Last night, my eight year old looked up from doing his homework at the dining room table and said, with tears in his eyes, "Mom, I'm not good at subtraction." It nearly broke my heart. My son loves math. He has been thinking about numbers with me since he could talk. We enjoy playing with numbers and ideas and he revels in it. This is my child who, two years ago, at age six, asked me if you could divide ten by three. I told him to give it a try. He thought about it and said that he didn't think it would work very well because you'd have three, three, and four. I responded that I agreed that the answer wouldn't be a whole number, but that I thought you could still split up ten into three equal parts. He thought for a few more minutes in the back seat of our car. I thought he had moved on to thinking about something else when he piped up, "It's three and one-third!" You should have seen my grin. Just a few months ago we talked about subtraction. My son had not yet been taught multi-digit subtraction in class but he had run across questions about it in an online software program the school uses. He hadn't understand the point the software program was trying to make and told me he could solve any subtraction problem without a number line. I decided to have him show me. I gave him the problem 472-96, written vertically. Remember, he had not yet been taught multi-digit subtraction nor regrouping in subtraction. My son started by looking at the ones column ,thought for a minute, and told me that 2 minus 6 was negative 4. He wrote -4 on the paper. He then looked at the tens place and thought for a really long time. He eventually said that 7 minus 9 was negative 2, but he said it with hesitation. I asked him if why he seemed hesitant. He said something wasn't right about that. I asked him if it would help if I reminded him that the 7 and the 9 weren't really 7 and 9, but 7 tens and 9 tens. "Oh!" he said, and wrote -20 on his paper to the left of his -4. Next he looked at the 4 in the hundreds place and wrote 400 on his paper. Now his paper read 400-20-4. He read this out loud as "four hundred and negative twenty and negative four." "That's 374!" he cried after a minute or so and wrote it down. I asked him to check again. I told him that his method was correct but that he made a tiny mistake. He looked at his paper and said, "Oh!" He erased the 4 and wrote 376. Now, a few months later, here I was listening to this same child tell me he was bad at subtraction. I went right over, took his hands in mine, looked him dead in the eyes and said, "You are one of the best kids I know at math. You think about math and figure things out and ask amazing questions; you are so good at subtraction!" His face crumpled and he started to cry. "What is giving you trouble that makes you think you are bad at subtraction?" I asked. "I just can't remember the three kinds of rule I am supposed to do," he said. When I asked him for clarification he added, "I know when the two numbers are the same I have to put a zero. When the floor is bigger I have to go next door, but I can't remember what else!" I pulled him into may lap, carried him to the couch, and we both cried. He cried out of frustration about feeling stupid. I cried because my child had been trained to ignore his number sense and memorize rules that had no meaning to him. I had had a tough day, had a headache, and just wanted to go to bed. This was the last straw. As I hugged my son I apologized to him. "Honey, I am so sorry that I have not yet been able to fix it so that all teachers know how to teach math well and help kids understand it." [An aside here. My son knows that I am "a doctor who teaches teachers to teach math." He knows I work with teachers to help them understand math better and help kids understand math better.] I probably could have chosen a better tack here, but in that moment I was so worried about my kid that I couldn't think further ahead than that. We both cried together, sobbed really, and hugged each other tightly. After we both calmed down a bit, I sat him next to me and said, "Let's look at this together. I want you to forget all about the rules. Ignore the rules and think about subtraction. You know subtraction and what it means. Think about what you know." The next problem on his homework paper was 54 minus 28. He looked at the problem and said, "Well, 4 minus 8 goes into the negatives so I can't do that. So I have to do 8 minus 4." He wrote four in the ones place. He then did five minus two and wrote a 3 in the tens place. "34," he said. "How can you tell if that's right?" I