Here is my proposal:
Based on Deanna Burney's "In Craft Knowledge: The Road to Transforming Schools":
"Learning those kinds of skills is not a solitary endeavor; rather, it needs to be a highly social one. It depends on continual discussion and demonstration. People learn by watching one another, seeing various ways of solving a single problem, sharing their different "takes" on a concept or struggle, and developing a common language with which to talk about their goals, their work, and their ways of monitoring their progress or diagnosing their difficulties. When teachers publicly display what they are thinking, they learn from one another, but they also learn through articulating their ideas, justifying their views, and making valid arguments."
We start with Walter's idea of proportional reasoning and flow it into fractions as an extension as per Karen’s suggestion. This then is universal in buy in as it is an enduring concept that is foundational to mathematics at every level. Elementary teachers will have as much to say as anyone as they are in on the ground floor of developing the ideas in students and secondary/post secondary teachers can ensure that our language is appropriate and transferable through the years and into other branches of mathematics. Mathematicians can strengthen our grip on any aspects we seem to internalized incorrectly ourselves due to our successful memorization and rule building through our schooling.
Anyone is welcome to participate because it is the articulation and justification of ideas that lead to our professional development while learning from one another and also lead to improvements in students’ learning. Those that want to ‘listen’ in may, and those who are more active learners can be as active as their time permits. Please respond whenever something strikes a strong cord or gives an ‘aha’ moment. Already we have had China, Ontario, Vancouver Island, Cranbrook, the Lower Mainland and elsewhere involved.
At PIMS on Saturday we had a fascinating conversation around “perception based perspective (ie. empirical based learning processes) versus reflective abstraction. These big words were made clear through the example: What is the sum of two odd numbers? Having students add a bunch of odd numbers to discover the ‘rule’ two odds add to an even would be the empirical. Having students think about a chess club (or any two person game group). With an even number of players, they could all be playing at one time. An odd number of players means that one person is waiting to play. But if two chess clubs with an odd number of players were to join (add) then the two waiting could play each other immediately. This would be the example of reflective abstraction. At what point can students ‘see’ an odd number as one more than an even? In any case I see this level of discussion on proportional reasoning as our goal. Katherine B. taught some of us about the Pirie-Kieran Learning theory (1994 I believe). The P-K theory describes the growth of student understanding through a series of ever expanding circles from primitive knowing, to image making, to image having, to property noticing, to formalizing, to structuring, to observing, to inventizing [and humourizing! - my addition].
Where do we start with proportional reasoning? I’m initially thinking it would be in the real world with sharing food with a friend, with three other friends, etc. before representing the food with an image like a rectangular chocolate bar. The key is for students to understand “Parts of a whole”. I’m aware that Singapore math would swear by the ‘bar’ approach and wonder why we are so taken with circles for teaching fractions. I’ve had good effect in expanding an idea from Jump Math of utilizing nested equivalent bars like the diagram below. I can use the biggest bar as “1” to show the parts of this whole but the proportional reasoning is exposed by providing different starting numbers in various parts of the diagram. The example here could show 16 in the biggest bar to start the puzzling out of division or I could supply the bottom little one as “1” and then the concepts of repeated addition and multiplying would be exposed. Providing one of the “4’s” would cover both ideas.
Would this type of problem with a simpler diagram be an example of proportional reasoning in younger grades? At what level can students do this thinking? We don’t want rote memorization to be our foundation but a conceptual depth to seat any memorization/learning that takes place.How long is the longest bar?
or
A strong understanding of –– three 3’s is the same as “9” is necessary before proportional reasoning can be seated (and remembered, Magnus.)
Comments? Other ideas around starting points? Any points made here that trigger a positive or negative response? I’m no expert on early numeracy but am learning from the gaps in understanding that emerge with the increased complexity of secondary and through professional associations like with yourselves here.
8 comments:
Without particular knowledge of what is expected here, I would suggest that many students would continue the top row starting with 3, by filling in 4 and 5, using the counting sequence. The fact that the boxes have equal length may not be seen as significant.
Would we do some bar work first to build in this correspondence?
