I was inspired to work with Chantelle Aquino to present the “Mindfulness: Its not what you think!” workshop at our Physical Education Conference mainly for two reasons. First, I have experienced the benefits of incorporating a mindful presence in my own life and take any opportunity to share those benefits with others. Second, I wanted to support Chantelle in her discovery of this way of being by co-facilitating a meditation practice that is effective in creating a more mindful presence. We operated from the definition that mindfulness is the intentional, accepting and non-judgmental focus of one’s attention on the emotions, thoughts and sensations occurring in the present moment. We introduced our concept—that mindfulness is not what you think—with an excerpt from Jon Kabat-Zinn’s “Coming to Our Senses: Healing Ourselves and the World Through Mindfulness” that concisely explains that meditation is a way of being in relationship to the present moment and to one’s own mind and experience. Our poster-board included a summary of the benefits including the role of mindfulness to build our capacity to recognize scripts and disengage from our scripted emotional tendencies. By developing mindsight we are able to observe thought processes and the role they play in reinforcing reactivity and associated scripts that distort reality. With this kind of awareness about ourselves, we become better able to observe our students’ feelings and behaviours more objectively and thus to respond with respect and acceptance. We conducted a 20 minute guided meditation that focused on the breath as an anchor to the present moment. It was a challenging session because of the noise next door in the weight room, nevertheless a fantastic practice and opportunity to watch the mind attempt to reject the loud sounds. After the meditation we asked participants to share their experiences. Based on what was shared Chantelle and I felt we had succeeded in dispelling the myth that mediation practice required having a quiet mind, and in instilling the concept that meditation is a practice that asks one to continue to return to an anchor point (the breath in this case) over and over again as soon as one found the mind wandering. It was a wonderful session. I recently read a book entitled, "Mindfulness for Teachers" by Patricia A. Jennings. Mindfulness isn’t a new concept to me, and yet it was still valuable to encounter it in this framework for teachers. I am so happy that the practice is being noticed for how useful it is and that the mystery surrounding its “esoteric” Eastern beginnings is being secularized. Everyone can benefit from this worthy pursuit, especially teachers for whom reflection should be an integral aspect of their practice. There are many free podcasts that offer guided meditations. One of my personal favourites is a podcast found through itunes by Tara Brach. She also has a website on which her podcast can be found. Follow the link below to visit her site.
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Stephen Covey’s “The 7 Habits of Highly Effective People” offers practical advice for succeeding in life. In short, the 7 habits are as follows: 1) Be proactive; 2)Begin with the end in mind; 3) Put first things first; 4) Think win-win; 5) Seek first to understand, then to be understood; 6) Synergize; 7) Sharpen the saw. Overall I agree with Covey’s notions that the way we see the world is based on our own perceptions and therefore the only way to change a situation is to change our perceptions. I would add that perceptions stem from beliefs; note that this idea may be covered in the book, but I only read the much abbreviated (yet very informative) summary of Covey’s book (a button link to said summary can be found at the bottom of this blog). Covey’s depiction of the evolution of society’s definition of “success” is intriguing (perhaps devolution is a more appropriate term). It appears that the shift from an ethic of “character”—where success hinged on integrity, honesty, courage, justice, and the like—to an ethic of “personality”—where public image, attitude and personality determines success—correlates to the success of the film and television age. In the 1920s the United States was producing more than 500 films per year. Again, perhaps this correlation is mentioned in the book. Regardless, it appears to mark a change in consciousness from one that is reflective and inward looking, to one that is projective, or outward looking. Covey appears to address this paradigm through the 7 habits since each habit focuses on achieving a deeper level of self-change. From the website:
The habit I have chosen to focus on for this blog is the first one: Be Proactive. Perhaps my disposition has carried over from a past life as a Delphic Priestess, regardless I believe it is of utmost importance to “know thyself.” In my opinion this habit is the foundation upon which all other habits rest and for this reason it is the most important. Reactive people see the problem as being outside of themselves and in this way feel victimized by the ways of the world. Alternatively, proactive people recognize they have responsibility. Proactive people are able to see choices: choices for attitude, choices for action, choices for responding and/or not responding. Much of the ability to choose lies with the focus point. If we are focused on areas we cannot change (namely those outside of ourselves) we begin to blame and emanate negative energy. Blaming demonstrates that we believe that others have power over us. This kind of belief system is often deeply