Hello everyone! The topic for our virtual class over Spring Break is:
"Examples, Counterexamples and Mathematical Concepts"
It's a topic that relates to the design of curriculum, teaching and assessment. Examples and counterexamples and the ways that we ask students to work with them is central to the design of classroom activities, textbooks and tests.
The two articles to read for our virtual class are
1) Goldenberg & Mason: Example Spaces
2) Swan: Collaborative learning in mathematics
(Note: I have updated the links to these articles -- please let me know if you are still encountering any problems linking to them!)
(Note that there is a new book out, based on an ICMI study, on Task Design in Mathematics Education. I haven't had the chance to get a copy yet, but you can preview chapters via this link, and it looks like a fascinating follow-up to the work started in these two articles! The image on the left is from another interesting book by Mason and Johnston-Wilder that deals with similar topics -- lots to explore here!)
For the virtual class, you should:
(i) Read both articles, and take notes on your own "stops" as you read.
(ii) Post your ideas as comments to THIS posting (on our class blog, here at <http://mathedbeyond.blogspot.ca>)
(iii) Respond to other people's postings and keep the discussion going.
By 9 AM on Monday, March 28, you should have posted AT LEAST four substantial comments to this online discussion.
I will dip into the discussion and add some of my own comments as the week goes on.
Hope you enjoy these & the discussion that comes out of them!
cheers,
Susan
"Examples, Counterexamples and Mathematical Concepts"
It's a topic that relates to the design of curriculum, teaching and assessment. Examples and counterexamples and the ways that we ask students to work with them is central to the design of classroom activities, textbooks and tests.
The two articles to read for our virtual class are
1) Goldenberg & Mason: Example Spaces
2) Swan: Collaborative learning in mathematics
(Note: I have updated the links to these articles -- please let me know if you are still encountering any problems linking to them!)
(Note that there is a new book out, based on an ICMI study, on Task Design in Mathematics Education. I haven't had the chance to get a copy yet, but you can preview chapters via this link, and it looks like a fascinating follow-up to the work started in these two articles! The image on the left is from another interesting book by Mason and Johnston-Wilder that deals with similar topics -- lots to explore here!)For the virtual class, you should:
(i) Read both articles, and take notes on your own "stops" as you read.
(ii) Post your ideas as comments to THIS posting (on our class blog, here at <http://mathedbeyond.blogspot.ca>)
(iii) Respond to other people's postings and keep the discussion going.
By 9 AM on Monday, March 28, you should have posted AT LEAST four substantial comments to this online discussion.
I will dip into the discussion and add some of my own comments as the week goes on.
Hope you enjoy these & the discussion that comes out of them!
cheers,
Susan
As I suspected, I am the first post! So, here it goes... :D
ReplyDeleteThe first article I read was Goldenberg and Mason's paper about example spaces. It's ironic that they continually emphasize the importance of examples in their work, but I found myself constantly wanting more examples of what they were talking about. Does anyone know what they meant by a "non-example"? I kept trying to find a place in the paper where they define this, but couldn't seem to find one.
Swan's article, on the other hand, is full of examples. In this paper, I found myself wanting a bit more theory with the examples and a generalization of how these strategies could be implemented in the classroom.
Swan's remark that students "often find it difficult to produce and extended chain of reasoning" definitely resonated with me. Since we do a lot of proofs in the courses I teach, I personally struggle with this a lot. I liked Swan's recommendation of cutting apart a card with an argument and placing it together to create a proof; this is something that I might try in my classroom in the future. Unfortunately, I question whether this will help students develop the intuition that one must develop in order to do mathematical proofs. I would argue that being given the pieces to the puzzle (as is the case in the example Swan provides) is much easier than constructing the pieces yourself. On the other hand, this could be a route for students who are new to proof and also requires students to distinguish what information is not necessary for the problem.
Indeed, we often assume students will be able to undersatand and produce proofs with very little scaffolding in teaching them how to do so. Students are often surprised when I prove things to them as they are often simply provided with formulas and methods. Even so, I rarely ask students to create or reproduce proofs, which means many get to university without ever having had to properly understand one. I always thought the idea of reproducing proofs on tests was a waste of time, and I resented memorizing them in the first years of university math studies. In retrospect, this probably did help me make sense of them and internalize mathematical reasoning to a degree. The method of putting a proof in order sounds like a great in-between step of not having proofs to mastering them.
