Several ideas resonated with me from these chapters. The first is around the power of mistakes. Our learners enter our classrooms with the notion that math is either right or wrong and you must be right to be successful. The “fear of failure” permeates and I feel holds back growth in understanding mathematics. I would agree with Eric that “mistake” might be a misnomer. One definition of mistake is “an error in action, calculation, opinion, or judgment caused by poor reasoning, carelessness, insufficient knowledge, etc.” I would rather we change that definition which seems negative to me to rather we hit a dead end, a wall and need to change direction. It might not be caused by “poor reasoning” at all but very sound reasoning that didn’t lead to the answer we were seeking. I was struck when I visited the Kennedy Space Center how many “failures” they had before a rocket was successfully launched. And my favorite movie Apollo 13, shows true perseverance with the scene of trying to figure out how many switches could be turned off to have the least amount of power used. The astronaut kept trying and trying several approaches until finally one path was found. When mistakes are honored, perseverance is strengthened. My favorite quote that Patricia Helmuth shared with ANN members was “mistakes are expected, respected, and inspected”. Imagine if every math classroom had this quote posted as learners entered.
A second topic was that of speed, that speed means smart. This is found directly in the tests our learners must face. Everything is timed giving no credence to the importance of thinking through for problem solving. One test, the Accuplacer, is at least untimed which would allow students time to truly think through finding a solution. Unfortunately the test itself is a lot of symbol manipulation but at least students could reason through. Speed is also focused on doing computation quickly which is such a small part of understanding and exploring mathematics. I loved the Fortune 500 most valued skills list. Computational skills were at the top in the 70’s when I first began teaching. Now that ranks near the bottom of the list today.
Lastly what I am most excited by Boaler’s and other’s research is what has been found through cognitive science research. My friend is a Wilson reader trainer and I remember being envious of those in the reading world who had mountains of evidence about how children learn to read but when I looked for articles on how children learn mathematics very little was found. I finally was introduced to David Sousa’s book “How the Brain Learns Mathematics” as my first book of evidence. Now more has been discovered especially around visualizing mathematics and how the brain functions using mathematics. This research I find exciting and hopefully will be further proof why we need to change our practice in the classroom.
My original education and training was in foreign language teaching. Our attempt to re-frame the concepts of "mistake" and "error" reminds me of William Nemser's idea of "approximative systems" in second language learning. Instead of comparing language learners' developing languages to either their native language or the perfect target language, Nemser suggested that there is a lot of value in looking closely at the learners' current, "approximative" system for all the patterns and stages of the progression that they necessarily must pass through. For example, it would be silly to consider "drived" as a "mistaken" past tense form of the verb "to drive" if the learner's progression hasn't included the concept of strong/irregular verbs. So, it's not an error, it's an approximation that will likely adjust closer to the target language as the student progresses. I, of course, don't know whether language and math are even comparable in terms of their development, but we do often refer to math as a new language and it is an interesting approach.
First let me say that the first two points you speak about also caught my attention. I'm very interested in the discussion I see happening about what constitutes a mistake and what exactly makes our brains grow. In chapter 2 on page 13, Jo Boaler gives the example of the entrepreneur that founded Starbucks. His first attempts at replicating Italian coffee shops did not take off, and it was his going "back to the drawing board" that eventually resulted in the Starbucks that became so successful. I don't really see his first attempts as mistakes, per say: rather learning experiences that led to success, similar to what you speak about in relation to space exploration, and what we see in our students when they try something that doesn't work and continue to persevere until they find something that does! One teacher at a workshop I facilitated worded this process really well when she said that every answer a person comes up with (when trying to solve a math problem) is honored as a thoughtful math step, which might not be leading to "the solution", but does lead to some more math thinking. Brain growth! For more on "mistakes are expected, respected, and inspected" follow this link to CollectEdNY.
I also appreciated the reference to Swartz's autobiography in chapter 3 where Boaler quotes Swartz as saying that "being quick or slow isn't really relevant". As a person who is a bit of a slow thinker, I always appreciate it when I see this in print somewhere. It's interesting to see what's happening in in NY now in regards to the time constraints on standardized tests in grades 3-8. If I'm reading this Washington Post Article correctly, it looks like time limits have been removed from the tests. Even at that, many parents have joined the opt-out movement.
