In the comment section below, share at least one "aha" moment (evidence) and "why." Read comments by other group members and respond to posts by at least two other teachers.
My "aha" moment is on page 19, Ease the Struggle. The importance of structuring experiences for students to discover concepts and learn by exploring and talking with each other, rather than traditional pedogogy model of explaining, demonstrating, and "going over" so that students are creating their own meaning....it strengthens their connection to it. Memorizing strategies or procedures is temporary and limited to only that specific type of problem.
My other "aha" is on page 14, explaining the dichotomy of "knowing" and "understanding." The map and subway anology is a perfect way to understand the difference. When we explore something on our own, our brain organizes material and having a complete map shows everything in relationship to each other...so that one always knows the connection around them and cannot get lost. One can always make a slight adjustment or use information to orient oneself. Subway dirctions (strategies or memorized procedures) are only helpful if they are remembered correctly, if there are no detours, and are only applicable in one and only one instance (getting from point A to point B). We want our learners to develop working maps of math concepts, not simply follow a memorized route!
I really liked how the author used this comparison too! We want our students to be able to apply what they know when they are in different situations rather than just be able to solve a problem following specific steps. It seems like it's all about equipping students to be able to problem solve when given a variety of problems and scenarios!
I too liked how the author compared knowing or understanding. For a student to understand why they do a problem is so much better that just to know how to do that problem.
Yes, great example. I think of those poor souls that do the correct algorithm on their GPS and end up in a field instead of their destination and can't figure out why. We have to teach kids the correct concepts and strategies so they recognize when and why an error was made.
I still remember the day in seventh grade when I realized WHY you divide, multipy, subtract, and bring down to solve a long division problem. I was euphoric!
I believe that it is important to introduce the math concepts in different ways to help the different learners to understand the best possible way to solve the problems. Instead of repeating the directions louder to the students we need to change our presentation of the lesson.
My first "aha" moment was when the author explained the difference between procedural and conceptual learning and how it really important to build and emphasize a foundation based on a thorough understanding of concepts side before introducing procedures and algorithms. This is especially important for students who struggle with math.
My second "aha" moment was how the author emphasized helping students make connections across math domains by finding ways to connect different concepts. I can see how this would help students to see the overall picture of a math concept and help with building schema.
This is just like with reading, making connections is the best way to comprehend. It does prove that we need to spend A LOT more time on one concept before moving onto another. From everything I have learned with the Common Core Standards, they will allow for more time for students to really understand the conceptual before the procedural.
I also liked the examples used in the book. Math concepts linked to things that students would find truely engaging, such as how many bubble can be blown in a time allotment and graphing data. Much more engaging for students to be engaged, collecting data, and discussing with partners than reading in a text book about someone else doing an experiment. Think of how strong those connections were made during that math lesson!!
It really reminds me of inquiry science. Students get to discover and experiment and make connections and have their own aha! moments. In my mind, this is what truly builds connections.
A connection to previous learning for me was on page 4, I have heard several times the need for more research and interventions for struggling learners in math. We have an abundant amount for reading, and once students are proficient readers they can build on those foundations to excel, but with Math there is always the conceptual demands and they are to know more information as they move up in grades. This makes for an "aha" moment that I had a few weeks back at a conference where the presenter shared with us the increasing number of students in college that are enrolled in remedial math classes. The numbers are staggering, even right here in Kansas.
The big "aha" for me was on page 18: very successful teachers who are moved to another grade level will often take a year or two to achieve the same level of success at the new grade level.
On Melisa's second "aha" that it may take even successful teachers a year or two to achieve the same level of success as previously when teaching a new grade level. I find this to be true as I delve into 4th grade math...trying to really learn the material in a way that I can best explain/show/model to students in several different ways (and not just louder and slower.....).
Great point about the expansion of math knowledge. This does set it apart from other subjects. It also makes it difficult to keep a continuum from year to year, school to school, state to state, where the same foundations need to be taught for all students.
My "aha" was when the author talked about using machines to teach math concepts. I have taught with that idea, so I was glad to see it in print. To me it seemed the students understood it better when it was working with something their already knew about.