asked, remembering that at one point earlier he told me he could check his answers to subtraction by adding. He added his answer of 24 to the subtrahend 28 and realized that he did not get a sum of 54. "That didn't work," he said. He then thought for a few seconds and said decided to start over. He erased what he had done and looked again at 54 minus 28. This time he decided to try regrouping. He decided he would "go next door" and take from the 2 in the tens place [of the subtrahend] instead of the 5 because "it's easier." He crossed out the 2 and rewrote it as a 1 and changed the 4 into a 14. Now his ones place was 14 minus 8. He wrote 6. His tens place was now 5 minus 1. He wrote 4. His new answer was that 54 minus 28 was 46. Again, he checked his work with addition and found it was not correct. This whole time I had been biting my tongue, watching him divorce the numbers from their meaning. I suggested that he stop thinking about each digit on its own and think about 54 and 28. "How can you do 54 minus 28?" I asked. My son proceeded to think about a number line and use the number line to count up, in chunks of easily compatible numbers, from 28 to 54. He now knew that the correct answer was 26: 2 plus 20 plus 4. "That is a great strategy for subtraction," I said. "Can you think of another strategy to use? Forget the rules, don't think about the rules, just strategies." My son thought for a bit and decided his other strategy should use regrouping but wasn't sure how to do that. I was surprised he didn't use the negative number strategy he had used in the past, but I think he was limited by the fact that he has clearly been told in class that he can't subtract a bigger number from a smaller one as part of the "rules." So, regrouping it was. I told him I could think of lots of way to group 54. I suggested two ways: 5 tens and four ones; and zero tens and 54 ones. My son thought that was pretty cool, but wasn't sure how to use it to help him. I asked my son for other ways to regroup 54. He sat there and finally told me he wasn't sure. I suggested 2 tens and 34 ones. I chose this grouping on purpose because his he was subtracting 28 and the tens would subtract out with no remaining tens. I rewrote the problem vertically and showed that regrouping in the work (see the lefthand problem in the photo). My son decided that was pretty cool because the tens was easy. He decided that 34 minus 8, though was "too many ones," and perhaps would take too long to figure out. I asked my son to think of other ways to regroup 54. He decided that he should use 4 tens and 14 ones because those numbers made it easy to do the subtraction and they weren't so big. I rewrote the problem that way and he told me what to write to solve the problem. After he mentally checked his answer using addition and declaring it correct I said, "Isn't this the way your teacher said to regroup it using her rule?" He looked at me with an amazed smile and said, "Yes!" I went on to explain that many people thought regrouping by taking only one of the tens and regrouping it ones was easiest because it kept the numbers pretty small but still worked. He thought that was a good idea. I asked him if he had used blocks to build subtraction problems with tens and ones. He said he had and that it took a long time to trade in the ten for ones. I suggested that he didn't have to actually build each one, but that he could imagine his blocks every time he did a subtraction problem. "Forget about the rules,' I said, "and use one of your strategies. When your teacher wants you to use regrouping, your strategy is to remember and imagine your blocks. Do you think you can do that?" "I know I can," he replied. He did the last few problems on his worksheet while imagining his blocks and I got him upstairs and in bed. After I kissed him goodnight my son looked over at me and said, "Mom?" "Yes, honey." "When I grow up, if you haven't fixed it so all the teachers know how to teach kids math really good, I want to help the rest of the teachers learn to do it, too." I couldn't be more proud.
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“What we’re really doing here [in a typical mathematics textbook problem] is taking a compelling question, a compelling answer, but we’re paving a smooth, straight path from one to the other and congratulating our students for how well they can step over the small cracks in the way. That’s all we’re doing here. … The textbook is helping you in all the wrong ways. It’s buying you out of your obligation for patient problem solving and math reasoning.”
