Having a strong number sense allows students to tackle any math concept so it is vital that they constantly practice and strengthen their skills in estimation, mental math, numerical equivalents, fractions or decimals, and place value. So many students are afraid of math but they don't really know why. Constant reinforcement of what has already been taught is a great way to regularly reinforce concepts and increase student's confidence. While I agree with Leinwand that repetition and spiraled learning is important, I also think a teacher has to keep in mind the age of the children she is teaching. Many of the questions he asks review concepts older students have covered but which haven't been learned by younger kids. First graders are just learning about place value and can determine which is the bigger number, but haven't tackled various ways to express numbers or fractions yet. Also, teachers need to keep in mind the varying levels of students within his/her class. My favorite part of the shift was the teacher who gave the students a large number and asked them to figure out what measurement of his age it represented. What a great way to get the students involved in math in a fun way. They don't realize that the work they are doing is reinforcing what they have learned, they just see it as a fun riddle to solve.
Leinwand suggests taking every opportunity to have children use number sense. In my opinion, students need more exposure to estimation activities. At the younger ages, you could even tell the students how many of a specific item is in one container and have them estimate how many they would think would be in a larger one. I liked how he encourages teachers to have their class compute mentally as often as possible rather than using calculators. I have had students in the past, when I worked in a 2nd grade classroom, enter the information incorrectly on their calculators only to come up with answers that did not make sense. This supports Leinwand's ideas nicely. As teachers, we should always encourage children to estimate what answer would make sense before actually solving the problem.
I agree Jeanne - estimation is a skill that most children are not very proficient in and it is likely due to a lack of practice. I think estimation is a great way for students to really get a feel for numbers though.
I like how Leinwand started this chapter off by talking about multiple choice questions and the use of them on math tests. I always thought that multiple choice tests were rather pointless for math. I think that either children get the right answer or they just get frustrated and pick a random answer. However, Leinwand brought up a good point that multiple choice questions can help children have a better understanding of number sense. When used appropriately, children need to use estimation in order to understand which choice best fits their answer (A, B, C, or D). I also agree with Leinwand, when he mentions in the summary at the end of the chapter, that children need to be provided with more opportunities to put their calculators down and estimate. Nowadays, especially with the technology available to us, children see route computation completely unnecessary because they have constant access to calculators. I almost believe that while calculators are good for quick computations and checking answers, they shouldn't be used at the elementary level. When you get to the higher grades I think they are more beneficial. By monitoring the use of calculators, children are forced to understand number sense and they need to rely more on estimation. This I think only better prepares them for the real world (ie. calculating their total when grocery shopping or a tip when at a restaurant).
Holly, I like how you mentioned real life calculations such as calculating totals and figuring out tips....If teachers use real life situations, students will be more apt to grasp the concept if they have a connection to it.
Jeanne, and Holly, To be honest with you my dad drilled number sense into my head using real world applications, which gave me those opportunities to not rely on a calculator but rather be confident in my math skills. To this day I use my mental math to figure out tips at restaurants quicker than many of my friends with their iphones all because my father taught me! I hope that our students won't rely on these calculators and new pieces of technology too much after having us as teachers!
Holly sums it up very well. Today it is all too easy to look for a calculator or a device to get the problem solved. But, is this really always a good idea. A good solid foundation and a strong understanding of number sense is vital. At the preschool level, the children are so excited by numbers and the mastery of counting. If teachers can build on the initial excitement and mastery, it can create stronger skills
I agree that children are initially really excited by numbers and counting. I was just in a Kindergarten class today and all the students wanted to do was count. As teachers it's really important to try and maintain that excitement and use it to our advantage when teaching math.
Liza, I completely agree with the conclusion you came to. Seeing the excitement at such a young age does dwindle down more once students get older because they are more excited about the newest version of their calculator that will do the math for them! I hope that by getting students excited about numbers and mathematics younger they will be able to say "I don't need the calculator for this," and figure it out in their head or on paper based on what we have taught them.
I agree Liza, I think that as the children get older and concepts and numbers become more abstract thinking, the excitement of learning in math is lost. I think the excitement would be great to help aide in learning. In addition, these two chapters are very closely related I feel, I feel like in both the more concrete a teacher can make number sense, the better students are going to grasp the concept and topic and the more excited the students will be about learning.
I have to agree with Liza, that Holly really summed this up well. With an excitement about number sense, teachers can help students to feel like they can use their knowledge to answer questions and not rely on technology to help them. I really liked that Leinward doesn't just want to give students opportunities to use their number sense to solve problems, but also to have conversations about the problems, and how they could come to conclusions and why. It is this deeper understanding that is found in conversation that brightens that light in our student's heads to make them remember how, why, and what when it comes to number sense.
I agree with Leinwand that it is very important, especially in today's day and age that students get the opportunity to put down the calculators and estimate possible answers for problems. In my class, I have experienced that students can easily become reliable on the calculators and other technology and then experience a mental block that they cannot complete a problem without the technology. The skill of estimation feels as if it is becoming extinct among students as technology becomes more advanced. I also agree with Holly, that calculators are a helpful tool for checking answers and finding answers easily, however as a teacher of middle school boys, I would not want them to use calculators in math even at the higher level. My students become dependent on calculators and shift to the mindset of, "why should I have to do out the work when I can just use a calculator." I think estimation is a skill students are losing and that every opportunity possible we should be developing and discussing number sense.
