Two articles by John Mason on attention:
1) Mason 1982
2) Mason 1989
Please read these two thought-provoking articles on mathematics, attention and abstraction.
Then, over the course of our virtual class (from today to our return on March 24), please post at least four substantive posts in the "comments" section of this posting. Make sure to read and consider the whole thread of the conversation to date and respond to what is being said, as well as adding your own ideas. Your four posts should be separated by at least a day each.
Enjoy this interesting conversation!
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ReplyDeleteOne aspect of learning that I find particularly interesting is the shift in perspective and conceptual repackaging that you experience as certain principles start to become automatised. The immediate example that comes to mind is learning an instrument. For me, recently, this was the accordion. Acquiring this skill was a constant repackaging of information: first understanding the location of notes, then transitioning between notes, then practicing coordinating both hands, and so on. Every stage of learning required that I had automotised the previous stages.
ReplyDeleteI am reminded of this after reading these two articles by Mason. In 'Attention' he speaks of how the allocation of a student's attention can be "totally occupied by one aspect of the task at hand, and so no attention is available for the remaining aspects" (p. 22). After practice, and gaining familiarity and experience, however, the student may no longer have to pay as much attention to that one particular task and can begin to venture into more more abstract areas. In 'Mathematical Abstraction as a Delicate Shift of Attention' Mason speaks of this conceptual repackaging, having the reader work through an example where each stage has them thinking about the pattern of numbers in more and more abstract, or generalizable, terms.
This concept of abstraction brings to mind Vygotsky's zone of proximal development: if a task lies too far away from some student's ZPD, then they may either not be able to afford enough attention to all of the relevant details, or will find that the amount of attention they need to spend is not great enough for them to have to pay attention.
It is interesting how many of us are drawn to how one's attention can be totally occupied by a task to a point there is no attention left for the other aspects. Maybe it is because we can all relate to is in many aspects of life. Take my 11-month-old for example, he pays all his attention to shift positions from sitting to standing and vice versa that there is often no attention available to navigate himself out of danger. As a result, I keep an out for my child's safety. Perhaps in like manner, we need to navigate some aspects of our student's learning such that their attention is not taken away from minute tasks at hand. For example, if a student has not mastered basic multiplication facts, a times table could be provided for the student.
DeleteYour mention of attention in learning an instrument is an interesting one. Since I've spent much more of my life doing ballet than math, I always try to put ideas in that context. During a ballet class, there are so many things that I need to pay attention to during a single movement. Even if that movement is a very basic one, I can be completely overwhelmed by everything I need to think about. When I dance consistently, all of these things are much more automatic. Unfortunately, now that I dance very infrequently, I have to pay attention to SO much. I often find myself paying attention to the aspects that I know I always used to struggle with in the past, in fear of falling into old habits. It's particularly scary to not pay attention to certain aspects of my dancing, as doing so could lead to injury!
DeleteI suppose that in a mathematical context, when you are doing mathematics often, you are much more aware of what you need to be attentive to. For students that don't do mathematics on a regular basis, are they not used to being mathematically attentive?
Vanessa your point makes me relate to my belief of teaching in general. I have colleagues who shy away from teaching math or say they are not good at educating their students in math, because they themselves are not good at it. I tend to think of this differently. I feel that it is often hard to teach things that you are good at because you may not have been attentive to the actual steps you are taking in order to progress. I have always found this tricky in my field of interest, field hockey. There were some skills that came naturally to me and when teaching them to athletes I struggle to break it down into enough steps to effectively coach them through it. I equate this to math in that if you do not know why you are doing the steps or 'pay' attention to the steps you are taking when solving questions, it is often difficult to explain what you have done, or teach someone how to do what you have done.
DeleteThis brings me back to the articles on proofs, where you have to convince yourself, then a friend, then an enemy. I like this idea because without explaining what you have done, you often do not fully know.
Phillipa, your comment regarding paying attention to the steps when solving a problem made me think of how teachers of each grade-level draw attention to particular aspects of a problem. I have noticed that if I have a student who is struggling with a particular concept, and I have to reintroduce them to some of the rudimentary concepts, they will often still struggle unless I have "drawn their attention" in the same way that their previous teacher did. It seems to me that when students learn how to solve particular styles of problems for the first time, their teacher creates a series of "dots". Each "dot" can be seen as a moment of attention that the teacher wanted the student to focus on. It is the goal of the teacher that the student will connect the dots on their own without the prompting of the teacher. Sometimes, as teachers, we try to focus a child's attention on something that we believe is important, but does not seem important for the student. I feel as if we need to connect the dots not only between subjects, but also between grades. If we can show the student how to make these connections, and shift their attention efficiently, they will become more effective learners.
DeleteIt is interesting to note that how many of us tend to quantify the term 'attention' as if this is something that is tangible. For instance, I notice phrases such as "amount of attention", "enough attention", "often no attention available", "pay attention", etc. in this conversation. We often forget the origin of this particular term. 'Attention' is a term that is rooted in the field of psychology. Cherry-picking terms from other fields of study to describe phenomenon in mathematical education removes the historical and symbolic meanings, environmental relationships, and explanatory or descriptive nature of the terms inherent in the realm of the field of origin. From our earlier reading, Kilpatrick sums up the process of borrowing terms from other fields of study to collectively "not add up to very much" for the progress of mathematics education.
DeleteMurugan, your mention of quantifying attention intrigues me. Very often, a teacher may ask his/her students to give him/her their undivided or full attention when he/she is about to teach a lesson or to do an in-class activity. I personally find it difficult to pay full attention not because I have a short attention span, but because certain factors inevitably reduce my attention. For example, when I explain a math problem or a math concept to my students both verbally and in writing at the same time, I need to divide my attention for both of these modes of presentation - 40% for oral and 50% for writing. I reserve 10% for keeping my ears open for my students' questions. What I am saying is that if I put 100% attention on any one of these tasks, I do not think I can accomplish much in a tutoring session. Hence, I do not demand full attention from my students, but I do expect them to pay most of their attention to the math task at hand.
DeleteKevin, our expectation of something, such as attention, from students may be a problem here. When we expect our students' attention, what are we really looking for? Are we expecting them to be aware of (what and how), notice (the teacher, process, or product), or observe what is going on? This is the that that fascinates me about attention. By expecting our students' full/partial attention, we accomplish our task to conveying (?) our message to the students. How do we find out if students got your message? Certainly, I don't think that our usual assessments will accomplish this task?
DeleteOK, hypothetically, let's say your students supply (pretending) most of their attention to your demand. What does it accomplish in terms of their congnitive abilities and activities? Jee, more and more questions. Here is a nice link related to psychology of attention:
http://www-psych.stanford.edu/~ashas/Cognition%20Textbook/chapter3.pdf
Murugan, you make a great point in your March 13th posting about the use of the word "attention". Mason (Mason, 1982) has a section of his article called "What is attention?", where he goes on to answer the question from a very introspective point of view. As you pointed out, the word "attention" has a meaning and a history in psychology already. His introspective writings don't address this meaning and body of existing research.
DeleteThe main takeaway for me from these two articles deals with the idea of what a teacher thinks paying attention or abstracting means and how that is often different from what students think paying attention and abstracting means. Mason notes in the second article “Mathematicians, wishing to help students, sometimes look for apparatus, for physical objects … in which they can see manifestations of their mathematical aware-ness, their abstraction. They then offer their representa-tion to students, perhaps forgetting that students may not be seeing things from the same perspective. Students stress and ignore according to their experience, and not according to the experience of the expert.” This idea is so true for any learning environment, but especially for math. This can often be shown through the use of manipulatives. I read an article for the short paper which discussed how often times teachers think manipulatives help students to understand a more general concept, but in reality it might be confusing students more because it was not their own construction of the manipulative usage and/or because they might not understand how the concept can be generalized.
