My submission
to: Walter Whiteley, York University
from: Richard Clausi, Elmira High
regarding: Your position on Calculus
date: January 30,2006.
Hello Walter:
I appreciated your email today inquiring as to “common ground” in our ongoing discussion of the future of Calculus. I have included both your email and my original email response to you (Dec. 21,2005) to remind you of my personal opinion. I would like to address my comments to you in your capacity as chairman of the
Council of Ontario Universities (COU) Math Advisory Group that is commenting to the Ministry of Education and other participants on the revisions and revision process.
The suggestions given in the Nicholls' petition to the Premier and Minister are a bottom line position that over 280 people have expressed support for. I assure you that the recommendations within that petition represent lengthy discussion and a starter solution to the problem of Math dilution.
Even at that, there is a major concession to mediocrity- there are precisely 6 courses in the suggested course menu (academic 9,10,11, MCB4UI, MDM4UI,MGA4UI) as
compared to 7 (academic 9,10,11,12, Calculus, Algebra, Finite/Functions & Relations) in previous years. This is a dilution, pure and simple, and very unsettling indeed given that these courses are OPTIONAL courses. I have wondered if the current Calculus could be taught in grade 11 if the algebraic skills relating to factoring and division of polynomials could be firmed up in grade 9 and 10. This is a radical notion but, my point is that rather than removing Calculus, that study could have an earlier presence in the secondary school. I quote from a letter Ed Barbeau sent in mid January: "It is not for nothing that Calculus is regarded as one of the great intellectual achievements of all time, and we have managed to come up with a denatured version where students treat it as a sterile operation with no inkling of its depth and power." The very thought that Calculus could be written entirely out of the curriculum or ,even, trivialized seems to be a horrible intellectual injustice.
I am sure we can agree that there is no such thing as bad Mathematics-our problem is that we have over 2500 years of it to fit into a secondary school curriculum.
Walter, we need to break away from the notion that “important” Mathematics distilled from all of the Math ever discovered can be taught in 4 years. Your group needs to re-visit the notion of students electing to stay in the secondary school (and, today, there is no regulation that forbids this) until they feel they are ready to step into post-secondary studies. Students do mature mathematically at different times. We really need to move our students to the leading edge of current knowledge by the end of secondary school (be it after 4, 5 or even 6 years). This breathing space might also address your concerns with success and stress levels among students and teachers.
The Calculus course which the Waterloo Math Heads have developed as a potential Locally Developed Course, as it now stands, could be an excellent course that fulfills Gord Nicholls' suggestions. If you pursue the 50/50 “Whiteley/Taylor” model as your proposal now stands then there is still a gap to be filled. Courses such as ours would still have to go forward --students need to see more Calculus than you are proposing before post-secondary studies. Perhaps this opens the door for a legitimate, province-wide adoption of the AP Calculus. Unfortunately, many issues of Equity and Access ,within boards and across the province ,emerge if we wish to publicly fund the AP. There must be a better way.
I like your suggestion of reviewing the entire continuum from grade 1 onwards. If this is to happen though, the Ministry must rethink the protocol. The process , quite frankly, has failed miserably this time around. It has generated frustration and bewilderment across the province. Mathematics education is too important to be left to the whim of any small group of insiders who may have captured or ambushed the attention of the government of the day. Historically, it is an Ontario tradition that each new government puts it stamp on education by “tinkering” with the curriculum. Mathematics and English are usually the first to be dissected. I imagine the goal is to ensure that as students graduate, those students become “ready” for whatever happens after graduation. It is a way to directly affect our future economic prowess. This wishful thinking takes a turn for the worst with the willful dismantling and dilution of a body of knowledge that is critical to the competitiveness of the nation. This damage could be pervasive, with serious local implications as we fall behind other jurisdictions.
I appreciate the problem your committee has.
Curriculum design means making wise choices in terms of what to deliver and how to deliver it. Clearly, to develop a continuum of Mathematics experientially or via personal, hands-on “manipulatives” would take a very long time…in fact, centuries! We do not have that luxury of time. It is equally obvious that every item cannot be represented in a curriculum that must span some 13 years. Tough choices have to be made that reflect priorities that in this 21st century must represent what best prepares our youth to participate in the professional community/local community/industry/job market.