Having a strong number sense allows students to tackle any math concept so it is vital that they constantly practice and strengthen their skills in estimation, mental math, numerical equivalents, fractions or decimals, and place value. So many students are afraid of math but they don't really know why. Constant reinforcement of what has already been taught is a great way to regularly reinforce concepts and increase student's confidence.
ReplyDeleteWhile I agree with Leinwand that repetition and spiraled learning is important, I also think a teacher has to keep in mind the age of the children she is teaching. Many of the questions he asks review concepts older students have covered but which haven't been learned by younger kids. First graders are just learning about place value and can determine which is the bigger number, but haven't tackled various ways to express numbers or fractions yet. Also, teachers need to keep in mind the varying levels of students within his/her class. My favorite part of the shift was the teacher who gave the students a large number and asked them to figure out what measurement of his age it represented. What a great way to get the students involved in math in a fun way. They don't realize that the work they are doing is reinforcing what they have learned, they just see it as a fun riddle to solve.
Leinwand suggests taking every opportunity to have children use number sense. In my opinion, students need more exposure to estimation activities. At the younger ages, you could even tell the students how many of a specific item is in one container and have them estimate how many they would think would be in a larger one. I liked how he encourages teachers to have their class compute mentally as often as possible rather than using calculators. I have had students in the past, when I worked in a 2nd grade classroom, enter the information incorrectly on their calculators only to come up with answers that did not make sense. This supports Leinwand's ideas nicely. As teachers, we should always encourage children to estimate what answer would make sense before actually solving the problem.
ReplyDeleteI agree Jeanne - estimation is a skill that most children are not very proficient in and it is likely due to a lack of practice. I think estimation is a great way for students to really get a feel for numbers though.
DeleteI like how Leinwand started this chapter off by talking about multiple choice questions and the use of them on math tests. I always thought that multiple choice tests were rather pointless for math. I think that either children get the right answer or they just get frustrated and pick a random answer. However, Leinwand brought up a good point that multiple choice questions can help children have a better understanding of number sense. When used appropriately, children need to use estimation in order to understand which choice best fits their answer (A, B, C, or D). I also agree with Leinwand, when he mentions in the summary at the end of the chapter, that children need to be provided with more opportunities to put their calculators down and estimate. Nowadays, especially with the technology available to us, children see route computation completely unnecessary because they have constant access to calculators. I almost believe that while calculators are good for quick computations and checking answers, they shouldn't be used at the elementary level. When you get to the higher grades I think they are more beneficial. By monitoring the use of calculators, children are forced to understand number sense and they need to rely more on estimation. This I think only better prepares them for the real world (ie. calculating their total when grocery shopping or a tip when at a restaurant).
ReplyDeleteHolly, I like how you mentioned real life calculations such as calculating totals and figuring out tips....If teachers use real life situations, students will be more apt to grasp the concept if they have a connection to it.
DeleteJeanne, and Holly,
DeleteTo be honest with you my dad drilled number sense into my head using real world applications, which gave me those opportunities to not rely on a calculator but rather be confident in my math skills. To this day I use my mental math to figure out tips at restaurants quicker than many of my friends with their iphones all because my father taught me! I hope that our students won't rely on these calculators and new pieces of technology too much after having us as teachers!
Holly sums it up very well. Today it is all too easy to look for a calculator or a device to get the problem solved. But, is this really always a good idea. A good solid foundation and a strong understanding of number sense is vital. At the preschool level, the children are so excited by numbers and the mastery of counting. If teachers can build on the initial excitement and mastery, it can create stronger skills
ReplyDeleteI agree that children are initially really excited by numbers and counting. I was just in a Kindergarten class today and all the students wanted to do was count. As teachers it's really important to try and maintain that excitement and use it to our advantage when teaching math.
DeleteLiza, I completely agree with the conclusion you came to. Seeing the excitement at such a young age does dwindle down more once students get older because they are more excited about the newest version of their calculator that will do the math for them! I hope that by getting students excited about numbers and mathematics younger they will be able to say "I don't need the calculator for this," and figure it out in their head or on paper based on what we have taught them.
DeleteI agree Liza, I think that as the children get older and concepts and numbers become more abstract thinking, the excitement of learning in math is lost. I think the excitement would be great to help aide in learning. In addition, these two chapters are very closely related I feel, I feel like in both the more concrete a teacher can make number sense, the better students are going to grasp the concept and topic and the more excited the students will be about learning.
DeleteI have to agree with Liza, that Holly really summed this up well. With an excitement about number sense, teachers can help students to feel like they can use their knowledge to answer questions and not rely on technology to help them. I really liked that Leinward doesn't just want to give students opportunities to use their number sense to solve problems, but also to have conversations about the problems, and how they could come to conclusions and why. It is this deeper understanding that is found in conversation that brightens that light in our student's heads to make them remember how, why, and what when it comes to number sense.
ReplyDeleteI agree with Leinwand that it is very important, especially in today's day and age that students get the opportunity to put down the calculators and estimate possible answers for problems. In my class, I have experienced that students can easily become reliable on the calculators and other technology and then experience a mental block that they cannot complete a problem without the technology. The skill of estimation feels as if it is becoming extinct among students as technology becomes more advanced. I also agree with Holly, that calculators are a helpful tool for checking answers and finding answers easily, however as a teacher of middle school boys, I would not want them to use calculators in math even at the higher level. My students become dependent on calculators and shift to the mindset of, "why should I have to do out the work when I can just use a calculator." I think estimation is a skill students are losing and that every opportunity possible we should be developing and discussing number sense.
ReplyDelete