DeleteI do often stress examples and non examples as students forget math rules. For example, if they have an expression such as (2x+5)/2, can the twos be cancelled. I ask them to consider examples with numbers to see if the terms can be divided.
I agree with Vanessa that Swan's suggestion to have students arrange the different parts of a proof in sequence may not automatically train students in doing mathematical proofs. I do think it helps students to think about the different possible parts of a proof, the relationships between the varied components, and like Vanessa pointed out, to discern between whether the information is needed, or just a red herring that will not lead to proving a principle or concept.
DeleteBesides mathematical proofs, it is also valuable that what Swan recommends is extended to classroom learning and problem-solving as well. I see value in having students think about what they may not usually think about (their usual focus is on finding the solution) - training their minds to pay attention to the logical reasoning or flow, the features that are essential and do not change and the parts that can be varied. Giving students the practice and experience in dabbling in such thought processes may also help in de-mystifying the need to memorise and always follow a single way of approaching a problem, and hopefully allow them to perceive more possible connections/relationships between concepts, formulate general principles and discernment on whether these may also be applied to other problems. Working it through collaboratively with classmates also broadens the range of perspectives, flows of logic and reasoning, that students may be exposed to.
About a year ago, I had an undergraduate TA who was in 2nd year math. She was a very good student and arrived to university with a lot more mathematical knowledge than the majority of 1st year students. She was marking proofs for the students I taught and was very good at discerning student proofs and flaws in logic. I was curious how she got to such a mature level so early in her academic career, so I just asked her "how did you learn to do proofs?" Her answer was quite fascinating to me, since she just said "I just did proofs." She took out her own personal time to do proofs and struggle through them on her own. We agree with each other that although reading proofs can be a useful exercise in seeing how an elegant proof should be written, the exercise of working through the struggle, realizing what the wrong turns are, and creating your own train of logic is the most valuable exercise.
DeleteOn that note, I think this is the difficult thing about "teaching proofs." At the moment, I'm of the opinion that you can't necessarily teach someone "how to prove." Perhaps you can teach them good practices, but there is no set method on how to approach any given proof. With the new curriculum emphasizing multiple ways of reasoning in mathematical problems, I think proofs could have an interesting place. Perhaps if we start students with non-mathematical proofs (i.e. logic problems), this could be a nice way to get them thinking in a way that is conducive to producing mathematical proof?
Finally joining the fray..
DeleteVanessa, I too read the articles in that order and had a similar reaction: why are there so few examples in this paper on examples? One things that stopped me in this article was a comment made on the second page: "In a mathematical context there is little difference between an example an a counterexample". This struck me as so fundamentally opposed to what I have come to believe I had a hard time coming around on this paper. Perhaps I'm splitting hairs, or misapplying these ideas, but: in the context of proofs (which arise in these articles!) a counterexample carries *so* much more weight! Even outside of proofs, coming up with examples seems simple in comparison: change a number, a context, a sign.. But counterexamples: those are so much harder to come up with, and I would assert really show a much stronger understanding. Personally, I found it hard to get past my own such experiences, and the discussion that followed where Goldenberg and Mason explained what they meant fell rather flat.
As for the cutting up of pieces of proof: I actually tried this once. It really did not work. I remember being surprised; I tried with a keen small class populated with students happy to humor me. But at the same time, it was third year graph theory and the proof was nontrivial - challenging to understand with it all laid out in front of them as whole. In hind sight, perhaps if I had added more details into the proof, and made each 'piece' more substantial things might have gone better? I got spooked away from that activity and didn't try again - after all, my students were doing proofs all the time.. They did not really need a potentially more confusing mechanism to get there.
Sophie,
DeleteIt's nice to know that you've tried this proof cutting technique before. As you mention, it was for third year students who were familiar with proof and I could see that it would be confusing to have all these statements out there that have so much meaning on their own. Then the question is, how would this technique work with novice proof students? If anything, it might at least foster the development of the thinking we want students to have when they approach a proof.