Your post reminded me of an article I read a while back about lesson study (a Japanese process of practitioner research and professional development). The article features a case study of a US school that is starting to use lesson study. The project involved teachers meeting to plan lessons and observe them together. Visiting Japanese teachers were involved in observations as well and shared their reactions:
"For example, during a post-lesson discussion, one Japanese practitioner asked why a student’s incorrect solution had not been discussed by the teacher in order to reveal similarities and differences among students’ solution methods. This question revealed an aspect of the Japanese teacher’s philosophy of mathematics instruction that was surprising to some of the US teachers and that is still talked about by the catch-phrase ‘'mistakes can be treasures.’'"
Perry, R. R., & Lewis, C. C. (2009). What is successful adaptation of lesson study in the US?. Journal of Educational Change, 10(4), 365-391. http://www.lessonresearch.net/success_adapt.pdf
This way of thinking about mistakes is part of a larger perspective on student learning, and I don't think it comes easily to us. I know my first instinct is to help students fix mistakes quickly and move on. For example, I taught a lesson recently on exponential growth which involved doubling numbers many times to find out a bacteria population after, say, 10 hours. Students got exhausted with the doubling pretty quickly.
This is fine, but at some point students want to say, "If there are 64 bacteria after 2 hours, then there would be 320 bacteria after 10 hours, since 64 x 5 = 320 and 2 x 5 = 10. I don't need to keep doing this over and over." When this happened in class, my instinct was to show students why this couldn't be true. That may or may not be the right way to respond, but either way it would be great for the class to look at this misconception afterwards. Even though it doesn't work here, proportional reasoning is a powerful tool. I don't want to discourage students from thinking about numbers that way. And in order to understand exponential growth, it makes sense to see how the growth is not happening in a linear, proportional way.
Although I was already familiar with the new brain research on the value of mistakes, what struck me was the fact that our brains grow even when we don't know that we've made a mistake! This is powerful. One of the phrases I have adopted with students is "not yet." I am fortunate to teach developmental math classes in which students can retake a test if they do not pass the first time. I never write "F" on their paper but rather "not yet". It makes a difference in their attitude about wanting to keep trying, which helps them move from a fixed to a growth mindset.I also do not correct students' mistakes. I circle the number of the problems in which there is an error and I allow them to review their work and fix their errors. It is a great tool for knowing the depth of understanding because students who have a good conceptual understanding can usually find their errors pretty quickly .
It will take time to change the culture of the mathematics classroom from being an environment of "getting answers" to one of thinking deeply and asking questions while engaging in solving problems, but I believe we are at an exciting time of being math educators and I am glad to be part of this movement!
Thanks for sharing this approach to giving feedback to learners. It makes so much sense in that it facilitates the continuation of learning. I know that as a student I would appreciate having the opportunity to correct mistakes.
I have read some of Jo Boaler and Carol Dweck's work in the past, but after reading chapter one on The Brain and Mathematics Learning, I asked my students a few questions. Out of the 12 students that were present today, only one thought that they were "pretty good" in math and the rest thought they were "bad" in math before they started their Adult Education classes. When I pushed and asked, "what does being pretty good look like," the student said he was fast, finished his work early and then waited for others to finish up. The others who labeled themselves as "bad" in math, said that they never had the right answers and used to make up numbers just to complete the work, and always felt behind because they didn't understand what happened the day before, so they were totally lost.
As a brand new elementary school teacher 20 plus years ago, I must say there was a sense of relief that I only had to teach "easy" math. My own instructional strategies were quite traditional until one day I walked past a neighboring classroom. A fourth grade teacher was teaching math and there were Base Ten blocks scattered all over her classroom. My students were in art at the time, so I asked to observe as I was curious about all the noise, enthusiasm and blocks. This teacher was showing what division looked like - she had a number like 867 divided by 4. Because she was dividing by four the students knew that four pieces of paper went onto the floor to represent the groups. Then she had a student go to the "bank" where the blocks were stored and pick up 8 hundreds, six tens and seven ones. (Great review on place value). Then in group so three the students talked over how to divide the 867 into 4 groups. By starting with the 8 hundred blocks, it fit perfectly into 4 groups - each group having 2. Then the 6 tens - 4 of them could go into the groups, but what to do with the "left over" 2 tens. They could trade them in for twenty ones. Well you get the point - this one lesson completely changed my teaching career. Hearing the students talk about why and how numbers work - had me hooked. Students had a conceptual understanding (varying degrees) of division and the lesson encouraged challenge, discussion and consensus - not to mention excitement for learning. Since that day, I have tried to make math resemble what I saw on that day over 20 years ago. Now that I'm in Adult Education, some of the challenges are the same, but many of our adult learners bring a lot more baggage to the classroom. This creates challenges, especially in the area of struggling and making mistakes as being a part of the process of learning.