My "aha" was when the author talked about the importance of making connections to other mathematical concepts. He made the analogy to children learning to swim outside of the pool. No matter how much they practiced the skill of swimming outside of the water, they weren't able to apply the learning to the actual act of swimming IN the water. Students have the same problem when they are taught math problems in isolation. Until they can make the connections to actual applications, they often struggle with the math. Teachers need to teach math within the problems in order for students to see the need to learn the math. He stated that the students will do a better job when they are allowed to explore a concept first and then practice skills that come from the problem. I think that we too often teach the skill and then give students problems that support that skill. We need to spend more time allowing students to 'discover' their own methods for solving problems and then allow them to use methods that make sense to them.
I liked the analogy as well because it still speaks to teaching explicitly while "in the water." We aren't just throwing them in and hope they can figure it out. We are still guiding their exploration and discovery.
I agree- it was a great analogy to explain. We want to prepare students but often it doesn't make sense when they don't know what we're "preparing" them for.
My "aha" is simply seeing the real world implications of environmental factors impacting student learning in general but specifically in math. My lowest students are the ones with the least support from home. I love teaching conceptually and searching for the why of mathematics, but if the student misses even one class peroid, it becomes difficult to send that conceptual lesson home for homework and expect mom or dad to help them. The students have to be present and for some families it doesn't seem to be a priority.
I also find a great deal of resistance from parents as we teach our "new math" to students. It takes a great deal of work with the parents to help them understand what it is that we are trying to do. Even then you run into parents who react negatively to the idea of changing the way that we teach as it "worked for them".
My "aha" moment was when the book talked about the use of algorithms in instuction. I totally agree with the book when it said that an algorithm is a shortuct that is not useful unless a student conceptually understands why it works.
I agree, too, and am looking forward to hopefully learning how to help students who are "stuck" to using algorithms without conceptual understanding. I have found that students, and not just teachers, have at times gotten used to shortcuts and want to just know how to get the answer without thinking about what they are doing in a broader sense.
I agree as well. I am noticing this now more than ever when my students learn to use different algorithms to solve basic addition and subtraction. We teach these to help students understand more about place value. However, it was instantly clear that my students had a fuzzy concept of place value. After we were able to make the connection between place value and why it is important in solving multi-digit equations, student understanding was improved.
My "aha" moment was on page 14 when it speaks about developing true understanding of a concept rather than the procedure for completing a process allows the student to use the concept as a tool. Through using the concept as a "tool" they can then develop a much greater understanding in deeper mathematical ideas.
I agree and actually think this is part of why I was only good at math for certain teachers or classes- if I didn't get "the steps", I didn't have deep enough understanding to transfer skills to other classes in high school and college (without a lot of help) ;)
"Memory difficulties, for example, might have more to do with instruction than with a student's memory." (Tapper p. 6) I found this VERY interesting because as an ESS Teacher, a lot of students appear to have memory problems, but the issue could be that math concepts are not connected with other information. The author goes on to say that what appears as "gaps" may be lapses in concept development. It can be challenging as a teacher to stick with students to enable them to understand concepts instead of rushing to teach procedures without them having the broader understanding.
I agree, Bethany. This is an idea that really made me think about what we need to do with our students who are struggling. I would like to see some sort of diagnostic test to see what concepts a student does not have....(I think that there is something like this out there, I just cannot think of what it is...and I have never used it!) This could possibly help us begin to fill those gaps and allow our students to begin to move forward.
Mrs. Masimula, I also found the information about making connections very interesting. I think that providing students with connections to concepts they already know will further there understanding rather than teaching concepts in isolation. Furthermore, I think that when we help students make those connections, we should explicitly model how that connection is being made. That way, they are able to recognize the connection and hopefully be able to begin making connections on their own.
I agree! It is so hard for our students to remember directions from one minute to the next, remember and memorize facts, etc. If they do not have a true conceptual understanding, then this is an issue of instruction, not memory. I also believe that exposure can be an issue, especially when a curriculum spirals and there is no time spent in mastering concepts.
I had many aha moments when reading this book. The biggest moment was on page 8 reading My Story. "When kids have difficulty with anything at school, it seems, people are always asking them to demonstrate their difficulty." I think it just mad eme realize how defeated I would feel if I kept having to perform for different people knowing that I didn't get how to do it. Also, thinking about algorithms this year, I think I will dump teaching lattice multiplication this year! How about everyone else who teaches that?