ReplyDeleteAnother thing that was very interesting to me from Mason’s first article was on the concept of split attention. He states, “the student's attention is totally occupied by one aspect of the task at hand, and so no attention is available for the remaining aspects which are essential to complete the task. “ This reminds me of the skill of note taking. We often times expect students to be able to take notes, listen to what the teacher is saying, and make connections between what the teacher is saying to what the notes on the board are saying to what their own notes should say. This is true for note taking in any subject, but especially true for a foreign language class, which it can be argued that math is of a different language.
Your comment reminds me of discussions in Ann's class about the appropriateness of manipulatives for children. Perhaps the introduction of manipulatives might hinder the process of abstraction. Indeed, using manipulatives could make a concept seem more concrete and applicable to the real world. I have been thinking of the appropriateness of introducing such physical objects in the learning process. Is it better to introduce them at the beginning of an abstract question, in the middle, or at the end of a section? What might be a particularly interesting application would be for students to come up with their own physical example of a concept, and share it with the class. This way, the student can develop their understanding on their terms, and then listen to the reasoning and abstraction of others.
DeleteI always find that I'm much less attentive to content when I'm taking notes. I think this is because my attention is on writing down everything that gets on the board so that I can try to make sense of it later. As you say, it is indeed very hard to pay attention to the teacher, the notes, and make connections. I think this is part of the reason a lot of university instructors are opting for the "flipped classroom." This way, students are engaged the instructor and the material in the class, rather than trying to frantically write down definitions and theorems for an hour and a half.
Vanessa, it is interesting what you're saying about taking notes, because I find almost the exact opposite to be true for myself: when I take paraphrased notes in class I retain so much more information! It is as if the act of translating the spoken words into written text through fine motor skills is helping to etch the information into my memory. However, the limit for this is when I am required to take lots of notes. At that point, all I am doing is copying! My attention for what is being spoken is drastically reduced, and I am instead simply trying to keep up with what I have to write. When I can take only a few notes, however, the amount of attention I have to give to writing is minimal and instead I can focus on the content of what is being said.
DeleteAs for when to introduce manipulatives, that is a question I can't answer! I think, though, that one can have very powerful discussions with students in regards to the inherent limitations that concrete models have for abstract concepts. Trying to find the facets of a concept that manipulatives excel at, and fail at can be a great way to help further understanding.
Keri, I fully agree with your statement about manipulatives. Yes, they can be useful for many students, but there are some who prefer to abstract information in their own way. I'm reminded of base 10 blocks. As an elementary teacher, we use (or are expected to use) base 10 blocks for many different concept attainment and reinforcement activities. I definitely see young children benefiting from using base 10 blocks to demonstrate regrouping in addition and subtraction. The blocks are also used in multiplication, which is an interesting way of representing the problem. I believe that most teachers, often due to the textbook being used, feel an obligation to use base 10 blocks for all sorts of problem solving. I wonder if the student's attention would be better suited focused on something other than manipulating the base 10 blocks. We often try to teach material in as many ways as possible, to try to hit all of the differentiated learning styles, that it's no wonder a student has difficulty knowing what to focus on. On top of not being able to establish a focused attention, there is little time for any learning to sink in. As Mason stated, "Unless time for reflection is provided, learning from experience does not take place." (p.22)
DeleteDavid, your comment takes me directly to my use of base ten blocks. What I find to be quite abstract for students is the switch between base ten blocks for whole numbers and then the use of them in different ways for decimals. The small/single square represents 'one' when looking at whole numbers and then switches to represent 'one hundredth' when exploring decimals. I believe in this circumstance some students find this switch difficult conceptually and may benefit from choosing their own representation for fractions.
DeleteAs for taking notes and paying attention, I have been looking into doodling as a tool for students to take notes and maintain focus on the learning. I am like Conrad in the fact that I need to be taking notes or doodling in order to stay focused on what is being presented. I have tried to teach a few of my students the doodling strategy in order to aid their attention span when listening to lessons. A great resource is Austin Klawman's book "Show Your Work," and also Brad Ovenell-Carter who works locally at Mulgrave and has a real focus on doodling as a form of notetaking. http://thejournal.com/articles/2013/11/13/these-gorgeous-ipad-notes-could-lead-to-the-paperless-classroom.aspx?=THEMOB
If people are interested I can bring in some of my very initial practices on this.
Philippa - Please bring them in!!! I would love to see this!
DeleteIt is interesting to see a discussion about how manipulatives influence children's attention. Some teachers that I know use manipulatives a great deal to bring math to life and to make math concepts more concrete. They think these objects are very effective in helping struggling students learn math in a visual way and in reversing students' math misconceptions. Others that I know have negative attitudes toward the use of manipulatives because some students treat these objects as toys that shift their attention from exploring math concepts. Hence, these teachers think manipulatives do not always guarantee success in guiding students through their learning of math.
DeleteI recall that last term, in EDCP 552, I administered task-based interviews with a 10 year old female student as a research assignment. In all the interviews, I provided her with manipulatives which helped maintain her attention to her problem-solving activities. However, in one interview, the given manipulative tended to draw her attention away from the math task she engaged in and prevented her from solving the problem. She decided to abandon the manipulative because it did not help her make sense of the math task at hand. It is difficult to say whether manipulatives distract students' attention from doing math or whether manipulatives help students focus their attention on it.
Kevin, thanks for sharing your experiences. I have also experienced the limitations of using manipulatives in my task-based case study involving a young child. Unlike your case, the child in my study liked the manipulatives, but tended to focus most of her attention on "unnecessary details" when attempting to solve problems. I thought that some of the details, such as the non-uniform size of unifix cubes, colors of objects, thickness of elastic bands, etc, were irrelevant in order to solve the problems at hand. Upon further probing her line of thought, fortunately, I was wrong. She perfectly understood the problems. What was on her way to solving the problems were the defects and scratches in the manipulatives or anything that distracted her attention. She focussed her attention on these defects. Thus, creating or building barriers around her ability to solve the problems.
DeleteI don't think that this was an isolated experience, where students create barriers around themselves when it comes to learning mathematics. It is fascinating to think about how students, mostly struggling ones, tend to focus on things that we consider to be irrelevant. Is it our limitation? How do we convey to students what is relevant or irrelevant when attempting to solve problems?
Murugan, great point you make about students paying attention to details that are irrelevant to the problem at hand. It's interesting that you were able to experience this first hand. Mason states this in one article (Mason, 1982): "…. the general phenomenon of students responding to "surface features" of a question rather than to a logical or deeper structure …". In answer to your excellent question "How do we convey to students what is relevant or irrelevant when attempting to solve problems?", I can think of two exercises which may help. One would be to ask students to classify problems based on what they feel are the similarities between problems. If the classification is based on "surface features", the "correct" classification is presented and then the exercise is repeated for a different set of problems. In this way the student could progressively refine their ability to "see" the deeper features and ignore the surface features. However, for such an exercise to be maximally effective I think the student would also have to solve some of the problems, so as to truly get a feeling for these "deeper" features.
DeleteAlain, thanks for pointing out the correct term, "surface feature." As you have suggested (and I am going to stretch it a little further), gaining a deeper understanding of mathematical structures may involve moving beyond problem-solving, use of manipulatives, or hands-on activities. Here is where we lose our students' attention and, in turn, lose our students' desire to learn mathematics. I think that it is more to do with our difficulties in conveying the message across to the students. At the basic level (most difficult), the message is that changing or modifying the surface structures (features) of mathematical problems does not alter the underlying conceptual or mathematical structures of the problem. That is, an abstract understanding of the problems may be necessary in order to generate, integrate, evaluate, and relate mathematical concepts and ideas (back to good old Bloom's taxonomy).