embedded in our consciousness. Depending on our upbringing we may have held this belief since we were two years of age. I point this out because it is important to realize that a shift in thinking—in beliefs—doesn’t usually happen over night. It requires time and effort and that is after the recognition that the belief exists in the first place. It takes courage to face the idea that deep down we don’t feel in control of our lives, that we blame others for our circumstances and in so doing give our power away. In pointing to the damaging quality of beliefs we do not address Rilke said, “Where I am folded in upon myself, there I am a lie.” Of course, we are only trying to protect ourselves. What we don’t realize is the falsity of this kind of protection. Hiding behind a wall of blame and lashing out with reactive action or language, both internally and externally, only causes more pain and suffering. It might be true that our society hasn’t done a great job at modelling for us how to focus on what we can control. We have been inculcated, to a degree, to expect convenience, to shun effort and, yet, to feel entitled to (fill in the blank). In general, we have misplaced those values that nurture intrinsic worth. By focusing on the first habit of mind—be proactive—at home, in school, in the community, and in the workplace we can begin to see where we have influence. To get there we must implement self-reflective practices. We may require support in the form of therapy, exercise, self-help books and the like to help us break down old belief systems and being to see how to control our own feelings and actions. Once we become adept at this we begin to serve as models to others whether we know it or not. As a teacher I feel it is very important to embody this habit. With effort and practice one begins operating automatically from this place; the habit becomes embedded in one’s pedagogy and models to young people how to feel in control of one’s thoughts, emotions, ethics and actions. To be proactive involves critical and creative thinking, attends to both social and personal responsibilities and aids in effective communication. In this way, the habit helps develop the curricular core competencies that are so important in helping people be successful learners and teachers. Because we are all teachers and we are all learners working together for the success of all. The video I’ve linked at the bottom of this blog shares a deeper-learning program that successfully operates at a US school. The overall, nutshell message is that if we believe in the capacity of students to achieve greatness they will be great. One way to push students' capacity for learning is through challenge and struggle. (I personally don’t agree with the word choice of “struggle” because to me that pushes so far beyond the comfort zone it becomes too hard to succeed; however, “challenge” I agree with). Students can be challenged with content material that requires deep levels of focus, contemplation, involvement, and the capacity to sit with ambiguity. They can also be challenged by processes such as group dialogue, debate, and creative means to demonstrate their learning that goes beyond simple regurgitation of facts and builds courage and confidence. Challenge begins to extend to all facets of life as students begin to grow meta-cognitively and challenge themselves. For instance, one student in the video shared his story of perseverance as it applied to his running; he affirmed the connection between his scholarly perseverance and his physical training, pointing to the underlying mindset, or belief, that he could go further in any endeavour to which he was engaged. I, too, have experience with endurance growing into a mindset that thinks anything is possible. Thus, in general, I agree that challenges help people grow as long as they aren’t consistently scaffolding for failure. I think it is important to remember that every individual will respond to various challenges differently and in order to scaffold for success these differences must be taken into account. A teacher who demands, for instance, that an extreme introvert stand in front of the class to lead a lesson on a topic they only just started, would be scaffolding for failure. This may seem like an obvious point, but I have witnessed a similar scenario. My point is that when presenting challenges, teachers must observe closely and know their students well in order to support them to be successful. Respectful relationships are therefore crucial. There are many practical ways that traditional systems can include a challenge program. Debates, for instance, that are set up in a way that each student must argue in rotation for both sides of the debate in order to see both perspectives deeply. Other forms of deep learning that challenge the students on many levels include cross-curricular projects that culminate in a project fair for presentation to another classroom or school, collaborative art projects that involve a contribution to the community such as a mural, dialogues seminar style where each student is responsible for tabling a point of interest and a question, group chats that involve the sharing of both personal information/feelings and academic goals/challenges etc (referred to as “crew structure” in video). Activities that encourage critical thinking such as student-lead experiments, and activities that provoke thinking about thinking such as reflective journals & guided meditation are both important to include as often as possible. I realize that integrating the concept of challenge for deeper