In regards to counterexamples versus examples, I completely agree with you. I'm pretty sure I remember the first time being asked to construct a counterexample and being really confused by it. Some of the hardest "quick" problems I have encountered have been counterexample construction problems. Constructing counterexamples really challenges students to understand the subtleties of a definition or a theorem. Moreover, it challenges them to understand the subtleties of various mathematical objects. How do we encourage students to get better at generating counterexamples? Is doing these sorts of questions the only way to do it?
Not to disagree with your discussion about examples and counter examples. Certainly, you both have a lot more experience with these than I do. However, I liked the author's articulation as depending on where attention is focused in that a counter example can help to articulate a concept. I do this all the time when presenting new terms such as examples and counter examples of functions, irrational numbers, polynomials, infinite sequences etc. This can be valuable and helpful to help students articulate the subtleties of a mathematical concept.
DeleteI loved Swan's article. While most of the ideas were familiar to me, I thought it did a great job of outlining and discussing possibilities for lots of easy-to-apply collaborative learning strategies at various levels. In particular, I like the chart on page 168 with the 'always, never and sometimes true' statements. I think these can be excellent lead ins to important topics such as proofs and logic building. I have enjoyed a few classes where students have gotten into heated arguing matches about questions such as these. Here are a few I have used... if you have any others, maybe you can add below...
ReplyDelete1) Is a square a rectangle?
2) absolute value of (x+a) + a (a not absolute valued) is positive.
3) x^2 > x
Like David, I also enjoyed Swan's article very much! For me, I gleaned many ideas that I may want to adapt and apply to lessons. Nonetheless, in the midst of this excitement, the practical educator side of me wondered about two things:
Delete1) Will Swan's recommendations work better for learners who are new to mathematics as a subject, meaning they haven't been schooled in the usual way mathematics is taught in classrooms and hence may need to unlearn and relearn some of their approaches to Mathematics; or will they work better with the older students, who have some prior knowledge of mathematics to tap on, so that it may be more meaningful because they have a wider range of prior knowledge of mathematical concepts to draw and generate multiple approaches from? Or does it not matter?
2) Securing buy-in from teachers and students is so essential, especially if trying out some of Swan's suggested strategies means changing the way mathematics is usually taught in classrooms. I think most educators should be able to see some value in what Swan is suggesting. However, to help overcome some of the practical constraints such as limited classroom time or the need to translate classroom learning into academic performance on tests/exams, I found it integral that support in the form of resources, as well as teacher exposure and training in these new thought processes and approaches mattered. In fact, some practical feasible steps include having teachers in a school collaboratively generate a possible list of questions that could probe their students' thinking more deeply, make students' reasoning more visible to the teacher and their peers, as well as generate multiple approaches for different mathematical topics; or like David mentioned, a list of problems that have made students question, argue and apply their reasoning to.
This could also be an essential segment in Pro-D sessions on particular topics or a course in a teacher education programme that explores the common misconceptions/errors in the students' reasoning process and further concept-application, coupled with possible questions that the teachers could use to encourage students to probe deeper and wider. I would think that such sharing of knowledge and exchange of perspectives may be very valuable for beginning teachers, and maybe even for experienced teachers who may see value in adapting some of Swan's strategies but who may not be used to teaching mathematics in a different manner from how they've been used to teaching it all this while.
I'm always a sucker for ready-to-go teaching strategies. Swan's article was chalk full of them. I agree with Vanessa that it perhaps could have used some more theory to back up the examples. I wonder if Swan would suggest using these strategies throughout a curriculum or perhaps only interspersed throughout standard textbook work.
ReplyDeleteTo address Audrey's first question, I imagine that these suggested strategies would be difficult to implement if one decided to try them all at the same time (or within the same school year). Perhaps it would be better to try just one strategy that intrigues the teacher, reflect on how it went, and go from there.
Many of these strategies involve both collaboration and learner-directed activities. In order for these to be effective, I would imagine that a teacher would have to have built a classroom environment where the students feel safe to express their opinions. If a student has been able to get through their mathematical education without ever talking to their peers or raising their hand to offer an opinion, it may take them some time before Swan's strategies work for them. They may not ever work for some students, which is why I would supplement a curriculum with examples such as these, rather than completely switch over. If a class fully buys into it, then I would try to implement more similar activities.