The third chapter laid out nicely the need for reasoning in math; looking at problems, posing questions, taking other people's perspectives, and adjusting your thinking as you collect more data and/or evidence. In this way, math becomes engaging and real in the lives of our students.
I also work in adult education and asked my students yesterday how they felt about math. Most replied that they hated it or didn't like it. We had a conversation about it an many have fears that they can't do it or that they won't be able to understand it.
I really liked the story about the Base Ten Blocks. I might have to dig around and find some of those. I like this idea and I know that I have students who cannot divide and I believe a big part of that reason is because they don't really understand what it means or they can't visualize it. I have found that many of my adult learners are very visual.
One thing that struck me in chapter two was the strong language against tests and grades. I've felt for a long time that neither tests nor grades are useful for helping students learn math. Last year I didn't give quizzes in the first half of the year and while I marked answers as right or wrong on homework, I also provided written feedback. When we asked students for feedback in the middle of the year, they uniformly said that they wanted to have quizzes so they would know how they were doing. At the beginning of this year I asked my students how they felt about quizzes and they all said they wanted them. After returning the first quiz on which I had reported their scores as a fraction of points earned out of possible points, several students were insistent on figuring out what their letter grade should be. I gave another quiz recently and one student got so few right answers that I couldn't bear to mark up the quiz. We had some time to process it together - at first he was extremely upset, but after a while he came to accept that anxiety had interfered with his problem-solving ability and the quiz didn't really reflect all that he had learned. I'm not sure that this was worth all the pain for him, but maybe it was because it helped him start to think about managing his anxiety.
I'm trying to create a classroom culture that values reasoning, mistakes, creativity, and flexible thinking, and quizzes and grades seem to go against that. I also want to respect my students' priorities and not deny the fact that the main reason they are in my class (for them) is to prepare for and pass a high-school equivalency test. I'm looking forward to reading more about this in chapter eight.
We've experimented with quizzes in science, but they are are more self-assessments for students. Some of them focus on reviewing the key points from the previous class, particularly the things you want to build off of. Students do them solo, then discuss in groups and then we talk all together if need be. Then everyone is back in the mindset of where the last class left off, including people who were absent. The other kind of quiz is a prediction guide, where we ask students some questions they will be able to answer by the end of the class, but we ask them in the beginning of class. The idea is to engage their curiosity and background knowledge and to get a sense of what students know. Then at the end of class we have students go back and correct any answers they need to and add a reflection. I think it would work just as well in math, I just haven't had my own math class to try it out with.
There is 7th grade teacher named Leah Alcala, who has a few videos on The Teaching Channel website. One of that that I really like is called, My Favorite No which is a strategy I have used with students and is a really great way to celebrate student mistakes in a way that everyone can learn from (and which can be anonymous, though students sometimes identify themselves). Basically the teacher does a quick assessment on index cards or something like that, collects them all and then shares a few of her favorite mistakes on the board, asking the class what the mistake is and why they think she liked it.
The other video she has is called Highlighting Mistakes: A Grading Strategy where she describes her process for providing feedback on tests. When she assesses tests, she does not put grades on them. The first thing she does is go through and highlight all the moments where students made a mistake. Then she she goes back through them all and does this thing called, "flow through credit" where she takes off points for the moment the student makes a mistake, but if they do all of the following steps correctly within the context of that mistake, she doesn't take off points. Also, if the student has an idea in their head and keeps repeating the same mistake, she only takes off points for the first time because they just need to have one conversation.
Then in class, she starts off doing a few Favorite No's and discusses them as a class. Then she returns the tests to students with no grades (though she makes the grades available to them online the following day). She asks them to try to understand their mistakes, with the option to ask their neighbors or the teacher. And she gives them an opportunity to retake the test.