learning into my own practice is equally as important as facilitating it in students. This requires similar if not the same techniques; through community involvement, collaborative projects that are creative, fun and purposeful and self reflective practices such as meditation I am able to grow my capacity for comfort with ambiguity, my courage, my confidence, my empathy and my ability to support others to do the same. Secret lair: a metaphor for the processes and activities that help a person survive and thrive in the challenges of teaching. I think of the secret lair as a place I can visit wherever and whenever I choose. The activities and/or mindset I employ differ from time to time based on my needs and the opportunities available, but I’ve come to appreciate some activities/mindsets more than others. Perhaps it is a combination of their usefulness along with my familiarity and adeptness with them that keeps me returning to the same trusted pursuits. Every now and then I think of (or am introduced to) another approach. Being the natural adventurer that I am, I’m usually happy to try something new. Practice has taught me that participating in new experiences, or diverging from routine, promotes creativity. For this reason, don’t be surprised if you see me juggling porcelain dolls as I skip around the local walkway. In no particular order, these are the pursuits I include in my secret lair: Talking it out with my partner, daughter or a friend—this includes debriefing the day to reflect on what did or did not work and the person to whom I speak may determines the content of the talk. Sometimes this a rant, but more often it is a serious quest into how I could improve a strategy, a relationship, and any number of things. Writing it out—this can be everything from goal oriented to-do lists compiled in a daybook, to journal entries and letters to the editor, to short fiction to poetry, essays and letters to a friend. Sometimes even just writing the events of the day is a way to let go of the past just like planning for the next day is a way to let go of anxiety or anticipation of the future. Date night—this can be done alone or with my partner/friend/daughter and it involves ordering food I wouldn’t normally cook myself and savouring every bite. This kind of celebration is somewhat of a reward after a long week or completion of a project. It doesn’t always include a special drink or dessert, but it can! Experience has taught me how important it is to celebrate my successes and accomplishments. Those don’t come easy, countless hours of hard work go into my successes and so taking a break from that hard work to celebrate is an important balancing mechanism! Of course it doesn’t have to be limited to supper. It can be a movie, a show, a concert, even a trip. After 4 long years of dedication to completing a degree I treated myself to a month long trip to Italy to visit all the places I’d read about and see, first hand, all the art I studied. That trip, though it put me in debt, was one of the most rewarding experiences of my life since it helped to consolidate much of the learning I had done! Reminiscing—often I enjoy watching my screen saver flash photos of past trips I’ve taken. Seeing a photo can take me right back to the place I visited and remind me, for instance, of what it was like to feel the course hair on the back of an elephant against my legs, how different ‘down-ward facing dog’ (a yoga pose) feels in a rural yet tropical outdoor setting. These memories may take me out of the present, but they are so vivid it is like a break from reality—and that can be just what the doctor ordered sometimes. Mindfulness meditation—on the opposite end of the spectrum, mindful meditation techniques help me become centered in the present. No denying what is, no rejecting what is, just being, sitting (or standing), breathing in the present moment and feeling the air softly swirl on the surface of my skin. I practice this as much as possible during the day when I must wait for something… a free washroom, a bank machine, to pay for groceries, or sleep to come. This exercise has potential to be the most enlivening of them all. Exercise is also enlivening! A brisk walk provides a much-needed refresher after an extended period of time spent in deep focus. Sometimes I get too close to an event to be able to think differently, to discover the vitality in the situation. A brisk walk out-of-doors with dogs running up to sniff and smile, the ducks quacking in the distance and friendly others passing by on their own walks is enough to stimulate new energy surrounding what ever I happened to be working on before the walk. Personal Development—this includes delving into the endless areas that interest me. I read/listen to books voraciously. With a glance at one of many bookshelves in my house I see books on brain science, on quantum energy, on mindfulness, on creativity, on art, on poetry, on gardening, philosophy books, cook books, language learning systems, novels, religious texts, books on fairies, gnomes and architecture. I may well have the most eclectic collection this side of Kansas. I also enjoy music: seeing it, hearing it, and taking lessons when money and time allow for it (I was taking piano lessons until a year ago). I want to learn the banjo next. The list goes on. I take any classes I can afford: clay, dancing, Jungian analysis, and drawing are a few I’ve done in the past. As a learner and a teacher, I know the value of having broad interests and going deep with them. I am constantly seeing new connections and building