DeleteIn reading Mason's two articles, thinking of what to include in a posting, and then reading both Conrad and Keri's entries, I am pleasantly surprised that a quote in each of their posts was one that had stood out to me and I was set to include it. Instead of going back and struggling, I figured I would present my take on it anyways. Much like Conrad and Keri, I was drawn to the mentioning by Mason (1982) that "the student's attention is totally occupied by one aspect of the task at hand, and so no attention is available for the remaining aspects which are essential to complete the task." This makes me think about my initial teaching experience with manipulatives. I was so excited to have made, what I thought were, incredible pizza's to practice fractional parts with the students. I let them play at first and then tried to move on with the lesson; however, much to my dismay, the student's attention was still on making pizzas, not on what math could be accomplished with the pizzas. I believe this was a clear indication of how attention can be fully occupied by something extraneous for students, in this case the pizzas, leaving no remaining attention for the students to hold on the fractional mathematics. I believe this often happens in skill progressions in math, for example moving into larger number multiplication. A student of mine this year was having difficulty with larger number multiplication, and upon closer inspection, was still worried about the fact that she hadn't reached automaticity with her basic multiplication facts. This had created an obstacle for her, as her attention would not focus on larger number multiplication, because all she could think of was her need for further practice on her basic facts.
ReplyDeleteI also appreciated how Bennett in Mason's article Attention (1982) calls "the field of my awareness," "present moment." Children's lives are so complicated and in order to teach students new and often challenging concepts, it is important to realize what is occupying their present moment state of mind.
Lastly, I was intrigued by Mason's mention of the fact that "students are interested, as are we all, in minimising the energy they need to invest in order to get through events. Thus we look for cues, in what the teacher says and does, for what they should do" (1989, p.7). It remains so important as educators to really be purposeful in all our teaching and help students guide their attention to specific areas that will aid them in taking their knowledge to a point where they are able to shift their attention from one skill to the next and begin to apply their knowledge to new situations. Application of a skill to new situations rather than repetition I feel is the end goal in mathematics education.
Phillipa - Love the quote you used in your last paragraph. In Ann's class last term, we talked a lot about students creating their own algorithms when they needed them to minimize energy For example, kids will count objects one by one until you challenge them with too many to count that way, and then they will come up with skip counting.
DeleteThis makes me wonder how much of what we understand of Academic Mathematics comes from people finding more efficient ways to complete tasks that were important for something, and how much is the opposite - people creating complicated concepts that lack purpose.
I imagine that for most people who have discovered something new, they see it as the former, yet most of our students likely see it as the latter.
David - as you mentioned in the second and third paragraph, it seems as if some mathematics we teach is often thought of as not having a purpose in the eyes of some of our students. I see this particularly in the pre-calculus class I teach on Saturdays. Since the students in this class are all enrolled in public school, but only taking math with me on Saturdays, it is hard to help them see the purpose of concepts taught in the class since I only see them one day a week. Not only is it difficult to keep them motivated to learn the concepts, it is difficult to get them to complete assignments and take tests, especially since I do not do any of the grading for the assignments and assessments (an online school does this). All of this leads me to wonder if the way our system is set up works against students finding meaning and purpose in the mathematical concepts we teach. I wonder if things like summer break or class all day or tests or ... the list could continue ... work against students finding true value in the concepts they learn at school.
DeleteI enjoyed the description and analysis about attention in the first article, but I kept waiting for the author to address the verb that goes with it... ;paying' attention.
ReplyDeleteI found myself pondering how the analysis of attention went with the idea of paying for something. I believe paying implies a transaction where something is exchange. Maybe the hope is that by asking someone to pay attention, I am promising them something back (knowledge? grades? something else?)
Also... Conrad, please bring your accordion to class next time.
I also pondered on the word 'pay' as it implies that something is exchanged. This could be considered in two aspects, internal and external. You mentioned an exchange of students' attention for knowledge or grades. I thought of attention in exchange for intellectual engagement, satisfaction, entertainment, self-improvement and etc.
DeleteIt may not be something that you promise the students in return. Perhaps it is something they promise themselves in return for paying attention to the mathematics. Maybe the attention they afford could ultimately transport them from an abstraction of bewilderment to an abstraction of wonder and excitement. Who know?
David, I got caught up on that word 'pay' as well. It does, as you say, denote some sort of transaction, however, it is interpreted as something very different. A study by Carson, Shih, and Langer (2001) asked teachers students what they thought it meant when they said to 'pay attention.' The overwhelming response had to do with stillness: if you are still, you must be 'paying attention.' This response is strange because it has nothing to do with having to pay for something, and it does not connote some sort of transaction. Paying, I think, is a bad word. It makes you feel like you are giving something that is yours away. A better monetary metaphor would be the word 'invest.' If you invest your attention then you are rewarded. What do you think?
DeleteCarson, S., Shih, M., & Langer, E. (2001). Sit still and pay attention?. Journal of Adult Development, 8(3), 183-188.
Wow! I really like the idea of using "invest your attention" rather than "pay attention" because it is exactly like you said. Using "pay" makes it seem as if students are giving something away whereas using "invest" makes it seem like a smart decision to do that will give positive outcomes back to them.
DeleteIn both articles, Mason basically argues that what is obvious to the teacher may not be obvious to students and that whatever attracts their attention first in a math task leads them to think in that direction. Mason (1982) points out, "Attention absorbed by basic skills triggered by a question can only cope with surface features. It does not permit any deeper contact (p22)". His statements really resonate with me. A lot of the students that I tutor in my learning centre have merely a superficial understanding of math concepts and are very good at manipulating algebra in a meaningless way. Even though their computational competency earns them high marks in math at school, they lack the necessary skills in analyzing the underlying logic of math concepts. Especially, when it comes to doing proofs in circle geometry, some of my students find this exercise abstract. Mason (1989) argues that students have such an opinion because they fail to have "a delicate shift of attention from seeing an expression as an expression of generality to an expression as an object or property (p2)". Such a problem that students encounter has something to do with their present moment which Phillippa has mentioned above. When doing proofs in circle geometry (which some teachers still do as recreational math), my students are aware of only the "rules" and "algorithms" at their present moment which prevent them from finding relationships among different ideas. For example, when asked to prove that opposite angles of a cyclic quadrilateral are supplementary, my students choose random numbers that add to 180 degrees. Normally, I need to provide them with guidance and draw their attention to some of the properties that they need to investigate. Through my intervention, the students gradually move from numbers to symbols to represent angles and shift their attention from seeing the numbers/symbols as a form of generality to an expression as objects. At the end, most of my students have developed a better understanding of the structure of cyclic quadrilaterals.
ReplyDeleteWhen I tutor students in math, I never use the word, obviously. As Keri has mentioned above, students don't always think in the same way as the teacher. Whenever I talk about and investigate a concept with students, I stay away from seeing it in the "obvious" way and think from the students' perspectives. This gives us opportunities to learn from each other.
I completely agree with your point Kevin! The way we teach math at our school (and I believe in most schools) involves a lot explanation of thinking, multiple strategies, and the showing of work. This is often difficult for the students that attend out of school math tutoring at places that purely involve repetition of facts. When asked to explain their thinking the students are often stuck, stating they just know the answer. I believe it is imperative for students to understand why they are doing the math. Without this understanding students are less likely to be able to shift their attention purposefully onto more complex procedures with understanding. It does take extra time, however, the deep level of understanding and the ability to switch attention focus is worth it!
