More responses to “Math is not linear”

Picking up where I left off… Now looking at the “What Can I Do?” section.

“Go on tangents”: I agree that occasional purely random tangents are fun and worthwhile. But on the whole I prefer being able to make meaningful connections to the core material of whatever class I’m teaching. I don’t want to make math seem like a random jumble of cool, but not particularly understandable, facts. So the kinds of tangents she describes need to not take over the flow of the course. And of course the specific example she describes (introducing the hyperreals of nonstandard analysis) is not an option for most teachers.

“Foreshadow”: I definitely agree with this suggestion, as it fulfills two goals: first, to ask challenging questions that motivate later material, and second, to make the connections that show students the coherence of the subject.

But it does bring up a tricky point. For example, we spend a good deal of time in calculus using derivatives (and algebra) to find the maxima and minima of functions. If that were the first time we ever tried to do such a thing, it might seem that we have found a “killer app” for calculus. But it’s definitely not the first time we find maxima and minima, since those are important, intuitive questions about a function, that are easily answered (roughly) by looking at a graph or (pretty accurately, but not exactly) with a few buttons on a graphing calculator. So the problem is, how do we motivate using calculus to do max/min?

The essential point is that the teacher needs to understand why it really is useful to use calculus to do max/min. Given that calculators and computers exist, it’s not because calculus will find the max of a single explicit function. It’s because calculus gives you a way to understand max/min which leads to general formulas (for whole families of functions and examples), to important (and often simple) theorems and scientific principles (e.g. Snell’s Law), and to generalizations where calculators are less useful (e.g. higher-dimensional max/min). These are all higher-order reasons, and require a fair amount of knowledge on the part of the teacher; they are also challenging to communicate to students, but honesty requires us to try.

“Relate material back to previous classes”: more coherence in the curriculum is always desirable. And isolated review problems, disconnected from the current topic, are not a good way to do that (even though they are easy to plug in without much effort). One of my goals is to keep refining questions to bring in, in a natural, interesting way, previous material that isn’t getting reviewed already. It does involve real work to do that!

“Be less helpful” (a la Dan Mayer): Absolutely. I got this notion first from Dan Teague at NCSSM, who didn’t use this phrase, but the “less helpful” motto is short and sweet. Don’t sell students short. Don’t hold their hand when they don’t need it. Teach higher-order thinking by requiring higher-order thinking before they even start the problem. Teach formulating problems, communicating math, and using common sense. All by just not laying it all out for them.

“Teach an elective”: This goes back to the dilemma I mentioned in the previous post about highly advanced students. I’d love to teach a discrete math course, for example (and I may get to do that someday in independent study settings). Our school does teach stats, which is absolutely right on target, and which many students take. So we do some of this, but it is hard for students to make time. What I really don’t want to see is students spreading themselves too thin and getting a superficial treatment of too many subjects. I’m a big fan of depth over breadth, so I think students (particularly at a place like where I am, where we have many, many electives when you include all disciplines) take too many courses, and get frazzled, when they could be calming down, and focusing on fewer things.

For example, we have had a logic elective in the past, and it was a great course, taught by an expert in the subject. But recently we needed to bump it as a result of a decision to do a slower, more thorough treatment of multivariable calculus, which ate up another semester. Alison Blank might decry this choice as favoring a “sequential” subject in favor of a more diverse elective, and I can see her point. But multivariable calculus is amazing math, no less than is logic, and we wanted to make sure that students taking the multivariable class would actually be able to learn it—something that few students do in a one-semester college course in the subject. And while the mode of thinking you get from a logic class is very valuable, there aren’t many courses that treat it as a prerequisite, as opposed to the many that require multivariable calc. So the calc class opens more doors for further study—not “further progress along a boring track”, further really cool math.

I don’t think that either way is necessarily obviously best, and I really do see the appeal of broader electives, where possible. But there are arguments both ways.

“Start a math circle”: Absolutely. I love helping out with the local equivalent of a math circle, and I actually would like to see it move in a more diverse and eclectic direction. I also help students practice for math contests, which, while it has its own narrowness issues, does expose students to real problem-solving and different ideas from the regular curriculum. If it were purely up to me I would move that in a more “circle” direction as well, but the kids really enjoy doing well on contests, and I don’t mind supporting that.