~ Dan Meyer in his 2010 Talk, Math Class Needs a Makeover, available at https://www.youtube.com/watch?v=NWUFjb8w9Ps I have been struggling a bit with my 8th grade math class this year. I love the kids. This is our third year together and I find them engaging, fun, interesting people with whom I greatly enjoy spending time. The other day one of my students said, “We don’t just learn math in here, we learn about life.” And we do. We talk math, we talk politics, we discuss current events in our lives and the nation, and we discuss how vital an understanding of mathematics is in this age of data. I should have responded that the life and math are often so deeply intertwined that it is impossible to have one without the other, but I think I was too busy grinning and soaking up the moment to do so. Despite all of this, I’ve been struggling, and as a result, my students have been, too. I have taught this course more than a dozen times over the last 20 years. I know the goals, I know the content, I have taken the High School’s midterm and final exams (that I am required to give to my 8th graders as a common assessment) a multitude of times and know the desired outcomes without having to think about them. In terms of content, I can do it all in my sleep, and I thought I did a pretty good job of helping my students understand the whys behind the calculation-heavy course that is Algebra One in American education. This year, though, it is not working as well as I’d like. And in truth, if I am honest with myself, it hasn’t really been working as well as it should for a while. But this year is worse. I think I know why this year is worse, but that answer has many levels. The simple answer is that my students have a different preparation than in years past. To understand, you need some history. The Early Years When I first started teaching the course, 20 years ago, my students entered my Algebra 1 class having taken a traditional Pre-Algebra course. They memorized computation strategies and learned how to get the right answers by following prescribed procedures. They did this well. I taught the Algebra course the way I was expected to teach it. I followed the book, page by page, and tried to teach my students why these computation strategies made sense through my verbal explanations. Despite the fact that I had been given the “lower-level algebra one class” [An aside here: How crazy is that, right? Having a regular level and an algebra level wasn’t enough, we had to further subdivide the algebra level according to computation facility], my students completed the text, scored well on the exams, and nearly half of them scored well enough to be placed in the honors level Geometry class for their 9th grade year. I was proud of them. I some ways my instruction of this course has not changed much. I still rely mostly on my verbal explanations and questioning to help students make sense of the computational strategies they are asked to “master” (read memorize and use to efficiently compute). I started the last paragraph, though, by saying my students’ preparation was different. Now that you have an image in your head, let me describe how my students’ sixth and seventh grade classes have changed over the years. When I first started teaching, we used a textbook series like most others in the United States. It was from a major textbook publisher, was called something like “Mathematics 6” or some other non-descript title, and contained a multitude of unrelated ideas separated into discrete chapters which, often, had little to no discernable theme amongst unrelated sections within the chapter. Teachers used the text “as a resource,” meaning they had it in their room, but really made up their own worksheets (this was before the age of easy internet access) and shared amongst each other. Frankly, the stated curriculum was a list of the textbook chapters and I felt like I could teach whatever I wanted, however I wanted. I tried to teach conceptual ideas because that is how I understand mathematics, but I wasn’t particularly skilled at doing so and had limited models to draw from. The Connected Math Years Not many years in to my teaching career, though, my school purchased a new textbook series. We were very lucky. Someone, somewhere in curriculum administration, was doing their job well. Our district purchased Connected Mathematics, one of the mathematics programs developed with National Science Foundation support that was developed using strong research on mathematics teaching and learning. Finally, I had a tool to help me teach conceptually. It was hard at first, because we teachers wanted to make sure our students had the tools to solve that problems given to our students. We wanted to pre-teach the math. By that I mean that we wanted to teach procedures to students so that when they encountered a rich problem they could figure out which procedure to use and know how to do it. After one unit, though, I learned to trust the program. I found that each unit began with a problem that students wouldn’t be able to solve efficiently. It would ask them to get their hands dirty and engage with the mathematics. Then, the unit would give students problems to work on whereby they would develop their understanding of the mathematical ideas and, though careful problem construction, lead students to the development of computational algorithms. Over the years we found that, for many students, we had to supplement with computational practice to develop the fluency we wanted in our students, but the problem-solving ability of our students shot up. More important to me, though, was that my students learned that computational strategies were logical processes that could be developed from their own ideas and honed to increase efficiency. They learned when their algorithms worked, when they didn’t, and when others’ algorithms (including the traditional algorithm) may be easier—or harder—than their own. Along the way, I made sure to