DeleteI think having students understand what they are doing is much more present in the primary level than it is in secondary school. This is inflated by the fact that mathematics becomes even more procedural once students enter secondary school. They are provided with dozens of polynomials which they need to factor or expand, without any real meaning behind what they are doing. For example, students become accustomed to the process of "FOILing" polynomials, rather than seeing the polynomial as an object which may be manipulated in multiple ways. This, however, is when we consider the language of process and object. True, secondary teachers do have a great deal of procedural content to teach, but what good is a procedure when you don't know what it's good for? As you mention Kevin, students do very well on tests, but when they are asked to explain why they do it, their responses can somewhat empty.
DeleteThen, when the students enter back into university where we want them to really understand the material conceptually, this is something they haven't done since primary school! I was particularly bothered by Mason's 1982 quote "unless time for reflection is provided, learning from experience does not take place." In a University setting (where he is placing himself during this quote), is it not expected that the majority of the reflection happen outside of the classroom? The material is at a deep enough level, that spending 15 minutes for reflection is certainly not enough. A student might need to spend hours reflecting on a topic before they can fully understand!
I think the issue of students not conceptually understanding the mathematical content taught at the secondary level is mainly due to the fact that teachers are required to teach so many concepts in a short about of time. Therefore, the task of making sure every student conceptually understands each concept is a very difficult one for teachers. It is almost as if the ways in which the school system is set up requires teachers to only teach procedural skills so that everything concept can be covered. Since students are not fully understanding what they are learning, no wonder it is hard for a lot of students to make abstractions and give their full attention to math concepts/lessons/classes.
DeleteThe school system I worked for in my first two years of teaching originally had a 180 day calendar year, but dropped to 160 days because of budget cuts. Not only did we lose 20 days, but they also changed the start time from 8 am to 6:50 am. As you can imagine, the math teachers in the school taught in a more procedural manner that year than in previous years. This was because of limited time and limited attention from students due to the early start time. Needless to say, it was a nightmare! However, my second year in the district they added 10 days back in making the school year 170 days long and moved the start time up to 7 am instead of 6:50 ... a lot of change that did.
Sometimes i think I misuse the word abstract when I talk to students about upcoming math topics to mean the same thing as difficult. For example, when teaching Math 12, I tell them they can look forward to the combinatorics unit as it will be less abstract than the trigonometry units, and most people find it easier.
DeleteBasically, if a math topic is abstract, it implies a student doesn´t understand it. When they understand it, it isn´t abstract anymore. Part of the reason this comes up is that teachers are concerned with teaching students the tricks they understand to solve problems efficently before the students have developed more detailed understandings of the underlying concepts. Trigonometry is probably really not any more abstract than combinatorics, there are just sometimes more steps that are sometimes skipped as you work through the problems.
I think its fair to say abstract can also describe something that is difficult. There's that old math joke where two mathematicians argue back and forth about whether something is obvious or not, one completely convinced that it is, and the other that it is not. After huge amounts of heated debate, the other suddenly sees it, and whole heartedly agrees that indeed, it is obvious. That can often be the way it is with mathematics topics that are classically considered more abstract, less grounded in reality: huge obstacles until suddenly, you just see right through it. Perhaps it's your attention finally being `delicately shifted'?
DeleteAn abstract concept to student in math usually relates to unfamiliar living experience. And it is also true that if a concept is abstract, that means students do not understand it.
DeleteBut in math we cannot expect everything to be straight forward. Sometimes the reason is hidden behind the seemingly unfriendly proving process. In math proof session we have read articles about how mathematical proofs help students understand abstract concepts, whereas those students mentioned in the articles are some top students who are willing to sit down and read the proofs. In reality, some students just do not want to make that efforts. Meanwhile, I also agree that there may be many different reasons for students who do not want to make that efforts. For example, their calculating ability (calculatility? :)), reading skills, etc.
So is it fair to say that most math concepts on and below secondary school level are straight forward, even trigonometry?
(For this article, I focused on the Mason 1982 article).
ReplyDeleteIt is extremely important to avoid saying that something is "obvious", I'm glad you brought this up, Kevin! I'm not sure if you have ever had this line used to answer you when you were a student, but I remember asking a question in an integral calculus class about the limit to obtain e, and I was told that, "I should have seen this already" - or in some proofs books, "this proof is extremely trivial and is left as an exercise for the reader". Oh, fun.
I am surprised that as of yet, nobody has written about the division between content, processes and technology. An additional dimension to mathematics education in the classroom is the environmental factor of technology exposure - in this case, the technology I mean is expressed in terms of electronic/social media. Could it be possible that the process of doing a task, if it is not engrained in a student's automated processes (as with Jubilee's 11 month old), could be detracting from the attention needed to learn a mathematical process? I can certainly say the same for myself. Rather than focusing on mathematics in any online app I have used, I find myself distracted by the logistics of the online mechanism. This could present a problem for students who do not have a computer at home and provide an additional obstacle for students in addition to their lack of access at home.
Not only this, but the use of technology entirely detracts from paying attention to anything in the students' surroundings. As a classroom management method, I used to stand a foot or so away from a student who was on their phone, and I waited (looking directly at them) until I got their attention. I have decreased this method not because it was effective in enforcing my expectations for the classroom, but because it would take too long! I would often have to wait a minute or even two before they would notice! What is worse, is that students believe they can "multitask" (divide attention two multiple tasks at once), which is debunked in most introductory psychology courses.
I definitely agree with you, Alex. Technology is both beneficial and harmful to students' attention depending on how it is used. Some parents do not allow some of my students to use calculators even when they are in senior grades. The parents think that calculators can reduce their children's computational skills. Even though I understand their concerns, I do not think it is completely reasonable. I mean, while young children do need some practice with their mental math, older students who demonstrate an excellent mathematical aptitude easily get bogged down by tedious computations using paper and pencil when they need to focus their attention on the process. In this case, technology is an asset to bring their attention back to their math task. On the contrary, technology can be a distractor. On a regular basis, I need to give my students tests and use the assessment results to write reports about their progress for their parents, If they get an unsatisfactory report, they will likely be reprimanded by their parents. Occasionally, I catch a few students using their phones to email the test questions to their friends for help. In this case, technology shifts their attention from its good use. So, students' attention is "controlled" by how they use technology in both good and bad ways.
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DeleteIt is interesting how technology is brought into the conversation. I am surprised that to this day people equate simple calculators to technology. I can see how a scientific calculator or a graphing calculator that possesses higher functions is considered technology. However, a simple calculator that performs basic operations to me is just a tool. Similar to what Kevin stated, a simple calculator can be a useful tool to help students redirect or shift their attention from the tedium of computing large numbers to the focus of the new math concepts at hand. However, I do agree that when the computations are simple, we should encourage the students to rely on mental math. For example, one year, I decided that I would teach the entire Math 9 curriculum without a calculator. Interestingly enough, instead of catching students texting on their cell phones under the desk, I was removing calculators from the students. That semester, I did not have any issues with the 'misuse' of technology (not counting simple calculators as one).
DeleteAttention is indeed a fascinating concept. It can be so malleable depending on how the instructor exercises "control" over it. I would argue that semester with the 9's was the most profitable in terms of the quality of mathematics students learned. The class scored the highest average on the common final exam out of 5 classes. I am not saying that there is no value in technology or the use of calculators (because this is an entirely separate topic); however, how attention is focused on the actual math being taught is what can make a true difference.
Kevin, I'm currently invigilating a test where the teacher allows them all to use graphing calculators. In a calculus class. What in the world. I almost didn't let them use it until I found out their teacher allows them to. Given the school where I'm subbing, I can't say I'm too surprised.
DeleteWhat is unfortunate is the dishonesty you have to deal with, Kevin. That reminds me; I heard through a list-serv that one of the students has recently received an Apple watch, which apparently they're able to use to communicate to other people? I know that was the concern voiced by the teachers in the email.