This doesn't answer your larger question about dealing with the disconnect between wanting to prepare mathematical thinkers and preparing students for problematic high stakes assessments that don't necessarily value or prioritize deep mathematical thinking. But I think it is an interesting way to try to have students learn from their tests as opposed to just looking at their grade and deciding from that whether or not they are good at math or not and I was curious what other people thought, or if anyone had tried anything similar.
I could relate to your post as I also work with students who are preparing for the HiSET tests in a special education setting. I have one student who consistently wants to know what his "grade" is. I try very hard to let him know that I don't "grade" anything but rather am working to close the gap from where he is currently in his understanding to where he needs to be for successful completion of the tests. I think students have been conditioned to this "grade mindset" rather than a conceptual learning mindset. This is hard to overcome. I was glad to read in your post that he was learning new strategies to working through his anxiety.
I definitely appreciate this approach to learning. In high school, in the 1970s I was very good with Geometry and not so much with Algebra. I found I understood it up to a point and then I "hit the wall." I am not sure if it was the teachers or the fact that Geometry just naturally is more visual. As I mentioned elsewhere I find myself in the position of not considering myself to be a "Math Wizard" (merely competent enough at the high school level) and trying to coach an adult learner through severe Math anxiety. For instance, at the beginning it was all pretty difficult for this student, but now I find the multiplication is easier for her as we have moved on to tasks that involve long division. Every time we move on to introduce new things, the old stuff becomes a little less sticky. However, the long division remains a sticking point especially when she is asked to try it in front of me (add to that having maybe learned a slightly different method in South America). All of this I can relate to, because I don't necessarily perform well with someone looking over my shoulder.
I definitely want to use things like the Paper Folding exercise to introduce Geometry (Triangles, Squares, determining area) because I think it might help to make sense of things that may seem difficult at first.
I am just hoping this will eventually lead to less anxiety on an actual timed test which is a mix of word problems, geometry, algebra and functions/linear equations. The good thing that the TASC seems to do is to rely less on memorizing formulas. In fact, I think they give each student a page of formulas for volume, circumference and area of spheres and circles.
Teaching in the adult correctional setting, I find that dealing with mindset is often more of my job than actually instruction in content areas. The young men that I work with have lost much of their desire to improve academically; they don't see any point in it and consider it being 'forced' to attend classes. Many of them have been told for years that they will never amount to anything or they are 'no good' to the point that they believe it to be true. I wholeheartedly believe in the statement made on page 9 regarding the link between the message teachers give and the achievement levels of the students. I tell my students that I believe they are capable of learning anything that is presented to them and they can achieve beyond where they are currently but this will only happen when they believe that themselves.
One practice I use in my classroom is to give the students the set of problems or work to do along with an 'answer key' sheet. I explain to them that it's not the answer I want, rather what I am looking for is that they understand how to get the answer. After considering the value of mistakes from the reading, I am considering adding an activity similar to one I had in graduate class where we needed to identify the computational error/misconception from a sampling of student work.
Another change I am considering for the upcoming academic year is to start by showing students the video from YouCubed referenced. I really liked how this approached the concept of learning math and abilities. I think it might mean more when I say how much I believe in them if they hear this concept from another source, not just me. Sometimes I feel that they believe I "have" to say that to them rather than believing that I actually think it.
Finally, I plan to do more activities similar to those presented with students to build understanding of concepts and maybe lighten things up to make learning fun again. If they can enjoy attending class, I believe they may begin to put more effort into actually thinking through things. It's at least worth a try.
I heard this past summer in a professional development class that I took that between 70%-80% of all people in the United States hate math. That statistic did not surprise me but it unreal when we thing about 7 or 8 out of 10 people do not like math. I think a lot of this stems from not having an elementary school teachers that does not enjoy math or teaching math. Also, not knowing their math tables. Plus, not having the proper mindset. Also, at any point we change this mindset and it is up to us to have this change occur. That is huge and we can change that statistic by just changing their mindset from fixed to a growth mindset. I tell my students what I want from them the most in my classes is that their self esteem and confidence increase the most. Not that they get a 18 on their HiSet test. I love the feeling of accomplishment if my students leave feeling better about themselves then when entered out program. I take pride in this. I think it totally starts with changing how they feel about themselves.