I thought about writing the same thing. It reminds me of when a friend of mine had serious back pain and the doctors kept asking her to bend forward and touch her toes so they could feel her spine. In tears every time! Testing over and over is like torture for those kiddos. I would just not even try on the parts I knew just because there will be another test another day.
No to lattice! I want students to "break apart" the place values and explore multiplication this way. I think that teaching them lattice along with having them develop a conceptual understanding, THEN building to the algorithm, would be too confusing. Too many students rely too heavily on this method, and they don't know how it works, and it tends to slow them down. THANK YOU for bringing this up!
As a beginning teacher, it was extremely helpful to read about the difference between procedural and conceptual learning. The example about the difference between knowing how to use a map (or set of directions) and understanding how to get somewhere by previous experience in the area, was a great analogy that will help me analyze my future planning and teaching of math.
My "aha" moment was students having very emotional responses when they can't see the connection between the concepts they are learning and their lives. I can see this happening clearly across all content areas. Students first need to see the connection with their own lives before they will engage and feel that what they are learning is important. Such a simple thing, make it real and students will be more receptive.
My "aha" moment was when Tapper compared how people do math to finding our way around a city: you can have a specific set of directions written out for you, but you won't learn how to find your way around the city the next time. Or, you could explore on your own with a map (the big picture) in front of you, and you will begin to see patterns, big ideas, and learn your way around. I feel that all too often, we as teachers fall into the written directions realm, setting out a "recipe" in solving problems instead of teaching students how to solve problems by thinking and trying on their own. Perhaps this can be blamed on time or curriculum constraints, but I have noticed that lessons that give students time to explore and discuss tend to be the most successful.
Ken Reever can't wait to begin!
ReplyDeleteThis is Stephanie Pappal....just testing.
ReplyDeleteHi everyone! Just testing.
ReplyDeleteThis is a test. Only a test=)
ReplyDeleteMy "aha" moment is on page 19, Ease the Struggle. The importance of structuring experiences for students to discover concepts and learn by exploring and talking with each other, rather than traditional pedogogy model of explaining, demonstrating, and "going over" so that students are creating their own meaning....it strengthens their connection to it. Memorizing strategies or procedures is temporary and limited to only that specific type of problem.
ReplyDeleteI agree that it is a better way to teach math with hands-on experiences than to stand and lecture. Linda Berges
DeleteI also agree that is a better way to teach. It is amazing what students can come up with when they are allowed to figure out problems by themselves.
DeleteMy other "aha" is on page 14, explaining the dichotomy of "knowing" and "understanding." The map and subway anology is a perfect way to understand the difference. When we explore something on our own, our brain organizes material and having a complete map shows everything in relationship to each other...so that one always knows the connection around them and cannot get lost. One can always make a slight adjustment or use information to orient oneself. Subway dirctions (strategies or memorized procedures) are only helpful if they are remembered correctly, if there are no detours, and are only applicable in one and only one instance (getting from point A to point B). We want our learners to develop working maps of math concepts, not simply follow a memorized route!
ReplyDeleteI really liked how the author used this comparison too! We want our students to be able to apply what they know when they are in different situations rather than just be able to solve a problem following specific steps. It seems like it's all about equipping students to be able to problem solve when given a variety of problems and scenarios!
DeleteI too liked how the author compared knowing or understanding. For a student to understand why they do a problem is so much better that just to know how to do that problem.
DeleteYes, great example. I think of those poor souls that do the correct algorithm on their GPS and end up in a field instead of their destination and can't figure out why. We have to teach kids the correct concepts and strategies so they recognize when and why an error was made.
DeleteI still remember the day in seventh grade when I realized WHY you divide, multipy, subtract, and bring down to solve a long division problem. I was euphoric!
DeleteI believe that it is important to introduce the math concepts in different ways to help the different learners to understand the best possible way to solve the problems. Instead of repeating the directions louder to the students we need to change our presentation of the lesson.
ReplyDeleteI so agree. Students learn in many different ways and it is important to introduce math concepts in as many ways as we can.
DeleteI agree too. I love how this book uses real life concepts to teach math.
DeleteMy first "aha" moment was when the author explained the difference between procedural and conceptual learning and how it really important to build and emphasize a foundation based on a thorough understanding of concepts side before introducing procedures and algorithms. This is especially important for students who struggle with math.