Jubilee, I thought what you said about shifting attention from "crunching" to the math concepts was very neatly put. How did teaching without a calculator turn out for you? Could you please elaborate? I have seen teachers do that at upper grades, but not so much in younger grades.
Jubilee - it is interesting to me how you do not consider a simple calculator to be technology. I have always thought of technology as being something like a computer or phone, but last semester I heard the word technology being used to describe something that works as an aid. I just googled the definition of technology and google says it is "the application of scientific knowledge for practical purposes, especially in industry". I think it still could be argued that a basic calculator could be (or not be) technology.
DeleteWhen I taught grade 8 in Georgia I had a student that had just moved to the states from Mexico and didn't have many of the basic math skills like adding, subtracting, multiplying, or dividing. Even though he did not have these basic skills, he could understand the concepts taught in grade 8. What was unfortunate was that he spent so much of his attention on learning the basic skills since he couldn't use a calculator in class that he began to fall behind on the current concepts. We ended up holding him back a year so that he could repeat grade 8. He passed the second time around and went on to high school where he was allowed to use a calculator. This is a situation were a shift in attention from basic skills to concepts by the use of a calculator would have been very beneficial had it been allowed. (Please keep in mind that this was not a rule of mine; it was a county wide - maybe even state wide - rule.)
I find the issue of technology and how it affects attention an interesting one. While I am a strong advocate for the use of technology in the classroom, I have definitely witnessed how technology can be misused. I define technology simply as a tool to aid in accomplishing a task. A pencil, ruler, paper are all forms of technology. With the introduction of any new technology, there will always be an inherent danger that it may detract one's attention away from particular learning outcomes. I can see how some may view the use of technology as a way of cheating, or bypassing important thought processes. This is an issue that must be addressed by the teacher. For example, if I am assessing my students' ability on whether or not they know their basic facts, I will not let them use a calculator. However, if I am assessing their ability to find the area of a rectangle, I may let them use a calculator. If they know to multiply the length by the width, then I know that they understand how to find area. I would not want their inability to recall basic facts be the sole reflection of whether or not they understand the concept of area.
DeleteAny form of technology can be seen as a tool. Like any tool, it may be used appropriately or inappropriately. If used inappropriately, it will detract one's attention, or negatively abstract, from the intended goal. If used appropriately, the tool will allow the user to free up their attention so that he/she can focus on, "the isolation of specific attributes of a concept so that they can be considered separately from the other attributes." (Mason, 1989)
It is a sensitive topic understandably. Unfortunately due to the common misuse of technology, there is a negative association between it and a student's attention. The reality is that once we leave the formal education setting, we are expected to know how to effectively use the tools that are available to us. Rather than completely shutting off access to technology, we need to teach and model for our students how to appropriately use technology as a means to supplement our education as 21st century learners.
Jubliee, I agree with you and your mention of a scientific calculator being more than a tool than a technology. However, I think there is a larger problem at hand. If we want students to understand the process of a problem so that their attention is on the concept rather than tedious arithmetic, why include tedious arithmetic? If the goal is to demonstrate understanding, is it not contradictory to provide a problem that detracts from what you want the students to pay attention to?
DeleteMoreover, this might be an even larger problem. As soon as sqrt(2) or some irrational number is included in a problem, I think a lot of students automatically think that a calculator is needed. I remember my first ever set of office hours when I was teaching in 2013. My students had some algebra and geometry problems as review, which were all on webwork. I had a number of students come to me very confused as to why the computer would not accept their solution of 1.732 for a problem where the solution was sqrt(3). The numbers in this problem were not complicated, but the students' attention was on using the calculator for a simple computation they didn't need it for. In fact, by using the calculator, they had an incorrect solution! As David mentioned, technology could be used inappropriately to the point where the students' attention has been detracted from what it means to "find a solution." Does this mean that the students do not have an understanding of what it means to be a solution to an equation? Perhaps, perhaps not. An interesting research question indeed....
Bringing it back to Alex's first comment about doing a task sometimes distracts from the attention needed to learn a mathematical process, I find tricky. I do agree that this can happen but I also believe that sometimes doing a task can hook students into the mathematics involved. I will often ask the students to engage in a task first before I present new topics, allowing students to make educated guesses about what we will be exploring next. Students tend to get excited about the exploration and further excited about the topic they will learn.
DeleteWith regards to the use of technology in the classroom, this issue can become quite complicated. I agree completely with David that it all depends on what you are trying to achieve, learn, and assess. I find it interesting that a lot of grade 4 mathematics curriculum involves gaining number sense, developing understanding of why and how we manipulate numbers and yet on the FSAs the students are allowed to use calculators. I often find that using a calculator to work on equations can take longer than doing mental math or writing out their work, demonstrating the need for students to practice the simple use of a calculator. I also agree that teaching students the appropriate use protocol is essential. This requires us as teachers to clearly set out expectations and in this ever changing world, anticipate problems that we may not have even imagined.
Alex, teaching grade 9 students without a calculator was very eye opening. Students can really surprise you with what they are capable of without constantly turning to the calculator as a crutch. As Vanessa suggested, in order to make the course non-calculator based, I had to make sure the numbers were either nice to work with or could be left in a reasonable expression. Both the parents and the students saw value in using mental math whenever possible. Instead of trying to focus on the final 'answer', the class was able to focus more on the process. It also forced the students to constantly ask themselves whether the end result made sense. We relied a lot on mathematical estimations which is a very useful skill. The students enjoyed learning math tricks so they could manipulate simple operations quickly and accurately. In fact, a lot less students asked when they would ever need the math in real life. Although it took a lot for the students to put away their calculators at the beginning, students including the weakest ones gained more confidence in math at the end of the term. It was a very profitable experience. If I ever get the opportunity, I would definitely do it again. Also, I always teach Math 11 and 12 without the use of a graphing calculator. Technology is advancing so rapidly. I do not see value in spending hours on teaching different key strokes to obtain a certain graph on a particular setting. I would much rather use online or cell phone graphing applications to teaching all the important behaviours of different types of graphs.
DeleteIn response to David, I do love to incorporate technology into my teaching. However, as you mentioned, it all depends on how technology is used to enhance student learning and their attention. However, I also see how in the absence of technology students are able to focus their attention on a different level, such as the mathematical process and number sense. Sometimes, in order to "[allow] students time to convert hard-won expressions of generality into confidently manipulable objects," we need to remove the calculators and "[help] students to experience the delicate shifts of attention..." (Mason, 1989, p. 8). Other times, we must make use of the available technology to free up time for the students' experience of mathematical abstraction.
In the course I teach we do not allow them calculators or any other technology on tests and quizzes. While in class the students will sometimes turn to their cell phones or calculators to compute or check an answer, which I certainly allow, but I try to also use those opportunities to show them how I would have done the arithmetic without a calculator. I have actually heard the words, 'oh, that's how you multiply/divide' from these university students. The attention they give/invest/pay at such moments is so much more than had I simply started the lesson with "Let's review how to multiple two numbers larger than 10 together."
DeleteThe other thing that can occur when you remove technology from the equation, is that you can highlight the relationships between numbers better. In my class (I'm not sure how much this transfers) we are trying to produce students with stronger problem solving skills, and arithmetic becomes somewhat secondary. I strongly encourage my students to leave answers in a form which includes operations: eg. 2^3*3^4*7. Knowing this is 4536 is less important and can actually impede understanding because suddenly they have this big number, instead of smaller, more manageable factors which are multiplied together. For them, performing the calculation is a waste of time, prone to errors and doesn't really provide any new information. The step right before such a final calculation often illustrates all of the skills we are interested in them learning, and allows students to focus their attention on problem solving, rather than on worrying about computation errors.