on past knowledge to create new knowledge, meanings, feelings and experiences. It is this zest for life, this insatiable wonder and curiosity that keep me from getting too serious. My hilarious friends help in that department also. Fortunately I don’t have to work too hard at developing a secret lair, for mine comes as naturally as nourishment: when I'm hungry, I eat! Technology fundamentally transforms a classroom; I don’t think this can be disputed. But I do wonder if it transforms a classroom in a positive way. I am not, as you may have deduced by now, a gung-ho tech-savvy sailor. Yet, I am not completely opposed to technology. I believe it has its usefulness by giving students more options in how they learn and express their learning. I can see technology being beneficial also by helping to make a lesson better: perhaps creating more interesting and interactive demonstrations. Technology provides tools that we can use, much like a ladder helps us to climb up to a place we couldn’t otherwise reach. I am concerned, however, about technology becoming a crutch for student engagement whereby the teacher fails to be able to effectively communicate directly. In this era of television, smart phone and computer screens, technology has become a danger in that it has infiltrated every aspect of life. You don't see everyone walking around with a ladder in their back pocket for reaching those unstep-up-able heights, but almost everyone has a device of some kind on their person. It is my observation that people are becoming more and more reliant on texts/messages and photos for communication. We aren’t getting as much eye-to-eye contact or even oral communication. The more we physically distance ourselves from each other, the more we become unable to read each others’ feelings and so we become prone to alienation. There are all kinds of studies demonstrating that children are missing out on important developmental milestones because their parents are more engaged with their devices than they are with their kids. eg: http://www.npr.org/sections/health-shots/2014/04/21/304196338/for-the-childrens-sake-put-down-that-smartphone How unhuman can we get!? We are social beings and like all social animals we require physical interaction with others to maintain a healthy system. Sure students are engaged in school work when they are creating using digital tools to make all kinds of podcasts and vlogs and experimenting with robotics, photo-shop and music programs etc., but in my opinion the analog tools are just as important to incorporate and practice to carry students to their full potential as citizens contributing their gifts to the world. Creating things does allow students to take responsibility for their learning and sharing these things on the internet provides an authentic audience so motivation is built in to many tech-based projects. However, I feel that motivation is an intrinsic force when completing any project that a person is invested in on a personal level. Therefore, if students aren’t interested in, say, blogging their summer vacation, should they be required to blog if they would rather write a poem, or compile a hard-copy photo-album. For me the answer is no. Students should let their interests guide their projects. There is no doubt that the future is going to demand tech knowledge and it seems obvious that students will need these skills more than they need to know ‘stuff’. Stuff can be accessed anytime, anywhere, as long as you know how and have the technology to do so. In a world that has grown exponentially in the last 100 years, there is more stuff to know than is even possible to teach over the course of the 12 years of school. Therefore, integrating technology has the potential to build important skills and expose students to equipment that will be required in the future job force. I’m not sure if it is necessary or even possible to integrate technology when it comes to teaching equally important future life skills such as critical thinking, discernment, and dexterity, along with the social human life skills of compassion, cooperation, perseverance and patience. I don't think its entirely unfair to to attribute cause to computers (and the ease of access to information) for the effect of the current populace's impatience, lack of discernment/critical thinking, and inability to express and intuit feelings. So while I am not opposed its use, I feel that technology needs to be balanced right along with our nutritional, physical, spiritual (connection to natural world), personal and social needs. As teachers we have to think about how much technology a child is getting in the home and then we need to balance this in our classroom. Perhaps some students need experience with tablets, for instance, more than others. In any case, by modelling balance in the classroom and careful consideration as to when to integrate technology and when not to, we have the opportunity to teach our students self-regulation and the equal importance and beauty of all the other aspects of life. During my investigation into why students dislike math I discovered that one of the main reasons for the hatred is that they feel incompetent. Understandable – no one likes to do something for which they lack the skills. The reasons underlying the incompetence are vast: one possible explanation is that much of math is not intuitive due to the lack of evolutionary need for it, and yet we have evolved in our acquired math capacity exponentially (pun intended). Another possibility involves the way math is taught. There seems to have been two opposing