Ex:
2 ?
|----|--------|
6
|-------------|
and/or
3 3 ? 3
|---|---|------|---|
15
|-----------------|
Oops, some font substitutions. Picture the two sets of bars being the same length and the numbers spaced out properly to be over the middle of their appropriate sections.
Dot says..I hope I am on task here... but I am thinking of two things I have done in classrooms.
I have had the most success using cuisenaire rods instead of circles. It is difficult to change what is = to one with circles. as showing half and circle and calling it one is generally confusing for students... but with the rods.. you can say the orange is =1 or the black is equal to one... or the blue... etc.... The real "aha" moment was when students recognized that the little white block... was the one that was useful no matter what other block was called one. All the information that the students needed was to know which colour was going to be = 1 they then discovered the other rods that would make equal parts and then name those parts. I now have a new idea Fred from your diagram. Instead of just using all the same colour rods they could use different colours and still define the proportional size... It is so difficult to explain this without showing... hope this makes sense..
In grades 2-4 the picture book "The Remainder of One" provides the opportunity to primary students to make equal groups with twenty five and discover what size groups would leave no remainders. The character Joe in the story is lovable... they are so happy at the end when a line of five allows Joe to be part of the pack.... I used coloured blocks to indicate the ant characters in the story... We extended this lesson to include the partitive and quotative method of division and ended with the students writing their own problems and identifying which type they were...
I was thinking about the clarity of stating the partitive and quotative method of division and did a quick search for any good examples for people who haven't heard of these (lately). My search led me to another blog:
http://rationalmathed.blogspot.com/2008/10/what-can-division-tell-us-about.html
and this led me to Devlin's
http://www.maa.org/devlin/devlin_06_08.html
where this mathematician is talking about conceptual understanding and multiplication is not repeated addition. This perplexed me and led me to read a bit more before bed.
It reinforces for me the need to push for clarity and how we get ideas from one another. In secondary, I haven't pushed for clarity around division and haven't considered this deeply enough.
Thank you. Of course, now we'll need clarity on what multiplication really is.
On our study, can we express Dot's ideas using the cuisinaire rods with interconnecting cubes? I have these at our secondary school but haven't seen the rods for years – (decades?).
I would like to contribute a starting point.
It is the article by S. Lamon, called "Presenting and Representing: From Fractions to Rational Numbers". It deals precisely with the development of proportional reasoning. She and her team conducted a 4 year logitudinal study in 6 classes of students and reached some important conclusions that we should all know about, at least as a starting point to set our course right from the start (because we do not have this time to spend on re-inventing the wheel).
For our next lesson study meeting at PIMS, which it is set for Saturday, January 24 - please mark your calendars, let's come prepared with this as a background. From here we can begin with a design of some pivotal or anchor lessons to teach this well.
Quote: Vergnaud (1883, 1988) best captured the challenge and complexity of the mathematics entailed in the formidable task of knowing rational numbers when he described a conceptual field: as set of problems and situations for which closely connected concepts. procedures, and representations are necessary.
Here I would like to acknowledge and thank Peter Liljedahl, Mathematics Education professor at SFU, for bringing the work of Lamon and this article to my attention. It is one of the most useful pieces of research I've come across in the entire field of mathematics education literature, and it has definitely impacted my instructional practices regarding this content. You get incredible clarity on why you should reject the (sole) use of the standard Part/Whole interpretation of
fractions, which is unfortunately still the most dominantly used
interpretation in most textbooks, and it opens up possibilities for
building up a repertoire of lessons to teach this more effectively.
Natasa
[Editor: I will need to research a bit to figure out how to attach the 12 meg article and find out if I'm allowed to. If you'd like the article, request one of me and I'll email it]
I was very interested in what people had to say about this particular topic because I am currently writing my master's paper on teaching strategies for mixed-ability math classes.
When I was obtaining my data, I taught a unit on fractions to a grade 8 class. I mixed in rectangular diagrams, number lines, as well as the algorithm to multiply and divide fractions. I was having a very difficult time teaching the conceptual understanding of fractions. They just couldn't relate the diagrams to the algorithm. However, when I gave them a worksheet of the same format four times!!! At least I got them to know how to do it but disappointed in myself for not being able to get them to understand "why." I know that a third if not half of the class will not remember how to multiply/divide fractions next year and much time will be spent on reviewing.