All of this must also include considerations of pedagogy and delivery. You need to ask: “How do we implement and phase in the rapid transfer of knowledge to young people so that courses seamlessly and transparently take students to a point where they are ready to “contribute” to the nation?”
I believe that Mathematics curriculum designers would be wise to remember the words of Karl Friedrich Gauss, one of the greatest Mathematical minds of all time.
He observed:
“MATHEMATICS IS THE QUEEN OF THE SCIENCES,
ARITHMETIC IS THE QUEEN OF MATHEMATICS...”
In effect, the language and abstract concepts of Mathematics are humanity's attempt to touch a creator and see a “master plan”. Occasionally the fog clears and the horizon appears briefly---magical mathematical moments when the Einsteins and Newtons push us forward-- then darkness again. We are driven by this special urge, unique among species on the planet, to understand, to change and to improve our world. Our secret weapon is the abstractness and the language of Mathematics and Science.
Significantly, Gauss then observes that all of this abstraction is built on a foundation of pattern and simplicity--the heartbeat of Mathematics is arithmetic; that is,
Arithmetic IS the Queen of Mathematical analysis.
Gauss seemed to realize that the secrets of the universe play out in the mechanical arithmetic behaviour than underpins what we concretely see. That pattern is the basis for our number system, and the first 1000 years of Mathematical thought. It is appropriate to deal with the “basics” in the primary and junior grades (1-6). We all can agree that any educational experience should begin “at the beginning” but the study cannot dwell too long on those “basics”. Sooner, rather than later, we must move beyond concrete experiences which can only deal with things we can touch. We must focus on a grander manipulation of ideas. Perhaps, the 21st century needs a broader and more powerful set of “3 R’s”: Recall (knowledge of basic facts and theory), Reasoning (the ability to manipulate basic facts and insights to extend knowledge) and Research (either knowing where to go and what to do, if you forget a tidbit or what do to if you need new knowledge to solve a new problem!).
There is certainly a great deal of Mathematics thought that could fill a curriculum!
For this reason, we need to clearly define what a student exiting elementary school must have to continue Mathematics studies. This must mean more than just arithmetic and warm, fuzzy feelings of math. Students need to be exposed to specialists who have excited them about the subject and ignited curiousity in the subject so they are primed for grade 9.
We also need to clearly define what a student exiting secondary school must have to survive in a technological world as a user of technology, as a worker in technology and as an innovator in technology. This means we need to look at the needs of the colleges, of the universities, of the hi-tech and research sector, of the manufacturing sector, of the trades sector and of the citizen.
Everyone needs to have a voice in what the soul of Mathematics Education should be.
This “backwards design” approach means that we need to start our thinking at where we want to end up. It means deciding what the exit knowledge for secondary school and elementary school must be, and then working backwards from grade 12 to make sure that each year becomes a solid link in the chain of events that leads to clearly defined outcomes. This soul-searching must define the needs for a student exiting secondary school for all the major career paths because there is a dignity in every role that contributes to the good of society. There is also some urgency to find innovative ways to move our youngsters as quickly as is appropriate to their individual abilities to their leading edge-- in effect, not only must the torch be passed on , but that torch should still be “burning” so that it can be used to illuminate the darkness beyond. This does not penalize those students who struggle with Mathematics. It is possible for students to feel good about Mathematics AND do good Mathematics at the same time. These notions are not mutually exclusive but
we cannot throw out content in optional courses to facilitate success and warm feelings. By reducing and diluting Mathematics courses, we are taking post-modern Ontario back to the pre-sputnik days. We cannot afford to teach Mathematics that is mired in the past.
Finally, I would like to include a set of suggestions that might help address the concerns that post-secondary institutions have expressed with respect to the “readiness” and skill of incoming students. I offer these as the personal observations of a classroom teacher and a department head. I hope your group will consider these:
1. Develop a “back-planning” protocol that uses extensive consultation with post-secondary and industry to develop a target knowledge set students will need at graduation to pursue their life ambitions. Use the test of utility to determine what stays or goes. If there is more content than can fit, rather than squeezing it in, create more courses and allow time for optional courses to prepare students for a productive post-secondary experience.