My question is this: If these strategies were the primary sources of learning activities for students, how would the students fair in standardized assessment? It's funny how everything seems to always come back to assessment.
Hi David, I agree with on the point that in order to effectively deliver successful collaborative activities, it is essential for teach to build the appropriate culture first.For example, teachers have to be aware of that teacher asking questions in class can be loaded with risk for students who fear of revealing that they don’t know or understand something in front of both peers and teachers. Similarly, in student posing question activities the pressure is not reduced automatically. Especially for students from different cultures who might be still under the impression of asking questions is a sign of " I am slower than others so I need more help".
DeleteLike David mentioned earlier, unfortunately, it does seem to all go back to assessment - while I see value in the pedagogies changing with the intent of developing competencies and not only content knowledge, as well as igniting passion for mathematics learning, I do think that it may be a "one-step-forward, two-steps-back process" if ultimately the teachers' hands are tied in terms of curriculum time and allocation of resources. After all, like David and Ting have both pointed out and I agree, establishing a classroom culture/environment that supports such collaborative, inquiry-based learning, takes time – whether because students may need to unlearn how they previously approached mathematics in a time-efficient solution-driven manner, or because exploration time is needed for the students to think divergently and generate multiple interpretations, approaches and solutions, or because teachers may need to take more time than usual to brainstorm and plan for a collaboration oriented lesson, in order to be prepared to ask probing questions that facilitate conversations for students to think widely and resist the urge to of direct them in a particular direction or towards a specific solution.
DeleteWith that said, it would really be valuable for teachers in a grade level/school/district to come together and brainstorm and generate these approaches/questions together, because their pooled knowledge may help reduce the time needed to reinvent the wheel individually and also reduce some of the possible anxiety in trying out collaboration-oriented strategies in the classroom.
Nonetheless, I am genuinely curious about the outcomes of adopting such an approach in the classroom, maybe for a start to pick a topic that lends itself better to this approach and type of strategies, and see what the outcome may be. I am hopeful that with sufficient time allocated and fidelity of implementation of this approach, the students will learn widely and deeply, and it will help them in concept-understanding and application for standardised assessment. I suspect time may be a concern on the standardised assessment, because the students may lack the practice for making a convergent decision under time constraints. Perhaps it would help for the teacher to also equip the students with test-taking strategies and introduce practice at such time-based assessment every few topics, so that students may also be exposed to this. I wonder if it is possible for the students to know when to switch strategies/approaches – collaboration-oriented for learning/understanding concepts; solution-driven for standardised assessment, or will they end up more confused (split personalities) when a mix of the two approaches are introduced?
An essential element of learning math is inquiry — asking questions and figuring out the answers without instruction. Inquiry-based math classes allow students to discover key concepts, instead of simply memorizing formulate and practicing for speed. But how do you turn a high school classroom into a collaborative working place? Swan offers a feasible solution: ask students to create problems (p.169-171). According to the author, students are given the task of devising their own mathematical problems. Students first solve their own problems and then challenge other students to solve them. During this process, they offer support and act as ‘teachers’ when the problem solver becomes stuck.
ReplyDeleteIn my own classroom, I have been using this method for a while. One of the issues I have encountered is in the evaluation of student-posed questions. Giving a mark to students may be difficult for teachers who have had no prior experience with student-developed questions. Another issue is that the whole classroom very quickly gets into a habit of making certain types of questions, which Swan also noted. In fact, as I have seen, around 90% students will change variables of existing questions to make their own questions. Often, the class loses engagement and inquiry-based learning turns into another routine.
I would argue that successfully teaching and learning math using the inquiry model requires both teachers and students to be flexible and to be comfortable with not knowing the answers. This is different from Swan’s suggestion to have students devise problems that they know they can solve correctly. However, when students are aware that the posed questions should be able to be solved “correctly”, this places pressure on students and is harmful to the collaborative learning environment. If the goal is to move students away from the traditional model in which they just sit in the classroom listening to a single teacher, we need to also consider in which ways the new system might not behave exactly as we expect it to. Shifting teaching from traditional transmission orientation to collaborative orientation requires teacher to “making misunderstandings
explicit and learning from them”(p.162), this is where most challenges are located and will need more exploration.