ReplyDeleteMy second "aha" moment was how the author emphasized helping students make connections across math domains by finding ways to connect different concepts. I can see how this would help students to see the overall picture of a math concept and help with building schema.
This is just like with reading, making connections is the best way to comprehend. It does prove that we need to spend A LOT more time on one concept before moving onto another. From everything I have learned with the Common Core Standards, they will allow for more time for students to really understand the conceptual before the procedural.
DeleteI also liked the examples used in the book. Math concepts linked to things that students would find truely engaging, such as how many bubble can be blown in a time allotment and graphing data. Much more engaging for students to be engaged, collecting data, and discussing with partners than reading in a text book about someone else doing an experiment. Think of how strong those connections were made during that math lesson!!
DeleteIt really reminds me of inquiry science. Students get to discover and experiment and make connections and have their own aha! moments. In my mind, this is what truly builds connections.
DeleteA connection to previous learning for me was on page 4, I have heard several times the need for more research and interventions for struggling learners in math. We have an abundant amount for reading, and once students are proficient readers they can build on those foundations to excel, but with Math there is always the conceptual demands and they are to know more information as they move up in grades. This makes for an "aha" moment that I had a few weeks back at a conference where the presenter shared with us the increasing number of students in college that are enrolled in remedial math classes. The numbers are staggering, even right here in Kansas.
ReplyDeleteThe big "aha" for me was on page 18: very successful teachers who are moved to another grade level will often take a year or two to achieve the same level of success at the new grade level.
On Melisa's second "aha" that it may take even successful teachers a year or two to achieve the same level of success as previously when teaching a new grade level. I find this to be true as I delve into 4th grade math...trying to really learn the material in a way that I can best explain/show/model to students in several different ways (and not just louder and slower.....).
DeleteGreat point about the expansion of math knowledge. This does set it apart from other subjects. It also makes it difficult to keep a continuum from year to year, school to school, state to state, where the same foundations need to be taught for all students.
DeleteMy "aha" was when the author talked about using machines to teach math concepts. I have taught with that idea, so I was glad to see it in print. To me it seemed the students understood it better when it was working with something their already knew about.
ReplyDeleteMy "aha" was when the author talked about the importance of making connections to other mathematical concepts. He made the analogy to children learning to swim outside of the pool. No matter how much they practiced the skill of swimming outside of the water, they weren't able to apply the learning to the actual act of swimming IN the water. Students have the same problem when they are taught math problems in isolation. Until they can make the connections to actual applications, they often struggle with the math. Teachers need to teach math within the problems in order for students to see the need to learn the math. He stated that the students will do a better job when they are allowed to explore a concept first and then practice skills that come from the problem. I think that we too often teach the skill and then give students problems that support that skill. We need to spend more time allowing students to 'discover' their own methods for solving problems and then allow them to use methods that make sense to them.
ReplyDeleteI liked the analogy as well because it still speaks to teaching explicitly while "in the water." We aren't just throwing them in and hope they can figure it out. We are still guiding their exploration and discovery.
DeleteI agree- it was a great analogy to explain. We want to prepare students but often it doesn't make sense when they don't know what we're "preparing" them for.
DeleteMy "aha" is simply seeing the real world implications of environmental factors impacting student learning in general but specifically in math. My lowest students are the ones with the least support from home. I love teaching conceptually and searching for the why of mathematics, but if the student misses even one class peroid, it becomes difficult to send that conceptual lesson home for homework and expect mom or dad to help them. The students have to be present and for some families it doesn't seem to be a priority.
ReplyDeleteI also find a great deal of resistance from parents as we teach our "new math" to students. It takes a great deal of work with the parents to help them understand what it is that we are trying to do. Even then you run into parents who react negatively to the idea of changing the way that we teach as it "worked for them".
DeleteMy "aha" moment was when the book talked about the use of algorithms in instuction. I totally agree with the book when it said that an algorithm is a shortuct that is not useful unless a student conceptually understands why it works.
ReplyDeleteI agree, too, and am looking forward to hopefully learning how to help students who are "stuck" to using algorithms without conceptual understanding. I have found that students, and not just teachers, have at times gotten used to shortcuts and want to just know how to get the answer without thinking about what they are doing in a broader sense.