Comments
Several ideas resonated with me from these chapters. The first is around the power of mistakes. Our learners enter our classrooms with the notion that math is either right or wrong and you must be right to be successful. The “fear of failure” permeates and I feel holds back growth in understanding mathematics. I would agree with Eric that “mistake” might be a misnomer. One definition of mistake is “an error in action, calculation, opinion, or judgment caused by poor reasoning, carelessness, insufficient knowledge, etc.” I would rather we change that definition which seems negative to me to rather we hit a dead end, a wall and need to change direction. It might not be caused by “poor reasoning” at all but very sound reasoning that didn’t lead to the answer we were seeking. I was struck when I visited the Kennedy Space Center how many “failures” they had before a rocket was successfully launched. And my favorite movie Apollo 13, shows true perseverance with the scene of trying to figure out how many switches could be turned off to have the least amount of power used. The astronaut kept trying and trying several approaches until finally one path was found. When mistakes are honored, perseverance is strengthened. My favorite quote that Patricia Helmuth shared with ANN members was “mistakes are expected, respected, and inspected”. Imagine if every math classroom had this quote posted as learners entered.
A second topic was that of speed, that speed means smart. This is found directly in the tests our learners must face. Everything is timed giving no credence to the importance of thinking through for problem solving. One test, the Accuplacer, is at least untimed which would allow students time to truly think through finding a solution. Unfortunately the test itself is a lot of symbol manipulation but at least students could reason through. Speed is also focused on doing computation quickly which is such a small part of understanding and exploring mathematics. I loved the Fortune 500 most valued skills list. Computational skills were at the top in the 70’s when I first began teaching. Now that ranks near the bottom of the list today.
Lastly what I am most excited by Boaler’s and other’s research is what has been found through cognitive science research. My friend is a Wilson reader trainer and I remember being envious of those in the reading world who had mountains of evidence about how children learn to read but when I looked for articles on how children learn mathematics very little was found. I finally was introduced to David Sousa’s book “How the Brain Learns Mathematics” as my first book of evidence. Now more has been discovered especially around visualizing mathematics and how the brain functions using mathematics. This research I find exciting and hopefully will be further proof why we need to change our practice in the classroom.
My original education and training was in foreign language teaching. Our attempt to re-frame the concepts of "mistake" and "error" reminds me of William Nemser's idea of "approximative systems" in second language learning. Instead of comparing language learners' developing languages to either their native language or the perfect target language, Nemser suggested that there is a lot of value in looking closely at the learners' current, "approximative" system for all the patterns and stages of the progression that they necessarily must pass through. For example, it would be silly to consider "drived" as a "mistaken" past tense form of the verb "to drive" if the learner's progression hasn't included the concept of strong/irregular verbs. So, it's not an error, it's an approximation that will likely adjust closer to the target language as the student progresses. I, of course, don't know whether language and math are even comparable in terms of their development, but we do often refer to math as a new language and it is an interesting approach.
First let me say that the first two points you speak about also caught my attention. I'm very interested in the discussion I see happening about what constitutes a mistake and what exactly makes our brains grow. In chapter 2 on page 13, Jo Boaler gives the example of the entrepreneur that founded Starbucks. His first attempts at replicating Italian coffee shops did not take off, and it was his going "back to the drawing board" that eventually resulted in the Starbucks that became so successful. I don't really see his first attempts as mistakes, per say: rather learning experiences that led to success, similar to what you speak about in relation to space exploration, and what we see in our students when they try something that doesn't work and continue to persevere until they find something that does! One teacher at a workshop I facilitated worded this process really well when she said that every answer a person comes up with (when trying to solve a math problem) is honored as a thoughtful math step, which might not be leading to "the solution", but does lead to some more math thinking. Brain growth! For more on "mistakes are expected, respected, and inspected" follow this link to CollectEdNY.
I also appreciated the reference to Swartz's autobiography in chapter 3 where Boaler quotes Swartz as saying that "being quick or slow isn't really relevant". As a person who is a bit of a slow thinker, I always appreciate it when I see this in print somewhere. It's interesting to see what's happening in in NY now in regards to the time constraints on standardized tests in grades 3-8. If I'm reading this Washington Post Article correctly, it looks like time limits have been removed from the tests. Even at that, many parents have joined the opt-out movement.