Ting, it's interesting that you mentioned that sometimes students can get bored/lose interest in inquiry-based learning if it is used too much in the classroom. Perhaps it shouldn't be relied upon too much, rather it should only be a single part of a multi-faceted teaching platform. The question I have is, which part of teaching/learning should the inquiry be grounded in? Should it be used as a hook for a unit to pique interest? Should it be used in the middle of a unit to switch things up to keep student interest levels elevated? Should it be used as a summative assessment piece? Or should we play each unit by ear and see what works? If a unit is quite conducive to inquiry-based learning, then perhaps we should use it as much as we can until students begin to lose interest.
DeleteAssessment will always be an issue, I feel, in regards to this topic. It is much easier to create an assessment piece when you know exactly what material will be covered in class.
I have observed that many teachers who are in favor of inquiry-based teaching start classes with a lecture on a particular topic that guides students through questions that the teacher knows the solutions to. This technique is useful when the purpose of the class is to reinforce concepts and to introduce students into following procedures and confirming “correct” answers. It might lead students to build knowledge that leads to academic progress but it limits the development of inquiry skills, which requires authentic open-ended questions. In other words, I am in favor of fully open inquiry classes where the teacher provides only research questions for students. The students are responsible for designing and following their own procedures to test their questions and sharing their results with the teacher in order to receive feedback. But the workload of such class for the teacher might be a challenge. I often felt exhausted trying to catch up with my students in classroom. However, adding robotics into the classroom took off a lot of the burden since students no longer need to ask the teacher simple questions which could be answered by testing it out on robots.
DeleteDavid, in answer to your question on when it can be used, I would go with play by ear, more because I don’t have an answer, and also because I do think it may be used as a hook, as a means of engagement, or as an assessment piece at the end of a topic/few related topics =) In fact, I had the same question and was wondering if having a course in the teacher education programme or Pro-D allowing the teachers themselves to first experiment and try out the collaboration oriented approach for different topics and in different ways, and intentionally think about how their classrooms may look like when this is implemented will help, for example looking at: What are the range of strategies possible? What are the questions that a teacher can ask to help facilitate student conversations and explorations? What are common student misconceptions/errors that teachers can share and devise approaches for students to realise these misunderstandings?
DeleteCould we, do you think, have a “hands-on, put our collective brains together” course in the teacher education programme or Pro-D based on the collaboration oriented approach across a range of topics in the BC mathematics curriculum? How will it look like for it to be meaningful and useful for teachers? Will it be well-received/attended by the teachers?
When I was a student many moons ago, my school had final exams starting in grade 6 - which seemed very rough at the time but (at least in my case) turned out to be good preparation for future higher stakes tests. I don’t remember a whole lot from back then, but I do know that a good deal of time was spent on review. To that end we had a couple of teachers who had us (in groups) create our own “practice finals” for each other - we even typed them up and chose weights, though they had zero impact on our grades (which likely helped teacher workload). Trying to guess what a teacher might ask was an excellent way of studying, and I remember being hugely amused that most of the created questions were much more challenging than those on the actual final. I know I was not necessarily a typical student, but I wonder about the effect of raising the stakes just a little when it comes to getting students to make their own problems. Further, do you think the timing with regards to student’s mastery of a subject matters? At the end you are likely to get problems that are familiar to you as a teacher, but you might be missing out on some interesting creativity and insight.
DeleteI’ve certainly experienced the situation Ting describes above - where students lose engagement in inquiry-based learning. It was exactly as David suggested: being relied upon too much, and students were getting distracted and tired. In such instances, even though the class is supposed to be problem based, I have deemed it entirely appropriate to switch back to more traditional methods. I tend to believe *change* is really the key; when inquiry loses its lustre, students tend to be more receptive to other pedagogies - even methods that might have failed them in the past. Perhaps with more concrete ideas, such as those suggested by Swan, we can have more options to switch between and get that really necessary change.
I've always perceived inquiry-based learning to be a very open ended term for getting students to question and wonder about mathematics. How we can get students to wonder about the mathematics they are learning is a pretty big leap if you think about it. I recall always being told to learn, and by learn I mean memorize, the mathematics at face value. There was nothing to wonder about. Mathematics class existed as it was and didn't feel very full of wonder at all; it felt much the opposite. The route that teachers take to encourage inquiry and wonder can change from day-to-day. Whether it is through examples, a problem, an engaging lecture, or a math fair, if students are questioning the mathematical knowledge at stake, then it is of my opinion that students and teacher are engaging in inquiry-based learning.