“Talk about math”: I wish I did this more in casual conversation. I love being asked about math by someone with genuine interest in it, and I’ll take the slightest excuse to go on and on. But I don’t usually bring it up unbidden, since it can make eyes glaze over. I really enjoy giving talks to community groups (mostly senior citizens) about math, and there are many people out there who are very interested and really appreciate it. So that’s one way to “talk about math” in a somewhat structured setting, but outside of school per se.

Whew! Hopefully I’ve guaranteed that no one will read this but me. Congratulations if you read the whole thing. Comments?

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12 Responses to More responses to “Math is not linear”

  1. Don Smith says:

    I want to respond to David’s thoughts about tangents. I am probably as bad a practioner of that as anyone – I refer to them as digressions – but I do not agree with what I think David is presenting as “either/or.” I think a “random” tangent – cleverly chosen – can indeed fold back on the material at hand. I delight in finding some “cool” thing that then allows me to improve the understanding of my students. The best tangents are precisely those that appear random but then return in the near future to help the students gain more insight into a particular topic (or topics). I also have to say some random stuff does occur to me at times, and if the time is right I’m going to show it to the students. Why? Because as much as anything I want them to understand that math can be interesting, unexpected, funny, sad, creative, surprising, but above all engaging. Especially with the younger students I find this a very powerful way to get them curious about the subject, who did what, and why – and how that helps them get into the next problem area. (As an example, right now we are doing the Pythogean theorem in my class and for the last several days I have been giving them little snippets about Fermat’s last theorem. They seem to be really enjoying it.) I agree very much with David that I find little use for what he calls “not particularly understandable facts,” but I continue to absolutely astonished by the students who come back to me years later and say “do you remember when you told us about____?” It’s awfully hard for me to gauge what is understandbale for all of them and what is not.

  2. Paul Demers says:

    I have a couple of issues I would like to address with regard to the Prezi and David’s comments. I think Alison’s point about electives is fundamental to a strong math education for a young student. That said, it is much easier to have electives for older students than it is for younger students. Since our school starts working with students when they are in 6th grade, what can we do? Our students’ daily schedules are so packed that we can’t fit electives into them. I think this is where math contests can and should enter the picture. I believe they offer great opportunities for talented young math students to stretch their boundaries with regard to the subject. Young students can begin to think about higher level concepts in Algebra, Geometry, Trigonometry and even Calculus at this stage. A good contest program does not need to specifically teach those topics, but it can foreshadow (another of Alison’s points) them to spark the curiosity of these young students. Contests also allow for young students’ explorations of topics like Number Theory which does not get taught in detail in a typical math curriculum. I love the fact that we have so many faculty members contributing to the contest program from 6th grade through 12th grade.

    The other point I want to write about is “be less helpful.” This is something I have been trying to incorporate more in my teaching over the last two years. Ultimately, to be good math students, our charges must be able to stand on their own two feet then utilize reasoning skills and mathematical skills in order to make progress and accomplishments in the subject. That will not happen if we stand at the front of the class illustrating multiple possible nuances to solving problems and then answering all questions that our students have on homework questions. Not only do our students desperately need to figure some of this out for themselves in order to truly internalize the concepts that are taught, they must speak about these ideas and write about them in front of their peers to become well-versed in the language of mathematics. That said, I am a teacher and part of the reason for my career choice is that I relish in helping my students to understand what is presented before them. I find it very challenging to take a step back and watch my students struggle in front of their peers when the easy route would be for me to step in and help. For the past two years I have taken a back seat (literally, a seat in the back of the room) in my calculus classes while student write out solutions to challenging homework problems and then explain the steps that they took. I find I still need to comment here and there while this is happening, but I usually do so in the form of a question posed to the student presenting a solution or to the entire class. I would like to explore some of the open-ended type problems that David is using in his classes to get at the idea of “being less helpful”, too. I view this as an area of potential growth in my teaching.