2. Restore the Mathematics content in secondary school to at least the post-sputnik levels; that is, ensure that there are at least 7 Mathematics courses at the advanced level and one of these should be a comprehensive introduction to differential and integral Calculus.
3. Expand the Mathematics opportunities, optional or otherwise, in the secondary school so “new” Mathematics can enter the secondary curriculum..
For example, Computer Science topics including repetition and selection structures could easily be introduced as early as grade 9 Mathematics by using the TI83 programmable calculator/computers which all schools already have. We are woefully behind in including this new branch of Mathematics in the mainstream Mathematics curriculum.
4. Mathematics is a lifelong learning activity. Semestering is a cheap wine that can “satisfy” by delivering higher marks but falls short on depth and quality. All Mathematics courses should be traditional length and non-semestered (ie 10 months long). Perhaps, semestering is the root cause of post-secondary complaints regarding “readiness” and “quality”.
5. Ensure that specialist Mathematics teachers are in the senior elementary school Mathematics to both support and lead Mathematics instruction. Similarly, only Mathematics specialists should teach secondary Mathematics courses.
6. Create “Greenhouse” opportunities: Place real limits on class sizes especially in grade nine. Class sizes of 34 in grade 9 are land mines in Mathematics education.
7. Use technology to deliver courses that may not be available because of the size of the class or the size of the school. In the 21st century, internet or video delivery of material facilitated by a local Master Mathematics mentor can ensure that ALL students have access and opportunity.
8. Students within a district must not be barred access to certain courses or be forced to travel from a home school because a funding formula cannot provide for a class of , for example, MGA4UI. Equity and access are benchmarks for a public system of education.
The Ontario funding formula must be adjusted to support Mathematics education.
9. With the quick growth of mathematical knowledge, consider retiring the notion of a textbook, and use technology to make new materials such as online worksheets and lessons course-agile. For example, constant changes require constant expenditures on textbooks. Curriculum revisions can be facilitated by using online “tables of contents” that can be changed to reflect curriculum changes quickly while saving financially on the purchase and disposal of “old” texts. It is entirely possible that wireless “Blackberry-like” technology may allow each student to have a laptop that is every text, and note needed for secondary education and beyond.
10. Develop wise copyright arrangements similar to those that exist in industry and universities so that teachers can develop materials for online use while retaining some ownership and shared royalty rights. We need to foster creativity so materials will blossom and not be at the whim of text publishers.
11. Phase in courses changes, year by year, rather than implementing several courses across different years at the same time. The “here is a hot potato, catch it” game hurts students. A protocol for curriculum evolution is vital.
12. Use sophisticated tools such as MAPLE to help students move beyond mechanical manipulation so they can focus on problem-solving. Problem-solving should permeate the secondary curriculum in non-artificial and meaningful ways. We need to accelerate the passing on of Mathematics knowledge so it can be acquired and used to extend knowledge. The mood for this research and exploration is set in the secondary school.
13. Avoid being seduced by the use of models that do not blend seamlessly into the concept being explored. A system that must be bent into shape is not an aid to understanding but rather a distraction and hindrance. Clarity of vision requires clarity of understanding. Play and conjecture with technological tools should not be used as focal points in place of mathematics but rather as starting points for learning mathematics. “Playing” with manipulatives must lead somewhere and quickly.
14. Beware of gimmicks that rob from content time. Ideas such as homework and tutorial time that count as part of the allocated course time are simply ways to get homework done, boost marks and provide time for students to do more extra-curriculars or perhaps have part-time employment. Maximum time must be provided to learn Mathematics within the limited time window that exists. Do not drop content to make courses fit abbreviated and artifical time slots.
15. Teach Mathematics as a human endeavour that seeks to explore the universe and improve humanity.
16. Respect the language and traditions of Mathematics. Do not trivialize Mathematics.
In conclusion, I am pleased to hear that a task force is being set up to examine Calculus, and, hopefully, these larger collateral issues, too. Serious steps are necessary right now to reverse the recent tide of Mathematics educational thinking that believes that good Mathematics is a game best played with knowledge that is centuries old. Too much is at stake to ignore the pitfalls of crippling our best students by denying them the opportunity and means to move quickly to the leading edge of Mathematical knowledge. There is an urgency! Moore’s Law predicts that technology is about to stall as it struggles to break through the physical limitations of chips and speed. I hope that history will not record that post-modern Mathematics, paralleling Moore’s Law, became just an “after math” as it failed to escape a groove that had become a rut.