DeleteNow, my definition could be completely wrong. Is there a consensus as to what constitutes inquiry-based learning? I certainly think that the ideas presented by Swan could be very applicable in an inquiry-based classroom, but as some people mention, it's probable that some students might be bored. If there are strong students who appear to be bored and finished the problems early, it can often be a challenge for them to present their solutions on the board to the class. This could be a good option if a teacher feels as though something is falling flat.
Swan mentioned five types of teaching activities that will encourage distinct ways of thinking and learning. Thinking about the last two activities namely creating problems and analysing reasoning and solutions, I was wondering how they could look like in a classroom setting.
ReplyDeletePerhaps to combine these two activities, student groups could create a problem for each topic/lesson as part of a group e-journal, and the teacher collects these questions, which can then be grouped together as an assessment for the topic/across several topics for other groups. The objective of the assessment is not only to get the answer, but focuses more on generating multiple strategies to solve the problem, and comparing/analysing as a group which approach is more efficient, when it may be better to use which strategies, perhaps form generalisations and principles for themselves, allowing them to internalise their understanding better that way. In turn, the test-setting groups could surface, analyse and explain errors and possible causes of these misconceptions of the groups that did their assessment, which has the two-fold objective of reinforcing their understanding when explaining to peers from other groups, and helping them to check their own learning/understanding whilst being careful of making similar errors, especially those that arise as a result of interpreting questions differently.
Perhaps this is a possible way of making misunderstandings explicit and learning from them in a collaboration orientation? I especially liked this point from Swan's article and I think it is valuable to the learning process and also not intentionally done often enough in mathematics classes, although I can see how the contrasting point of correcting misunderstandings in a transmission orientation may be valuable under time constraints.
Nonetheless, I do think an activity like the aforementioned, can be more smoothly done if students are familiar with the process of creating problems, analysing and comparing them, among other strategies; and this familiarisation alone, will take time. With real time constraints in the classroom, perhaps such strategies can also only be used periodically. Perhaps this may first happen in out-of-class enrichment time more so than regular class time, before the more successful strategies can be imported into lessons. As earlier mentioned, I believe there is value and greater practical feasibility of teachers themselves experiment with these approaches, whether in combined teacher planning time, pro-D or teacher education programmes.
I really like the idea of having students create problems based on what they have learned, but I worry that the many of the problems will be almost identical to examples from the textbook. I suppose this is unavoidable, unless a teacher chooses not to use a textbook as part of their lessons. In this scenario, I would imagine that a teacher would still teach the year through units of the curriculum, however most of the work done would be creating problems that used newly learned skills. Another way to look at it would be to start with the student-generated problem, and then decide which units to focus on while solving the problem. In my mind this would be a little too chaotic for my liking.
DeleteI like the idea of keeping an e-journal as well. The problem with most schools, is that these sorts of resources are not always available, and if they are, they aren't always functional. Assuming that there are the necessary resources available, I imagine that a class would have to first practice journaling online, and learn how to appropriately respond to their peers. E-journals are also an easy way for parents to "check in" and see how their child is doing.
David, I want to jump off of your point about not using textbooks as part of a lesson. Earlier in this course we were tasked with bringing in problems from textbooks that we were to critique and open up. Do not many of the issues we see in textbook problems also arise in textbook examples? Do you think the methods we employed then to open up the problems could be used to minimize the effort of coming up with new examples all the time? Or does the nature of an example bring with it expectations of closed, singular lines of thinking?
DeleteI think in university we might run across courses taught without textbooks more often. The effect of this ranges a whole spectrum. My most frustrating memory of this was in an ODEs class. I had limited sense of what was going on, and due to no textbook, little in the way of examples to help me grasp the abstractions that seemed to be flying by. Upon asking the professor if he could please give us some more examples to work through, I was met with “I must guard my examples carefully; there are only so many.” Complete hogwash in hindsight, but the helplessness I felt in that moment drove me to the library where I checked out several texts - and thereby found his example sources and many more. Perhaps besides the point, but that was probably one of the best lessons I ever learned. It does go to illustrate the perils of trying to escape the confines of textbooks - it is not for everyone.