teach my students how others, who hadn’t learned conceptually, might verbalize about these procedures. I felt that it was important that they understand that when others said “is over of is percent over 100,” for example, it was a trick to help them memorize if they didn't understand what my students understood: that percents are an expression of a part to whole ratio where the whole is one hundred units and that other part to whole relationships could be expressed as percents by creating equivalent ratios. I tried to teach my students mathematics conceptually and yet also give them the tools to navigate the algorithm-driven world of school mathematics. I loved teaching with Connected Mathematics. I enjoyed the journey my students and I embarked on each time we started a new unit. I enjoyed hearing their ideas as they discovered some new mathematical idea. I loved the sense-making we did as a class as we talked about their methods and why they thought what they thought. I loved watching my students become and think about themselves as mathematicians. It was hard sometimes, because I had to think critically about their methods, figure out what they were thinking, help find holes in their thinking, draw out patterns in what students were discovering, and all the while make sure I was honoring their thinking rather than just my own. But the difficulty, and the different thinking that developed depending on the students involved, is one of the things that kept it interesting and challenging for me as a learner and teacher. During this time, I continued to teach my Algebra 1 class in much the same way as before. My focus was to help my students transition from a classroom where math was learned conceptually to a classroom where math was learned more traditionally. I wanted my students to learn how to still be conceptual learners in a traditional classroom. I knew that the rest of their math career may, unfortunately, be traditional in nature and that I needed to arm them with the ability to ask conceptual questions of each other and to remain conceptual thinkers despite their environment. I hoped they had gained an understanding and belief system about mathematics that they could carry with them as armor through their traditional world—armor that would withstand the traditional barrage of “memorize this” and “just do it this way” statements that would assault them. I hoped their armor would protect their view of themselves as mathematicians so that, when given the happy respite of a classroom or a world that valued their understanding, they would have enough of it intact to flourish. I viewed my Algebra 1 class as a necessary step in that armor-building. I would use the traditional text and yet talk the conceptual talk. We would wonder and ask questions about the mathematics while still solving traditional problems and discussion the computational strategies outlined in the text. We would try to backfill derivations and wonder about why the book was approaching problems the way it did. The Last Few Years I loved teaching with Connected Mathematics, but not everyone in my building agreed with me. Over the years, as teachers came and went, fewer and fewer of the teachers received any training on the Connected Mathematics Program and the style of teaching and learning it required. Teachers were relatively isolated and we entered the era of high-stakes testing in earnest. Many felt that, despite the data showing otherwise, we weren't teaching enough mathematics and that all this problem solving took too much time. More and more teachers began using the Connected Math books “as a resource” and taught the content using the procedural methods they were more comfortable with. Our school had been using the Connected Math series for more than ten years and our typical textbook replacement time came and went. Meanwhile, the NCTM frameworks, a voluntary set of national standards, was being replaced with the Common Core State Standards, a less-voluntary set of national standards. Our district decided it was time for a new textbook. After much searching, many school visits, and a lot of deliberation, we settled on the Math in Focus series for our Middle School. I served on the committee and played a part in the decision, but the views and perspective portrayed here are my own. We wanted a continuous program for grades K-8 and we knew the elementary schools were due for a new textbook also. We eliminated any textbook series that did not have an elementary component. We had elementary teachers on our committee to help us pick our Middle School program, and actively sought their opinion on the elementary components of the program as well. We made compromise; we knew that our schedule would not change and so eliminated textbooks that relied on block scheduling for implementation, despite the fact that these tended to be more conceptual textbooks. I was and still am happy with our choice of the Math in Focus series. It is nowhere near as conceptual as I would like, but it has enough conceptual mathematics in it in ways that make it difficult for teachers to just not do the conceptual components. I believe the Middle School will have more conceptual teaching happening as a result of the Math in Focus series than it did before simply because I think everyone will be comfortable using it and it will simultaneously push teachers towards more conceptual ideas. That said, the series does too much telling and not enough discovering. I need to take a moment here to talk about training. One of the downfalls of our implementation of the Connected Mathematics program years ago is that we trained our staff before the first year of teaching and never again. With teacher turnover, there were very few of us left who received any training on the program in our later years of using it. Without training, it was very difficult to implement the Connected Mathematics Program correctly and many teachers stopped using it because