Another thing I noticed being the only full-time math teacher in my high school is that some of the teachers lack the knowledge of deeply understanding mathematics themselves and instead of teaching units to the fullest they worry about 'kissing' each topic to finish the curriculum. That's part of the reason students can't connect concepts learned moving from unit to unit-they see it as learning seperate entities.
Just my bit - I'm interested to know what you think?
Kaljit
Fred:
Regardless of Deanna Burney's finger-pointing wrt the alleged reluctance of teachers to learn, share, etc., this has patently NOT been my experience teaching in a number of schools, at every level from primary through senior high. In my experience, teachers almost universally WANT to collaborate, share, learn from each other, work together to improve their practice and solve problems. They definitely WANT to have the satisfaction and joy that these opportunities bring. Many, many teachers live in a constant state of low-level grief and frustration caused by the difficulties of getting any of this going in the present system, where, frankly, the highest value at the moment is ultra-maximization of contact time (minimization of FTEs) and marks on standardized tests. If teachers aren't seen standing in front of LOTS of students in class, they are not considered "working". The 50-60hr weeks that virtually all teachers are calling "normal" is unhealthy, unsustainable and pretty unproductive for the amount of effort. A few hours of prep time a week (when experts tell us we need 1 to 2 hours to prepare properly for EVERY contact hour), with some teachers having NO prep time for up to 5 months of the year, is not providing the standard of education that the public is telling us it expects. That is no surprise to most teachers, who know perfectly well that more than half the job is the prep, and it is getting short-shrift along with their sleep.
Until the system VALUES the collaboration of educators enough to allow them to attend each others' classes, thoughtfully to evaluate each others' lessons, to work in groups to plan and refine instruction, to consult with experts, parents and colleagues, to develop new methods and materials, and to reflect in depth on their own practice, we are just spinning our wheels most of the time.
"[teachers] can develop their expertise only if they are willing to experiment, make mistakes, and analyze those mistakes -- with everyone else and in front of everyone else. There is no other way for new knowledge to infuse the system and create stronger instructional practice. This will require everyone in the process to place a high level of trust in everyone else -- colleagues, leaders, and outside experts." [my italics]
Deanna Burney
The only thing wrong with this statement is the nonsense about teachers needing to become "willing", as if teachers are the ones holding everything back. What rubbish!!! If teachers aren't willing to jump on every bandwagon, one should take a good, hard look for burned fingers and recent scars. One should observe the teacher falling asleep, drooling gently on his marking, or rising at 4:00am to prep classes every day. One should make note of the alarming rise in stress-related illnesses and absenteeism over the last 10-15 years, not to mention "deaths in the saddle", disabilities and very short retirements. Teachers are dancing as fast as they can. They would LOVE LOVE LOVE to change how they work and be ALLOWED to do what's needed to improve the learning of their students!
The most soul-withering experience is to spend hours, days and weeks of one's own time to develop and refine a really effective learning program, with associated practices and procedures, one that really work for students, only to have the system (AOs) refuse to make room for it, refuse to provide the essential conditions to run it, and ultimately flick it off like lint, citing administrative reasons such as "the computer system can't handle a course that doesn't begin and end on the official dates", or "we are not willing to police the few kids who don't comply with the rules", or "we can't afford the FTEs to run your program without losing elective blocks." or "It's too much trouble to deal with the insurance." Everyone's in a straitjacket. No one can move, even if teachers are ready, willing and more than able.
At a staff meeting several years ago, with the Superintendent in (RARE) attendance, our principal actually admitted, in response to a pointed question, that without having a significant number of senior students on "spares", we couldn't possibly keep class sizes anywhere near the "suggested" maximum of 30 students. The school was allowing students to opt out of all but the absolute minimum course loads required to graduate, IF they passed everything. Of course, some of those students were back the next year taking one or two courses in order to grad. Others just resigned themselves to the bottom rungs. Final grad transcripts showed that students had received the absolute minimum amount of instruction that the system was legally allowed to get away with. This was the strategy being used by our school, and probably others, to deal with the chronic underfunding that is at the root of many of our problems and frustrations in BC education. Lots of money is going to distance ed., open school and other online modalities, home schooling, private contractors, private schools, etc. But the workhorses of the system, ordinary public schools, are limping along on starvation rations, or being closed down.