Sophie, I agree, and disagree with some points in your comment. That you can highlight the relationships between numbers better, I think, is disputable. I frequently have my students derive the exponent laws through experimentation and extrapolation, by using their calculator to compute many examples they can immediately see the larger picture without having to wade through the calculations. Pushing numbers and variables around a page to derive a law is ok, but for so many students mathematical symbols are already so abstract, that, students might just take your word for it without really understanding what it means (similar to our discussion about proof the other week: are you trying to mathematically prove it, or simply convince them to believe you?).
DeleteKnowing that 2^3*3^4*7 = 4536 is, you're right, unimportant so long as your result doesn't need to be interpreted! Which result is 'better', the factored or numerical, entirely depends on the context of the question, a powerful discussion to have with your students. A similarly interesting discussion to have with students is regarding improper vs mixed fractions and which is 'better.'
It is interesting to note how we, technology immigrants, readily reduce or interpret the use or abuse of 'technology' as a tool targeting lower order thinking skills inside or outside the classrooms. Wikipedia defines technology as a "collection of techniques, methods or processes used in the production of goods or services or in the accomplishment of objectives." Furthermore, technology can also be thought of as the knowledge of techniques and processes embedded in devices that can be "operated by individuals without detailed knowledge of the workings."
DeleteOperating technological tools and devices for solving mathematical activities without a detailed knowledge of the underlying processes has always been an area of heated debate. For instance, (Keitel, C. (1989). Mathematics education and technology. For the Learning of Mathematics, 9(1), 7-13.) referred to the process of hiding mundane functions (adding fractions or multiplying large numbers) as further mystification of mathematical processes and ideas. Is this concern still relevant? Just because we learned all of these processes does not mean that we should expect ( or force) our students to learn and master. If a process can be accomplished through the use of technological tools and devices, why not take full advantage (or profit) of this?
So for students, digital natives, in today's classrooms, educational technologies may mean something totally different to what we have come to understand. I think that for current students, technological tools and devices would probably mean ways of interacting with people or objects or seeing things in their mind. I will stop my grumbling now.
Murugan, I had not heard that term before: technology immigrants. I love it. I think you are certainly right, that we are more prone to dismissing technology because it only carries out lower order thinking. This, of course, is less and less the case as computers get smarter and smarter. A naive question: how much computer science is taught in school? Is it relegated to post-secondary education? I have tended to begrudge my less 'techy' upbringing because I never felt I could master computers and calculators and other technologies the way my peers could. I never had cause or opportunity to focus my attention on learning how computers worked. So much technology is behind the glossy scenes now, but still: maybe some basic programming tasks might help bridge a gap or two for students mystified by some mathematical process?
DeleteSophie, as you may know, there is not only so much technology behind these glossy things, but there are also so many beautiful mathematical concepts and ideas embedded in these tools and devices. Hiding the actual mathematics as black boxes (functions) in these glossy things trivializes the desire to learn mathematics through 'standard practices.' This may create issues when it comes to mathematics education and attention. When there is reduced interest in learning something, how do we expect our students to concentrate and master the material, especially when students know that they don't need to learn what is being taught, i.e students' may be aware of easier ways to finding solutions or answers? This is the dilemma that we have to live through on a daily basis. Why are we trying to teach difficult routines for finding answers when easier options are available for finding the answers? Is it because we (educators and curriculum developers) think that students SHOULD learn these techniques?
DeleteIt is interesting to note how we readily dismiss emerging technological tools and devices, because we falsely assume that these are capable of only targeting lower order thinking. Unfortunately, our limitations play a big role and hinder exploiting these tools and devices for higher order thinking. What we end up doing, instead, is often make slides, blogs (Susan, no offence here), and spreadsheets and then claim that we are using technology. Let's face it, I don't think that these are called "using technology" any more.
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ReplyDeleteIt is interesting coming to this after one week's worth of discussion. Seeing topics ranging from connections between the readings and other contexts outside of math, to discussions of manipulatives and divided attention, to the use of the word 'pay' in 'pay attention'. The danger of teachers using the word 'obvious' was highlighted - an important point! What is obvious for us is not necessarily obvious for a student. And now there is an active discussion going regarding the use of technology and how that impacts students' attention: my attention is (ironically?) being divided between all these different topics!
ReplyDeleteRather than responding to each, I will take an easier route and muse that this is a lovely example of my own thoughts on attention - that we are more willing to 'pay' attention to a topic we enjoy. Mathematics, by its very nature, requires concentration, can we call it sustained attention? Some students thrive on the concreteness of learning to FOIL as mentioned above, or carry out other mechanical operations and algorithms: but what about that leap(?) to abstraction that is so critical to mathematical success? Mason argues this can be accomplished by an attention shift. He uses an example of a sequence, a pattern. It's a well chosen example in my mind. Can I be so bold to say that people like patterns? And making sense of them? I've taught similar, and never have any difficulty piquing student's interest. I feel like this is a relatively easy topic to explore, but why is that the case? Because it is so concrete to begin with? That repeated manipulation allows students to realize what is going on, which in turn motivates them to want to articulate it? I very much identified with Floyd's helix.
Is Mason then simply advocating for well structured problem based learning? If we can draw our students in with concrete questions, allowing for enjoyment and mastery of the topic before bringing up a more abstract question, would students be more willing to 'pay' attention? Could more students be motivated from their own curiosity, rather than simply the next looming assessment?
I think that your mention of people liking pattern is not as bold as you might think! Pattern makes common appearances in design, art, clothing, and so many other aspects of everyday life. There is something so satisfying about seeing objects organized in a specific pattern. People like things that look aesthetically pleasing, and I think that patterns add to this aesthetic.
DeleteI also liked the sequence Mason provided in the article. I'm teaching series and sequences right now and have considered including this example in either a workshop or in class as a project in groups. I caught myself using the word "attention" in class today when referring to all of the convergence tests that they need to remember, as well as just paying attention to the attributes of the series itself. I think that many student might struggle with series just for this reason; not only do they have to pay attention to the series, but to all the tests that are floating in their mind and the conditions that go along with them!
If Mason is advocating for problem based learning, it's not completely obvious to me. I think he is advocating for less problems that are completely mechanical and some problems that require students to think abstractly. I suppose this is "problem based learning" in a sense. Do you think that students are drawn in by concrete questions? Personally, I'm not so sure. I think a lot of students could be drawn in by a question which is quite open ended, but seems somewhat obvious. These types of questions, although hard to come up with, might be one of the keys to sparking students' curiosity with the subject matter.
I really appreciate the notion of patterning and the appealing nature this provides for individuals. I completely agree with Vanessa in stating that aesthetics play a large part in the appreciation for patterning but I also think it is something that most students find a lot of success in due to the fact that they can easily identify break in pattern when it is clearly laid out. As Vanessa states, most likely because they see it everywhere in their daily lives.
DeleteI also have a great appreciation for Sophie's mention of curiosity rather than looming assessment. I believe this is a huge piece to certain areas of study. Student's love tasks that get them thinking and doing, I guess this bring in Mason's problem based learning which Vanessa mentions. The times when I have engaged my students in this way, I find them self-motivated to solve problems and ultimately holding their attention for longer durations than normal. I always give the students a chance to show off their thinking and learning at the end which ultimately encourages them to have things to show. I believe that curiosity is the ultimate attention holder and cannot replace anything else.