they could not, due to lack of training, implement it effectively. When we were choosing the Math in Focus series it became clear that money was tight and that, in all likelihood, we would not receive all the training we needed. For that reason, I wanted a program where the conceptual components were so integrated that they were not removable by teachers, and yet one that looked traditional enough that teachers would use it and not just ignore it. [It turns out that we were right, and we did not receive all the training needed to implement the series in full. My statements, then, represent only my experience with the limited training my school provided, not an ideal implementation.] I felt, and still feel, that the Math in Focus program may allow our District to take small steps toward conceptual teaching, which is better than just ignoring a more conceptual program. I began using the Math in Focus textbooks with my students two years ago. This year’s 8th graders were in 6th grade. I planned every chapter with the goal of having my students discover what the books were telling students. I planned warm-ups where I carefully developed problems that guided students from previous lessons through ideas and asked them to find patterns or develop ideas about the mathematical content. Only then did I show them what the textbook had in it relative to the content. I took the textbook’s paved path (to use Dan Meyer’s metaphor from the quote at the beginning of this piece) and turned it into a gravel path. But it was still a path that my students could ride a bicycle on – and we rarely went off-roading. Why Isn’t My Algebra 1 Class Working? The simple answer is that my students have a different preparation than in years past. This year’s 8th graders have had wonderful experiences in math class. They have had those times when the tell me their minds are blown—they even have a hand signal where they hold their hands on their heads and move them out to the side while making an explosion sound. But this, in and of itself, may be part of the problem. If every day were a mind-blown day, they wouldn’t need a hand signal. It would be so common place that it would become normal. These students were not a part of the Connected Mathematics years, and when I made the Math in Focus path more gravelly, I still made it a straight path. There weren’t enough side roads or pitfalls, there weren't twists and turns, and—most importantly—I was the one leading them along the path. This year, I have been teaching my Algebra 1 class the same way I have for a long time; but my students have not journeyed along the same path as their predecessors. They are not at a place where they need armor around their self-concept as a mathematical thinker. They still need to discover themselves as thinkers. And that is the more complicated answer to my question. My Algebra 1 class isn’t working because I need to tailor my instruction to meet my students where they are, and not where their predecessors have been. My students have travelled a different path than their predecessors and this years’ students are not yet where I want them to be. It is not their fault. It is mine. I may need to let someone else build their armor; I need to build the mathematician inside that armor, and to do that, I need a rockier path. Or better yet, maybe I just need the mountain and have my students build the path with me. Author’s Note: I wrote this piece a number of months ago, while school was still in session, but was not brave enough to post it. This morning, while sitting in the airport waiting for my delayed flight to board, I decided it was time. I was reading Chapter 6 of Tracy Zager’s book, Becoming the Math Teacher You Wish You’d Had, and was reading about developing problems with open beginnings, middles, and ends. Such problems provide hiking huts on the mountain described in my piece. They provide waypoints and students determine their path from one waypoint to the next. I decided my experience, posting about my struggles, might help others. If I can admit that I still need help, too, maybe others can as well. Maybe my experience can help us all move forward. If you haven’t run across Tracy’s book, it is worth a read (or two or three). You can access it, and the online resources for it, here: http://tjzager.com/
What makes Flowers Grow All in a Row stand out is the way it subtly develops multiple aspects of number sense, and does so better than other counting books I have seen. Lisa Houck adds a new flower on each page of the book, but what makes this especially unique is that each flower is added to the right of the flower before it, equidistantly spaced. Houck essentially creates a flower number line, reinforcing the linear aspect of number. By using all flowers, and the same flowers in the same order, on each page, Houck is showing the idea that the next number in the sequence has a value one greater than the number before it. In other words, to get from one number to the next, we add one more item to our set, one more block to our string of counting blocks, or go one unit further to the right on our number line. When Houck has filled her pages with a flower number line she adds butterflies and a bug to the page, but not as part of the number line, and not with a numeral attached. This allows the reader to decide what to count, and how to describe it. Are there 7 flowers, 2 butterflies, and 1 bug, or are there 10 living things? Better yet, why choose one way to count? How many ways can we group items on the page, and what unit describes those items? These conversations about sets and units are part of what the book so wonderful. I’m also loving the three addends that sum to ten. I look forward to hearing what my child comes up with when we read it together. I am imagining some great "Which one doesn't belong?" discussions. Needless to say, I left the store with two copies, one for the baby shower gift and the other for my own library.