approaches to math education that put more effort into warring each other for the victory crown than attempting to understand what learning development and research says about the worth of their strategies. Turns out that the hands-on conceptual approach and the procedural approach that includes rote memorization are equally important and work together to effectively teach math skills. Imagine that?!? All facetiousness aside, I unearthed a further factor contributing to math competency. Since mathematics knowledge depends upon cumulative gains, it turns out that children who are exposed to “more numeracy-related activities at home show greater proficiency at [school].” In a study by LeFevre et al., parental “reports of numeracy activities were correlated with their child’s math performance.” The study looked at the correlations between activities like card games that directly involved skills such as counting and recognizing digits and activities such as baking that indirectly involve numeracy skills. The findings reinforced the hypothesis that the prevalence of direct and indirect numeracy activities at home is related to children’s “fluency with basic numerical skills, such as addition or number-line knowledge.” An interesting finding of the study is that a child’s “involvement in games predicted unique variability of the addition fluency measure,” that is, these children showed “substantial gains in their knowledge of number and magnitude.” While I don’t mean to pretend that certain math knowledge (such as LCM) is not important, the implications of this study with regards to what I witnessed in the elementary school classroom, are twofold. First, the students that I worked with must not have had the opportunity to engage in games and other numeracy activities at home for their unfamiliarity with numbers and basic math skills was obvious. (I blame it on television, but that's another blog.) Secondly, if games are a valuable tool for imparting math knowledge, teaching the class how to play cribbage and involving everyone in a round robin tournament may be a much more effective way to spend math class with a group of beginners than asking the same students to answer what is the lowest common multiple between two numbers on the chalk board. It is important to remember that as their teachers we must make the best choice for the direction of their learning. If they are two or more steps behind, it makes no sense to try to go forward. Playing a card game such as cribbage is a fun and sneaky way to slip math concepts into a child’s frame of reference! Something to think about anyway! References LeFevre, J., Skwarchuk, S., Smith-Chant, B. L., Fast, L., & Kamawar, D. (2009). Home numeracy experiences and children's math performance in the early school years. Canadian Journal Of Behavioural Science, 41(2), 55-66. doi:10.1037/a0014532 According to Dr. Daniel Ansari, there has been a fearsome ongoing debate about the most effective way to teach math—rote learning tactics versus discovery-based strategies—that has failed to take into account the vast research findings about how children learn math.
The side that advocates “procedural knowledge” emphasizes explicit teaching of strategies and encourages students to memorize facts while the side advocating “conceptual knowledge” focuses on student construction of knowledge through hands-on materials, strategic-invention, and solving of open-ended questions that do not involve memorization. Neither side, in this polarized and emotional debate, has any use for the other. Keeping the previous blog post in mind (that some mathematical procedures are not intuitive and the subsequent symbols are not mental representations), it would seem that conceptual knowledge could only carry us so far. It seems we require the environment (in this case the math symbols as objects and procedures) to carry some of the cognitive weight. But any observer in today’s classroom can witness the limitations of rote memorization as well. Indeed, Daniel Ansari’s investigation found that children learn best when procedural and conceptual approaches are combined. “Researchers [like Bethany Rittle-Johnson] have demonstrated that an effective use of instructional time in math education involves alternation of lessons focused on concepts with those concentrated on instructing students on procedures.” It turns out that different parts of the brain are used in the various methods such that both approaches are interrelated and mutually determine successful outcomes in math acquisition. One of the specific arguments by the discovery-based camp targets time limits in math education: they oppose tactics such as the commonly known as “mad-minutes” wherein students must calculate multiplication products quickly. Ansari points out that there is no evidence that “speeded instruction necessarily has negative consequences.” In a study to which Ansari contributed, he found that young adults who performed better on high school math tests had more areas of the brain active: specifically, the fact retrieval area located in the left hemisphere was active whereas the students who achieved lower scores “recruited brain regions associated with less efficient strategies, such as counting and decomposition in areas of the right parietal cortex.” The “data suggests that math fluency and its neural correlates contribute to higher-level math abilities.” Since speeded practice in combination with other approaches results in “larger gains” it is thus useful for helping “low-achieving students in overcoming their math reasoning