  3. Valerie Petrillo says:

    I want to respond to David’s comments on foreshadowing and relating material back to previous classes. In my mind these are just opposite ends of the same idea. I agree with his comment the earlier the better to introduce questions/ideas leading to calculus. This can and should be done regularly. For example, in my 9th grade Geometry class, when we look at regular polygons we notice that as the number of sides increases and approaches infinity, the shape approaches a circle, and we start talking about limits. Students at this level get really excited as soon as you start mentioning calculus. Not all students will follow you, but it sparks ideas in those who are ready, and it certainly doesn’t hurt the others to hear them. But foreshadowing is (or should be) done on a smaller scale as well. If my lessons are well planned, every day should foreshadow the next few lessons, and every new lesson should be related back to previously learned material. Isn’t this just a hallmark of good teaching?

  4. Jean says:

    I love going on tangents in class and developing ideas from student comments. However, I feel as though I do this less here than at other places where I’ve taught; I have less freedom in the classroom. This comes with teaching at a larger school where there is a standard and very full curriculum and where there are common exams. I simply cannot take time for many of the meaningful experiences that I think students would carry with them for a long time. As fantastic as this is, there is also a sadness for me in that I am not developing larger projects for students because I know I won’t be able to do them.

    A long time ago as a new teacher, I was told that if I was working harder than the students, then I wasn’t doing my job right. As a teacher, I needed the students to be doing the work in class. Helping less in the short run is actually helping more. Another way wise teachers word the same idea is to be a “guide on the side” instead of a “sage on the stage.” But it takes more time to draw ideas from students. The structure of our program is to give kids the material and then try to infuse meaning into it and to find a way for students to take ownership of it.

    Don’t get me wrong; we do a heck of a good job with our students. But I do think we need to continue the conversation of how to infuse meaning and creative thinking into our program. People will be able to access more and more ideas through technology. We need to teach how to use information, how to evaluate information, and how to create new ideas.

  5. Shirley Gonzales says:

    In education we often do an about face and head in the opposite direction when a method we are using does not appear to work well for all students; when really all we may need to do is a little fine tuning of the current technique. After reading the response of David, Don, and Paul to “Math is not linear,” it appears that a majority of the topics are really about balance. We need to consider balance and variety so that we reach students at all levels, interests, and motivation. Tangents can be interesting and thought provoking for some students, occasionally. However, tangents on a regular basis could indeed make it difficult for other students to understand the concept you are trying to relate. Being less helpful can be just what some students need to take the initiative and really become independent learners. Other students, however, may be inclined to give up and quit trying because they are not as successful, since they are at a different level of achievement and maturity. Presenting a challenging problem that shows a practical application of the topic being covered can capture the interest of some students, but perhaps not all students. Other students are sure that they will never have a career that involves using higher level mathematics and it seems that no matter what you present to them they are sure they will never need the material; while still others are motivated by the “puzzle” type of problem or challenge of a unique problem that they have never seen before. As students move through school and realize that their life may not turn out exactly as they predict, they begin to understand they should not shut the door on mathematics. The students start to appreciate the fact that they may yet end up in professions, some that may not even be currently available, that will require mathematics. I agree with David that although there may be different directions you can choose when taking math classes, there are basic concepts that need to be mastered before others can be taught. This too shows that there is not just one approach, but a combination. As he stated, the idea of a tree shows that shows there are choices that can be made in a horizontal direction as well as vertical direction. Throughout a student’s mathematical education, students are exposed to teachers that present material in many different ways; those that focus on interesting problem solving, those that focus on skills, those that focus on mathematical communication both verbal and written, and those that enjoy taking tangents which inspire students to think about possibilities. Students are captivated by different things and can learn much from each type of teacher. Hopefully, we can all attempt to provide the variety that would keep a majority of our students interested, working hard, and thinking about mathematics.

  6. Danielle says:

    After reading David’s comment about the prezi, it struck me that a lot of what David talks about pertains to coursework and options at the upper high school level. Relating some ideas to students in the middle grades poses a challenge. I want to specifically address two ideas: “teach electives” and “be less helpful”.