respectfully yours,
Richard Clausi,
Elmira District Secondary School
ENCLOSURE #1
=========================Dec. 21, 2005 email=====================
to: Walter Whiteley, York University
from: Richard Clausi, Elmira High
regarding: Your position on Calculus
Dear Walter:
Thank you for your email today. I am pleased that the ministry discussed the possibility of locally developed Calculus courses on December 1,2; however, you forgot to tell me the outcome. Does our course which you now have in your hands have a
chance of approval or is that door closed to us?
I appreciated your "rant.doc" as it answered several questions which had been posed to me by members of the community. As you can see, I am including them in my response with your attachment since, it seems, to provide an insight into the
reasoning behind the removal of Calculus from the curriculum.
Walter, we disagree in many fundamental ways, but given the influence you have had in this decision, and perhaps given that yours may be the underlying reasoning behind the decision to vacuum Calculus out of the secondary curriculum, I must draw
your attention to several vital points:
1. The Calculus is an OPTIONAL course! It is beyond the 3 credits required for graduation. You point out in "rant.doc" that you care about "the damage done to students as they hit a wall in first year calculus" yet, engineering , mathematics and
science faculties will not drop Calculus from their curriculum because Calculus is a fundamental part of what they do and how they do it. If you advocate dropping Calculus, it would appear you are denying students any hope they have of surviving
at all. I am sure this is not what you intend. Of course, as our students attempt to compete with students from other provinces and countries who do have Calculus, they will be the underdogs. Our young people will have the unenviable distinction of
being the worst-prepared students for advanced Mathematics study in 40 years.
Again, I am sure this is not what you intend.
2. You are ignoring the need to prepare creative innovators by providing them with the opportunity to study high-end mathematics. Premier Dalton McGuinty, in a recent visit to K-W, indicated that we are in a technological battle for survival in
which innovation is the key to our ability to compete. Today, in the Kitchener RECORD (Dec. 21, 2005) in a separate news item, U of Guelph president Alastair Summerlee says: "with the right resources we can deliver answers to the very serious
21st century problems facing humanity." In the same article, Tom Jenkins, chair of Open Text Corp. says "if you want to harvest corn in August, you had better put in the seeds in the spring". Walter, the solutions to those problems do not lie in
the concrete domain... they require the ability to create mathematical models that can be analyzed using elegant and abstract techniques. We need to ensure that we provide these tools to our innovators as early as possible. We need more not less
Mathematics in the secondary schools of Ontario.
3. You suggest that teachers are "obsessed" by Calculus and that somehow sudents are denied a "richer and more sensible experience than the techniques of differentiation". Walter, You under-estimate our students. They are bright and eager, and our
best excel. Rather than starving them of substantial mathematics and offering playful manipulatives and games , we should be exploring ways to expand and enrich and accelerate the curriculum. Clearly, this is hard to do if we are looking at
removing Calculus, and if I read your covering note correctly, diluting the grade 12 Geometry and Algebra course, and for good measure simplifying the Grade 12 Data Management course.
These actions all point to a diminishing of the secondary Mathematics curriculum... or if you wish a "dumbing down". If we are going to count on our young people to solve big problems , we need to provide them with current and powerful mathematics
in the secondary school. Your "refined and playful approach to problem solving and reasoning" may not cut it in the 21st century. I would like to draw your attention to an exceptional editorial in today's RECORD (Dec. 21,2005) in which a layman
offers an insight into Calculus. I assure you that this opinion is matched by email I have been receiving since this story broke last week...
In short, the current Ministry plan IS "bad"..... but .....
there is no such thing as bad Mathematics- our problem is that we are trying to squeeze 2000+ years of it into a 4 or 5 year period. Napoleon noted that the "advancement and the perfecting of Mathematics are closely joined to the prosperity of the
nation". The solution to the ills of the country is not less Mathematics.
I appeal to you to use your influence to reverse this "done-deal"- this backslide to a pre-Sputnik time where Canada was a technological light-weight. We do not want to go there again.
sincerely,
Rich Clausi
Mathematics Department
Elmira District Secondary School
669 5414
from: Richard Clausi, Elmira High
regarding: Your position on Calculus
date: January 30,2006.