DeleteI agree as well. I am noticing this now more than ever when my students learn to use different algorithms to solve basic addition and subtraction. We teach these to help students understand more about place value. However, it was instantly clear that my students had a fuzzy concept of place value. After we were able to make the connection between place value and why it is important in solving multi-digit equations, student understanding was improved.
DeleteMy "aha" moment was on page 14 when it speaks about developing true understanding of a concept rather than the procedure for completing a process allows the student to use the concept as a tool. Through using the concept as a "tool" they can then develop a much greater understanding in deeper mathematical ideas.
ReplyDeleteI agree and actually think this is part of why I was only good at math for certain teachers or classes- if I didn't get "the steps", I didn't have deep enough understanding to transfer skills to other classes in high school and college (without a lot of help) ;)
Delete"Memory difficulties, for example, might have more to do with instruction than with a student's memory." (Tapper p. 6)
ReplyDeleteI found this VERY interesting because as an ESS Teacher, a lot of students appear to have memory problems, but the issue could be that math concepts are not connected with other information. The author goes on to say that what appears as "gaps" may be lapses in concept development. It can be challenging as a teacher to stick with students to enable them to understand concepts instead of rushing to teach procedures without them having the broader understanding.
I agree, Bethany. This is an idea that really made me think about what we need to do with our students who are struggling. I would like to see some sort of diagnostic test to see what concepts a student does not have....(I think that there is something like this out there, I just cannot think of what it is...and I have never used it!) This could possibly help us begin to fill those gaps and allow our students to begin to move forward.
DeleteMrs. Masimula,
DeleteI also found the information about making connections very interesting. I think that providing students with connections to concepts they already know will further there understanding rather than teaching concepts in isolation. Furthermore, I think that when we help students make those connections, we should explicitly model how that connection is being made. That way, they are able to recognize the connection and hopefully be able to begin making connections on their own.
I agree! It is so hard for our students to remember directions from one minute to the next, remember and memorize facts, etc. If they do not have a true conceptual understanding, then this is an issue of instruction, not memory. I also believe that exposure can be an issue, especially when a curriculum spirals and there is no time spent in mastering concepts.
DeleteI had many aha moments when reading this book. The biggest moment was on page 8 reading My Story. "When kids have difficulty with anything at school, it seems, people are always asking them to demonstrate their difficulty." I think it just mad eme realize how defeated I would feel if I kept having to perform for different people knowing that I didn't get how to do it.
ReplyDeleteAlso, thinking about algorithms this year, I think I will dump teaching lattice multiplication this year! How about everyone else who teaches that?
I thought about writing the same thing. It reminds me of when a friend of mine had serious back pain and the doctors kept asking her to bend forward and touch her toes so they could feel her spine. In tears every time! Testing over and over is like torture for those kiddos. I would just not even try on the parts I knew just because there will be another test another day.
DeleteNo to lattice! I want students to "break apart" the place values and explore multiplication this way. I think that teaching them lattice along with having them develop a conceptual understanding, THEN building to the algorithm, would be too confusing. Too many students rely too heavily on this method, and they don't know how it works, and it tends to slow them down. THANK YOU for bringing this up!
DeleteAs a beginning teacher, it was extremely helpful to read about the difference between procedural and conceptual learning. The example about the difference between knowing how to use a map (or set of directions) and understanding how to get somewhere by previous experience in the area, was a great analogy that will help me analyze my future planning and teaching of math.
ReplyDeleteMy "aha" moment was students having very emotional responses when they can't see the connection between the concepts they are learning and their lives. I can see this happening clearly across all content areas. Students first need to see the connection with their own lives before they will engage and feel that what they are learning is important. Such a simple thing, make it real and students will be more receptive.
ReplyDeleteMy "aha" moment was when Tapper compared how people do math to finding our way around a city: you can have a specific set of directions written out for you, but you won't learn how to find your way around the city the next time. Or, you could explore on your own with a map (the big picture) in front of you, and you will begin to see patterns, big ideas, and learn your way around. I feel that all too often, we as teachers fall into the written directions realm, setting out a "recipe" in solving problems instead of teaching students how to solve problems by thinking and trying on their own. Perhaps this can be blamed on time or curriculum constraints, but I have noticed that lessons that give students time to explore and discuss tend to be the most successful.
ReplyDelete