the left.
In this instance the first student has matched three of the measurements correctly and gets three points. Then the students trade places to give the second student a chance to match the slope measurements with the lines. Feedback should be timely. Feedback should be specific to a criterion [as opposed to norm-referenced]. Students can effectively provide some of their own feedback.
Such immediate feedback is certainly timely, and also has the advantages of being criterion- referenced, of coming from a source other than the teacher, and of being non-judgmental.
When students work in pairs or small groups, additional feedback comes from other group members—also a particularly effective kind of feedback. The student receives immediate feedback as the rabbit goes beyond the target, with no need for the teacher to intervene, and the image makes the required corrective action easy for the student to determine: Because the rabbit actually hit the target but continued on, the number of jumps was too great.
In another example, calculus students can begin to learn about antiderivatives by constructing a slope probe: a short segment whose slope is determined by a given function. The figure on the left shows the trace left by a student dragging such a probe in the direction of its slope a slope which changes according to the value of the given function to trace out the approximate antiderivative of the given function.
The probe can be seen at the right end of the trace. During the dragging process, the slope of the probe gives the student constant feedback as to the direction of the antiderivative, and the thickness of the trace gives the student feedback as to the accuracy with which they are following the correct direction.
From the research, here are some useful tips for teachers: 1. Hypothesis generation and testing can be approached in a more inductive or deductive manner. Teachers should ask students to clearly explain their hypotheses and their conclusions.
In inductive hypothesis generation, students manipulate the sketch to generate a large number of cases and use their direct observations to form a conjecture. In deductive generation, students consider what they already know about the mathematics embodied in a sketch and form a conjecture based on that knowledge.
In either case students go on to test the conjecture in the same sketch or in a different sketch. Properly used, the generation and testing of hypotheses leads students to see the importance of presenting logical arguments for their conclusions. Seeing the value of and feeling the need for proof is an important consequence of hypothesizing and testing. A student has positioned the two red markers to add two secret numbers, with the sum shown below, also in code.
The student drags the markers to different positions, observes the sum, and uses the observations to makes hypotheses. Additional dragging is required to test the hypotheses, and the student continues generating and testing hypotheses until she has broken the entire code. By playing multiple games and improving their strategies, students improve their ability both to generate hypotheses and to test them.
Crack the code. They construct the triangle and one exterior angle, measure the exterior angle and the remote interior angles, and form a conjecture about how they are related. They test their conjecture by performing a calculation and the dragging the vertices to vary the angles. Finally, they do a rotation and a translation of the original triangle that suggest the path to a proof.
Various studies point to the kinds of cues and questions that are most effective: 1. Cues and questions should focus on what is important as opposed to what is unusual. Questions are effective learning tools even when asked before a learning experience. The questions presented in activity worksheets, and the questions teachers use both with individual students during the activity and with the entire class in a summary discussion, should tend toward an appropriately high level.
Teachers often get impatient and either call on the first student to volunteer or answer a question themselves if no student volunteers quickly. This is usually a mistake; students need time to think about a question and formulate their own answer.
Providing this wait time will elicit better answers from a wider variety of students, and will make it possible for those students who are not called on to compare their own answer with the answers given by other students. An effective related strategy is to call on several students to answer a question in turn, allowing each to express the answer in their own words. The activity Cartesian Graphs and Polar Graphs provides an excellent opportunity for teachers to use this strategy.
First, make a wild guess about what it will look like, and write down your guess. There is growing sentiment that classroom teachers … are almost impervious to change…. We believe that this is an overly pessimistic view not only of staff development, but of the profession of teaching in general. We agree, however, that substantive change is difficult. Busy teachers who have been doing things the same way for a fair amount of time will have many valid reasons for not trying a new strategy.
What is clearly required to alter the status quo is a sincere desire to change and a firm commitment to weather the inevitable storms as change occurs. The adoption of GSP and GSP activities into the Malaysian mathematics curriculum presents a unique opportunity for teachers to consider using new instructional strategies to accompany the introduction of new instructional technology.
When teachers are trained to use GSP activities, they can also be trained in how to use the most effective instructional strategies with this new instructional tool.
In conclusion, Sketchpad activities provide us with a unique opportunity both to change the way students learn mathematics promoting deeper learning, better retention, and a sense of the excitement and beauty of mathematics , and also to change the way teachers present mathematics to their students taking advantage of similarities and differences, summarizing, nonlinguistic representations, cooperative learning, effective feedback, generation and testing of hypotheses, and high-level cues and questions.
Pickering, D. J, and Pollock, J. Hopefully, this information will help you solve your problem. If you have any other issues related to the Zoom app, feel free to comment in the comment section below. We will try to solve your problem as soon as possible. Zoom Error Code is a problem related to connectivity in the Zoom app. First of all, you need to check your internet connection.
Update the Zoom app to tis latest version, clear cache, disable antivirus, configure network firewall, wait for a while, etc. If your network is secured by a firewall or proxy, you can contact a network administrator to check the Firewall and Proxy settings. Zoom Pas sword Not Working? ViralTalky’s team of content writers regularly helps write high quality content. Its a party over here. Your email address will not be published. Save my name, email, and website in this browser for the next time I comment.
Skip to content. What is Zoom Error Code ? In this article, we briefly explain the ways to fix these issues with the step-by-step guide. Check your Internet connection First of all, you need to check your internet connection, if there is any internet connection problem, please solve it first. Refresh the web page or click on the Retry button.
Turn off all your devices that are connected to your router. Try to keep your router near your device. You can try connecting your device to the router by using an Ethernet cable. You can also contact your service provider to fix it.
Also, you can restart your router for better speed results. Check Sim Settings Open Settings on your device. Click on SIM cards and mobile networks.
Click on your Sim and tap Mobile Networks. Tap on Select network automatically. Update the Zoom app Sometimes an outdated version of the application is responsible for various problems in the application.
If you are using an outdated version of Zoom, you can update it from Play Store or Zoom. Clear Application Data You should also clear the app data and allow all permissions, after doing both, restart and login again. See that the problems have been fixed or not. Uninstall and Re-install the app Uninstall your Zoom app and wait for a while.
Make sure to temporarily stop the antivirus while reinstalling the Zoom application.
Zoom: Fix Error Code – Public Knowledge – Rowan University – Rowan Support Portal.
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