Hello Walter:
I appreciated your email today inquiring as to “common ground” in our ongoing discussion of the future of Calculus. I have included both your email and my original email response to you (Dec. 21,2005) to remind you of my personal opinion. I would like to address my comments to you in your capacity as chairman of the
Council of Ontario Universities (COU) Math Advisory Group that is commenting to the Ministry of Education and other participants on the revisions and revision process.
The suggestions given in the Nicholls' petition to the Premier and Minister are a bottom line position that over 280 people have expressed support for. I assure you that the recommendations within that petition represent lengthy discussion and a starter solution to the problem of Math dilution.
Even at that, there is a major concession to mediocrity- there are precisely 6 courses in the suggested course menu (academic 9,10,11, MCB4UI, MDM4UI,MGA4UI) as
compared to 7 (academic 9,10,11,12, Calculus, Algebra, Finite/Functions & Relations) in previous years. This is a dilution, pure and simple, and very unsettling indeed given that these courses are OPTIONAL courses. I have wondered if the current Calculus could be taught in grade 11 if the algebraic skills relating to factoring and division of polynomials could be firmed up in grade 9 and 10. This is a radical notion but, my point is that rather than removing Calculus, that study could have an earlier presence in the secondary school. I quote from a letter Ed Barbeau sent in mid January: "It is not for nothing that Calculus is regarded as one of the great intellectual achievements of all time, and we have managed to come up with a denatured version where students treat it as a sterile operation with no inkling of its depth and power." The very thought that Calculus could be written entirely out of the curriculum or ,even, trivialized seems to be a horrible intellectual injustice.
I am sure we can agree that there is no such thing as bad Mathematics-our problem is that we have over 2500 years of it to fit into a secondary school curriculum.
Walter, we need to break away from the notion that “important” Mathematics distilled from all of the Math ever discovered can be taught in 4 years. Your group needs to re-visit the notion of students electing to stay in the secondary school (and, today, there is no regulation that forbids this) until they feel they are ready to step into post-secondary studies. Students do mature mathematically at different times. We really need to move our students to the leading edge of current knowledge by the end of secondary school (be it after 4, 5 or even 6 years). This breathing space might also address your concerns with success and stress levels among students and teachers.
The Calculus course which the Waterloo Math Heads have developed as a potential Locally Developed Course, as it now stands, could be an excellent course that fulfills Gord Nicholls' suggestions. If you pursue the 50/50 “Whiteley/Taylor” model as your proposal now stands then there is still a gap to be filled. Courses such as ours would still have to go forward --students need to see more Calculus than you are proposing before post-secondary studies. Perhaps this opens the door for a legitimate, province-wide adoption of the AP Calculus. Unfortunately, many issues of Equity and Access ,within boards and across the province ,emerge if we wish to publicly fund the AP. There must be a better way.
I like your suggestion of reviewing the entire continuum from grade 1 onwards. If this is to happen though, the Ministry must rethink the protocol. The process , quite frankly, has failed miserably this time around. It has generated frustration and bewilderment across the province. Mathematics education is too important to be left to the whim of any small group of insiders who may have captured or ambushed the attention of the government of the day. Historically, it is an Ontario tradition that each new government puts it stamp on education by “tinkering” with the curriculum. Mathematics and English are usually the first to be dissected. I imagine the goal is to ensure that as students graduate, those students become “ready” for whatever happens after graduation. It is a way to directly affect our future economic prowess. This wishful thinking takes a turn for the worst with the willful dismantling and dilution of a body of knowledge that is critical to the competitiveness of the nation. This damage could be pervasive, with serious local implications as we fall behind other jurisdictions.
I appreciate the problem your committee has.
Curriculum design means making wise choices in terms of what to deliver and how to deliver it. Clearly, to develop a continuum of Mathematics experientially or via personal, hands-on “manipulatives” would take a very long time…in fact, centuries! We do not have that luxury of time. It is equally obvious that every item cannot be represented in a curriculum that must span some 13 years. Tough choices have to be made that reflect priorities that in this 21st century must represent what best prepares our youth to participate in the professional community/local community/industry/job market.