Patricia,
Your post reminded me of an article I read a while back about lesson study (a Japanese process of practitioner research and professional development). The article features a case study of a US school that is starting to use lesson study. The project involved teachers meeting to plan lessons and observe them together. Visiting Japanese teachers were involved in observations as well and shared their reactions:
"For example, during a post-lesson discussion, one Japanese practitioner asked why a student’s incorrect solution had not been discussed by the teacher in order to reveal similarities and differences among students’ solution methods. This question revealed an aspect of the Japanese teacher’s philosophy of mathematics instruction that was surprising to some of the US teachers and that is still talked about by the catch-phrase ‘'mistakes can be treasures.’'"
Perry, R. R., & Lewis, C. C. (2009). What is successful adaptation of lesson study in the US?. Journal of Educational Change, 10(4), 365-391. http://www.lessonresearch.net/success_adapt.pdf
This way of thinking about mistakes is part of a larger perspective on student learning, and I don't think it comes easily to us. I know my first instinct is to help students fix mistakes quickly and move on. For example, I taught a lesson recently on exponential growth which involved doubling numbers many times to find out a bacteria population after, say, 10 hours. Students got exhausted with the doubling pretty quickly.
time (minutes) bacteria 0 1 20 2 40 4 60 8 80 16 100 32 120 64 140 128This is fine, but at some point students want to say, "If there are 64 bacteria after 2 hours, then there would be 320 bacteria after 10 hours, since 64 x 5 = 320 and 2 x 5 = 10. I don't need to keep doing this over and over." When this happened in class, my instinct was to show students why this couldn't be true. That may or may not be the right way to respond, but either way it would be great for the class to look at this misconception afterwards. Even though it doesn't work here, proportional reasoning is a powerful tool. I don't want to discourage students from thinking about numbers that way. And in order to understand exponential growth, it makes sense to see how the growth is not happening in a linear, proportional way.
Eric
Although I was already familiar with the new brain research on the value of mistakes, what struck me was the fact that our brains grow even when we don't know that we've made a mistake! This is powerful. One of the phrases I have adopted with students is "not yet." I am fortunate to teach developmental math classes in which students can retake a test if they do not pass the first time. I never write "F" on their paper but rather "not yet". It makes a difference in their attitude about wanting to keep trying, which helps them move from a fixed to a growth mindset.I also do not correct students' mistakes. I circle the number of the problems in which there is an error and I allow them to review their work and fix their errors. It is a great tool for knowing the depth of understanding because students who have a good conceptual understanding can usually find their errors pretty quickly .
It will take time to change the culture of the mathematics classroom from being an environment of "getting answers" to one of thinking deeply and asking questions while engaging in solving problems, but I believe we are at an exciting time of being math educators and I am glad to be part of this movement!
Sarah,
Thanks for sharing this approach to giving feedback to learners. It makes so much sense in that it facilitates the continuation of learning. I know that as a student I would appreciate having the opportunity to correct mistakes.
I have read some of Jo Boaler and Carol Dweck's work in the past, but after reading chapter one on The Brain and Mathematics Learning, I asked my students a few questions. Out of the 12 students that were present today, only one thought that they were "pretty good" in math and the rest thought they were "bad" in math before they started their Adult Education classes. When I pushed and asked, "what does being pretty good look like," the student said he was fast, finished his work early and then waited for others to finish up. The others who labeled themselves as "bad" in math, said that they never had the right answers and used to make up numbers just to complete the work, and always felt behind because they didn't understand what happened the day before, so they were totally lost.
As a brand new elementary school teacher 20 plus years ago, I must say there was a sense of relief that I only had to teach "easy" math. My own instructional strategies were quite traditional until one day I walked past a neighboring classroom. A fourth grade teacher was teaching math and there were Base Ten blocks scattered all over her classroom. My students were in art at the time, so I asked to observe as I was curious about all the noise, enthusiasm and blocks. This teacher was showing what division looked like - she had a number like 867 divided by 4. Because she was dividing by four the students knew that four pieces of paper went onto the floor to represent the groups. Then she had a student go to the "bank" where the blocks were stored and pick up 8 hundreds, six tens and seven ones. (Great review on place value). Then in group so three the students talked over how to divide the 867 into 4 groups. By starting with the 8 hundred blocks, it fit perfectly into 4 groups - each group having 2. Then the 6 tens - 4 of them could go into the groups, but what to do with the "left over" 2 tens. They could trade them in for twenty ones. Well you get the point - this one lesson completely changed my teaching career. Hearing the students talk about why and how numbers work - had me hooked. Students had a conceptual understanding (varying degrees) of division and the lesson encouraged challenge, discussion and consensus - not to mention excitement for learning. Since that day, I have tried to make math resemble what I saw on that day over 20 years ago. Now that I'm in Adult Education, some of the challenges are the same, but many of our adult learners bring a lot more baggage to the classroom. This creates challenges, especially in the area of struggling and making mistakes as being a part of the process of learning.