    Teaching a 9th grade course allows for some special “electives” work since there are 4 experimental education trips per year. Each trip, approximately 75% of the students remain and we have the freedom to introduce various topics. Some examples from prior years include: constructions, calculator programming, intro to excel, logic puzzles, and Cramer’s rule. These periods away from the regular curriculum are meaningful in several ways. Students are introduced to topics that won’t be tested so learning is less stressful and some topics tie well into enriching the curriculum. Hence, I agree we should introduce elective, where possible.

    Currently, algebra/geo classes are covering quadratic functions. One lesson was on completing the square when a = 1. I spent a second day on CTS when “a” was not equal to 1. For the third day, I had students CTS for x^2+Px+R= 0 then for HX^2+x+1=0. Completing the square with variables posed more of a challenge but the students saw the process was the same. Both problems were a warm up to solve Ax^2+bx+c=0, which is the derivation of the quadratic formula. My best attempt to be “less helpful” worked for the majority of students but there were some who got bogged down in notation and detail. If I had only given them the third example to attempt on their own, the breakdown of those succeeding might have been closer to 50/50. I feel it is important to introduce the quadratic formula like this, but for some the exercise did not work. Hence, with a younger age group, there has to be balance with how much subject matter is approached in this manner because students can quickly become overwhelmed.

  7. Mark Drexel says:

    To”be less helpful” is an idea that I embrace yet sometimes struggle to fulfill. I know that in order to “be more helpful” with respect to helping my students think independently in the long run, then I need to “be less helpful” in the short term. I often blame the lack of time to get through the curriculum to defend the way I explain ideas. Ultimately, I need to change the way my students need my help. They need me to pose interesting problems as opposed to just solving them with and for them. They need me to require higher-order thinking as David suggests which includes letting them use their common sense and allowing them to communicate math verbally. I can still help at the end to demonstrate the formal way to show the work, but I also need to remain open to alternative solutions. Some of the ideas that I remember the most have happened in class when a student has suggested a less formal yet elegant way of solving a problem. If I had shown them all the algebra that it might have taken me to come up with the same result, then the more direct, student method might have never come to be. In order to empower students, I need to share the power in the classroom. I need to ask more questions and give fewer answers.

  8. Austin Murphy says:

    I can certainly get frustrated when my students seem strangely unwilling to engage with a problem. I say strangely because they are, as a rule, hardworking and curious. But I guess their curiosity, and willingness to focus, has limits. Limits, however, that I help establish.

    I think being more helpful by being less helpful is a great idea. A way to foster an increase in their willingness to engage with material over a prolonged period of time, even if no answer is in sight. I can, as teacher, establish a bigger playing field, limits further off.

    The presi talked about motivation. I have worked in schools where motivating the students was the measure of the teacher. This motivation was measured under a classroom management rubric. This is not the sense that the presi had in mind.

    The presi talked about how needing rational functions in some future class was poor motivation for learning about them in this one. This is motivation in the sense of providing context in which learning the material has some meaning to the students. The motivation is provided by the context set up by the teacher. Some teachers are more successful at this than others, regardless of the textbook or the curriculum or whether the teacher is traditional or adventurous, linear or nonlinear. But here especially I think being more helpful by being less helpful is a rich avenue to explore.

  9. Janet says:

    I’m one of those weird folks who was (and is) motivated solely by the patterns and beauty of math. In college I hated real-life examples; they just got in the way. So I struggle with including any at all in my teaching. I try to encorporate more and more as I find examples that seem relevant. I also think, however, that the beauty and connections in the material can act as motivation for some students. Ideally, students could be led to make those connections and extensions for themselves. These are difficult lessons to craft and they take more time, typically, but the payoff is in depth of understanding and length of retention.

    On a different note, I hate the fact that certain colleges expect students to have taken calculus in high school. It becomes the criteria from which students and schools make decisions about curricular options. Ideally, students would make those decisions based on what is best for them personally, but that is not to be. I do think that having seen calculus once, often in a smaller setting, it is easier to succeed in it later in college.