All of this must also include considerations of pedagogy and delivery. You need to ask: “How do we implement and phase in the rapid transfer of knowledge to young people so that courses seamlessly and transparently take students to a point where they are ready to “contribute” to the nation?”
I believe that Mathematics curriculum designers would be wise to remember the words of Karl Friedrich Gauss, one of the greatest Mathematical minds of all time.
He observed:
“MATHEMATICS IS THE QUEEN OF THE SCIENCES,
ARITHMETIC IS THE QUEEN OF MATHEMATICS...”
In effect, the language and abstract concepts of Mathematics are humanity's attempt to touch a creator and see a “master plan”. Occasionally the fog clears and the horizon appears briefly---magical mathematical moments when the Einsteins and Newtons push us forward-- then darkness again. We are driven by this special urge, unique among species on the planet, to understand, to change and to improve our world. Our secret weapon is the abstractness and the language of Mathematics and Science.
Significantly, Gauss then observes that all of this abstraction is built on a foundation of pattern and simplicity--the heartbeat of Mathematics is arithmetic; that is,
Arithmetic IS the Queen of Mathematical analysis.
Gauss seemed to realize that the secrets of the universe play out in the mechanical arithmetic behaviour than underpins what we concretely see. That pattern is the basis for our number system, and the first 1000 years of Mathematical thought. It is appropriate to deal with the “basics” in the primary and junior grades (1-6). We all can agree that any educational experience should begin “at the beginning” but the study cannot dwell too long on those “basics”. Sooner, rather than later, we must move beyond concrete experiences which can only deal with things we can touch. We must focus on a grander manipulation of ideas. Perhaps, the 21st century needs a broader and more powerful set of “3 R’s”: Recall (knowledge of basic facts and theory), Reasoning (the ability to manipulate basic facts and insights to extend knowledge) and Research (either knowing where to go and what to do, if you forget a tidbit or what do to if you need new knowledge to solve a new problem!).
There is certainly a great deal of Mathematics thought that could fill a curriculum!
For this reason, we need to clearly define what a student exiting elementary school must have to continue Mathematics studies. This must mean more than just arithmetic and warm, fuzzy feelings of math. Students need to be exposed to specialists who have excited them about the subject and ignited curiousity in the subject so they are primed for grade 9.
We also need to clearly define what a student exiting secondary school must have to survive in a technological world as a user of technology, as a worker in technology and as an innovator in technology. This means we need to look at the needs of the colleges, of the universities, of the hi-tech and research sector, of the manufacturing sector, of the trades sector and of the citizen.
Everyone needs to have a voice in what the soul of Mathematics Education should be.
This “backwards design” approach means that we need to start our thinking at where we want to end up. It means deciding what the exit knowledge for secondary school and elementary school must be, and then working backwards from grade 12 to make sure that each year becomes a solid link in the chain of events that leads to clearly defined outcomes. This soul-searching must define the needs for a student exiting secondary school for all the major career paths because there is a dignity in every role that contributes to the good of society. There is also some urgency to find innovative ways to move our youngsters as quickly as is appropriate to their individual abilities to their leading edge-- in effect, not only must the torch be passed on , but that torch should still be “burning” so that it can be used to illuminate the darkness beyond. This does not penalize those students who struggle with Mathematics. It is possible for students to feel good about Mathematics AND do good Mathematics at the same time. These notions are not mutually exclusive but
we cannot throw out content in optional courses to facilitate success and warm feelings. By reducing and diluting Mathematics courses, we are taking post-modern Ontario back to the pre-sputnik days. We cannot afford to teach Mathematics that is mired in the past.
Finally, I would like to include a set of suggestions that might help address the concerns that post-secondary institutions have expressed with respect to the “readiness” and skill of incoming students. I offer these as the personal observations of a classroom teacher and a department head. I hope your group will consider these:
1. Develop a “back-planning” protocol that uses extensive consultation with post-secondary and industry to develop a target knowledge set students will need at graduation to pursue their life ambitions. Use the test of utility to determine what stays or goes. If there is more content than can fit, rather than squeezing it in, create more courses and allow time for optional courses to prepare students for a productive post-secondary experience.
2. Restore the Mathematics content in secondary school to at least the post-sputnik levels; that is, ensure that there are at least 7 Mathematics courses at the advanced level and one of these should be a comprehensive introduction to differential and integral Calculus.