Vanessa and Phillipa´s comments bring up a dilemma I sometimes come up with when teaching the Sequences and Series unit. Usually, we teach arithmatic and geometric sequences, and then get into recursive sequences (where terms are defined based on previous terms). The Recursive sequences lesson is usually a pretty quick one that is skimmed over to get to focus on the formulas. This is a unfortunate as it´s really a fascinating branch of math, and a great deal of problem solving intuition can come out of getting students to mathematically articulate patterns using this method yet I have found myself guilty of skimming over it to get students to focus on mastering the formulas required for the course. Maybe students would ~pay more attention~ if we let them develop their own formulas and mathematical expressions for numerical sequences. It´s also interesting to note that these are the types of questions often appear in IQ tests and tests for government job promotions.
DeleteDavid, have you had a chance to explore the Fibonacci sequence with your students? There's a lot you could do with them while exploring recursive patterns (but not necessarily having them to come up with their own formulas and expressions?). Coming up with equations for arithmetic and geometric sequences, though, are great exercises for students - and I'd argue, given a basic foundation, can be easy to do. (On that note, the geometric series formula is a bit tricky, maybe a bonus problem for students). Even just starting with students having to add up numbers from 1 to 100 as a problem for students to figure out in groups (give them 20 minutes or so to work through it on your own) could open up a lot of avenues for students. Groups of students with whiteboards all around your room could probably do it together. Geometric series is a little harder, though, I'll say.
DeleteWhen I was typing this article, I just finished reading the 1982 article, Attention (Mason, 1982, p.21). I kept thinking if I was fully involved in this activity: composing this article. Because when I was thinking this, I was not focusing on what I was doing.
ReplyDeleteSo the answer should be, “Attention, Shan!”
Well, this was just not true. I was quite enjoying composing this article, because what I was doing, even though they were not directly related to typing/composing the article itself, helped me generate ideas and arrange logic of this article. They were inevitable and should be part of it.
So is this situation similar to students’ cases when they are in a math classroom?
I believe so. In the case that a student is in a math class, it is more or less unfair to say s/he is not concentrating on the class when s/he generates images or pictures of things outside of class. His/her experience inevitably brings him/her these thoughts, and more or less these thoughts help him/her understand the material by relating the taught knowledge to the student’s own experience. But these thoughts should be limited in a range that the student does not totally go off the track. What I understand here is what Mason (1982) says the motivation and concentration (p. 23).
Notice this article was published in 1982, and the next one, which I have not read at this moment, was published in 1989. The teaching equipment and styles of that era were significantly different from now. For now, Internet resources provide students opportunities to compensate their dis-concentration in class by reviewing the class materials after class. In this setting, the origin and reason we consider attention must be redefined, at least from the scope of this, “Why students must 100% concentrate in class?” If they are self-motivated, the reciprocal learning process, i.e., attending class (may not be fully but mostly concentrated) self-learning after class on materials missed in class ask questions base on self-learning, may be a more efficient way of learning math.
You bring up some excellent points Shan. I also agree that just because a student is not modelling the classic image of "paying attention", it does not mean that they are actually not paying attention. True, it often means that they are not paying attention, but not always. I have experienced first hand in my classroom, as well as been to keynote speeches, where different signs show that a student can be focussed on the lecture. For example, some students find eye contact uncomfortable, yet we always demand it while lecturing. If a student is forced to always look at the person who is talking to them, they may be more focussed on how uncomfortable they are, rather than on what is being said. Many students are at ease when they are drawing, or fiddling with something. These are also things that teachers tend to associate with inattentiveness. For some students, drawing or fiddling can create a sort of white noise to drown out all of the other distractions in the classroom, so that the rest of their attention can focus on the lecture. It is a fine line however, as some students can be overly consumed with the drawing/fiddling that they leave zero room to pay attention to the lecture. These are examples of what "paying attention" sometimes looks like, yet what does it sound like?
DeleteAnother classic example of what most teachers demand in their classroom is silence while working. While this may work for the type of student who also benefits from studying in a library, it does not work for all. The type of student, who later in life enjoys reading in a coffee shop, may also benefit from some white noise in the classroom. I have even read on some IEP's (individualized education plans) that it is recommended to allow the student to hum while they work. When I taught in high schools I know that whether or not students could listen to music while they worked was always an issue. Different schools had different policies, and even within a school some teachers chose to abide by the policy or not. From what I understand the entire premise of no electronics in the classroom is to stop students from texting each other, playing games, or making phone calls during class. However, if students are simply wanting to listen to music while they work, as long as the teacher does not require their attention, I don't see a problem as long as the music can't be heard by neighbouring students. I find that sounds are only ever distracting when they are not anticipated. For example, a pencil case falling off of a desk, a child with the hiccups sitting next to me, or someone falling off of their chair (it happens a lot in elementary school). If all of those unexpected sounds could be blanketed with music, or even drawing/fiddling, then perhaps the student could focus more on the task at hand.
Things are definitely changing and I find it incredible how quickly! There is always something new that students are trying to respectfully fiddle with in order to keep their attention on lessons. First were the stress balls, then the wobble cushions and now stools that encourage correct posture and gentle movement. All students are different and unfortunately it is trial and error until you find the best fit for each one. Having been the student who needs nothing going on around me, I tend to impose that on my students with classical music playing when they are doing strenuous work, in order to maintain a calm setting for them. In some years I have found that even this, which I thought was no fail, seems to create anxiety in some students who are pressured to play and practice piano all the time at home. I would definitely agree that anxiety and attention seem to be inversely related. I wonder what is going to happen as we move more and more into an instant gratification filled world, where attention spans minimize for certain things and increase for others? I always wonder what will hold students' attention next?!
DeleteDavid, I am glad that you've raised the point about drawing in the classroom. From time to time, I notice that a small number of my students draw pictures during class. In the past, I thought their drawing habit arose from a motivation-related issue and tried to work it through with them. Having had informative conversations with them, I realized that most of them used drawings as a way of maintaining their attention to their math tasks at hand. Some said that they experienced an information overload in their minds and had difficulty processing further data after 30 minute continuous concentration on problem-solving. Others stated that drawing pictures helped them incorporate their mathematical thinking into their problem-solving activities. In both cases, my students may seem to be off-task, but their drawings (which are full of geometric properties) serve as a means of redirecting their attention from one form of mathematical thinking to another. I am always surprised that they can come up with better problem-solving strategies after a temporary shift in their attention. Unfortunately, my employer does not buy into the idea that drawing keeps students' mathematical thinking active and focuses their attention on their math activities. However, I have to admit that not all students in my learning centre draw pictures for the benefits of learning mathematics.
DeleteSomewhat in this vein: does anyone have any thoughts on students falling asleep? There is a complete lack of attention in such cases- but they are `still' as Conrad mentioned above... I suspect this is mostly a symptom of large lecture halls, but the lecture itself tends to be rather snooze inducing.. I have certainly used doodling and daydreaming in order to keep myself upright in case something particularly compelling/followable came along. It's hard to pay attention in class if material is way beyond you.
DeleteSophie- it's funny that you mention falling asleep in class, as a friend and I were discussing it the other day. Personally, I've never fallen asleep in class - ever. I have an absolute fear of doing so, not just because I don't want to miss something important, but because it has always seemed upright rude to me! I can't help but laugh when I go to seminars (in the math department particularly) and there are professors asleep left and right! Even the professor that is hosting the speaker can sometimes be asleep!
DeleteI cannot recall where I read it, but there was a study where researchers looked at students pulse during a university course. During a "traditional" lecture, pulse was low and pretty constant throughout the class, whereas more interactive lectures saw more fluctuations in pulse and alertness (could we equate this with attentiveness?). This isn't entirely surprising, but it does remind us teachers that being engaging will help keep our students engaged. A completely separate issue is that a lot of students just don't get enough sleep...