difficulties.” With his emphasis on research-based evidence, Ansari advises us to consider what is appropriate for the developmental level of the student when considering the sequence and content of learning. He leaves us with this important reminder: “[l]earning math is a cumulative process—early skills build on the foundations for later abilities.” If students enter the classroom without the necessary background in mathematics, if numbers and patterns are largely unfamiliar to students, then we must go back to square one with them. The next blog takes a peak at what that might look like in the classroom. References Ansari, D. (2015). No more math wars: An evidence-based, developmental perspective on math education. Education Canada, 55 (3), 16-19. Are mathematical symbols merely symbolic notations that exist separately from the abstract mathematical truths that underlie their actions? Some would say yes, that mathematical cognition occurs after we have converted the symbols into meaning, a meaning that exists outside of space-time, and a meaning that is as equally contained in an ancient mathematical manuscript as it is in a recent explication. Others think that perhaps mathematical symbols do something more than simply express mathematical concepts: that they “enable us to perform mathematical operations that we would not be able to do in the mind alone.” By assuming part of the cognitive load, mathematical symbols become “epistemic actions.” From the perspective of a mathematics philosophy, a causal account of how we obtain math knowledge is complex. Elementary numerical knowledge represents our ability to perceive natural numbers. From a young age we can see when there is more or less of something. Likewise we can “match the number of voices [to] the number of people speaking,” or the number of items in a hand to the number of items on a screen. However, many mathematical truths don’t commonly originate from direct sensory observation. Rather, they are attained from cultural sources. The mechanisms taught require cognitive reconstruction in order to understand the abstract structures underlying the concept. In many instances, mathematical mechanisms are not closely matched to our intuitions. For this reason we must build an entire mental framework around math concepts in order to make sense of them. An obvious example can be found in the number zero which equals both nothing and something at the same time. On its own it equals nothing. As a placeholder it means much more. All the properties of zero--for instance that it is the neutral element in addition (1 + 0 = 1), or that any number elevated to zero potency equals one, or 0! = 1! = 1--are not inferable. One of the ways we support mathematical cognition is through external devices such as calculators and slide rulers. Any device outside of the human that assumes some of the cognitive load is an epistemic action in that its “primary goal is to obtain information about the world.” When coins are sorted by value and stacked in tens, for instance, it is much easier to count them than if they are in a disorganized pile. By designating part of the task to the environment in this way, performance improves. In comparison, a notebook serves as an external memory device for someone with short-term memory loss. The extended mind thesis presents the notion of active externalism wherein objects within the environment function as part of the mind. The mind and the environment are a coupled system that function together with the same purpose. According to De Cruz and De Smedt, “[a] way to interpret the extended mind without contributing cognition and agency to artifacts is to argue that not all concepts are mental representations. One must then suppose that not all concepts can be entertained by human minds, due to intrinsic limitations of human cognition.” Using the example of ultraviolet light, which humans cannot perceive due to our lack of receptor cells for ultraviolet light, De Cruz and De Smedt show that we can entertain the concept of ultraviolet light without actually having a mental representation of it because of the special instruments we’ve developed to capture it. Similarly, “mathematical symbols can represent objects that are not representable with our internal cognitive resources alone [and this implies] that not all mathematical concepts are mental representations.” Through their research De Cruz and De Smedt found that the use of symbols is cognitively demanding on students despite their familiarity with symbolic notation systems. They prefer to use lengthy descriptions that imagine particular situations to solve problems such as “how to obtain the number of girls in a class when the number of boys is known, and you know that boys outnumber girls by four.” They also discovered difficulty for students shifting from one symbolic representation to another. Students consistently failed to represent 1+7 in blocks; rather than laying them out in two separate groups one of one block and the other with seven blocks, they arrange the blocks such that they spell 1+7 by their configuration. Students require explicit instruction to use the symbolic objects successfully. To understand symbols requires that we “decouple meaning from materiality.” While this is a process that occurs early on in development—by the age of two most children can discriminate between representational objects (e.g. a photo of a bottle of milk) and the actual object—the parts of the brain occupied with this endeavor differs from task to task and culture to culture based on the way the symbolic