3. Expand the Mathematics opportunities, optional or otherwise, in the secondary school so “new” Mathematics can enter the secondary curriculum..
For example, Computer Science topics including repetition and selection structures could easily be introduced as early as grade 9 Mathematics by using the TI83 programmable calculator/computers which all schools already have. We are woefully behind in including this new branch of Mathematics in the mainstream Mathematics curriculum.
4. Mathematics is a lifelong learning activity. Semestering is a cheap wine that can “satisfy” by delivering higher marks but falls short on depth and quality. All Mathematics courses should be traditional length and non-semestered (ie 10 months long). Perhaps, semestering is the root cause of post-secondary complaints regarding “readiness” and “quality”.
5. Ensure that specialist Mathematics teachers are in the senior elementary school Mathematics to both support and lead Mathematics instruction. Similarly, only Mathematics specialists should teach secondary Mathematics courses.
6. Create “Greenhouse” opportunities: Place real limits on class sizes especially in grade nine. Class sizes of 34 in grade 9 are land mines in Mathematics education.
7. Use technology to deliver courses that may not be available because of the size of the class or the size of the school. In the 21st century, internet or video delivery of material facilitated by a local Master Mathematics mentor can ensure that ALL students have access and opportunity.
8. Students within a district must not be barred access to certain courses or be forced to travel from a home school because a funding formula cannot provide for a class of , for example, MGA4UI. Equity and access are benchmarks for a public system of education.
The Ontario funding formula must be adjusted to support Mathematics education.
9. With the quick growth of mathematical knowledge, consider retiring the notion of a textbook, and use technology to make new materials such as online worksheets and lessons course-agile. For example, constant changes require constant expenditures on textbooks. Curriculum revisions can be facilitated by using online “tables of contents” that can be changed to reflect curriculum changes quickly while saving financially on the purchase and disposal of “old” texts. It is entirely possible that wireless “Blackberry-like” technology may allow each student to have a laptop that is every text, and note needed for secondary education and beyond.
10. Develop wise copyright arrangements similar to those that exist in industry and universities so that teachers can develop materials for online use while retaining some ownership and shared royalty rights. We need to foster creativity so materials will blossom and not be at the whim of text publishers.
11. Phase in courses changes, year by year, rather than implementing several courses across different years at the same time. The “here is a hot potato, catch it” game hurts students. A protocol for curriculum evolution is vital.
12. Use sophisticated tools such as MAPLE to help students move beyond mechanical manipulation so they can focus on problem-solving. Problem-solving should permeate the secondary curriculum in non-artificial and meaningful ways. We need to accelerate the passing on of Mathematics knowledge so it can be acquired and used to extend knowledge. The mood for this research and exploration is set in the secondary school.
13. Avoid being seduced by the use of models that do not blend seamlessly into the concept being explored. A system that must be bent into shape is not an aid to understanding but rather a distraction and hindrance. Clarity of vision requires clarity of understanding. Play and conjecture with technological tools should not be used as focal points in place of mathematics but rather as starting points for learning mathematics. “Playing” with manipulatives must lead somewhere and quickly.
14. Beware of gimmicks that rob from content time. Ideas such as homework and tutorial time that count as part of the allocated course time are simply ways to get homework done, boost marks and provide time for students to do more extra-curriculars or perhaps have part-time employment. Maximum time must be provided to learn Mathematics within the limited time window that exists. Do not drop content to make courses fit abbreviated and artifical time slots.
15. Teach Mathematics as a human endeavour that seeks to explore the universe and improve humanity.
16. Respect the language and traditions of Mathematics. Do not trivialize Mathematics.
In conclusion, I am pleased to hear that a task force is being set up to examine Calculus, and, hopefully, these larger collateral issues, too. Serious steps are necessary right now to reverse the recent tide of Mathematics educational thinking that believes that good Mathematics is a game best played with knowledge that is centuries old. Too much is at stake to ignore the pitfalls of crippling our best students by denying them the opportunity and means to move quickly to the leading edge of Mathematical knowledge. There is an urgency! Moore’s Law predicts that technology is about to stall as it struggles to break through the physical limitations of chips and speed. I hope that history will not record that post-modern Mathematics, paralleling Moore’s Law, became just an “after math” as it failed to escape a groove that had become a rut.
respectfully yours,
Richard Clausi,
Elmira District Secondary School
ENCLOSURE #1
=========================Dec. 21, 2005 email=====================
to: Walter Whiteley, York University
from: Richard Clausi, Elmira High
regarding: Your position on Calculus
Dear Walter:
Thank you for your email today. I am pleased that the ministry discussed the possibility of locally developed Calculus courses on December 1,2; however, you forgot to tell me the outcome. Does our course which you now have in your hands have a
chance of approval or is that door closed to us?