The third chapter laid out nicely the need for reasoning in math; looking at problems, posing questions, taking other people's perspectives, and adjusting your thinking as you collect more data and/or evidence. In this way, math becomes engaging and real in the lives of our students.
I also work in adult education and asked my students yesterday how they felt about math. Most replied that they hated it or didn't like it. We had a conversation about it an many have fears that they can't do it or that they won't be able to understand it.
I really liked the story about the Base Ten Blocks. I might have to dig around and find some of those. I like this idea and I know that I have students who cannot divide and I believe a big part of that reason is because they don't really understand what it means or they can't visualize it. I have found that many of my adult learners are very visual.
One thing that struck me in chapter two was the strong language against tests and grades. I've felt for a long time that neither tests nor grades are useful for helping students learn math. Last year I didn't give quizzes in the first half of the year and while I marked answers as right or wrong on homework, I also provided written feedback. When we asked students for feedback in the middle of the year, they uniformly said that they wanted to have quizzes so they would know how they were doing. At the beginning of this year I asked my students how they felt about quizzes and they all said they wanted them. After returning the first quiz on which I had reported their scores as a fraction of points earned out of possible points, several students were insistent on figuring out what their letter grade should be. I gave another quiz recently and one student got so few right answers that I couldn't bear to mark up the quiz. We had some time to process it together - at first he was extremely upset, but after a while he came to accept that anxiety had interfered with his problem-solving ability and the quiz didn't really reflect all that he had learned. I'm not sure that this was worth all the pain for him, but maybe it was because it helped him start to think about managing his anxiety.
I'm trying to create a classroom culture that values reasoning, mistakes, creativity, and flexible thinking, and quizzes and grades seem to go against that. I also want to respect my students' priorities and not deny the fact that the main reason they are in my class (for them) is to prepare for and pass a high-school equivalency test. I'm looking forward to reading more about this in chapter eight.
Sarah,
We've experimented with quizzes in science, but they are are more self-assessments for students. Some of them focus on reviewing the key points from the previous class, particularly the things you want to build off of. Students do them solo, then discuss in groups and then we talk all together if need be. Then everyone is back in the mindset of where the last class left off, including people who were absent. The other kind of quiz is a prediction guide, where we ask students some questions they will be able to answer by the end of the class, but we ask them in the beginning of class. The idea is to engage their curiosity and background knowledge and to get a sense of what students know. Then at the end of class we have students go back and correct any answers they need to and add a reflection. I think it would work just as well in math, I just haven't had my own math class to try it out with.
There is 7th grade teacher named Leah Alcala, who has a few videos on The Teaching Channel website. One of that that I really like is called, My Favorite No which is a strategy I have used with students and is a really great way to celebrate student mistakes in a way that everyone can learn from (and which can be anonymous, though students sometimes identify themselves). Basically the teacher does a quick assessment on index cards or something like that, collects them all and then shares a few of her favorite mistakes on the board, asking the class what the mistake is and why they think she liked it.
The other video she has is called Highlighting Mistakes: A Grading Strategy where she describes her process for providing feedback on tests. When she assesses tests, she does not put grades on them. The first thing she does is go through and highlight all the moments where students made a mistake. Then she she goes back through them all and does this thing called, "flow through credit" where she takes off points for the moment the student makes a mistake, but if they do all of the following steps correctly within the context of that mistake, she doesn't take off points. Also, if the student has an idea in their head and keeps repeating the same mistake, she only takes off points for the first time because they just need to have one conversation.
Then in class, she starts off doing a few Favorite No's and discusses them as a class. Then she returns the tests to students with no grades (though she makes the grades available to them online the following day). She asks them to try to understand their mistakes, with the option to ask their neighbors or the teacher. And she gives them an opportunity to retake the test.