That said, I have also seen text-less courses taught masterfully, where instructors readily admit to drawing on several source materials for content and examples. But barring a high school level math class, I have never seen student derived examples used in any manner. It is challenging to imagine: when classes are not enormous, the content tends to be proofs. In the latter case perhaps the counterexample needs to be brought to the fore; it could be a fantastic tool for grasping why certain qualifiers are present? In the former, perhaps peer assessed e-journals might help.
Going off of the idea of an e-journal, I tried something similar on a small scale with my students this year. I have them maintaining a blog, and on every other written homework assignment, they have to write a blog post. The purpose of the blog posts is for students to have an opportunity to "abstract" the concept from the week. It's been shown that many students don't have opportunities to go through the abstraction process and that open-ended, written exercises can help in this process.
DeleteI actually had one of the problems as a "make your own example for this concept" post. Some students took to this example very well, but others just pulled from examples in class and changed the numbers. I think the exercise has the potential to help students understand the important features of a examples for a particular concept, if they approach it in a meaningful way. Unfortunately, a number of students created cookie cutter examples, but they were marked accordingly.
I only took one textless course in my undergrad, but I was lucky enough to have an extremely supportive professor for that class. He compiled a lovely collection of notes for that course and pulled from a variety of sources. That class was my first "proofs" course and I was completely overwhelmed with the material, but my professor was nice enough to provide some recommendations for textbooks. In my Master's, I had a number of classes with "recommended texts," but I was mathematically mature enough at that point to make an educated decision on which texts to use.
Returning to high school students, I think it is a massive responsibility of the teacher to go "textless." In that case, a teacher must be extremely confident with the curriculum, what students need to learn, the progression they want to take, etc. I think a text book definitely provides some sense of security for teachers, particularly new ones. But, one question does come to mind through all of this: do examples have to be exercises? It seems as though we often equate these two, but I'm not convinced that they are the same.
I agree that students may have difficulty creating problems that look different from textbook examples, especially if these may be all the example pantry space that they have known. I believe it may probably start from there for some time. I wonder if it will help this process to work in a group where everyone brings different understandings/ perspectives, and the teacher in a K-12 class explicitly gives instructions that the problems could not look like textbook examples, and must have multiple solutions / open-ended questions. Would the process only be more meaningfully carried out by the more advanced math students, who then may have that required flexibility of mindset to manipulate and adjust - does this process require strong math foundations or backing; or would the younger children as yet not steeped in mathematical principles be able to creatively devise problems, if instructions were given as a means of a game?
DeleteI also agree that examples do help understanding, especially a wisely-selected range of examples that illustrate various principles in the area to sufficiently give students a holistic picture. However, I wonder whether it is possible for teachers to discern when and how to draw the line between providing sufficient examples that are perhaps representative of processes and principles; and when the core principles have been sufficiently represented, and manipulation is needed to adjust questions for the application of these principles/processes.
Gildenberg and Mason argue that simply ‘giving’ examples and construction techniques is rarely sufficient for most learners in order to populate their example space, which is defined as the set of experiences that link mathematical objects together with construction methods and associations. I like the examples illustrated in the paper which give me an opportunity to look at them from different perspectives.
ReplyDeleteIn the following example, the author claimed that the two questions trigger different experiences with regard to the number dimensions of allowable variations and associated ranges of changes.
A. Please construct a number which, when you subtract 1, leaves a number divisible by 7.
B. Please construct a number that is 1 more than a multiple of 7.
At first glance, these questions represent identical algebraic expressions, but the first question tends to require a larger “example space” in order to fully comprehend. Even though the sets of answers to the two questions are the same, the first question requires slightly more manipulation to arrive at an answer as it requires “backtracking”, an additional abstraction in order to come up with a viable solution. This is equivalent to the algebraic manipulation we do in our minds – the step that takes students from form A to form B is simply the step
x – 1 = 7n
x = 7n + 1
and requires at least a broad understanding of the manipulation of addition and subtraction from “both sides” of the equation. Indeed, students who struggle with these concepts may resort to brute force when presented with form A, checking each number onwards until they arrive at a suitable response. However, giving a thought of implementing the approach in the classroom, I realize that there would be enormous challenges facing teachers and students. It requires teacher be aware of many possible variations and limitations and at the same time it requires a transition of classroom culture to a more abstract academic setting which is not "fun" for many students.