Disclaimer: I have no known association with the author or publisher and am not being paid or compensated in any way for this post. I was moved to share this great find with others interested in exploring number with young children. Image credits: Photos by Ann Gaffney. Original copyright 2016 Lisa Houck. Used with permission from Pomegranate Communications, Inc. To help girls see themselves as scientists, I decided to make posters from the many tweets using the #ActualLivingScientist and #DressLikeAWoman hashtags. This ensured I got tweets of women scientists doing their thing. This is not a political statement, but rather an easy way to filter for women scientists. I tried to choose tweets that showed the diversity of women in science. The files, which you can download here (just right click on each image and save the ones you like), are scaled to be printed on 24x36 inch paper. My local copy center is printing my set now. Can't wait to get them posted around my classroom! I recently ran across some really interesting research on women and STEM. Linda Carli, a researcher from Wellesley College, my alma mater, has conducted research on gender stereotypes and STEM. In a nutshell, she found that people’s mental image of a scientist (including mathematicians, engineers, doctors, and those in other scientific fields) is of someone whose personal traits are traits that we tend to think of as very masculine—more masculine, in fact, than we picture men! (To read the paper and listen to a podcast interview with Carli, see http://journals.sagepub.com/doi/suppl/10.1177/0361684315622645.) So what does this mean and why should we care? The short answer is that this mental image of a scientist does not match our mental image of ourselves. Why would we choose to become something that does not match how we see ourselves? This is especially true for women and girls. If we see scientists as hyper-masculine then we do not see ourselves as scientists, even if we see ourselves as good at science.
I found this fascinating. As a math (and sometimes science) teacher, I now see my role as something different than before I heard about this research. Before, I thought my job was to help my students, including my girls, see that they had the skills to be mathematicians and scientists. This is still true. Now, though, I see my role also as helping my students see what scientists (including mathematicians) are. Scientists work on problems that help people. They work in groups, they communicate with each other and bounce ideas off each other. They collaborate. Scientists are logical, but also work on intuition. Scientists are creative and passionate about their work. Scientists are not just geeky men who don’t communicate well and like test tubes more than people. Scientists are as diverse as the science they study. My role, as a teacher of the maths and sciences, is to help my students see the diversity of scientists just as I help them appreciate their own diversity. We have come a long way, in education, towards helping students understand what science is. Now we need to help them see who scientists are. Perhaps then we will not only be able to close the gender gap, but also increase the number of men and women choosing to become scientists. I’ve been meaning to start a blog for a long time, but something keeps getting in my way. Sometimes I blame my lack of starting on time, or rather lack of time. But how I use my time is simply a matter of priorities. If I want to be heard, I need to prioritize saying something. Sometimes I blame my lack of starting on the technology. Sure, I know I can start a blog on any number of sites, but I don’t want to have to move it later and I want to link it to my webpage and I need to fix my webpage first and I want my own domain name and I need that first and…. You get the idea. If I am really honest with myself, though, I need to admit that my lack of starting my blog is my lack of confidence that I have anything worthwhile to add to the conversation.