  10. Alan says:

    To see what we value in teaching mathematics to our students, consider our de jure three year graduation requirement: Algebra I, Geometry, Algebra II and Trigonometry. For half a century this curriculum has prepared and continues to prepare students for college work in mathematics. College admission has added a de facto fourth year graduation requirement, which many students fulfill by taking calculus. I wonder what percentage of our graduates actually takes another mathematics class in college, and what percentage takes two classes in mathematics. It might be much smaller than I would guess.

    Any well-taught mathematics class, regardless of content, is valuable to our students. It provides them the opportunity to learn to make distinctions, to use precise notation, to reason soundly, to justify steps, procedures, and arguments, to discover, to problem solve, and to prove. In short, while the mathematical content will fade with time, they have learned to think more clearly for life.

    The single most important change we can make to our curriculum to benefit our students would be to add Statistics as a graduation requirement. A citizen of today’s society is well served by knowing more about data collection, data analysis, data use and misuse, and statistical inference.

  11. Heather Sullivan says:

    When I was first viewing the Prezi and read the statement about teaching an elective, my mind jumped to electives outside of the realm of mathematics. I thought, “Yes, I would love to teach an art class…a biology class….” I circled that I agreed. Soon, it became clear that she meant mathematical electives. I still agreed, but it made me think about why I love teaching math even though I can certainly not call myself a mathematician and am in my heart more of a generalist. Over the years, I’ve often heard (from people who meet me outside of school settings), “You don’t strike me as math teacher…You seem more like an artist or counselor or a (fill in the blank).” I assure them that there are all sorts of math teachers, and that we don’t always see in black in white…we do see shades of gray…we do seek balance. When I start the year, I assure my students that I know they are much more than math students and that I will do my best to keep that in the front of my mind; and I want them to know that I am much more than a math teacher. Perhaps it seems obvious that we are all more than that, but do we really keep that in focus? For me, I love teaching math because I find it to be powerful vehicle for relating to and connecting with students. I actually find math to be a very emotional realm in our society. Whether you are great or not so great at math, the pressure and expectations that parents and colleges and other groups build up around the subject are intense. If you really love math, what sort of social stigma does that create for a middle-schooler at a typical school? I want to use math to encourage people to push themselves while still accepting themselves, to embrace challenges and stretch their minds, to develop their communication skills, to collaborate, to work through frustrations, to seek simplicity in times of confusion, to recognize that there are many different paths to a correct answer or a desired goal. While I loved (love) many subjects as a student and could never decide if my favorite was art, math, or science; I know that when it comes to teaching, I LOVE TEACHING MATH. Math affords us as teachers a unique and special realm for challenging, empowering, and supporting people. I’m sure teachers of other subjects feel the same, so we each have to find the right fit. So while I would enjoy teaching another subject as an elective, the point would be to keep in mind the fullness and wholeness of each person even when we’ve specialized in one subject area. As for how we teach, we need to be open to experimenting with different methods outside of our comfort zones…perhaps stepping back and helping less…perhaps mixing up our routines…perhaps using more technology. However, we also need to be honor the fact that it’s wonderful that we have so many different styles and that we don’t all have to embrace the same trends to be the best teachers that we can be. Whether we as teachers are more motivated by the pure mathematical beauty, the applications, or something that doesn’t fall into those easily defined categories, I think we should be honest and genuine with our students about why we love about math AND WHY WE LOVE TEACHING. It’s not only about understanding what motivates students. It’s also about understanding what motivates teachers, because we want motivated classrooms and schools and communities. It’s about modeling interest in the math, interest in learning, and interest in living full lives. I certainly can’t say I’ve got that all sorted out. I struggle with the day to day stresses, and I am often in need of stepping back to get perspective on the big picture. I know this: Students’ learning and retention depend on more than the sequencing of topics that we THINK we are teaching. They need to FEEL something. I don’t mean they don’t need to feel warm and fuzzy, but they need to be emotionally engaged…and not just about the math problems but about themselves, their peers, and their teachers. Emotional engagement allows for the formation of lasting memory. It’s not fluff; it’s neurology!

  12. Heather Sullivan says:

    Make that third to last sentence start with, “I don’t mean they need to feel warm and fuzzy…”

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