systems were learned to begin with. Chinese-speaking students who learned math with an abacus rely more on motor-related brain areas when performing mathematical tasks than English natives whose language-related brain areas are more active during the same task. Likewise, there are differences in cognitive organization between literate and illiterate people. Brains of illiterate people show more white matter, the function of which is “to connect different functional areas of the nervous system [indicating] that literate people have less connections in their brain.” De Cruz and De Smedt explain that literate people can afford to forget since they can use daybooks, notes and the like to store memory. Mathematical symbols refer both to procedures (the value of pi is the ratio of a circle’s circumference to its diameter) and to objects (structurally π represents 3.14159265…). Studies indicate that the process of moving from procedure to mathematical object occurs in students as they learn new math skills. In a study of adolescents’ working with negative numbers, many mistakes were made due to “a studious application of rules, rather than an intrinsic understanding of negative numerosities; [thus] the best predictor of success in individual students was their ability to use the minus sign correctly, not their conceptual understanding. Adolescents aren’t the only ones struggling with negative numbers. As it happens, negative numbers stumped mathematicians for eons and were largely rejected until well into the 1800s. De Cruz suggests that negative numbers are non-intuitive because they lack evolutionary import; “there seems to be no compelling reason why natural selection should have equipped [us] to deal with negative quantities.” Thoughts such as these just wouldn’t endure without mathematical symbols to carry them, for the demand on our cognitive resources and attention is great in the course of the problem solving and cultural conditioning that occurs moment to moment in life. Could it be that students find math difficult for the simple reason that we weren't designed for it? Perhaps we didn't evolve to intuit higher mathematics equations, but when we look how far we've come along in our understanding of the universe through mathematics we cannot deny that something has evolved: perhaps it is our capacity for evoking mental representations from symbols. One thing is for certain, we didn't come to this kind of understanding overnight. Understanding mathematics requires skillful integration of a variety of innate and acquired numerical processes beginning at the simplest level. Therefore, the next step in understanding why students perceive math as difficult involves looking at the ways in which math is taught. References De Cruz, H. H., & De Smedt, J. (2013). Mathematical symbols as epistemic actions. Synthese, 190(1), 3-19. During a recent shared learning experience in a grade 6/7 class, I was discouraged by the number of students who moaned and groaned when asked to take out their math books in preparation for a lesson. My position in the class gave me the flexibility of working one-on-one with several students and I took advantage of this opportunity to try to discover what exactly was underlying the contempt. I was surprised by the certainty with which students asserted their math incompetence. They hated math because they “couldn’t do it.” Since I relate to the feeling of not wanting to work on something at which I am consistently unsuccessful, the explanation made sense to me. But why can't they do it? What is so difficult? Taken in small steps, math logically unfolds. Or does it? During the first lesson, the students were supposed to be working on lowest common multiples (LCM). The student with whom I worked closely wasn’t very familiar with the concept of multiplication, so the idea that some numbers shared something in common to do with multiplication confused her even more. I ended up spending the entire lesson one-on-one with her, first reviewing the idea of multiplication through arrays, and then working on a multiplication chart in order to show her how she might use the chart as a tool to help with LCM. Later that week, in considering why math is difficult for students, I read several articles which shed light on the topic in different ways. For the next few entries of my math blog I would like to share these ideas. The first article suggested that the trouble might lie in the fact that many math concepts aren’t easily translated into mental representations (this idea will be fleshed out further on). The article didn’t dig into the depth of brain development research, so there wasn’t a focus on the relationship between formative development and math readiness. However, due to my own experience with mathematics, both personally and throughout my daughter’s course of learning, I am convinced that math readiness is an important consideration in the elementary classroom where learners possess a wide range of differences in learning styles, numeracy familiarity (previous knowledge) and levels of cognitive development. Perhaps this is an obvious reflection, yet the observations I have made in recent classrooms confirm my suspicions that math readiness is often unaccounted for—that is to say, it appeared that all students were expected to be able to do the work presented in the lesson, and yet they were not all able to complete the tasks. I realize the issue is more complex than student readiness; there are also considerations such as how well the teacher knows the subject, and how thoroughly it is taught. The