One of the biggest challenges of teaching mathematics is deciding which order students should attain certain concepts. The times when students find Math easy and most meaningful are often the times when they realize that a concert there learning is one they already knew. The example you bring up here illustrates this challenge effectively in that it shows the two complementary topics can be thought of as the same thing by people with full understanding of mathematics but might be thought of as two separate things as students are learning and creating new example spaces in their minds.
DeleteI remember when I took my psychology 101 course we talked about young children gaining schema - new concepts which they needed to make sense of in their mind which was how they formed understandings of the world around them. Similarly it is helpful to understand the concept such as the second equation you list perhaps sooner than it is to understand the first equation you list. Our job as educators is to help students build the schema they need to understand both of these equations and how they relate to each other.
Throughout my math career I relied heavily on examples. On my own though, I often ran into the exact problem Ting highlights: an insufficient example space, and on test day something would surely show up which challenged my understanding. As an instructor now, it can be hard to notice when a slight adjustment changes student’s ability to understand since we have such a complete picture of the material. However, I think such instances can become opportunities to (first sympathize and) exploit the difficulties. Indeed, students may find example A more challenging, but is it reasonable for them to grapple with why a number they get in B also works for A? If they get their x through B, can they not test that it also works with A? Some students might try more numbers, some might figure it out in general? I would be hesitant to remove example A based purely on the fact that it is tougher.
DeleteI know order is a big deal in teaching math, but we will inevitably make wrong decisions, no? I’m not suggesting anything ground breaking here, but when an example that arises that is surprisingly difficult I always try to return to it when my class might be better ready to absorb and note the subtleties that made it trickier in the first place.
This research is particularly interesting in relation to examining student-posed questions. Since the form A is abstracted from form B, students may often think of form B first, as it is more “natural” and is more accessible from their “example space pantry”. But in the context of student-posed questions, it is natural for students to try to make their questions as challenging as possible, so many students will attempt to perform as many abstractions as possible (in a similar fashion to the B -> A transition).
DeleteI am very glad that other people have discussed the Goldenberg & Mason article, and that I have had a chance to read through the posts before posting my own. I found the article itself difficult to read as I couldn't really gain any momentum. Just when I thought I understood what was being said, some other jargon was thrown in. That said, by reading other people's responses, most of my confusions have been alleviated.
DeleteWe can see how important examples are in mathematical understanding. Because of this it seems that teachers should take extra special attention when choosing examples to pose to their class. I believe that everyone will take different things away from examples, and because of that we should not just focus on the type of example, but we should offer many examples. Hopefully, by throwing several examples at the class, each of the students will make a connection with at least one of them.
I agree with Sophie that slight changes to how problems are posed can completely throw a student's ability to solve the problem off. I have even noticed that when something as minute as the type of font is changed, a student may be thrown off a bit. For this reason, several differing examples are a good idea. A teacher could create a general problem, and then the students could each re-write the problem in their own words. Then the class could share a variety of ways on solving the problem as well. I think that this all comes down to developing an ability to be flexible with different perspectives and ways of presenting and processing information. This is surely a skill that is useful outside of mathematics as well.
I think some teachers are aware of students' prior schema, especially the more experienced ones who have gathered student misconceptions and errors over time. For teachers who may not be as experienced, or who may lack this adequate range of examples, do you think there is a possibility of starting a topic allowing students to figure out how they would approach and solve a problem using their own informal strategies, without having first learnt the formal methods in the topic?
DeleteWould this be help make visible some of the prior schema and understanding that students are drawing from before the topic begins? The teacher could use this as a quick gauge of possible prior conceptions that he/she could build on and make links to formal methods that he/she will teach in the topic; and perhaps use it to surface or highlight possible misconceptions/errors in understanding that may arise. The choice of the problems to pose at the start may be challenging as well - are there characteristics of problems that may work better in drawing out some of students' prior knowledge/schema?
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