In my last school, a valuable daily tutorial program was scrapped after several years by admin. (supt. and principal). They didn't like the "leakage" when some students chose to skip class during the tutorial block. Admin. wasn't interested in putting into place attendance monitoring structures and meaningful consequences, nor in following through with them, in order to keep a truly educational program working well for students. It was too much trouble, apparently, to commit to something educationally powerful that required ADMINISTRATORS to change THEIR practices and attitudes. It was much easier just to dump it.
This is the same school (different AO) that scrapped flexMath, after no end of hard work, collaboration, and 7 years of careful refinement by teachers WORKING TOGETHER, mostly on their own time.
I really believe that the System viciously suppresses, frustrates, wears down and even actively prevents achievement of the standards of practice envisioned in the books you cite. I retired early because it was nearly impossible to do the job I wanted to do and knew I should be doing for students, I was professionally lonesome despite my own efforts and the good will and efforts of nearly our entire teaching and non-teaching staff, and it was making me sick. I loved the work; I hated the job. I felt kneecapped every time I tried to get up, and I became exhausted just trying to deal with the workload and goat-brained policies in a system that truly did not value professional practice. I don't think my experience is unique. There is a good reason why many new teachers are NOT teaching after 5 years, and many more have quit after 10 years in the trenches. The system wrings all the good juice out of them.
I am aware that the Union line is all about autonomy and creativity. The problem with this language is that we are talking to a public that thinks autonomy means teachers get to do whatever the heck they want with no curriculum, no oversight, no standards and no accountability. Joe-Six-Pack thinks that creativity means using his kids as guinea pigs. Along with many things for which we have the Union to thank, we can add a singular insensitivity to the world views of the taxpayers. (It did not go unremarked that my Professional Year at UVic did NOT include a course in communications, nor that representations by many PDPP students to the Department regarding this deficiency remain ignored to this day.) Our Union has been UNhelpful in articulating the nature of teaching, communicating the true requirements for effective and powerful learning, and earning respect for the teaching profession. Some willingness to change and new insights are urgently needed there, too.
I think a much better term for what we really need to fight for is "professional authority". When I analyze a learning situation on the basis of my long education, experience, good research, insight and intelligent reflection, I expect my boss to have respect for that and generally to defer to my legitimate authority in matters of professional judgment. I expect administrators and the Board to listen carefully and to turn themselves inside out to create and maintain for me and my colleagues the conditions in which we can do our very best possible work to serve our students. If I say I need time to prepare properly for my classes, to confer with parents, to conduct research, to create and produce good materials, to design and refine lessons, to coordinate and collaborate with colleagues and other experts, I expect a functional system to make that possible, with the focus of ALL of us firmly on the welfare of our students. If teachers judge a Math class of 30 kids with 10 special needs students, unworkable, then admin. needs to fix it. If teachers say that more than 24 or 20 or 18 students in a science lab is dangerous, we should believe them. School principals, the members of the Board, the Superintendent or even Gordon Campbell are NOT competent to be making that call.
If we want classroom hours to be of the highest quality and yield great learning, of COURSE we HAVE to do the prep, with all that entails. We HAVE to work together. We HAVE to have an atmosphere of trust and the ability to take risks and make mistakes without fear of reprisals by supervisors or the likes of that bastion of deep ignorance, the Fraser Institute. We HAVE to be able to take things that work and run with them, share them, continue to develop and refine them. We must communicate the true nature of our work far more clearly to the public so that our authority is seen as legitimate. We have to EARN the respect of the taxpayer, not by working ourselves to an early grave in constant, sniping frustration, but by redefining clearly what teaching is all about and what we really stand for. We need to jettison these awful misrepresentations called "autonomy" and "creativity". They are too easy to use as scapegoat language by power structures that find it advantageous to keep the teaching profession overworked, exhausted, underfunctioning and under general suspicion, available to blame for all the shortcomings of government policy and bureaucratic intransigence.
Yours wholeheartedly for Professional Dialogue,
now that I have time to think!
KT Pirquet
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