second article speaks to the so-called “math wars,” a debate between “procedural knowledge” and “conceptual knowledge” as the most effective approach for math instruction, offering an evidence-based, developmental perspective on math education. The third article presents evidence that early home experiences that expose children to quantitative activities and indirect numeracy activities (such as card games) provide a foundation for better acquisition of mathematics in school. The study thus implies that students who may have missed out on home numeracy experiences could benefit from similar types of experiences in school. Perhaps, playing cards instead of learning about lowest common multiples will be a better use of time for the development of numeracy skills. Before we come to any definitive conclusions, we should look a little closer; please follow my math blog to see how the evidence unfolds. After watching a video about teacher stress I came up with a few ideas that might make teaching easier. But of course, I’m still in my anticipatory phase, which means that I’m a shameless idealist, an excited romantic. I wasn’t surprised to discover that novice teachers experience various phases during their first year of teaching that move from anticipation and excitement to the daily struggles of survival mode followed by utter disillusionment. Having had a variety of life experiences, this sounds fairly par for the course. Especially if one has great expectations about the endeavor she is about to embark on. Herein lies the key to avoiding disillusionment. Have no expectations. Easier said than done, but worth a try! Also worth remembering is the lesson that Pip learned: kindness and conscience are more important than scholarship (props to Charles Dickens). But wait, there’s more to the phases and they go like this: if coping strategies are properly attended to, novice teachers can bounce back from the depths of despair feeling rejuvenated, ready to reflect on mistakes and problems which will be followed by another anticipation phase that looks forward to a new year: one that will be faced with new coping and planning strategies. (Cue the sound effect: needle scrapes across the record.) Since I am still in my first anticipatory phase, the one characteristic of brash assertions and risky system-bucking behaviors, I propose that the upward turn of phase in this case, is due to obsequious adherence to convention. Rejuvenation thus stems from the hope that conformity will yield better crops; and reflections consist of gazing over the map in order to avoid the same dirt paths and dead ends one might have been lucky enough to avoid in the first place if only… But I ask you, what place does convention hold in our education systems? Is it our fate to repeat over and over the failures of the past? I don’t disagree that acquiescence may have its place in the overall scheme, even Galileo had to admit he was ‘wrong,’ but the matter is delicate, and these children are the future. Most of what I have just proclaimed may have sounded inconceivably conspicuous, and that is as it should be. For now, let us move on to my first brilliant idea (one of many more to come) that is both ahead of and behind my time. I speculate that much of the frustration that teachers experience stems from year after year of new classes full of new students all of whom arrive with different learning levels and abilities that they must get to know before they can help. It takes an enormous amount of time and energy to build the kind of trust, understanding and emotional bonds that facilitate real growth in learning. The longer this takes, the less time a child has to do the serious work of progressing along the learning continuum. Therefore, my proposal for this blog—based on the intuitive thoughts that rose to the surface of my cerebral cortex during the video—is that teachers continue to teach the same students year after year, grade after grade. (Ideally we would drop the whole “grade” thang too, but for now I’ll stick to one suggestion for easing the novice teacher into her vocation.) By growing and progressing from grade to grade right along with the students, teachers would have the opportunity to best manage their classrooms, develop relationships, understand student needs and be innovative with lessons. This strategy is called “looping” in some progressive circles and, if I remember correctly, it was regularly practiced eons ago in the first one-room schoolhouses that speckled the land. The classroom community would evolve and change, yet connections would be easier to facilitate and work would be most efficient. The main downfall of this restructure, as I see it, is the risk that teachers and students would become patterned. That is to say, familiarity breeds contempt. It is sometimes the case that we become “comfortable” in our relationships and/or settings. Often we strive for the comfort of the habitual only to become bored with certainty and its stagnant waters. Without awareness we can end up typecasting based on preconceived notions that can be harmful to others and ourselves. However, a well-trained teacher knows how to avoid this pitfall. She knows that the boundaries of a person are limitless and that everyday we can come face to face with riddles that force us to change the way we think. A teacher well trained in the art of living and teaching, knows the value of ambiguity and spontaneity and she uses tools like these regularly. |
AuthorNatalie Nickerson; that's me. Archives
March 2016
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