This may sound strange if you’ve read my bio. After all, I have plenty of academic credentials and awards. It may also sound strange if you know me in person. I am a talker. I am not afraid to state my opinion even in a formal public setting and even when I know others will disagree with me. So what is it about writing online that holds me back? Confidence. In this blog post, my first ever, I’d like to focus on confidence. As a math teacher, I see one key part of my role as that of a confidence-builder. Often I have students who know more than they think they do. They see mathematics as a subject built around finding answers in the ways teachers tell them to. I try to get them see mathematics my way: as a way to understand the world. I want them to see themselves as mathematicians, as people with mathematical ideas that make sense and are built on sound reasoning. I want my students to see that math class is not about finding answers my way, but rather about understanding systems and expressing their ways of thinking about these systems. After all, mathematicians discover (or create, depending on your viewpoint) new ways of thinking about and explaining our world. As a math teacher, one of my jobs—perhaps the most important one—is to convince students that they have the power to do this and then convince them to use this power. In other words, I have to help them build their mathematical confidence. There are a number of ways I help students build confidence in themselves and begin to see themselves as mathematicians. The first way is by creating an environment in which their thinking is honored. My students and I talk about “logical mistakes.” When we make mistakes, there is often thinking behind them that is logical. Sure, it's a mistake, but the mistake is based on an idea that we partly understand, misunderstand, or misuse. Given what we do understand at the time, our answer or process is rational, though flawed. Thinking about mistakes in this way helps my students realize that a mistake is not a sign of being bad at math, but rather can illuminate their understandings and rationality. When my students begin to think about themselves as people who are logical mathematical thinkers, they become more willing to believe they have something to share. They begin to see themselves as mathematicians. Another key component of changing students’ belief about themselves as mathematicians is providing opportunities for success. If we only talked about their mistakes, no matter how logical they all were, students would not see themselves as mathematicians, but rather as failing mathematicians. When provided with opportunities for success, students see that they not only can develop logical, mathematical argument, but that this way of thinking can lead them to a solution that makes sense and that others agree with. Notice that I didn’t say a correct solution. Their solution should also be correct, but if my goal is a correct answer, I am taking the role of mathematical arbiter upon myself. By putting the emphasis on a solution that makes sense and that others agree with, the onus is upon my students to determine if their answer is correct. My student body is the mathematical community that vets each other’s work and determines if the solution is correct. Only after my student community has decided an answer is correct do I also tell them they are. Eventually, they no longer need me to validate their work. They see themselves as mathematicians who can judge the validity of an argument based on sound mathematical ideas. This confidence building process takes time. I find that this can happen much more quickly with students who have often been told they are good at math, and happens much more slowly for those who have not felt successful in the past. Some students who leave me without ever truly feeling successful. I do think, though, that they leave believing, at least sometimes, that they do have something of value to share and are further along than they were when they first set foot in my classroom. Let’s jump back to me and my blog. (It’s related, I promise.) Why is it that I don’t have the confidence to believe I have anything worthwhile to say? I think the issue is that, just as my students come to me not seeing themselves as mathematicians, I come to blogging not seeing myself as a writer. I am willing to share my beliefs orally, but not as a writer. Sure, I can write—I’ve even been told I do it well. But I don’t enjoy writing and I don’t see myself as someone who writes. I talk, I teach, I explain; but I rarely write unless I am forced to. In order to overcome my lack of blogging confidence, I need to begin to see myself as a writer. I have plenty of evidence to show that I can do this. I made it through many degrees by adapting paper topics to fit something I cared about and was able to create pieces of writing I was proud of. So why don’t I trust myself to do that here? Writers do two things (related to today’s discussion). Writers decide what to write about, and determine what they have to say about that topic. The second part isn’t so scary. When asked my opinion, I always have one. When a discussion is happening, I can enter it and participate. The first part, deciding what to write about, seems to be the sticking point. I often feel that the things I am interested in are either things that everyone already knows or that no one cares about. So as a result, I don’t talk about anything. I have decided that in order to overcome this lack of confidence in my own ability to have something worthwhile to say, I need to try speaking and give myself opportunities for success. So I am going to try to reframe this task. Instead of choosing to write, I choose to have a voice. I choose to say what I think about whatever comes to mind. Sometimes, it may be something everyone already knows. If so, perhaps my readers will get validation from my thoughts. Sometimes, I will say something my readers are not interested in, but then they can just ignore it and wait for the next post. No harm, no foul. But sometimes, just sometimes, maybe I will have something to say that makes my readers think about things differently. Through writing, perhaps I will gain confidence that my voice is a meaningful voice. Perhaps I will begin to see myself as a writer as well as a mathematician. Wish me luck! |
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