This doesn't answer your larger question about dealing with the disconnect between wanting to prepare mathematical thinkers and preparing students for problematic high stakes assessments that don't necessarily value or prioritize deep mathematical thinking. But I think it is an interesting way to try to have students learn from their tests as opposed to just looking at their grade and deciding from that whether or not they are good at math or not and I was curious what other people thought, or if anyone had tried anything similar.
yours in productive struggle,
Mark
Sarah,
I could relate to your post as I also work with students who are preparing for the HiSET tests in a special education setting. I have one student who consistently wants to know what his "grade" is. I try very hard to let him know that I don't "grade" anything but rather am working to close the gap from where he is currently in his understanding to where he needs to be for successful completion of the tests. I think students have been conditioned to this "grade mindset" rather than a conceptual learning mindset. This is hard to overcome. I was glad to read in your post that he was learning new strategies to working through his anxiety.
I definitely appreciate this approach to learning. In high school, in the 1970s I was very good with Geometry and not so much with Algebra. I found I understood it up to a point and then I "hit the wall." I am not sure if it was the teachers or the fact that Geometry just naturally is more visual. As I mentioned elsewhere I find myself in the position of not considering myself to be a "Math Wizard" (merely competent enough at the high school level) and trying to coach an adult learner through severe Math anxiety. For instance, at the beginning it was all pretty difficult for this student, but now I find the multiplication is easier for her as we have moved on to tasks that involve long division. Every time we move on to introduce new things, the old stuff becomes a little less sticky. However, the long division remains a sticking point especially when she is asked to try it in front of me (add to that having maybe learned a slightly different method in South America). All of this I can relate to, because I don't necessarily perform well with someone looking over my shoulder.
I definitely want to use things like the Paper Folding exercise to introduce Geometry (Triangles, Squares, determining area) because I think it might help to make sense of things that may seem difficult at first.
I am just hoping this will eventually lead to less anxiety on an actual timed test which is a mix of word problems, geometry, algebra and functions/linear equations. The good thing that the TASC seems to do is to rely less on memorizing formulas. In fact, I think they give each student a page of formulas for volume, circumference and area of spheres and circles.
Teaching in the adult correctional setting, I find that dealing with mindset is often more of my job than actually instruction in content areas. The young men that I work with have lost much of their desire to improve academically; they don't see any point in it and consider it being 'forced' to attend classes. Many of them have been told for years that they will never amount to anything or they are 'no good' to the point that they believe it to be true. I wholeheartedly believe in the statement made on page 9 regarding the link between the message teachers give and the achievement levels of the students. I tell my students that I believe they are capable of learning anything that is presented to them and they can achieve beyond where they are currently but this will only happen when they believe that themselves.
One practice I use in my classroom is to give the students the set of problems or work to do along with an 'answer key' sheet. I explain to them that it's not the answer I want, rather what I am looking for is that they understand how to get the answer. After considering the value of mistakes from the reading, I am considering adding an activity similar to one I had in graduate class where we needed to identify the computational error/misconception from a sampling of student work.
Another change I am considering for the upcoming academic year is to start by showing students the video from YouCubed referenced. I really liked how this approached the concept of learning math and abilities. I think it might mean more when I say how much I believe in them if they hear this concept from another source, not just me. Sometimes I feel that they believe I "have" to say that to them rather than believing that I actually think it.
Finally, I plan to do more activities similar to those presented with students to build understanding of concepts and maybe lighten things up to make learning fun again. If they can enjoy attending class, I believe they may begin to put more effort into actually thinking through things. It's at least worth a try.
I heard this past summer in a professional development class that I took that between 70%-80% of all people in the United States hate math. That statistic did not surprise me but it unreal when we thing about 7 or 8 out of 10 people do not like math. I think a lot of this stems from not having an elementary school teachers that does not enjoy math or teaching math. Also, not knowing their math tables. Plus, not having the proper mindset. Also, at any point we change this mindset and it is up to us to have this change occur. That is huge and we can change that statistic by just changing their mindset from fixed to a growth mindset. I tell my students what I want from them the most in my classes is that their self esteem and confidence increase the most. Not that they get a 18 on their HiSet test. I love the feeling of accomplishment if my students leave feeling better about themselves then when entered out program. I take pride in this. I think it totally starts with changing how they feel about themselves.