In regards to Kevin's comment, I think it brings up an important point. The arts and mathematics are always considered such separate entities by students, and this genuinely makes me sad. I think that incorporating some "mathematical art" into the classroom would be a beautiful idea! Not only could it get students feeling like they can be creative when they are in math, but also shifts their attention in a positive way (as Kevin mentions). Why can't art be a mathematical activity? Both of my parents are artists, and when I was young, all I wanted to do was draw and paint. My parents took that as an opportunity to talk about line, space, and perspective; something that I think kept me very open to what I thought mathematics was.
The falling asleep issue---- this is a very common issue in China, and I personally students who sleep in my class as long as it is an occasional case. I still remember the moment when I was in a high-school music class. I really could not keep awakening at that time, and I could not help but falling asleep. What touched me till now was that my teacher, after noticing I was asleep, did not directly wake me up----instead she went close to me and put my coat on me. Unconsciously, I even pulled the coat to cover my head. At this moment, the class was bursting into laughter, and I was totally awoken but pretending to be sleeping.
DeleteWhat I am thinking here is that all these “improper” behaviors in class, lose attention, falling asleep, or anything else, are just some normal human behaviors students need to do at that time. As long as they are not in a regular base, it may be OK for teachers to accept them. At least those who present in class are those who want to be in class. Class moment is just not a very good moment for them at that time.
Well, I think in this way because I experienced an event that a teacher overly reacted to a student’s “improper” behavior. Both the teacher and the student were hurt. I am not trying to argue whether a behavior is improper or not; rather, do we have some better ways to deal with these behaviors we could not bear at that moment?
What I am thinking here is that all these “improper” behaviors in class, lose attention, falling asleep, or anything else, are just some normal human behaviors students need to do at that time. As long as they are not in a regular base, it may be OK for teachers to accept them. At least those who present in class are those who want to be in class. Class moment is just not a very good moment for them at that time.
Oh, I yearn for the days when I could tell my body not to sleep and it wouldn't. I miss it. I practically shut down at 10:30 now.
DeleteShan, couldn't agree with you more. I don't generally wake my students. I just check in with them at the end of class to ask them if they can get their notes from a friend. I never really fell asleep in high school, but I was quite adamant about getting at least eight hours of sleep. University, on the other hand..... I'm transcribing my undergrad notes now, and there are certainly some notes I can't read because of where my hand trailed off the page!
I think it depends on the layout of the class, however. One of the classes where I've been team-teaching has high stools with students constantly getting up and moving around. Classes aren't all that stimulating sometimes.... and that can really even just depend on the professor's voice (not matter how interesting the content is!). I wouldn't say that it's the teacher's or professor's duty to make class entertaining, but there's something to be said for showmanship in a lecture or presentation that makes a huge difference.
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DeleteInteresting discussion about the issue of attention in class. Mason brings this up: "One of the first functions of any teacher with a class must
Deletebe to get the students involved in some relevant task. …. It seems to me that this is really a question of focusing or attracting attention."
I think the model of focusing or attracting attention is perhaps incorrect. What is needed is engagement, not attention. Unlike Mason, I don't believe that the teacher's role is to actively focus the student's attention. Rather, it is the role of the teacher to increase student engagement. It is impossible to be engaged and not paying attention. On the other hand, a student can pay attention and be disengaged.
Right now I am a lecture TA for physics 102, and every three classes or so we have a researcher come into the class and do an "observation". The researcher records levels of student engagement/attention at 10 minute intervals, mainly by looking at what is in front of the student on the desk/table (e.g. computer open with a video game vs course notes or worksheet, etc..). The goal of the study is to see which classroom activities (worksheets, clickers, group discussion, lectures) correlate with student engagement/attention.
In an attempt to avoid long stretches of lecturing, I have taken the habit of relaying student questions to the instructor while she is lecturing. We have found (anecdotally) that when she and I have a brief conversation (with me relaying student questions) class attention is increased.
Questions for the reader:
Is there anything you do in your class that immediately draws the students' attention?
Do you allow your students to be "off-task" in your class (e.g. on Facebook)?
Do you limit the time you spend lecturing to a certain amount (e.g. max 10 minutes)?
When lecturing, do you use voice inflection, or phrases such as "Watch out for this mistake!" to draw students' attention at key points?
Alain, I think you raise a good point of teachers' role in class. I agree with you that a teacher must be the one that help students engage in class activities rather than simply draw attention. On the other hand, the questions you ask also suggest that drawing attention in class is a starting point of creating engagement.
DeleteRegarding your questions:
Q1) Is there anything you do in your class that immediately draws the students' attention?
Well, normally everything that is "out of topic" may draw students attention. For example, telling a story (no matter whether it is about the lecture topic or not), watching a video, comparing two examples, etc. But I also believe teachers should do those things with good reasons. If these are truly "out of topic", lecturing efficiency may be low. By this I reject your suggestion in Q2.
Q3) Do you limit the time you spend lecturing to a certain amount?
Yes, I do. Normally students cannot focus on class materials for a long time (20 minutes to 30 minutes?), so in a 90-minute class I prefer to divide it into 2 or 3 parts, each with a short reflective discussion session. But in university level courses I think longer period may be possible.
Q4) These changes are essential in class lecturing as they draw students' attention. Moreover, they truly tell the students some key points.
Alex, speaking of the voice of a professor, a teacher and the like, I remember having a Science teacher in grade 8. He had a "special" way of using his voice to draw his students' attention. He was very strict, demanded his students' full attention, and suppressed every disturbance he noticed. Most of us had our attention distracted by his excessive strictness and found his commanding voice somewhat rude. He never missed a day without saying, "shut up" at least 7 times when he tried to direct the noisy students' attention to his lessons. His exclamation seemed to be more distracting than my classmates' noise. His classroom mangement methods seemed to get our attention to his teaching. But, I do not know how many of us were really engaged in his lessons. In hindsight, I feel that he could have considered a more respectful way of requesting his students' attention instead of being so authoritative.
DeleteAlain, I really love your use of the word 'engagement' over 'attention'. SFU's slogan used to be 'thinking of the world', then they paid a lot of money and thought really hard and it turned into 'engaging the world'. I had always dismissed it as a bit of a lateral non-move, but there is something stronger about engagement, no?
ReplyDeleteI tend to allow students to be off task, or to at least not embarrass them or make a big deal about it. With a problems based class, its easy to sidle up to someone and pointedly ask how their work is coming along as a means of getting them more on task. That said, a while ago I was employed as a technology enforcer for someone with a philosophy more similar to Kevin's science teacher. It was my job to make sure students with technology sat in the back row, and if they did not comply, to ask them to leave the class. Not a job that came easily to me, I don't think I was ever so sweaty. It created an atmosphere I will not been keen to replicate of 'us vs them' and I question whether that cost was worth the added attention that might have been gained. That's the thing, right? It's one thing to distract yourself, it's a whole other to siphon off the attention of your classmates. I don't have an easy answer, but I have found that keeping students engaged in class time through very limited lecturing tends to limit such instances.
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ReplyDeleteIn response to Alain's prompt about maintaining attention and engagement with respect to Facebook, I've been most successful in classes where I use cellphones in the lesson for feedback, research, etc for a very specific task. One of my favorite resources is Socrative, which is an online (free! really!) program that allows students to text-in (via Wifi connection) answers to quizzes available through the website. Students are stuck on the page answering questions and can't exit out of the app to do other things until they're done.
ReplyDeleteMaintaining attention with my students is easy to regulate. If I'm bored with my own teaching and voice, so are they. I'm generally quite conscious about my tone and student energy/interaction, but sometimes I tend to get focused on the lesson.
On the note about learning and the shift of attention in Mason's article, there is much discussion on the attention paid by students. Does this type of attention change? Is our type of learning changing now? How does the shift of attention in teachers when re-learning concepts, for example, come into play? Is the success of learning based on this shift of attention?