I appreciated your "rant.doc" as it answered several questions which had been posed to me by members of the community. As you can see, I am including them in my response with your attachment since, it seems, to provide an insight into the
reasoning behind the removal of Calculus from the curriculum.
Walter, we disagree in many fundamental ways, but given the influence you have had in this decision, and perhaps given that yours may be the underlying reasoning behind the decision to vacuum Calculus out of the secondary curriculum, I must draw
your attention to several vital points:
1. The Calculus is an OPTIONAL course! It is beyond the 3 credits required for graduation. You point out in "rant.doc" that you care about "the damage done to students as they hit a wall in first year calculus" yet, engineering , mathematics and
science faculties will not drop Calculus from their curriculum because Calculus is a fundamental part of what they do and how they do it. If you advocate dropping Calculus, it would appear you are denying students any hope they have of surviving
at all. I am sure this is not what you intend. Of course, as our students attempt to compete with students from other provinces and countries who do have Calculus, they will be the underdogs. Our young people will have the unenviable distinction of
being the worst-prepared students for advanced Mathematics study in 40 years.
Again, I am sure this is not what you intend.
2. You are ignoring the need to prepare creative innovators by providing them with the opportunity to study high-end mathematics. Premier Dalton McGuinty, in a recent visit to K-W, indicated that we are in a technological battle for survival in
which innovation is the key to our ability to compete. Today, in the Kitchener RECORD (Dec. 21, 2005) in a separate news item, U of Guelph president Alastair Summerlee says: "with the right resources we can deliver answers to the very serious
21st century problems facing humanity." In the same article, Tom Jenkins, chair of Open Text Corp. says "if you want to harvest corn in August, you had better put in the seeds in the spring". Walter, the solutions to those problems do not lie in
the concrete domain... they require the ability to create mathematical models that can be analyzed using elegant and abstract techniques. We need to ensure that we provide these tools to our innovators as early as possible. We need more not less
Mathematics in the secondary schools of Ontario.
3. You suggest that teachers are "obsessed" by Calculus and that somehow sudents are denied a "richer and more sensible experience than the techniques of differentiation". Walter, You under-estimate our students. They are bright and eager, and our
best excel. Rather than starving them of substantial mathematics and offering playful manipulatives and games , we should be exploring ways to expand and enrich and accelerate the curriculum. Clearly, this is hard to do if we are looking at
removing Calculus, and if I read your covering note correctly, diluting the grade 12 Geometry and Algebra course, and for good measure simplifying the Grade 12 Data Management course.
These actions all point to a diminishing of the secondary Mathematics curriculum... or if you wish a "dumbing down". If we are going to count on our young people to solve big problems , we need to provide them with current and powerful mathematics
in the secondary school. Your "refined and playful approach to problem solving and reasoning" may not cut it in the 21st century. I would like to draw your attention to an exceptional editorial in today's RECORD (Dec. 21,2005) in which a layman
offers an insight into Calculus. I assure you that this opinion is matched by email I have been receiving since this story broke last week...
In short, the current Ministry plan IS "bad"..... but .....
there is no such thing as bad Mathematics- our problem is that we are trying to squeeze 2000+ years of it into a 4 or 5 year period. Napoleon noted that the "advancement and the perfecting of Mathematics are closely joined to the prosperity of the
nation". The solution to the ills of the country is not less Mathematics.
I appeal to you to use your influence to reverse this "done-deal"- this backslide to a pre-Sputnik time where Canada was a technological light-weight. We do not want to go there again.
sincerely,
Rich Clausi
Mathematics Department
Elmira District Secondary School
669 5414
posted by Richard Clausi | 9:09 AM
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